Skip to content

Richard Behiel Physics Science

The Twin Paradox: Time Travel, Explained Visually

A near three hour visual masterclass that derives the twin paradox from one strange fact: light has a single speed for everyone. Behiel builds time dilation from a bouncing light clock and the Pythagorean theorem, then generalizes it until it applies to all matter, which he frames as bound energy that has less time for time when it moves. He resolves the paradox not with acceleration but with the tilting of the present moment when the traveling twin turns around, and grounds the whole thing in real evidence: GPS, muon rain, the LHC, and the Hafele-Keating atomic clock flights.

Published Jun 12, 2026 2:55:39 video 20 min read Added Jun 14, 2026 Open on YouTube →

At a glance

The twin paradox is the one where a twin flies off in a rocket, comes home, and finds the twin who stayed on Earth has aged more. It is real, it is time travel into the future, and it is not actually a paradox. Richard Behiel spends nearly three hours showing you exactly why, and the remarkable thing is that he builds the entire edifice from a single strange fact: light moves at one fixed speed, the same for everyone, no matter how they move. From that one rule he derives time dilation with a bouncing light clock and the Pythagorean theorem, generalizes it patiently until it applies not just to clocks but to matter itself, and then resolves the paradox with the idea that turns out to be the whole key, the way the "present moment" tilts when the traveling twin turns around.

This is a derivation, not a summary of conclusions. It earns every result. Below is the full argument, in order, with the evidence he uses to prove it is how reality actually works.

The puzzle, stated honestly

Two twins. One stays on Earth, the other flies to a point a few light years away at a large fraction of the speed of light, turns around, and comes home. When they reunite, the traveler is younger. The traveler has, in effect, fast forwarded into the future.

Behiel is careful up front: the word "paradox" is a misnomer. There is no logical contradiction here, only a deeply counterintuitive result. As he puts it, it should really be called "the twin very counterintuitive thing," but that does not roll off the tongue. The actual puzzle has two layers of weird:

Why does going fast slow down time at all? Speed should not have anything to do with aging, yet it does.
Why is the aging asymmetric? From the Earth twin's point of view the traveler left and returned. But from the traveler's point of view, it was Earth that rushed away and came back. So why does one of them, and only one, end up younger? Naively you would expect the effects to cancel.

The first layer is time dilation. The second layer is the real heart of the paradox. The video tackles them in that order.

The one rule that breaks everything: light has a single speed

In a vacuum, light travels at exactly c, 299,792,458 meters per second, and that number is exact because the meter is now defined to make it so. The radical principle of special relativity is that this speed is the same for every observer regardless of their own motion. That is nothing like a train, where a car driving alongside sees the train barely moving. Run alongside a beam of light as fast as you like and it still recedes from you at the full c.

This is not a theory anyone talked themselves into. The Michelson Morley experiment went looking for the "luminiferous ether," a medium light was supposed to wave through, by measuring the speed of light in different directions relative to Earth's motion. It found no difference. Light had the same speed every way they looked. That null result is what forced physics to take relativity seriously.

Behiel then drops a beautiful escalation: gravitational waves travel at the same speed as light. We know because in 2017 the neutron star merger GW170817 sent gravitational waves and a gamma ray burst across 130 million light years of space, and they arrived just 1.7 seconds apart. That pins the speed of light and the speed of gravity to within about one part in 2.4 quadrillion. The philosophical payoff is large: c is not really "the speed of light." Light just happens to travel at it. It is better understood as the speed, the speed of cause and effect, a structural fact about our universe. Light and gravity are two utterly different things (the wobbling of the electromagnetic field versus the wobbling of the geometry of spacetime), and they share this speed because the universe has one.

Because the universe has a the speed, physicists often set c equal to 1, so all speeds become fractions of light speed. In those units everyday speeds are absurdly tiny numbers (100 mph is about 0.00000015), and that smallness is the whole reason we never notice relativity in daily life. Relativistic effects scale with speed as a fraction of c, so at our speeds they are essentially zero. One more fact to file away: matter cannot reach c, because matter is bound energy tangled up on itself, and the closer you push it to c the more energy it takes, without limit. That theme returns later and turns out to be intuitive rather than arbitrary.

Where time dilation comes from: the light clock

Now the centerpiece of the first half. Imagine a clock that keeps time by bouncing a single photon between two mirrors. Down, up, that is one tick. Build two identical clocks, leave one at rest, and set the other moving sideways, perpendicular to the bounce.

Here is the move you can see with your own eyes. Because light travels at a fixed c, the photon in the moving clock has to travel a diagonal path, sideways while also bouncing up and down. A diagonal is longer than a straight up and down. Same speed, longer path, so the moving clock takes longer to complete a tick. The moving clock runs slow. You do not have to take it on faith; the geometry shows it.

Put numbers on the geometry and you get a right triangle, and a right triangle means the Pythagorean theorem. A few lines of algebra later, the ratio of a moving tick to a stationary tick comes out as:

gamma = 1 / sqrt(1 - v^2/c^2)

This is the Lorentz factor, written gamma. It is the single most important number in the whole subject. It is 1 for things at rest, stays essentially 1 for any ordinary speed, and then blows up toward infinity as v approaches c. Some reference points he uses: at 10% of c, gamma is about 1.005, barely off 1. At 60% of c, gamma is exactly 1.25 (five fourths), a convenient number the rest of the video leans on. At 80% it is 5/3. At 99% it is about 7.09, meaning the stationary clock ticks roughly seven times for every one tick of the moving clock, which is busy moving and, as Behiel says, does not have much time for time.

Is this real? Yes, and here is the proof

The light clock is a contrived thought experiment, so the video grounds it in measurements across the whole range of the gamma curve:

GPS. Satellites move around 9,000 mph, a tiny gamma, but GPS depends on clocks staying synchronized to nanoseconds. Over a day the time dilation amounts to microseconds, thousands of nanoseconds, and if the system did not correct for the Lorentz factor (plus a separate gravitational effect), GPS would simply not work.
Muon rain. Cosmic rays hit the upper atmosphere and produce muons, heavy unstable cousins of the electron that should decay in about 2.2 microseconds. Yet muons created kilometers up survive all the way to sea level, living five, ten, twenty times longer than they should, exactly because their high speed gives them a large gamma. About 10,000 of them fall through a square meter every minute. That is time dilation measured on real particles, far out on the right side of the gamma curve.
The Parker Solar Probe, the fastest object we have ever built at around 430,000 mph, still has a gamma of essentially 1. Even our fastest machines barely touch relativity.
Relativistic electrons inside heavy atoms. The inner electrons of elements like gold scream around the nucleus at 10 to 30% of c, enough for gamma to deviate by a few percent, and that small deviation has visible consequences: it is why gold is gold colored, why mercury is a liquid at room temperature, and where the voltage in a lead acid battery comes from. Relativity is hiding in a car battery.
The Large Hadron Collider. Protons there move so fast you can describe their speed as c minus about 7 mph, with a gamma in the thousands. Relativity is not a correction at the LHC, it is the dominant fact of every collision.

So the curve is not a story. It is how matter behaves, from the gentle left edge (GPS) to the violent right edge (the LHC). (As a fun aside, that same muon rain is what the Scan Pyramids project uses to x-ray the interiors of the Egyptian pyramids without touching them, by measuring how muons are absorbed on their way through the stone.)

From a toy clock to all of matter: more move, less swirl

This is the most ambitious stretch of the video, and the most satisfying. A skeptic could say: fine, a sideways light clock slows down, but real clocks and real bodies are not photons bouncing between mirrors. Behiel answers by generalizing the argument in steps until it swallows everything.

First he replaces the single bounce with an arbitrary closed loop of light in a plane, then with a three dimensional "time cube" of three perpendicular beams. Analyzing the beam that runs along the direction of motion forces out a second effect for free: length contraction. A moving object shrinks along its direction of travel by exactly the same factor, its length becoming L divided by gamma. (You never feel your own contraction, because if you could you would be able to measure your own absolute speed, which is forbidden.)

Then he lets the loop become any closed path in three dimensions, with any number of points, even a continuous curve. The same algebra, helped by the fact that a closed loop returns to its start so the net displacement is zero, gives the same gamma. As he says, that which is true for the thread is true for the sweater: weave many loops into a bundle and the whole bundle time dilates and contracts too.

The final, quiet leap is the deepest. Because light and gravity share the the speed, nothing in the argument actually required the swirling things to be photons. They can be any "specks of unspecified essence" that travel at c. And matter, it turns out, is exactly that. Chemistry is electrons bound to nuclei by the electromagnetic field, mediated by photons; a rock sitting still is made of particles moving at enormous speeds whose net motion happens to be zero. Matter is bound energy, cause and effect swirling at the speed inside a stable pattern. E = mc^2, better read as m = E/c^2, says mass is just bound energy (verified by nuclear power, the Sun, and the electron positron annihilation behind every PET scan, which spits out two telltale 511 keV photons).

That annihilation signature is not just a lab curiosity. The telltale 511 keV peak has been detected coming from the center of our galaxy, and there is more of it than expected (probably not aliens, probably colliding stars or some intense cosmic process, but a genuine open question). And Behiel is refreshingly honest about how much remains unknown beneath all this. We do not know why the standard model has the particles it has, why its gauge group is SU(3) cross SU(2) cross U(1), why there are three generations of matter (every particle has two heavier cousins for no known reason), or even what the Higgs field really is. As he puts it, science is not over, the story has not come to an end, we are still figuring things out. The one thing we do know, and the thing that matters here, is that mass is energy.

Once you see matter that way, time dilation stops being bizarre and becomes obvious. If a thing is a pattern of stuff swirling at c, then moving the whole pattern means that stuff has to spend some of its fixed speed budget on the bulk motion, leaving less for the internal swirl. More move, less swirl. The pattern evolves slower. It is too busy moving to have as much time for time. This is the epiphany the video is built around, the moment your intuition stops fighting relativity and starts pulling you toward it.

It is all relative: both clocks run slow

Everything so far was told from the stationary clock's point of view. But each clock is at rest in its own frame, so the situation is symmetric: each twin sees the other's clock running slow and the other's ruler shrunk. That sounds impossible. How can each be slower than the other?

The resolution is that space and time are not separate. They are one thing, spacetime, and you have to reason about it geometrically. Behiel builds the tool for that: the spacetime diagram, with time drawn as an axis so a history becomes a static picture, and from it the Lorentz boost, the correct way to shift from one observer's point of view to another's.

A few ideas carry the rest of the argument:

The light cone. Draw the paths of light through any event and you get a cone (at 45 degrees in natural units) that divides all of spacetime into what can affect the event (inside, timelike) and what cannot (outside, spacelike). A Lorentz boost slides events along hyperbolas whose asymptotes are the light cone, so the cone never tilts. The speed of light stays the same for everyone precisely because of this.
Time dilation and length contraction fall out automatically. Boost between two drifting observers and each sees the other's pulses slower and ruler shorter, with the same gamma, with no extra assumptions. Relativity is internally consistent, not a pile of separate effects.
The spacetime interval is invariant. Boosts change space and time but leave the combination s^2 = c^2 t^2 minus x^2 unchanged (sign conventions vary). Geometry does not vanish in relativity; it just moves into Minkowski spacetime, where intervals, not separate distances and durations, are the real, observer independent quantities.
Rapidity, for the curious. A Lorentz boost can be written as a rotation through a hyperbolic angle phi, the rapidity, where v/c equals the hyperbolic tangent of phi. Its tidy payoff: successive boosts simply add their rapidities, the way ordinary rotations add their angles.

Behiel offers a clean way to swallow the "both slower" weirdness. If you borrow money and then drift through space forever, never meeting the lender again, you never have to settle up. The contradiction only appears if you reunite. Two observers can each see the other as slower for as long as they stay apart; if they ever come back together, they must agree on who aged less on average. Relativity allows the disagreement, but never an actual contradiction.

The crux: the present moment tilts

Here is the single idea that resolves the paradox, the relativity of simultaneity. Your "now," the set of events you call simultaneous with this instant, is a slice through spacetime, and the angle of that slice depends on your speed. In natural units the rule is memorably simple: the slope of your now slice equals your speed (more precisely, slope = v/c^2). Change your speed and your definition of "right now, far away" swings.

This is exactly what the broken Galilean (everyday) picture gets wrong. Under the old common sense boost, everyone keeps the same horizontal "now" and the same universal time, but then the light cone tilts, which directly contradicts the constant speed of light and every experiment that confirms it. You cannot have it both ways. Keep the light cone fixed for everyone, as reality demands, and the now slice must tilt instead.

Putting it together: the 60% trip

Now the concrete example. A ship travels to a cool spot 3 light years away at 60% of c and comes straight back. Gamma is 1.25, the tidy five fourths.

From Earth: it is a 6 light year round trip at 0.6c, so 10 years pass on Earth. The traveler's clock runs slow by 1.25, so the traveler ages 10 / 1.25 = 8 years.
From the traveler: their clock is fine; instead, length contraction shrinks the 6 light year round trip to 6 / 1.25 = 4.8 light years, and 4.8 / 0.6c is, again, 8 years. Two different accounts, same answer. That is the coherence relativity demands.

The slippery question is why both twins agree that Earth aged the full 10 years. Behiel sharpens it with birthday signals: each twin sends a light speed pulse once a year. Because of the Doppler effect alone (before time dilation), the rates change with direction. On the way out the factor is sqrt((1+v/c)/(1-v/c)) = sqrt(1.6/0.4) = 2, so outbound signals arrive every 2 years; on the way back it is the reciprocal, every half year. Crucially, on both legs the picture is symmetric: each twin sees the other's clock running slow by 1.25, and each receives the other's birthday signals at the same rates. If it is symmetric on the way out and symmetric on the way back, where on earth does the asymmetry come from?

The resolution: the turnaround

It comes from the turnaround, and only the turnaround. Watch the traveler's now slice.

Outbound, in the traveler's frame, when they reach the cool spot they have aged 4 years, and their now slice says Earth has aged 3.2 years (4 / 1.25). Perfectly symmetric so far. Then they fire the engines and reverse direction. Their speed relative to Earth flips, so their now slice tilts the other way, and the slice sweeps across a huge swath of Earth's history in an instant. The present moment on Earth, from the traveler's point of view, jumps from 3.2 years to 6.8 years. That fast forward, 3.6 years of Earth time skipped over during the turn, is where all the asymmetry lives. Add the inbound leg's 3.2 more years of Earth aging and you land on 6.8 + 3.2 = 10 years on Earth, against the traveler's 4 + 4 = 8.

The Earth twin never has to do any of this gymnastics, because they never change frames. They sit in one inertial frame the whole time and tell the simplest possible story. The traveler is the one who switches frames, and switching frames is what tilts the now slice. That is the asymmetry, made of geometry, not of who felt more tired.

It was never really about acceleration

A natural objection: maybe the turnaround's g forces are what cause the time travel. Behiel dismantles this with the "twin exchange program," a version with no acceleration at all. Your twin flies past Earth at 0.6c and just keeps going forever. Out at the cool spot, an alien who happens to be cruising inbound at 0.6c passes your twin, reads their clock (4 years elapsed), and carries that count home, arriving 4 years later by their own clock. Nobody accelerates, yet the relay still totals 8 years against Earth's 10. The g forces were never the cause. They are only what it takes to make one object follow the bent path.

The deepest framing comes last. In ordinary Euclidean space a straight line is the shortest path between two points. In Minkowski spacetime it is the opposite: between two events, the straight (inertial) path has the most proper time. Any bent path, any going out and turning around, has less. The stay at home twin took the straight path through spacetime and aged the most. The traveler took the bent path and aged less. As Behiel says, if you see that, you understand the twin paradox.

It has actually been measured

The video also leaves the chalkboard. In 1971 the Hafele Keating experiment flew atomic clocks around the world, east and west, and compared them to clocks at the US Naval Observatory. Because Earth rotates, the eastbound flight moves fastest (its motion adds to the rotation) and should lose the most time, while the westbound flight is the slowest twin and should gain. Folding in both the kinematic twin paradox effect and gravitational time dilation (clocks higher in the gravity well run faster, the same effect dramatized in Interstellar), they predicted the eastbound clock would end up about 40 nanoseconds slow and the westbound about 275 nanoseconds fast. They measured about 59 nanoseconds slow and 273 nanoseconds fast, both consistent with the predictions, and the experiment has since been repeated at higher accuracy. Real clocks, real airplanes, the twin paradox in actual nanoseconds.

Chapters

Timestamps are clickable. Click one and the player jumps there and keeps playing while you read.

0:00 Twin Paradox Intro
3:59 Speed of Light, c
11:43 Simple Light Clock (time dilation from geometry)
20:03 Lorentz Factor Chart (GPS, muon rain, the LHC)
30:08 Planar Light Clock
35:37 3D Light Clock: Time Cube
38:44 Parallel Light Clock (length contraction)
49:34 Fully Generic Light Clock
57:52 Unspecified Essence
1:02:34 Matter is Swirly (more move, less swirl; E equals mc squared)
1:16:05 Multiple Perspectives (it is all relative)
1:25:39 Spacetime Diagrams (light cones)
1:34:08 Boosting the Grid
1:39:32 Carbonated x-Axis
1:42:53 Tilting of the Now Slice
1:45:35 Time Dilation
1:50:33 Length Contraction
1:52:48 Lorentz Boost Equations (rapidity, the spacetime interval)
2:00:12 The Three Ingredients
2:05:23 Galilean Boost (wrong)
2:11:28 Hafele-Keating Experiment
2:18:47 Setting the Stage (the 60% trip)
2:23:10 Birthday Signal Animation (the Doppler effect)
2:27:01 Earth POV
2:34:40 Space POV, Outgoing
2:44:03 Space POV, Incoming (the turnaround, 3.2 to 6.8)
2:47:57 Twin Exchange (no acceleration needed)
2:55:06 Patreon Credits

Notable quotes

The so-called twin paradox is actually not a logical paradox. It's just a very counterintuitive thing. Richard Behiel, 0:25

The speed of light C is not really the speed of light per se. But no, it's just the speed. Richard Behiel, 7:00

More move, less swirl. More move, less swirl. Richard Behiel, 1:14:50

That which is true for the thread is true for the sweater. Richard Behiel, 57:10

Spacetime is flexible enough to accommodate a disagreement about whose clock is ticking more slowly. Just so long as if those observers ever reunite, then both of them have to agree on whose clock ticked more slowly on average. Richard Behiel, 1:34:10

G forces are not a causal agent. They're just a requirement of having to put the same thing on that path. Richard Behiel, 2:50:00

In Euclidean space, a straight line minimizes the distance between two points. In Minkowski spacetime, a straight line maximizes proper time. Richard Behiel, 2:54:30

Resources mentioned

Introduction to Elementary Particles, David Griffiths, chapter 3, section 3.1 on Lorentz transformations.
GW170817, the 2017 neutron star merger detected by LIGO and Virgo, with its 1.7 second gravitational wave to gamma ray delay.
The Hafele-Keating experiment (1971), atomic clocks flown east and west around the world.
The Michelson-Morley experiment, the null result that killed the luminiferous ether.
The Scan Pyramids project, which uses muon rain to image the interiors of the Egyptian pyramids.
The Angels Camp Museum in California, which has an exhibit on Albert Michelson.
Behiel's companion videos: Electromagnetism as a gauge theory and Superconductivity and the Higgs field.

The one idea to walk away with

Time is not a universal clock ticking the same for everyone. It is the length of your particular path through spacetime, and a straight path is the longest one. Go fast, turn around, come back, and you have taken a shortcut through time itself. You really do arrive in the future, having aged less, and there is nothing paradoxical about it once you see that "now" is a slice whose angle is yours alone.

Full transcript

Hey my friends, today we're going to talk about the twin paradox. The twin paradox is one of those things in physics where you're like, "Huh? What? What is this? How does this work?" Because it's very strange and it involves actual time travel into the future. Now, I should point out right from the start, the so-called twin paradox is actually not a logical paradox. It's just a very counterintuitive thing. So, it should be called the twin very counterintuitive thing, but that's a mouthful, so we call it the twin paradox. And so, the puzzle of the twin paradox is to see how it's actually not a paradox. But we're getting ahead of ourselves here. First of all, what even is the twin paradox? Well, here's what it is. Suppose there are two twins. Now, one twin is going to stay on Earth and the other is going to get in a rocket ship and go way out into space at super high speed and then come back. So, if you imagine the Earth and also some place very far away from Earth, then the Earth twin is going to stay on the earth and the space twin is going to fly way out there at a very high speed where their speed V is going to be a large fraction of the speed of light. See? So they go there, they turn around, they come back, and when the twins reunite, amazingly, they find that the Earth twin is older than the space twin. That is, somehow the space twin has aged less. So from the space twin's point of view, they've fast forwarded into the future. So it's like time travel. And this is real. Like this is legit physics. It's actually true. There's tremendous theoretical and experimental evidence that this is the case. as we'll talk about in this video. But even though it's true, it's still super weird. And it's weird in a couple of different ways. First of all, why does going fast make time slow down? See, because from the Earth twins point of view, the space twin is not aging as fast, so they're slowing down. And so apparently, just because the space twin is going fast, that means their clock is going to tick more slowly. So that's weird. What's up with that? Well, we'll address that in the first part of this video. We'll talk about time dilation and the Loren factor and Loren boosts and how space and time get a little weird when you factor in that the speed of light is the same for everyone. But then the second weird thing and this is actually even weirder than the first weird thing is that why do we have an asymmetry in aging even though both of the twins moved relative to each other? That is to say, we're thinking about it from the Earth twins's point of view, where the space twin goes away and comes back. But from the space twin's point of view, it's the Earth twin that moves away and comes back. So, at first glance, we would expect there to be symmetry and for all this weird time dilation stuff to balance out, right? So when you think about relativity and the situation from each twin's point of view, that really makes you think, wait a minute, how can it be that one twin ended up aging more than the other even though from each other's point of view they both go out and they come back in? And that asymmetry is really the essence of the so-called twin paradox. Because the puzzle is to explain how this asymmetry in aging that facilitates this fast-forwarding into the future can possibly exist in a physical framework where everyone's point of view has to cohhere into a sensible description of reality. And so we'll address that asymmetry in the second part of the video after having warmed up our relativity skills. But first of all, let's go ahead and do a physics warm-up where we talk about some of the concepts from special relativity having to do with the speed of light and space and time and all that. Okay. So, I'd like to say a few words about the speed of light, which we call C because C is short for the speed of light. So in the vacuum of space, light travels at exactly the constant speed C= 299,792,458 m/s. And that is an exact number because the meter is defined such that the speed of light is exactly that whole number. But we often think of the speed of light as about 300,000 km/s or about 186,000 m/s. That's a pretty fast speed. Light can go around the Earth in just 0.13 seconds and it can go all the way out to the moon in just 1.3 seconds or to the sun in about 8 minutes. So, light is pretty fast. Now, the fundamental principle of special relativity is this strange idea that the speed of light C is the same for everyone regardless of their speed. And that is a very strange concept cuz normally you think if something is moving and you're moving in the same direction as it then its speed is going to be slower from your point of view than from the point of view of someone who's not moving. You know like if there's a train and if you're standing there and you're like oh wow there's a train there it goes. But if someone's driving a car on a road next to the train from their point of view it's like oh the train's not moving that fast. But light is not a train. light is always moving at sea regardless of how fast you're moving. That idea is completely incompatible with conventional notions of space and time. And so therefore, special relativity forces us to update our notions of space and time. And just on a historical note, you know, at the beginning of the 20th century when Einstein was coming up with this idea, people used to think that maybe light is a wave in the so-called luminiferous ether. You know, like a wave on the water is a wave in the water. People were like, "Okay, light seems kind of wavelike. Maybe there's like a ether that light travels through." But then if you look into the Michaelelsson Mley experiment, in that experiment, they measured the speed of light in different directions relative to the Earth's motion with a sensitivity that would have been able to see the differences in the speed of light in each direction if the luminiferous ether existed. But what they found was that the speed of light was the same in both directions. So that was the first big experiment that made people think, "Oh jeez, we have to deal with special relativity for real, like it's actually real." Oh, and by the way, if you ever happen to find yourself in Angel's Camp, California, check out the museum cuz they have a whole room, a whole exhibit on Albert Michaelelsson, including the Michaelelsson Mory Experiment, and it's really cool. So, just a little shout out to the Angel Camp Museum. All right. So, as weird as it is that the speed of light is the same for everyone, let me hit you with another amazing fact. And that is that gravitational waves also travel at the speed of light. Huh? What? How do we know that? Well, there's actually beautiful evidence for this fact, which is that in 2017, the neutron star merger GW170817 was detected by two gravitational wave detectors, LIGO and Virgo. And then just a mere 1.7 seconds after the gravitational waves came to the earth, a gammaray burst came in as well from the same event. So what happened was 130 million light-years away, 130 million years ago when dinosaurs roamed the earth, two neutron stars collided in a cataclysmic event which sent out gravitational waves, ripples in spaceime throughout the universe as well as a burst of gamma rays, high energy light. And these waves traveled for 130 million years through the vacuum of space until they hit the earth just 1.7 seconds apart. So that tells you that the speed of light and the speed of gravitational waves only differ by at most one part in 2.4 quadrillion. So that is an extreme precision measurement that light and gravitational waves travel at the same speed. Now then of course there is the question of how do we explain the 1.7 second difference and I believe the leading hypothesis on that has to do with something called a late breakout where you can imagine neutron stars colliding and smashing together and it's like a crazy mess and there's like ejecta expanding around and then after like a second or two a relativistic jet punches through that ejecta and emits gamma rays. So something like that where the neutron stars collide and then a flash comes out a moment later. That could explain the slight difference in the arrival time between the gravitational waves and the light. So why do I bring that up? This is not a video on gravitational waves. So what's the big deal that gravitational waves travel at the speed of light? Well, actually this is a huge deal. This has tremendous philosophical implications the likes of which should blow your mind because this means that the speed of light C is not really the speed of light per se. But no, it's just the speed. That's my speed sound effect. Cuz when you think about it, it's like a light is the transverse flopping around of the U1 gauge field. Gravitational waves are the transverse flopping around of the metric tensor. We have no app priority reason to think that those things should propagate at the same speed other than the fact that apparently our universe is such a universe that has a the speed. So this thing about the speed of light and the speed of gravitational waves being exactly the same that's really a statement about reality. It's a deep statement about motion and space and time and it is a fundamental aspect of reality as we know it. So that's neat. Now because our universe has a the speed naturally it's often convenient to set C equal to 1 by definition. So in that convention all speeds are fractions of the speed. So for example if something is moving at 10% the speed of light we would say it has a speed of 0.1. And that's a very zen way to think about speed. It's also very mathematically convenient. The only problem with natural units is that regular speeds that we're used to are super small numbers. So, for example, 100 mph is 0.0000 lot of zeros 149. But that's not really a problem. I mean, a small number is not problematic. It's more of just an inconvenience if you want to describe everyday speeds. But the fact that everyday speeds are such a small number in natural units is actually a really insightful observation. Because what it means is that the speeds we're used to in everyday life are all approximately zero relative to the speed of light. And so that's why we usually don't notice relativity. Relativistic effects depend on speed as a fraction of the speed of light. So if it's basically zero, all of those relativistic terms in our equations are going to go away. So even on a very fast airplane, you're still basically standing still compared to the speed. Oh, and also one more thing to say on this topic. Matter cannot go faster than C. And this is because matter is bound energy. It's all tangled up on itself. And so at most, if you put a ton of energy into it, you can get it to approach the speed of light. But the nature of matter is incompatible with traveling at or beyond the speed of light. And that's a theme we'll explore throughout this video. And it's something that even though it sounds weird at first, you'll see it's actually very intuitive when you think about the nature of matter as bound energy. So we'll talk about that later on. Okay. So now that we've talked about the speed of light, I want to show you the classic experiment for seeing how a moving clock is going to tick more slowly than a clock at rest. And this is the famous light clock thought experiment. So the thing to imagine is that we have a clock that measures time by bouncing light between two mirrors. So the light bounces down, it bounces back up, and when it comes back up, we count that as one tick of the clock. So then with that in mind, all we have to do is imagine the same clock moving in a direction that's perpendicular to the axis of the light. So you see here the green clock is the light clock at rest and the magenta clock is totally identical except for the fact that it's moving. So when you think about this and you think about how the speed of light is constant all over the place and then you realize that the light in the moving clock has to go on a diagonal path in order to bounce up and down while also moving to the side then you see that the moving clock has a longer path for the light. And so therefore it's going to tick more slowly than the clock at rest. So right there by looking at this and thinking about how it is you can see with your eyes with your own two eyes why it is that a moving clock is going to tick more slowly than a clock at rest at least for this specific light clock which is moving in this specific direction. So this thought experiment is very contrived. It's very specific. It's very idealized. In reality a clock is a messy thing and it's not so simple as light bouncing up and down. But let's set those concerns aside for a moment because later we're going to generalize this concept and show that it applies to all clocks and in fact to all things. So that's really cool and we'll prove all of that later on in the video. But for now, the fact that this thought experiment is so contrived and idealized and specific is actually a good thing because it gives us something really tangible and really clean to analyze. So what we're about to calculate is that factor of how much the moving clock slows down as a function of its speed relative to the stationary clock. To do this analysis, let's go ahead and draw the figures shown here in which we draw one full tick of the stationary clock as well as a full tick of the moving clock. Bearing in mind that these two ticks take a different amount of time. So we'll go ahead and use the letters T and T prime to be the time that it takes for the light to bounce back and forth in the stationary clock and the moving clock respectively. And then all we have to figure out is what is t prime divided by t. That is what is the ratio of the duration of a moving clock tick relative to a stationary clock tick? Well to do this all we have to do is a little bit of geometry. First, let's notice that for the stationary clock, the light simply bounces down and then bounces back up. The total distance that the light travels round trip is going to be the speed of light time t, the duration of the tick, because distance is speed * time. So, the distance between the two mirrors of the stationary clock is going to be 1/2 c * t. Now then, because the moving clock is identical to the stationary clock except that it's moving, then the vertical distance between the two mirrors is also going to be 1/2 C * T. But then if you think about what is the length of this diagonal, well, that's going to be 1/2 C * T prime, where T prime is the duration of the tick of the moving clock. So that's the same reasoning as we saw before. The diagonal distance here is just half of the distance that the light travels as it bounces down and comes And then finally, if you think about what is that horizontal distance of this triangle, well, you realize that's going to be the speed V of the moving clock* t prime / 2 because the light goes from the top mirror to the bottom mirror in time t / 2. And so the distance that the clock travels during that time is going to be at speed * t prime / 2. Now then this triangle shown here contains all of the insight that we need to calculate the ratio of t prime and t because if you look at it it's a right triangle. And what do we know about right triangles? We know the pythagorean theorem. So you take all of these things and do a^2 + b^2= c^2. And that gives us the equation shown here. Now, we can clean this up by making the twos go away and then moving the VT prime term on over to the right side of the equation. That gives us the equation shown here. Next, we go ahead and divide everything on both sides by c^ 2. So, on the left side, we have a t ^2. And on the right side, we have a factor that's a function of the speed of the clock* t prime^ 2. And because what we want to know is the ratio t prime over t. All we have to do is divide everything by t ^2 and then take a square root. And you see that the ratio t prime / t is simply 1 over the<unk> of 1 - v / c^ 2. So there it is. That is the equation that tells us by how much a moving clock is going to slow down relative to a stationary clock. And that ratio is very special and it has a special name. This is called the Loren factor, also known as gamma or the gamma factor. So in relativity, when you see gamma, that's the time dilation factor. It tells you how much a moving clock slows down. And as you can see in this equation, the lorren factor gamma is a very simple function of the speed of the moving clock. Now, as we're going to show later on in the video, this concept, the Loren factor, applies to all clocks, not just this idealized example of a light clock moving perpendicularly to the axis of the light. So, in just a moment, we're going to generalize this reasoning and show that the Loren factor that we just calculated from the Pythagorean theorem is actually the time dilation factor for anything that's in motion. But before we do that generalization, let's take a moment to look at a few specific examples of a moving clock at different speeds. So we can build up some intuition for the lorren factor. Well, first of all, the animation here shows a clock which is moving at 10% the speed of light. Now, a speed of 10% the speed of light corresponds to a loren factor of about 1.005, which is almost 1. And as you can see, clock are almost synchronized. But if we go ahead and fast forward, you can see that eventually even a small lorren factor is going to cause the clocks to be out of sync because the moving clock is ticking ever so slightly more slowly than the stationary clock. Now if you compare 10% the speed of light to the 60% that we were just looking at you can see that at 60% which has a loren factor of 1.25 the time dilation effect becomes noticeable we have this cool 54 resonance between And now if we go all the way up to 99% the speed of light, you can see that the moving clock is so busy moving to the right that it doesn't have time for time. You know, it doesn't tick nearly as fast as the stationary clock. And in fact, by the time the moving clock ticks one full cycle, the stationary clock has already ticked about seven times because for a velocity of 0.99 C, the Loren factor is about 7.09. So you get about seven ticks of the stationary clock for every one tick of the moving clock. All right. So now that we've seen a few examples of the Loren factor, let's go ahead and plot it as a function of all possible speeds. So on the y-axis we have the Loren factor gamma and on the x-axis we have speed as a fraction of the speed of light. So any possible speed is going to be somewhere between 0 and one on this chart. So on the left side of this chart we have a thing with zero speed totally at rest and on the right side of the chart we have something whose speed is approaching the speed of light. The purple curve is a plot of the equation that we've derived for the loren factor. That is 1 over the<unk> of 1 - v / c ^ 2. As you can see gamma is basically one for any speed that's not super fast. But then as the speed gets super fast and you approach the speed of light, gamma blows up and approaches infinity. Now later on in this video when we examine an example of the twin paradox, we're going to use a speed of 60% the speed of light. And that's a nice convenient example because it corresponds to a loren factor gamma of 1.25 or 5/4s as we've seen. But don't take my word for that. You can see for yourself that 1 / the<unk> of 1 - 0.6^ 6^ 2 that's going to be 1.25 because 0.6 2 is 0.36 1 - 0.36 is 0.64<unk> of 0.64 is 0.8 and 1 / 0.8 is 1.25. So the math checks out and that also matches what we see in our chart. And by the way another convenient example would be 80% of the speed of light because that has a lens factor of 5/3. But anyway, the ideas we're exploring in this video are going to be totally generic for any speed between zero and the speed of light. So, we'll use convenient numbers just for fun, but the equations apply generically even for inconvenient numbers. Now, while we're here, I want to look at a few things in the real world that have some speed and therefore have some loren factor so we can orient ourselves to the physical effects of the lorren factor. And also it's important to see these things so that we know that our Pythagorean reasoning is describing the real world and we're not just getting high on. All right. So first of all, if you're in a car and you're driving at 60 m hour, your speed is a fraction of the speed of light is going to be super small and your lens factor is going to be basically one. So for all intents and purposes, if you're driving in a car, time dilation is totally irrelevant. And even if you're on an airplane going 600 mph, well, your speed is still basically nothing compared to the speed of light and your lens factor is still basically one. And the same is true even if you're on a crazy fast jet like the SR71 going thousands of miles an hour. Well, your lorren factor is still basically one. But now then, if you think about a GPS satellite, those things tend to travel around 9,000 mph or so. And so even a GPS satellite has a lens factor that's pretty close to one. But here's where things get interesting. The way GPS works is your phone or whatever is bouncing signals off of these satellites and based on the response time and the speed of light, your device is able to triangulate its position based on the satellites. And it's amazing that that actually works. But in order for it to work, you need to have clocks that are super precise and you need everything to kind of remain synchronized and not drift, you know. And so even with the small Loren factor here, you think about multiplying that number by a day. Well, the result is still basically a day plus about 7 micros. Now 7 micros that doesn't sound like a lot, but that's like thousands of nanconds. And so actually GPS satellites have to use the Loren factor to correct for time dilation and keep their clocks from drifting and therefore keeping GPS working as a thing. And if they didn't take into account the Loren factor, GPS wouldn't work. It would be way off like it wouldn't work at all. And by the way, there's also a very subtle gravitational time deation effect because the satellite is higher up in the Earth's gravity well. But that's a story for another day. But the point is GPS satellites give us a real example of a thing that directly has to contend with time dilation. So this is very good experimental evidence that the Loren factor is actually a real thing at least towards the left side of this plot where the Loren factor starts to deviate ever so slightly away from one. But let's go even faster and see what happens. Well, if you think about the fastest satellite ever made, the Parker Solar Probe, which reached a speed of 430,000 mph, even still, it had a Loren factor of basically 1. So, even the fastest satellite ever made, it didn't experience a tremendous amount of time dilation. However, if we want to think about stuff that's crazy fast, imagine some of the inner electrons of heavy elements like gold and stuff. Well, those electrons are so close to the nucleus and things are so crazy intense inside those atoms that their kinetic energy is insanely high corresponding to speeds of somewhere between 10 and 30% the speed of light. And at those speeds, the Loren factor starts to deviate from one by like a few%. which still isn't that much, but it does have some real world effects like pertaining to the color of gold and the fact that mercury is a liquid and also where the voltage comes from in a lead acid battery. All of these things have to do with relativistic effects that are based on the Loren factor being greater than one. And we can go off on a whole tangent about that, but let's not for now. I just bring it up in passing. But the next thing I want to mention is actually a more significant example of a thing that's real and has a big loren factor and that is muon rain. Now this term muon rain it's not like standard like I haven't heard it used a whole lot but that's what it is. So what happens is some high energy particles come in from space you know super high crazy energy cosmic rays. They smash into the upper atmosphere and those collisions end up producing muons. A muon is basically identical to an electron in every way, except for some strange reason, it's about 207 times more massive. No one knows why. The muon is the second generation version of the electron. For some reason, there's three generations of matter. Every elementary particle has two heavier cousins. Why that is, no one has any idea whatsoever. It's one of the deepest mysteries in physics. Some would argue it's evidence that God exists because only a sentient creator would have the sense of humor to torture us with a mystery like this. In any case, muons are very unstable. They rapidly decay. So, they are not long for this world. So, as the muons come raining down from the upper atmosphere, some of them are decaying along the way. So, imagine that muons are raindrops coming down. Well, as they're coming down, a certain number of them are decaying. And so there are fewer muons hitting the ground at sea level than would hit the ground on top of a mountain for example. And by the way at sea level if you imagine one square meter a meter by a meter square every minute about 10,000 muons will fall onto that square. Now one of the things that people have done is they've measured how many muons are coming in and hitting the ground at sea level versus hitting the ground on a mountain versus at various different altitudes. What is the flux of muons that are coming in and therefore how fast are they decaying? Now, what these experiments show is that these muons that are raining down to the earth and traveling very close to the speed of light are living way too long. Okay, a muon is supposed to decay in about 2.2 micros. But these things are living 5 10 20 times longer than they should. And so that right there is a profound piece of evidence that the Loren factor affects even elementary particles and it's also a wonderful experimental data point that's way out here to the right of the plot for the Loren factor. So these measurements of the muon rain give us really good evidence that the Loren factor is a real thing and that our curve is correct, our equation is correct even way out there near the speed of light. Oh, just as an aside, the muon rain is what the scan pyramid projects uses to scan the pyramids. Look up that if you haven't heard about it. That's a really cool thing. They're finding all kinds of cool stuff in those pyramids. Anyway, one more thing before we move on. The ultimate test of the super high speed limit of particles is at the Large Hydrron Collider where they're colliding protons with energies in the terra electron volts, which is crazy high energy. And these particles are going so fast that you can think about their speed as the speed of light minus 7 mph. The speed of lightus 7 mph, that's just 7 m less than the maximum number of miles that could even theoretically fit in an hour. So that's like 99% the speed of light and it corresponds to a loren factor of like 7,000. So these experiments that are happening at the Large Hydron Collider, these are deeply in the relativistic limit way on the far side of this plot where the Loren factor is not just greater than one, but it's like insanely high. It's like in the thousands. So it's extremely relevant to everything that's happening in those interactions. And this is why people say that special relativity is extremely well tested. Because even though in everyday life the Loren factor is basically one, in everyday particle accelerator experiments, the Loren factor is very much not equal to one. It's like a really big number and it has really noticeable consequences. And with that in mind, we can rest assured that the curve shown here for the Loren factor based on our Pythagorean reasoning is actually real. Like this is how reality actually works. All right. So now we've seen the Pythagorean light clock argument for deriving the Loren factor and we've also seen some real world experimental data which shows us that the Loren factor and time dilation is a real effect that really affects matter in our actual universe. But at the moment there's a disconnect between our simple idealized mathematical derivation and the messiness of material reality with all its complicated stuff. So to fully understand the nature of time dilation, we have to generalize the light clock argument and show that that reasoning applies to more and more generic scenarios until finally we see that it applies to matter itself with all its messiness. So to that end, the first thing I'd like to point out is that if we imagine the light clock experiment that we've been talking about so far, notice that this thought experiment is already somewhat generic in the way that the beam of light is oriented. Because so far, all that we've assumed is that the beam of light in the clock is perpendicular to the direction of the clock's motion. So, if the clock is moving along the x-axis, then the beam of light could be oriented along the y-axis or it could be oriented along the z-axis or in general, it could be oriented along some arbitrary axis in the YZ plane. And with that observation, we can go ahead and generalize our simple light clock thought experiment into a generic pattern of light in a plane which is moving in a direction perpendicular to that planer pattern. We can go ahead and define our planer pattern in the YZ plane as being a set of coordinates where the letter X with a vector arrow on top of it stands for a coordinate triplet XYZ and the requirement that all of these points be in the plane is simply the requirement that all of the X coordinates be zero. So our stationary clock is going to be based on some arbitrary set of points in a plane. 0 y1 z1 0 y2 z2 dot dot dot 0 y i zi where i is some number dot dot dot 0 yn zn and so the pattern is comprised of n points where n is totally arbitrary. We put no restriction on the number of points in the pattern. And in fact if we want to we can let n approach infinity. So the pattern in the plane we can even imagine as a continuous curve if we want to. So then the way this planer light clock works is that a beam of light is going to move through all of the points in the pattern reflecting from one point to the next and then when it reaches the nth point it's going to go back to the first point and start the loop over again. So you see this is a fully generic definition of a pattern in a plane that closes back in on itself and repeats. Now then if we go ahead and move this planer light clock in a direction perpendicular to the plane. So in this case along the x direction then the set of points in the moving clock is going to be identical to the stationary clock except the x coordinate is no longer zero. See because the clock is moving along the x-axis with speed v coordinate at each point is going to be the speed v * the elapsed time t. where by the way the time t is measured in the reference frame of the stationary clock. Okay. So just by looking at this and thinking about the way it is you can see that spiritually the moving clock is going to have a slower loop as the photon goes around. It's going to be a slower process because the clock is moving and so the light has to move to the side in addition to swirling around. But because the speed of light is the same, therefore it's going to take longer for the moving clock's light to loop around. So then the question becomes, are we sure we end up with the same lorren factor as before? That is, does this planer clock slow down in the same way as a function of its speed as for the simple light clock example we looked at earlier? And the way to answer that question is to just consider one leg of the loop. So pick some arbitrary point in the loop as well as the point that comes right after it. And then you see from one point to the next, it's a straight line in the YZ plane for the stationary clock. And for the moving clock, it's the same line, but it's moved along X by some distance VT prime / 2 as we saw before. And in fact, the same exact diagram and analysis we did before with the triangles and the Pythagorean theorem, that is exactly the calculation we would do for some specific leg of the loop because that analysis never specified the orientation in the YZ plane of the light. And so the argument fully generalizes to every single leg of this loop. So when you add that all up and you think about the whole loop, you find that we end up with the exact same lurens factor as before. And so the phenomenon of time dilation also applies to the planer light clock. Okay. So having generalized our calculation of the lorren factor to a planer pattern which is perpendicular to the direction of motion. Next of course we want to generalize it to a three-dimensional pattern. And that's a little tricky because if we have light moving along the direction of motion, the calculation becomes a little bit more nuanced. And to understand those nuances, I'd like to introduce the time cube. Now, this is not the time cube of internet fame. There's a whole another time cube out there that's like crazy. So, this is not that. But no, the time cube, as used in this video, is simply a three-dimensional light clock of the kind we were looking at earlier, where the light bounces down and it comes back up. And when it comes back up, we count that as one tick. But now the time cube is going to have three perpendicular axes along which light is going to bounce. So then what we're going to imagine is a time cube moving along the x-axis. Now we already know when we do that the y and the z beams are going to exhibit time dilation with the loren factor gamma that we've already calculated. But the x direction is going to behave a little bit differently because it's going to slosh back and forth along the direction that the clock is moving. And to figure out how to think about that, the thing you have to imagine is that if you are holding a time cube and looking at it so that it's stationary from your point of view, then assuming it's a perfectly machined time cube, all of the three beams of light are going to be perfectly synchronized with each other. That is, they're going to go out, reflect, and come back. And when they come back, they're all going to come back at the same time. But now bear in mind that light is weird. No matter how fast you're moving, if you're not accelerating, if you're staying at a constant speed, you cannot use the time cube to measure your speed. So if light were like a wave in the ether, then you could measure your speed because the light's going to go faster along some directions than along others. And that's the essence of the Michaelelsson Mley experiment. But in reality, light doesn't work like that. And no matter how fast you're moving, if you have a perfect time cube, it's going to remain synchronized. The reason I bring that up is because that gives us the key to understanding the behavior of light along the direction of motion of the clock. See, because we can impose as a constraint that as we do our analysis of the moving clock, the light beam along X has to return to its starting point in the same amount of time as the beams along Y and Z. So in other words, we should expect the same time dilation factor along the X direction as we've already calculated along Y and Z. But now the question still becomes how exactly can it be that the X beam is going to have the same time dilation factor as the Y and Z beams given that the motion is oriented differently than the case for Y and Z. Well, let's go ahead and draw some diagrams and see what's going on. All right, so we want to figure out what is going on with the X beam of the time cube that is with the axis of light that is parallel to the motion of the clock. So to analyze that suppose for the moment we imagine a simple light clock with one axis where that axis is parallel to the direction of motion. So then the time cube taught us that a light clock moving parallel to the photon axis necessarily slows down by a scale factor of gamma which is the same lorren factor as for the perpendicular clock relative to a stationary version of the same clock where gamma is the same lorren factor whether we're dealing with a perpendicular motion light clock or a parallel motion light clock and that is because all three axes of the time cube are synchronized. So a tick of the moving light clock t prime is going to be the lorren factor gamma times the duration of a tick of the stationary light clock. However, the parallel motion light clock works a little differently than the perpendicular motion light clock because of the way that the light is moving. And as we're about to see as we do this analysis, that has a profound geometric implication, which is known as length contraction. That is to say, from the stationary clock's point of view, it appears that the moving clock is going to shrink along the direction that it's moving. But don't just take my word for it. Let's get mathematical and we'll see why length contraction is a thing and how exactly it works. All right. So, much like we did before, let's go ahead and draw a full tick of the stationary clock as well as the moving clock. Now because the stationary clock is not moving the analysis for that is pretty much the same as we saw before. So if we imagine that the photon leaves the left mirror at time t kn then it hits the right mirror at time t kn plus t /2 because t /2 is half a tick of the stationary clock. So in that time the light travels a distance of c * t /2. Distance is speed time time. Then likewise on the way back it travels the same distance ct /2. All that's to say for the stationary clock the length L between the mirrors is ct /2. Or if you want to think about that in terms of t the time that it takes for the clock to tick is 2 * l / c. So now we can go ahead and try to apply very similar reasoning to the moving clock. But this time let's think about the three moments t kn t1 and t2 which corresponds to the photon leaving the left mirror hitting the right mirror and then coming back to the left mirror respectively. And so we're about to analyze what is the length between the mirrors of the clock when it's in motion which we can figure out based on the fact that we know the tick of the moving clock t prime is gamma * t and the fact that light travels at c. that will let us solve for the distance between the mirrors of the moving clock that is L prime as a function of L. And when we calculate that L prime to L ratio, we're going to see that the moving clock is going to shrink along its direction of motion. So to do that, the first thing we want to do is solve for the time t1 when the light hits the right mirror. We can solve for that by writing out the equation shown here. On the left side of the equation, we have the speed of light time the difference between t1 and t kn that is the travel time between the light going from the left mirror to the right mirror times the speed of light. So the left hand side of the equation is going to be the distance in space that the photon travels across as it goes between the left mirror and the right mirror. And what is that distance? Well, it's going to be the length between the two mirrors. That is L prime plus however much the clock moved during that time. And the distance that the clock moved during that time is going to be the speed of the clock v times that time interval t1 minus t kn. So in this equation all we've done is we've written in two different ways the distance that the light travels as it goes from the left mirror to the right mirror. From there it's just a matter of algebra to show that t1 = t plus l prime / cus v. Right? Cuz you subtract from both sides the v t1 - t term. Then you go ahead and divide by c minus v and then you add t kn. Well, okay. So now we have the time t1 relative to the starting time t kn as a function of the distance between the mirrors and the speed of light and the speed of the moving clock. We can then go ahead and repeat the analysis to solve for t2. Here again on both sides of the equation we write the distance in space that the photon travels as it goes from one mirror to the other. So on the left hand side we have the time difference t2us t1 * the speed of light. And then on the right side we have the length between the mirrors but this time we're subtracting the speed of the clock times the time difference because the clock is moving to the right and the photons moving to the left. And so the motion of the clock is going to effectively shrink the distance that the photon has to travel in order to get to the left mirror. And then we can go ahead and solve this equation for t2. And we see that t2 = t1 + l prime / c + v. Now these two equations for t1 and t2 are basically the same equation because in both cases that time step is going to be the length l prime between the two mirrors divided by the speed of light c but where that speed is adjusted by the speed of the clock v. Now between there and back the sign of that adjustment is going to differ because on the way there the motion of the clock is going to increase the time step because the light has to catch up and then on the way back the speed of the clock is going to decrease the time step because the clock is moving towards the photon but the speed of the clock is the same both there and back and so the equations are identical other than that sign difference. Well then now you know we can do something really interesting which is go ahead and write t prime the duration of a tick of the moving clock simply as t2 minus t kn which if you look at these equations shown here you substitute t1 into the t2 equation and you end up seeing that t2us t is simply l prime time this 1 / cus v from the first equation plus 1 / c plus v from the second equation. So all that is is we've just added up the two time steps that we just calculated and that gives us the total duration of a tick of the moving clock. But now if you look at this equation here and you do a little bit of pattern matching and you recall the definition of gamma as 1 over the<unk> of 1 - v / c^2. Well as it turns out this equation here simplifies very nicely to the expression 2 * lime c * gamma 2. But remember, we also know that t prime is going to be the lorren factor gamma * t because we already know from the time cube that the moving clock is going to have to slow down by the lorren factor gamma in order for the whole time cube to remain synchronized for every inertial observer. So we can bring that equation in as another constraint on what t prime has to be. Now then the next move is to replace capital T with 2 L / C because as we calculated earlier the time that it takes for the light to go there and back is going to be twice the length between the mirrors divided by the speed of light. So that tells us that T prime is going to equal 2 * L / C * gamma. But now look what happens if we bring together these two different constraints on what T prime has to be. On the one hand, t prime is 2 L / C * gamma. But on the other hand, T prime is 2 * L prime / C * gamma 2. And at first glance, those are not the same thing. But they're both T prime. So they have to be the same thing. And the only way for these to be the same thing is if the distance between the mirrors of the moving clock L prime absorbs a factor of 1 / gamma. Because if L prime equals L over gamma, then L prime * gamma^ 2 is going to be the same as L * gamma and then T prime= T prime and it's all good and there's no problem. But then when you think about it, gamma is a number that's 1 in the non-relativistic limit and then blows up as you approach the speed of light. So that is to say that 1 / gamma is going to be one in the non-relativistic limit but then is going to approach a very small fraction as you approach the speed of light. So if we have a clock that's moving with some speed v and therefore some loren factor gamma, it's going to shrink along the direction of motion by a factor of 1 / gamma. And that has to be the case when you think about the synchronization of the time cube and when you think about the amount of time it takes for the light to bounce around between the mirrors, which is of course the derivation that we've just done. So what have we figured out so far? Well, based on what we now know about the perpendicular and the parallel light clock, let's go ahead and reassemble the time cube. And we see that from the perspective of a stationary light clock, the moving light clock is going to tick gamma times slower and is going to be gamma times shorter along the direction of motion where of course gamma is the famous Loren factor. So just to make this very concrete, imagine we have a time cube that's 15 cm or about 6 in. Then each tick of that clock is going to be about a nancond. Then if you take the same time cube and move it with some speed v and some lorren factor gamma then it'll tick once every gamma nonds and along the direction of motion its side length is going to shrink to 15 / gamma cm. Now one thing I'll mention in passing and we'll think about this in more depth later is that from the moving clock's point of view it doesn't notice that it's shrinking because if it did then you could use a time cube to measure speed and that's not allowed. So from the moving clock's point of view, it doesn't notice any shrinking at all. It's only from the stationary clock's point of view that the moving clock shrinks along its direction of motion. But we'll talk more about that later when we get into Loren boosts. But before we talk about Loren boosts, we still have more generalizing to do because even though a time cube is three-dimensional, it's still not the same thing as matter. So we still have to make our argumentation more generic. So the next thing we want to do is to analyze the fully generic light clock which we can define as an arbitrary closed loop in three dimensions. And this is very much like the planer light clock that we analyzed earlier but now we're lifting the restriction that the x coordinates have to be zero. And so now the x y and z coordinates can all be totally generic. And yet again there's no restriction on the number of points n in the loop. So if you like to we can let n approach infinity and imagine this as a continuous closed curve. As before the same rule applies where as the light reaches the nth coordinate it then returns to the first coordinate and so it's going to loop back over on itself over and over again. Now I should point out that the following argument does not require the light loop to be perfectly identical every time. That is, it doesn't have to return perfectly to X1 because in reality, if you actually build a light clock, it's going to have some manufacturing flaws and no loop is perfect. But it just has to be close enough that despite the manufacturing flaws, you could still look at the thing and say, "Okay, yeah, that's light going around in a loop, at least well enough that it's more or less consistent so that you can count the number of loops over time." Well, all right. Then if you go ahead and define some set of points for the stationary light clock, then the moving light clock is going to have the same set of points, but they're going to be adjusted based on the velocity that the clock is moving from the stationary clock's reference frame. Now, in the following argument, there is rotational symmetry such that we have no loss of generality if we imagine that the moving clock is moving along the x direction because we're going to be considering a totally arbitrary set of points and the set of all arbitrary sets of points includes all possible orientations of every set of points. So, for the sake of simplicity, we can pick out the x-axis as the direction of motion for the moving clock. Now then for the stationary clock let's go ahead and define d sub i as the distance between the i coordinate and the i + 1 coordinate. That is d subi is the distance of the leg between those two points. So then because distance is speed time time we know that the light is going to go from one point to the next in a time step that we'll call delta t subi which is going to equal d subi / c. Because if you know the distance then we know the time. And so based on the set of coordinates we can construct a set of distances between each point as well as a set of time steps between each point. And also if we want to think about this in terms of the distance formula, we can say that c * delta t sub i^ 2 is going to be delta x sub i^ 2 + delta y sub i^ 2 plus delta z sub i^ 2. where delta x, delta y, and delta z are the steps in x, y, and z respectively between the i and the i + 1 point. So that's all just math. So now let's think about the moving clock. And what do we know about the moving clock? Well, first of all, we know that delta x prime subi is going to be delta x subi divided by the lorren factor gamma. That is to say, between any two points in the loop, the x separation between those two points is going to be identical except for the one over gamma scaling that comes from length So the whole set of points for the moving clock is going to be squished as seen from the stationary clock by a factor of 1 / gamma. However, delta yp prime and delta zprime are going to be the same as deltay and delta z because as we saw when thinking about the perpendicular light clock, there's no squishing along the directions perpendicular to the direction of motion. So those are the same. When we put this three-dimensional set of points in motion, it's not going to stretch or shrink in the yz plane. Now, another thing we know about the moving clock is that in the time step delta t prime sub i, the point x prime sub i + 1 is going to drift by an amount v * delta t prime sub i. Right? Because if you imagine you're at the point x prime sub i and you're a photon trying to get to the point x prime sub i + 1, well, the whole set of points is moving with speed v. And so that point x prime sub i + 1 is going to drift in the time that it takes you to get to it by that times the speed of the clock. And that's very much what we saw earlier when we were analyzing the parallel light clock. So now let's go ahead and write out the distance formula like so where on the left hand side we have the total distance between two points that is the speed of light time the time step between those two points squared and that will equal the sum of the squares of the separations between the two points along x y and z respectively. Notice that this formula is pretty much exactly the same as the distance formula we saw earlier for the stationary clock. But the step along x is shrunk in accordance with length contraction with that factor of 1 / gamma and it's also offset based on a factor of the speed of the clock time the time step. So in that delta x term we can see the relativistic considerations coming in. So then suppose we look at this equation like a mathematician and we just say you know let's go ahead and solve for deltat t prime sub i. Well that can be done because what we have here is just a quadratic equation in the variable delta t prime sub i. So we can go ahead and solve that by doing a bit of algebra and applying the quadratic formula. And then we end up with the equation shown here. delta t prime sub i is equal to gamma * delta t sub i + v * delta x sub i / c ^ 2. Now to see why that's the case, it's just a lot of boring algebra. And so we really don't have to get into the details here, but I'm going to show them on the screen so that if you're curious or if you're skeptical or if you just want to do this as an exercise, you can go ahead and pause the video and go through all of this algebra and you can see that the math checks out and it's all good. But we really don't have to spend much time on that because the moral of the story is that we have a quadratic equation in delta t prime subi. We go ahead and solve that using some algebra and then we end up with an expression for delta t prime subi in terms of gamma and delta t subi and v and delta x subi and c. And that is a very interesting equation because we can use it to calculate the total time that it takes the moving clock to tick. You see because t prime that is the time that it takes the moving clock to tick is simply the sum of all the time steps delta t prime sub i over the closed loop. So then using the equation we just derived we can write that as a sum over delta t sub i and a sum over delta x sub i like so. Now then if you look at that term sum over delta t subi, you can recognize that that is the time that it takes the stationary clock to tick because it's the sum over all the time steps between all the points in the stationary clock. So the sum over delta t subi is nothing more than capital t. And then also notice that for a closed loop, the sum over all of the x distances between all of the points is going to equal zero cuz those are the stationary x differences. is the delta x sub i. And as you go around the whole loop, you end up back where you started. So there's a net change of zero along the x direction. And so that is going to make that second term completely go away. And we see that t prime equals gamma * t just as expected. That is to say, for an arbitrary closed loop in three dimensions, if you put the thing in motion, you find that it will exhibit time dilation with exactly the same loren factor as we calculated when looking at the simple example of a perpendicular light clock. So, wow, that's quite a generalization. All right. So, we've now extended our light clock reasoning to show that light going around an arbitrary closed loop in three dimensions is going to exhibit time dilation and length contraction with the same Loren factor that we calculated in the more simple analyses. But now, my friends consider this. That which is true for the thread is true for the sweater. So let's weave together a whole bunch of arbitrary loops in three dimensions into a bundle of light. So you see a bunch of loops makes a bundle. And then if you ask yourself, hey, would this whole bundle of light exhibit time dilation and length contraction if it were set in motion? Well, the answer is yes, because every individual loop in the bundle exhibits time dilation and length contraction, as we've now proven. So if you apply that reasoning in parallel to each loop in the bundle, you see that yes, the whole thing is going to exhibit time dilation as well as length And in fact, the loops in the bundle don't all have to have the same period. We can stretch and warp it a little bit and have loops that take longer and shorter. And when you set the thing in motion, of course, a shorter loop is going to be a shorter loop, but it's still going to slow down by the same loren factor as any other loop in the bundle. So even if you have a kind of complicated breathing loopy bundley thing of light and you send it in motion, it'll exhibit time dilation and length contraction. By the way, what I'm showing here, this is just a donut. But you can imagine all kinds of more complicated ways that a bunch of light could be swirling around in a kind of bundle or like a super complicated system comprised of many bundles. It's a fun exercise for the imagination to try to think about all the different kinds of swirly shapes you can imagine that we've now proven will exhibit time dilation and length contraction when set in motion. And just to clarify, this particular donut is just an abstract image of some stuff swirling around. It is not a representation of like a particle or something, right? So this is just an animation of a mathematical idea. Specifically, I made this by taking the hop vibration, projecting it from four dimensions into three, and then wiggling the parameters of that projection to make the donut come to life a little bit. And if you want to learn how to do that, guess what? You can download the code that made this very animation on my Patreon. The way that works is if you sign up on Patreon with a paid subscription to support my channel, you get access to all of the animation codes that go into these videos. And those codes often contain extra and more specific mathematical insights that go a little bit beyond what we talk about in the video itself. So, it's a kind of bonus content and it's not pay per code. It's anyone who's a paying member gets access to all the codes. Also, if you sign up on tier 2, I'll put your name in the end credits of the videos. But anyway, back to the ideas. There's one more step that I want to take to generalize these ideas that we've been talking about. And this is a subtle but profound step. And all it is is the realization that the speed of light and the speed of gravitational waves in our universe travel at the same speed. And so therefore, we live in a universe where there is such a thing as a the And with that in mind, you realize that in all the reasoning we've been talking about so far, it doesn't matter that we think of the things swirling around in this bundle as being photons of light. because instead these could be any kind of massless spec that swirl around at And so the final step in our generalization is to relax the constraint that these swirly specs represent photons and instead we'll let these be specks of uh unspecified essence. That's kind of a strange term, you know, unspecified essence, but uh hey, you know, I think it works. So yeah, we'll go with that. That's all good. All right, then. So let us now transform these photons into a bunch of specs of unspecified essence. A bunch of specs of unspecified essence. Unspecified essence. Wow. So generic. I bet it will exhibit time dilation if set in motion because that's what a swirly light speed thing would do. All right. Well, I suppose the next thing we should do is to think about how matter is a swirly phenomenon. And you know, the easiest way to see this is to think about chemistry. Chemistry, when you get right down to it, is all about how particles of matter interact with the electromagnetic field, right? Because in chemistry, you're thinking about atoms and their bonds and molecules and all that. And all that really is is atomic nuclei with electrons bound to them via electromagnetism. and then the various complicated patterns that arise from the different orbitals and chemical bonds and all of this kind of stuff. So, chemistry of course is a very complicated thing. But at the end of the day, if you have a bunch of atoms and you think about the interactions of those atoms with the electromagnetic field as being mediated by the exchange of photons, which is one way to think about it, then you see that matter, at least at the descriptive layer of chemistry, is comprised in part of swirly patterns of photons. And so if that's all that matter were, never minding the elementary particles themselves, but just thinking about the electromagnetic interaction as swirly photons, well then that aspect of matter should exhibit time dilation and length Oh, and by the way, in chemistry, it's not necessarily the case that you can trace out photon loops, like loops of photons, where a photon is going to swirl around in a perfect loop. But if you think about our mathematical reasoning we've done so far, you realize that the photon doesn't really have to loop back in on itself. The complicated messy bound state where you have a bunch of particles flying around at light speed and staying in roughly the same area would still totally exhibit time dilation and length contraction. In any case, we see that chemistry is in large part the swirling of photons. And if you zoom out from there and you think about material science and just stuff, just regular stuff, then you see that the matter that we're made out of and the matter that's all around us is indeed a swirly phenomenon where you have this bound energy. And when you see that, you realize that if you have an object at rest, it's not that the object isn't moving, right? Right? So, if you have a rock on the ground, you think about that rock, the particles that make up that rock are moving at crazy high speeds. They're just not moving very far because they're bouncing around and the net motion is zero. And so, if you have an object that's just sitting there, the pattern is stable, but the things that make up the pattern are still moving around at unfathomably high speeds. The photons, of course, are moving at the speed of light. But then of course the question comes up, what about the electrons? And what about the atomic nuclei which are comprised of protons and neutrons which themselves contain quarks bound by gluons? It's like what's going on with that? Like what are those things? Are those things like really solid real actual bits of matter that exist permanently? Well, not exactly. I mean, yeah, kind of, but not really. Sort of, but no. Usually, but sometimes, yeah. So, well, okay, here's the thing. Let's talk about it cuz this is actually really important for this topic. So the thing is even elementary particles are less solid than they seem. So I want to say a few words about the fundamental nature of matter that is of the elementary particles. But it's important to point out first and foremost that the fundamental nature of matter still remains largely unknown. For example, we have no idea why the standard model has the particles that it has and why those particles have the properties they do. We also don't know what the deal is with the gauge group of the standard model. That is SU3C cross SU2L cross UNY. We have no idea why that's a thing. We know its consequences, but we don't know where it comes from. We also have no idea whatsoever why there are three generations of matter. That's like absurd. Like why would that be? I don't know. No one knows. And etc. etc. There's many other things we don't know. We don't know why there's a Higsfield. We don't even really know what the Higsfield is. Our current best understanding of the Higsfield is pretty similar to the Ginsburg Landau model of superc conductivity with a relativistic upgrade and a few other nuances. By the way, I talk about those in my video superconductivity in the Higsfield. But anyway, I'm derailing on a tangent here. The point is there are so many things we still do not know about physics. The standard model, even though it describes so much, is still to this day full of many very deep mysteries. And science is not yet over. Okay, the story has not yet come to an end. We're still figuring things out. But one thing we do know, which is super relevant for the topic of this video, is that a particle of mass m contains E= MC^ 2 of energy when it's at rest. Now, when the particle is moving, it's going to contain even more energy because it's moving. But in the particle's inertial frame where it's not moving, it'll still contain a pretty enormous amount of energy just by virtue of having mass. Makes you wonder what mass is. In fact, you know the famous E= MC². Really, you should think about it as M= E C^2. That is the phenomenon of mass can really be thought of as the energy contained in a particle divided by the speed of light. Now, this famous equation E= MC², it comes right out of the postulates of special relativity. But we also have plenty of experimental evidence that this is true. Nuclear power, nuclear bombs, the sun, posetron emission tomography. There's all kinds of natural phenomena and technology that demonstrate the fact that E= MC². As a case in point, let's go ahead and examine how an electron and a posetron can annihilate and in so doing they turn into pure light, typically two photons. Now this interaction electron and posetron becoming two photons, this has been studied extensively in all kinds of particle accelerator experiments. But what's cool about this is this is also the basis for how a PET scan works. If you've ever had a PET scan, how that works is they inject you with some radioactive sugar and then that sugar disperses into your various tissues and things. And because it's just a little bit low-key radioactive, what happens is every now and then that sugar puts out a posetron via a decay process. And then because that posetron is in your body surrounded by matter, it very quickly is going to encounter an electron. And then the electron and the posetron annihilate each other and two photons come flying out. The two photons are then measured by the detector that circle around you when you're in the PET scan. And using the trajectories of those photons, an image of your body can be reconstructed. This is truly a magnificent technology. I mean, how amazing, really. And it just goes to show you, in case there are any skeptics out there about antimatter, no, it's a real thing. It's really real. Like, for real. And one of the reasons that we know this is real, is that electrons and posetrons both have a rest mass of about 511 kilo electron volts divided by C^ 2. And so each outgoing photon is going to have an energy of 511 KVs. And they're going to come out equal and opposite as well by momentum conservation. So the signature of this kind of annihilation is very specific. You get two backtoback 511 keV photons coming out. Well, usually that happens. Every now and then the situation is messier. You might get three photons coming out, that kind of thing. But that's rare. more than 99% of the time you get two photons coming out. So yeah, this has been a pretty wellstied phenomenon. And in fact, you know, something cool just while we're on the topic, this 511 keV peak has been observed in particle accelerators of course, but has even been seen coming from the center of the galaxy. In this chart that I'm showing here, and you can read the paper cited here for more information, but if you look at this chart, you can see that it's photons versus the energy of the photons in this observation. And you see right there, there's a peak at 511 KVs. So they looked at the center of the galaxy and they saw this peak in intensity of the kind of light that's produced in this annihilation. And one of the cool things they've noticed from this is that hey, there's more of this than there should be based on what people were expecting. Now, it's probably not aliens. It's probably some stars colliding or some intense cosmic phenomenon or whatever. But anyway, the reason I bring up this antimatter thing is to show you that they seem. They can be deleted or untangled or canceled out or however you want to think about the interaction shown here. The truth is, no one really knows exactly what's going on when this kind of annihilation occurs. A more thorough explanation of this would require a more advanced understanding of what matter even is, but that's a problem for future generations to figure out. For today, all we know is what we know today. So now let's segue our meditation into the observation that nothing is indestructible and that therefore time dilation is not surprising. What I mean by nothing is indestructible is that every particle has an antimatter version. So if you look at the particles of the standard model, for every single one of these, you could also have an antimatter version of the same particle. And when the matter particle encounters its antimatter evil twin, they're going to annihilate and send out a couple of high energy photons. Usually it's two photons depending on which particles and all that. There's nuance there, but doesn't really matter. The point is any particular particle of matter under the right circumstances could be turned into light. And so therefore, there's no such thing as matter which is truly indestructible. Now, we don't yet know what the future theory is going to be for elementary particles. It's going to have to go a level deeper than what we know today. But of course, in the appropriate limit, it's going to have to cohhere with what we already know. But you can imagine that some future theory might be pretty bizarre. It might involve different kinds of mathematical structures. You might have fields. You might have strings. You might have marov chains or cellular automa or whatever mathematical structure you want to have. But let's go ahead and just make one very modest assumption, which is that in this hypothetical future theory of elementary particles, cause and effect are going to travel at C, at least for all intents and purposes. So, one specific example of this is there's this theory going around that maybe an electron is like a donut of light. Now, that theory doesn't work because a donut of light is unstable. And besides, it doesn't match experiment. But even still, you know, it kind of feels like there might be a grain of truth to the idea that elementary particles are some kind of soliton configuration of some field or whatever. And if that's the case, or if something like that is the case, then you can imagine sprinkling into the theory a bunch of our patented specs of unspecified essence. And you can watch them flow around like sawdust on a river to see the flow. And then however the particle is described in this theory, if cause and effect is swirling around as a bunch of specks of unspecified essence traveling at sea, well then you come to realize that time dilation and length contraction are to be expected. They're not surprising. They're what we would expect. But the reason it's important to see that it's to be expected is because when you first encounter relativity, your intuition fights you every single step of the way. And maybe you think differently, but if you think kind of like how I think, then the first time you hear a moving clock ticks more slowly than a clock at rest, you're going to be like, "No, come on. No way. Get out of here. What are you talking about? That doesn't make any sense. You're going to be this uh serial um skeptical serial munchin guy.jpeg." And it just doesn't make any sense. And you can try to learn the math, but the whole time your intuition is going to say, "No, this doesn't feel right. this doesn't make sense fundamentally. Even if you know the equations, maybe it doesn't make sense. That's totally normal when you learn physics. That's like one of the stages of the process. But then you think about it for a while and you think about the swirling and the math and the light clock and then all of a sudden it hits you all at once and you realize, oh, more move, less swirl. More move, less swirl. Oh, whoa. Hold on now. This is making way too much sense. You see what I mean? you you get this moment of, oh, of course, if matter is a bunch of swirly cause and effect, a bunch of bound energy, then yeah, if it's swirling around, if you move it, it's going to be moving more and therefore swirling less and therefore it's going to slow down because it's mo it's moving. It's too busy moving. It doesn't have as much time for time, you see? And then you see it. And that's the thing because now your intuition doesn't hold you back. your intuition pulls you forward and it pulls you forward strongly and powerfully because once you see that more move equals less swirl now there's no going back and from then on out in your journey in relativity it becomes way easier because then it's just math once you see it it's just math so that's why it's important to see it and that epiphany might be worth a like and a comment but to earn your subscription my friend we have to go deeper because you see relativity gets even a little weirder than what we've talked about so far Because as the hippies like to say, it's all relative, dude. Everything we've just been talking about is true, but we've been thinking about everything from the point of view of the stationary clock. So, we've been imagining this green stationary thing and then this moving magenta thing, which exhibits time But here's where it gets extra weird. Because if you think about it, both clocks are at rest from their own point of view. So, can't we think about it the other way around where the magenta thing is stationary and the green thing is moving and time deating and length contracting? Well, yeah, we can. Both clocks see each other as ticking more slowly and as being shrunk along the direction of Now, that seems extremely paradoxical. How can both clocks be slower than each other? That doesn't make any sense at all, right? It seems downright impossible. But this is where relativity breaks our conventional notions of space and time and we have to think about space and time as a unified spaceime. Now, in order to do that, we're going to have to talk about this concept of a lorren boost. And so, in just a moment, we're going to derive the Lorren boost and we're going to talk about space-time diagrams, and we're going to see how this all actually makes sense when you think about it. And in the process, we're going to actually rederive time dilation and length contraction from scratch without even trying to. So that'll be fun. So yeah, you know, hey, without any further ado, let's get into it. Let's derive the lorren boost. All right. So picture this. Imagine we have a green dot moving to the left and a magenta dot moving to the right. And those are the only things. There's no other things. Suppose this is happening way out there in space. So each dot is just floating along. And imagine that right at that intersection point, they somehow miss each other by just a little bit. So they don't collide. It's a near miss. And they keep on floating along the x-axis. Now, if you were to ask the green dot, "Hey, green dot, what happened?" it would say, "Oh, I was just out here in space just minding my own business and all of a sudden this magenta dot came on in and went right by me." And then if you ask the magenta dot, "Hey, magenta dot, is that true?" The magenta dot would say, "No way. That's not true. I'm the magenta dot. I was just out here minding my own business and all of a sudden this green dot came floating right by me." So the dots would disagree on what happened, but it's like dots, dots, come on now. You're both describing the same reality just from different points of view. And that's the nature of reality. It looks different depending on your point of view. But also, regardless of your point of view, there are certain things about everyone's description of the world that all have to cohhere into a sensible description of reality. So that when you put all of the points of view together, you end up with a story that doesn't contain any logical contradictions or paradoxes. So this dot example is a good illustration of this fact because we've seen that there are at least three different ways of describing the same situation. And you can imagine also there are infinitely many other points of view where you're letting your perspective drift along the x-axis by some amount other than what's shown in these three examples. So for example, you might be kind of close to the green dots point of view but drifting a little bit and you could still describe what happened. So that all makes sense. I mean that's pretty intuitive, right? Like it makes a lot of sense when you think about it. And so on that very intuitive foundation, let's go ahead and construct this idea of a space-time diagram. Oh, but you know what? Let's actually go ahead and make one upgrade to the situation, which is where when the two dots almost intersect, you know, they have that near miss and they keep on going. Imagine at that moment, at that place, and that time, a couple of photons are emitted, which travel each way along the x-axis. And suppose also that the relative speed between the green and magenta dots is a good fraction of the speed of light. So in that case we have the situation shown here where you can see that right as the two dots come together a couple of photons come flying out. Now this animation shows the very strange very bizarre fact that the speed of light is the same in every inertial frame. And by the way, an inertial frame, that's just the point of view of someone who's not accelerating. So they're drifting along at some constant velocity. And from their point of view, they can say, "Hey, I'm not even moving. It's the universe that's moving." Okay? So the point is for all observers who are just drifting through the universe under their own inertia. From all of those points of view, the speed of light is the same for everyone. Now, this is an incredibly bizarre and weird fact, and I don't want to downplay its weirdness by giving you some explanation for why this has to be the case. The truth is, nobody really knows why the speed of light is the same for everyone. Asking why the speed of light is the same for everyone is sort of like asking why there are three spatial dimensions in our universe. It's just a brute fact of reality that we have to contend with. Now, we know that it's a fact because it was confirmed by the Michaelelsson Morley experiment and since then it's been confirmed by many experiments to extreme precision. So, we have very good reason to believe that the speed of light is indeed the same for everyone. But we don't have any theoretical explanation for why this is the case. It's something that we have to take on as a first principle in our present understanding of reality. I can hear some people objecting, but wait a minute. Maxwell's equations. You know, famously, one of the first things you do when you're learning Maxwell's equations is you calculate the speed of light from the permeability and permitivity of free space. And that's sort of a right of passage. And when you're studying physics, when you calculate the speed of light, you're like, "Aha, I've tapped into something deep about the nature of things." But the problem is that calculation should really be taken as a demonstration that electricity and magnetism are two sides of the same coin and that light Magnetism is just special relativity combined with local U1 symmetry. And all of Maxwell's equations and the Laurent's force law and all of that, all of the structure of electromagnetism is all inherently built on top of special relativity. And so that speed of light calculation is still very profound. And it does show you that the speed of light is the same in every inertial frame within the theory of electromagnetism. But what you have to bear in mind is that fact was baked into electromagnetism from the start. Anyway, if you want to learn more about that, in my video electromagnetism as a gauge theory, I show you how you can derive all of electromagnetism from special relativity and local U1 symmetry. But that's a bit of a tangent and I only bring it up to highlight the point that the speed of light being the same for everyone is genuinely something that we have to take as a first principle that is imposed upon us by the results of very many experiments that are saying, "Hey, this is how it is." And even though it's confusing, we just have to deal with that. We just have to figure out how that works and how that makes sense. And when you think about it, you realize, well, the only way that could possibly make sense is if space and time can bend to accommodate everyone's perspective under the constraint that the speed of light has to be constant. Just to make that point very concrete, let's go ahead and pause the animation here and think about how far these photons are separated according to everyone's perspective. Well, the distance between the two photons, call that L, is going to be 2 * the speed of light time the amount of time since the photons were emitted. And all that is is speed* time 2 because you have the two photons going different ways. Now, in all of these different frames, everyone has the same speed of light. And so, everyone agrees that the photons are separated by that distance L. But now, you can very clearly see what the problem is. because each point of view is telling an apparently contradictory story. See, in the green dots frame, that left photon is pretty far away from the green dot, and the right photon is pretty close to the magenta dot because the magenta dot is moving in that direction, chasing after that photon. But then contrast that with the magenta dots frame where the left photon is pretty close to the green dot and the right photon is pretty far away from the magenta dot. So those things seem incompatible. It seems like different observers are telling different stories about the separation of things in reality. And it seems like those different stories should not be able to cohhere into a unified picture of So when you think about this diagram, you realize, wait, what? The speed of light can't possibly be the same for everyone unless spacetime is able to bend to accommodate this weird fact about the speed of light. And as we know that is in fact what happens. So what we're about to derive is something called the Laurent boost which is a way of boosting our imagination from one point of view into another that's moving relative to it in a way that takes into account the constant speed of light and therefore the bending of space and time as we boost from one point of view to another. And once we figure out the loren boost, it'll make everything click and we'll have a really coherent picture of time dilation, length contraction, and the relativity of simultaneity that is the tilting of the present moment from different observers point of view. So this is a very deep foundational concept in special relativity. And in order to figure it out, we're going to use something called a space-time diagram, which is a diagram where we draw time as one of the dimensions so that we can imagine things happening in space and time without having to animate our imagination. Because when you draw time on the page, you can just look at it. And an unfolding of events along a single axis of space now becomes a static image in a two-dimensional diagram with space and time. So let's go ahead and draw a space-time diagram of the situation shown here. And first we're going to start from the center of mass frame. And then we're going to figure out how to transform that is how to loren boost the space-time diagram into the green dots and the magenta dots different points of view. All right. Then now to make a space-time diagram, all we have to do is sweep the horizontal spatial axis, the x-axis, along a vertical direction, which in this case represents time. So this is a diagram in the xt plane. And as you can see, the dots approach each other, the photons are emitted, and the dots go flying away. Now, by the way, if you don't mind, let's go ahead and add one more thing to this diagram just to make it balanced. You know, it looks better if it's balanced. So, we'll go ahead and add a couple of lines corresponding to photons that are coming in and then meeting up where the dots intersect and then flying out. So, it's the same thing we imagined before, but just imagine that there's now photons coming in as well. That gives us the nice cool looking symmetrical space-time diagram shown here. Let's go ahead and say that the point where the two dots intersect is going to be the origin of the xt plane. So x= 0 and t equals zero. And this is of course an arbitrary choice because we're able to put our coordinate system wherever we want it to be. So we might as well center it at the interesting point where all of these things are coming together. Now, when you think about the nature of the space-time diagram, you realize that a dot moving along X at constant speed is going to be a line in the XT plane. Because to drift along at constant speed is to trace out a straight line of constant slope in the space-time plane. And as you can tell based on the slopes of these lines in the space-time diagram, the speed of these two dots relative to each other is a good fraction of the speed of light. Because in a space-time diagram, the speed of a thing can be seen in how tilted its trajectory is in the space-time plane. And by convention, let's go ahead and use natural units where the speed of light is one. And so therefore, these lines for the photons are going to be at 45°. Now, the reason I wanted to include incoming and outgoing photons along with the green and magenta dot is so that they'll trace out something known as the light cone. So, for any event in spaceime, you can draw the light cone by thinking about photons coming into the event and also the photons going out from the event. Now, in this example, we have motion along the x-axis. So, we're thinking about one dimension of space and one dimension of time. And so our light cone is going to be this two-dimensional thing in the xt plane. But if instead we were examining motion happening in two dimensions of space, then we would have a 2 + 1dimensional space-time diagram. And the light cone would actually look like a cone in that three-dimensional space-time diagram. And then in the full three spatial dimensional context, the space-time diagram is this 3 + 1dimensional thing. And the light cone is like a bubble of light that starts off really big, comes in, shrinks to zero at the event, and then expands and goes out towards infinity as a growing bubble, like a bubble that grows and grows forever. Now, one of the cool things about the light cone is that it divides up all of the possible events in the universe into two different categories. So relative to the collision event here where x and t are both zero, anything outside of the light cone, we call that space-like separated from our event. And anything inside of the light cone, we call that timelike separated. Oh, and then any event that's perfectly on the light cone, we call that lightlike separated from the event. And what's interesting about this is you realize that all of the possible things that could have possibly affected our event are all within the past light cone of the event. And also all of the things that our event could possibly have a causal influence on in the future are all going to be within the future oriented light cone. That is to say that all of the events that our event might have some kind of cause and effects connection with are going to be within or on the light cone. And anything outside of the light cone that is anything space-like separated is going to be too far away to have any effect on our event or for our event to have any effect on it. So the light cone is a really interesting concept and it arises in any theory where you have a maximum speed of cause and effect. Now then when you think about the basic ingredients of special relativity, you realize that we should be able to tilt from one reference frame to another without tilting the light cone. Now this has some weird implications about the nature of space and time that we'll talk about in just a moment. But before we get to that, I just want to focus on the two concepts that we're merging together here. So the first one is that the same physical situation can be thought of from different inertial frames. That is different observers that are drifting along at constant speed are going to have different descriptions of what happened. And yet all of those descriptions have to be able to cohhere into a unified picture of reality. So in a space-time diagram, an inertial observer by definition is going to drift along a straight line in the diagram because a straight line is going to have some constant rate of change of space versus time. That is some constant velocity. Okay. So, because we're going to have a bunch of different perspectives on what happened with these dots and these photons, that means there will be a bunch of different space-time diagrams which describe the same situation from different points of view. And if you want to find out whose point of view a space-time diagram corresponds to, well, just think about who is the vertical line in the space-time plane. Because a vertical line in a space-time plane is something that's not drifting in space. it's moving purely along time. That is to say, it's standing still and just waiting around. So, for example, here we're in the green dots point of view. And you can tell that because the line is staying at x=0 the whole time. And the magenta line is moving from left to right as time goes forward. Likewise, this is the magenta dot's point of view in which they're not moving. But now the green dot is moving from right to left as time goes on. And then of course if you're in the center of mass frame, the point of view of the center of mass of the situation, you see that both dots are going to come in and go out. So, we need some theory of spaceime that can accommodate all these different perspectives without tilting the light cone. So, how can we possibly do that? It seems counterintuitive. Well, let's go ahead and hop into the center of mass frame and we'll draw a grid like so. Now, in this grid, every horizontal line corresponds to the x-axis at some constant time. So if you think about moving along horizontally in our space-time diagram, you're moving purely along x and you're staying at a constant moment in time. So at that moment in time, you can think of a horizontal line as the now slice of that moment. That is the slice in spaceime which is simultaneous with that moment. Now then likewise a vertical line in this plot corresponds to staying at a constant point on the x-axis and not moving. So you're just sitting there and enjoying the passage of time without moving along the x-axis. Now then one more thing I'd also like to add to this grid is a whole bunch of 45° lines that correspond to the motion of photons between all the corner points of this grid. You see, because in the grid shown here, if you think about these corner points where the vertical lines and the horizontal lines and these 45° lines where they all intersect, you can think of those as an array of pulsing things along the x-axis. So the things are uniformly spaced along x and they're pulsing at some constant rate in time. So this grid of events given by the corners of these vertical and horizontal lines give us a nice uniform array of events in spaceime. And for convenience, we're going to have the same spacing between horizontal and vertical lines in our grid so that we can draw these 45° photon lines between all of these various events. And the reason I want to draw this out like so is so that we can see what has to happen to space and time as we hop from one inertial frame into another. So first let's go ahead and hop into the reference frame of the green dot. So, as you can see, space and time just got pretty weird because what was in the center of mass frame, a grid of events uniformly spaced in space and time, is now this weirdly warped kind of thing. And if you look at it, a number of things stand out as being pretty unusual. First of all, you'll notice that what were horizontal lines in the center of mass frame are now tilted in the space-time plane such that these pulsing dots which in the center of mass frame would be pulsing all at the same time are now going to be pulsing at different times from the green dots point of view. And this is the relativity of simultaneity which we're going to talk about in more depth in just a moment. But for now, I'm just mentioning that in passing as one of the things we see about the weirdness of the grid in the green dots frame. The horizontal lines have become tilted. Another thing that's kind of weird is that the intervals in space and time between these events on the corners of these grid lines have been stretched and skewed and are not the same square boxes that we had before. And that pertains to time dilation and length contraction. And those are also topics that we're going to explore in detail in just a moment. But before we get into those details, the purpose of this grid is not so that we can examine the details of the relativity of simultaneity or the details of time dilation or length contraction. but rather the purpose of this grid diagram is to provide an intuitive bridge between relativity and a constant speed of light for everyone and the concept of a loren boost which warps space and And so the way to think about this grid diagram is to not overthink it, but imagine that the grid is a material object that you're holding in your hands. And imagine that as you think about the situation from different observers point of view, you with your hands on this grid are tilting it and pulling it and stretching it and making these lines tilt into whatever reference frame you want to imagine. Like physically imagine actually doing that with your hand so that this grid is a material object, a real physical thing that you can feel. Now the nature of this material is that all of these diagonal lines, these photon lines will always have to stay at 45° no matter how you stretch and tilt the fabric of spaceime. So imagine that as the material constitutive property of this kind of grid. You can tilt it into different points of view, but all of those photon lines are always going to be at 45° no matter which frame you hop into. Now the spacings between them might grow or shrink in order to accommodate this. So this grid animation, it's all about making us feel that relativity combined with a constant speed of light implies that space and time have to get a little bit warped as we boost into different inertial frames. And if you see that and if you feel that, then you'll be ready for everything we're about to do. The other day I was drinking a sparkling water and I was having a delightful time sipping on my perier indulging in that bougie hydration and then a light bulb went off and I realized hey if we can carbonate water then we can carbonate Now you might be thinking rich of what possible utility is a sparkling dimension? Well, let's go ahead and draw a space-time diagram. As we extrude the x-axis along the time dimension, you can see that each one of these little fizzy bubbles popping is an event in spaceime. And so, when we draw the space-time diagram, we end up getting a sprinkling of bubbles in the space-time plane. And that makes it easier to see what's going on as we do a loren boost. This transformation is exactly the same transformation that we saw earlier with the grid lines along time and space and the rays of light connecting all the corners of the grid. So the space-time diagram here is undergoing that same transformation. But I find this easier to look at. You see, the point of the transforming grid diagram was to show us how relativity and a constant speed of light for everyone constrains the way in which the space-time plane has to transform under a boost. And that really compels you to realize that if the speed of light is the same for everyone, then necessarily space and time have to transform in this way. But that said, the grid is a little bit cluttered and kind of hard to look at. And personally, I find it easier to just have the sprinkling of bubbles in the space-time plane. So, as we're about to get into time dilation and length contraction and the tilting of the now slice, we're going to use the sprinkling of bubbles just as a way of giving a bit of texture to the space-time plane so that it's easier to see what's going on in spaceime under a loren boost. But before we dive into those specific details, let's take a moment to linger here and just absorb this visual spark plane. So the main thing to notice here is that as we do a loren boost, our sprinkling of events, our bubbles in the space-time plane are all moving along hyperbolic curves. That is to say, every inertial observer agrees on which hyperola any given event is. But as we hop from one inertial frame to another, the points in spaceime are going to slide along these hyperbolic curves. And you'll notice that the asmmptotes of these hyperolas are the light cone. And so you see right there that the light cone is not going to tilt under a loren boost because everything's moving along these hyperolas. So that is the first lesson of the carbonated x-axis. When we do a loren boost, the points in spaceime around our event are going to slide along these Now, the next lesson of the carbonated X-axis is that under a Loren's boost, the now slice is going to tilt from one reference frame to another. So, let's go ahead and boost into the Now, in this reference frame, the green dot is traveling vertically in the space-time diagram. That is to say, it's not moving along X. It's just waiting around and moving in time. Therefore, from the green dots point of view, the present moment at the moment of the near collision event is given by this horizontal slice in the XT plane. See, because all of the events along that horizontal line occur at the exact same time from the green dot's point of So, that now slice is the set of events that the green dot would say is simultaneous with its near collision with the magenta dot. But now then look at what happens to that set of events as we hop from the green dots reference frame into the reference frame of the magenta dot. Whoa. The green dots now slice tilted in spaceime. What what's going on with that? How can that be? Well, that's just how it is. Because when you do a Loren boost, things are moving along these hyperbolic curves. And so the set of events that the green dot considers simultaneous with the collision event is different than the set of events that the magenta dot would say is simultaneous with the collision event. simultaneity. It's one of the stranger things about relativity. And it shows that space and time in special relativity are really a unified thing that can't be cleanly separated out because what you consider to be the present moment depends on your speed. And if you have a different speed than someone else, you're going to disagree on what set of events are in the present moment. By the way, this is the crux of the twin paradox. This observation right here that the now slice tilts when you do a loren boost. And we'll see that later on in the video as we analyze the twin paradox. But for now, the important thing is to develop a comfort with the mental image that as you're doing a loren boost and all of these events are traveling along these hyperbolic curves, then the set of events which is simultaneous with any given event is going to change depending on your point of view. And so different observers are going to slice the present moment of spaceime at different angles such that for any given event, the two observers have a different set of events that they consider to be simultaneous with that event. All right. Now, I'd like to say a few words about time dilation and length We've already talked about those ideas in depth, so I don't want to linger on this too much, but I just want to show you that time dilation and length contraction are already embedded in the Lauren's transformation that we've derived purely from thinking about the fact that the speed of light is the same for everyone. All right. So, first of all, let's go ahead and hop into the center of mass reference frame and imagine that along each of the green and magenta lines, we put some dots that are spaced uniformly in time. So, what this diagram represents is someone who's in the center of mass frame of the dots and they see them coming together and going out. But now, we're also imagining that each dot is pulsing in time at the same rate as viewed from the center of mass frame. So we have a couple of pulsing dots floating together and then floating apart. Now then if we hop into the green dots point of view, you can see that now the magenta pulses take a longer amount of time relative to the green pulses. That is to say from the green dot's point of view, the magenta dot is pulsing more slowly just because it's moving relative to the green dot. And as it turns out, this is exactly the same calculation and the same time dilation factor that we derived earlier just by thinking about the light clock and the bouncing photon and all of that. So all of that reasoning is naturally embedded into the concept of Loren boost which is really cool. But now my friends, it gets even better because check this out. If we boost into the magenta dot's point of view, you see now it's the green dot that's pulsing more slowly. And so the magenta dot says, "Hey, I'm not going more slowly than green." No, the green dot is going more slowly than me. So you see that with this concept of the Loren's boost, which remember came purely out of the constraint that the speed of light has to be the same for everyone, we naturally see not only time dilation, but also the fact that it's all relative, dude, and that both of these things see the other's clock ticking more slowly. So that's wonderful. We have just reerived time dilation from the constant speed of light and have done so in a way which naturally accounts for all possible inertial observers. Wo now if you followed along with all the reasoning so far then you'll see why this has to be the case. But even still there's something deeply counterintuitive about the idea that two observers can both think the other one's clock is ticking more slowly. But the thing to bear in mind is that when we expand our understanding of spaceime with the concept of a loren boost, you realize that as long as these two dots are just drifting along, there's actually no logical contradiction with them both seeing the other as slower. The idea that that's contradictory comes from an old-fashioned Galilean notion that time has to pass the same for However, your intuition is not entirely wrong. Because if you had a situation where two dots start together and then fly apart and then come back together that is for example the twin paradox then in that case both observers would have to agree on whose clock ticked more slowly on average throughout the entire journey. So that is to say imagine the twin paradox where you have an earth twin and a space twin. The space twin goes out and comes back and when they come back the earth twin is older and the space twin is younger. But both the Earth twin and the space twin agree that the Earth twin is older and the space twin is younger. So even though it's weird that one twin aged more than the other, it's not logically contradictory. What would be contradictory is if the space twin comes back and the Earth twin says, "Hey, I'm the older twin." And the space twin says, "No, I'm the older twin." Or vice versa. Unfortunately, nowhere in the theory of relativity does that kind of thing ever occur because relativity is logically coherent. So maybe this is a helpful metaphor. Imagine that someone lends you a lot of money. So now you're in debt and you have to pay back the debt, right? Well, not necessarily. Not if you drift along through space forever and ever, never to reunite with the person who lent you the money. It's only when the lender catches up to you that you have a problem and you may or may not have to pay back your debt depending on your karate skills. So, it's the same thing in special relativity. Spacetime is flexible enough to accommodate a disagreement about whose clock is ticking more slowly. Just so long as if those observers ever reunite, then both of them have to agree on whose clock ticked more slowly on average. All right, then let's go ahead and hop into the green dot's point of view. And now let's go ahead and fatten up the green dot into a green line segment along the x-axis. So a dot moving along x is going to trace out a line in the space-time plane. And therefore, a line segment moving along x is going to trace out a strip in the space-time plane. The reason I bring this up is to show you that length contraction is also naturally embedded in the concept of the Loren boost. You see, because in this diagram, the green and the magenta line segments both have the same length in their own inertial frame. Cuz if we hop into the magenta dots frame, you see that now the magenta line segment is just as long as the green line segment was from its So, if we go ahead and boost back and forth between these reference frames, you can see that purely because of the way our carbonated space-time bubbles have to transform, it's necessarily the case that both the green and the magenta things are going to see the other one as shrunk along the x-axis. So, just like we did for time dilation, we've now reerived length contraction from the fact that the speed of light is the same for everyone. And now just again as we saw for time dilation, we now see the relativity of length So there it is. That's why length contraction is a result of the speed of light having to be the same for Oh, and by the way, you might be wondering, hey, so we've been talking about X and T, but what about Y and Z? Well, for the purpose of today's video, since we're imagining the twin paradox where the space twin goes out and comes back in along the same axis, we only need to consider the XT plane. And during a Loren boost along the X and T axis, nothing changes about Y and Z. So the spatial plane perpendicular to the Loren boost does not transform as we boost from one inertial frame to another. And you know what? That's actually a good segue. Let's go ahead and finally talk about the actual equations of how to do a loren boost. All right. So, first of all, I want to actually go into a little bit more detail than we need to understand the twin paradox, but I want to include these equations for reference just to be a little bit rigorous and so that those of you who want to learn more about the Loren boost will be familiar with some of the equations and what to study. By the way, in my electromagnetism video, I recommended the book Introduction to Elementary Particles by David Griffiths. And if you happen to have a copy of that book, well, chapter 3 is all about relativistic kinematics. And in particular, section 3.1 is all about Loren's transformations. So that's a great resource if you want to study this in more depth. But also, if you just Google Loren Boost, you can find all kinds of information about it online. This is one of the more famous and central concepts of special relativity. Well, anyway, suppose you have an event in spaceime with some coordinates t, x, y, and z relative to some inertial frame. And then if you ask the question of suppose we hop into a new inertial frame that's moving along the x-axis with some speed v. Well then if you consider the same event that had coordinates txyz in the old frame, what are the new coordinates of that event t prime x prime yprime zprime in the new frame? Well, the answer to that is given by the equations shown here. t prime is going to be the lorren factor gamma * the quantity of t - v / c ^2 x prime is going to be gamma * the quantity x - vt and y prime and z prime equal y and z respectively and so there's no change in coordinates whatsoever in the direction perpendicular to the relative motion between the two frames. The only thing that transforms is the space-time plane whose spatial component is the axis along which the two frames are moving. So the equations shown here for t prime and x prime. This is a way of mathematically encapsulating the concept of the loren boost that we've been looking at. All right. So if you look at these equations for t prime and x prime, there's actually another way that these equations are often written which uses a parameter phi which is called the rapidity. And the rapidity is related to the speed of the frame that we're boosting into. But it's not exactly the same thing as the speed. You see, because the rapidity is defined such that the speed v / the speed of light is the hyperbolic tangent of the rapidity ph. So v over c equals tangi. And the reason this parameter is interesting and useful is because we can use the rapidity phi to write our t prime and x-pime transformation equations in terms of hyperbolic functions of the rapidity ph. Now if you're a hyperbolic trig function enthusiast and you're familiar with oilers's formula you can look at these equations and say hey a lens boost is a rotation through a hyperbolic angle in the space-time plane. And so if you're familiar with this kind of math, you can see in this equation that a loren boost involves these hyperbolic rotations in spaceime. This form of the equation also highlights something interesting about the group structure of Loren boosts because it shows that successive boosts are going to add their rapidities. But anyway, we're drifting a little bit into the mathematical weeds here. I just wanted to bring this up in passing and I hope it provides some useful context. But the main thing I want to focus in on is that under a loren boost, there's a quantity c^2 t^2 + x^2 which remains totally unchanged. equal to c^2 t ^2 + x^2. So the first step is just substituting in for t prime and x prime those transformation equations in terms of t and x and then you work through the algebra and you show that in fact the two sides of the equation are the same. Now this quantity c^2 t ^2 + x^2 we can call this whole quantity s^2 where s is the space-time interval and the space-time interval s does not change under a lorren boost. In other words s is lorent invariant. It does not vary when we do a lorren boost. And by the way sometimes in this s squ equation you see a positive t term and a negative x term. And that's totally fine too. That's just a different convention. Oh, and also in the full free spatial dimensional picture, instead of just plus x^2, we would have plus x^2 + y^2 + z ^2. In any case, the reason I bring up the space-time interval, is that the fact that the Loren's boost keeps light rays at 45° for everyone, that is the same speed for everyone, is a special case of leaving the space-time interval unchanged for the case when s= 0, that is zero space-time interval. The reason being for a beam of light, we're going to have x= ct. That is, if the beam of light is traveling along the x-axis, then the distance it travels is the speed of light time the time that it's been traveling. And if you plug that into our s^ squ equation, you see that that corresponds to a space-time interval of zero. And in fact, if two events are separated in spaceime by a beam of light, then we say that those events are light separated and that corresponds to a space-time interval of zero. So the space-time interval is not a strictly positive thing like a distance but rather the sign of s^ squ can vary depending on whether the interval is space-like timelike or lightlike. Spacelike and timelike are going to have different signs on s squ whereas a lightlike separation is going to have a space-time interval of zero. So anyway, by showing that s remains unchanged under a loren boost, this is a way of proving that the t prime and x-pime transformation equations are in fact the correct equations which preserve the speed of light from everyone's point of view. Oh, and also you often see the space-time interval written in terms of delta s, delta t, delta x. And when you see it written like that, that's a really explicit way of writing that you're talking about the space-time interval between two events. In the equations shown here without the deltas, we're just thinking in terms of an event that's some interval away from the origin of the coordinate system. But more generically, if you think about any space-time interval between any two events in spaceime, then under Loren's boost, all of those space-time intervals are going to be the same from everyone's point of view. And that is an example of the coherent structure that all inertial observers have to agree on about the nature of reality. So even though in relativity space and time get a little bit warped and a little bit weird, that doesn't mean that geometry just goes out the window, but rather what it means is that we're working in this expanded notion of space and time, which is called Manovsky spacetime. That's the spaceime that special relativity lives in. And in Minkovski spacetime, space-time intervals remain constant under Loren's boosts. Okay, but that's a level of detail which is maybe a little bit more than we need for today's video. So, I want to focus in now on the three ingredients of the twin paradox. These are the three concepts where if you understand these, then the twin paradox is going to make perfect sense. Like, these are really the only three things that we have to know in order to fully understand how to time travel into the future and why that's not actually a paradox. All right. So, the first two ingredients, time dilation and length contraction. I think by now we should be pretty much experts on those things. Whenever you have a thing that's moving, it'll have a loren factor, gamma, and the moving things clock is going to tick more slowly by a factor of gamma. Likewise, in the direction that the thing is moving, it's going to shrink by a factor of gamma. So, we have t prime equals gamma t and l prime= l gamma. So that's all good. You know, we've talked a lot about time dilation and length contraction, and these things are pretty easy to see, even if at first they're a bit weird. But the third thing, which so far we've only briefly explored, is the tilting of the now slice. That is the relativity of And this ingredient is really the central key to resolve the apparent paradox. Because when the space twin turns around and comes back to Earth, their now slice is going to tilt in a way that's going to warp time. And that warping of time is going to explain why the space twin comes back having aged less than the Earth twin. So the thing we have to know about the tilting of the now slice is that if you have a thing that's moving with some speed v, then what it considers to be the present moment is going to be a slice in spaceime which has a slope delta t over delta x of its speed vid by the speed of light squared. So that is the equation. And you know, if you want to think about this in terms of natural units, never mind the c^ 2 cuz that's one. And in that case, the slope is just the speed that the thing is moving. And that's pretty easy to remember. The slope is the speed. But even though it's easy to remember, I owe you an explanation as to why that equation is what it is. Because we need to see that and we need to know where that comes from in order to truly understand everything there is to understand about the twin paradox. Fortunately, the Loren boost gives us a quick and easy way to calculate the tilting of the now slice. And that is based on the observation that the world line and the now slice always have the same angle with the light cone. By the way, the world line that's just the trajectory of a thing through spaceime. So in this diagram, that would be the thick magenta and the thick green lines. And when you think about what that trajectory represents, you can see that its slope is related to the speed of the thing by definition. Right? If a thing is moving along the x-axis with some speed v, then v is going to be the ratio of how far it moves along x relative to how far it moves along t. So by definition, v is delta x delta t. And if we wanted to, we could be all calculus about it and say v is dxdt. But we're thinking about lines here. So, may as well think about finite intervals, delta x and delta t. It doesn't matter either way. But then remember that in the space-time diagram that we're drawing here, t is the vertical dimension and x is horizontal. And so the slope of the line is going to be delta t over delta x. That is rise over run. So because v is delta x over delta t, therefore the slope delta t over delta x is actually going to be 1 / v. So now we're using natural units where C equals 1. And so the light cone is at a 45° angle in the space-time plane. Now from algebra, we know that reflecting across a 45° line is going to swap delta x and delta t because this kind of mirror reflection is going to exchange the rise and the run of a line. So then the slope of the now slice delta t delta x is going to be 1 / 1 / v which is v. So there it is. That's the memorable formula that the slope is the speed. Easy to remember. Slope equals But then of course if we want to generalize this from natural units to units where the speed of light is not equal to one, then all we have to do in this derivation is replace the time interval delta t with c delta t. and we replace V with V / C. These substitutions are generally how you would go from natural units to a more generic system of units. So when we do that, we find that C delta T over delta X= V / C. And then all you have to do is divide both sides by C and you find that the slope of the now slice delta T delta X= V / C ^2, which is what I showed you earlier. But now you know why that is. Hey, while we're here, I want to say a few words about the Galilean boost, which is the conventional wisdom about what happens when you hop into the point of view of a moving thing. But the Galilean boost is incorrect. Now, with everyday speeds, it's incorrect in a way that doesn't really matter, but at relativistic speeds, it's incorrect in a way which really matters. And so I think it's helpful to see the ways in which the Galilean boost breaks down at relativistic speeds so that we can disabuse ourselves of this conventional wisdom and can appreciate instead the elegance of special relativity and the Loren boost. All right. So once again we'll start off in the center of mass frame. And once again, a horizontal line in our space-time diagram corresponds to a now slice with constant time everywhere on Now, what makes a Galilean boost different than a Loren boost is that under a Galilean boost, all of the now slices remain horizontal. That is to say, within Galilean relativity, all of the different inertial observers are going to agree on the rate at which time passes. And also there's not going to be any tilting of the now slice. And this is in contrast with the Loren boost where depending on your speed, you're going to have a now slice at a different angle in spaceime. All right then. So let's do a Galilean boost into the green dots frame and see what happens. Well, as promised, you can see that all of the horizontal now slices remained the same throughout the boost. But if you look at our light cone, you see that it's no longer at 45°. Ew, disgusting. You can't have a light cone that's not at 45°. That's wrong. That is in direct violation of the fact everyone because according to the Galilean boost, depending on your speed, you're going to have a different apparent speed of light. But that contradicts the Michaelelsson Morley experiment and that contradicts all of the many many experiments that have validated special relativity. So that's wrong. It's not right. It disagrees with Well anyway, as far as space is concerned, you'll notice that space under a Galilean boost does not stretch. We have no length contraction. And you can see this because we have the same distance between all of these points that are uniformly separated in space. But under a Galilean boost, they just drift together in time in accordance with the speed of the observer whose frame we boost into. And that totally aligns with intuition. Because in the center of mass frame, if we have these dots that are all uniformly spaced along the x-axis, well then in the green dots frame, if we just apply common sense and do the Galilean boost, we would expect to see the same array of dots with the same spacing, but they're just going to drift uniformly over time because the green dot sees the center of mass frames x-axis as moving. And if we want to go ahead and put an equation on this Galilean boost transformation where x and t are space and time in the center of mass frame and x prime and t prime are space and time in the green dots frame. Well, there's no change in time and so t prime equals t. And along space, we just have a drift with uniform spacing. And so x prime is just x plus v * t. that is the x- axis are the same but the new one just drifts in time with a constant speed. Now one more point I want to make before we move on is that there's another problem with Galilean relativity which is that it has no consistent definition of the light cone. You see, because if we draw the light cone from the center of mass's reference frame on top of the light cone in our magenta dots reference frame, you can see that these are not the same light cones. And so in the Galilean picture, which events are space-like versus timelike separated depends on your speed. And when you think about it, that actually totally undermines the very concept of space-like and timelike separated events, right? That's no longer like an objective fact about the separation between two events. By contrast, in special relativity, an event is going to have the same light cone for all inertial observers. So regardless of their speed, all observers are going to agree on which events are inside versus outside of the light cone of any given event. So in special relativity, there really is an objective fact about whether any two events are timelike or space-like separated. That is whether there can or cannot exist a cause and effect relationship between the two events. And that's a very elegant feature of special relativity. It's very nice. And by the way, it's also more generic than just space-like versus timelike separated. In general, in special relativity, the space-time interval S given by S^2= C^2 T ^2 plus the magnitude of X^2 between any events is the same for all inertial observers. So the light cone being the same is the special case when S is zero. Long story short is that in special relativity, the space-time interval between any two events is the same for all inertial observers. But in Galilean relativity, we don't have that because the tilting of the light cone makes it so that different observers are going to disagree on what's space-like and timelike separated and the whole thing is just a mess. And it's not right. And you can really feel that it's not right because the light cone should not be tilting. I mean, come on. What is this? A tilted light cone? No, I don't think so. That's not how it works. So anyway, I hope this little tangent about the Galilean boost has helped to set your intuition against what previously most people would think of as the conventional wisdom. But now let's go ahead and return to actual reality by looking into one of the most interesting and famous experiments relating to the twin paradox which is known as the Haya Keiting experiment. And I hope I'm pronouncing Haya correctly. I don't know. I've heard a few different pronunciations so I hope that's right. Anyway, in 1971, Joseph C. Hea and Richard E. Kading flew twice around the world. First, they flew around the world east. Then, they flew around the world again west with atomic clocks. And as they were flying around, they also had clocks on the ground at the US Naval Observatory. So, this was actually a real genuine test of the twin paradox. Now there is a subtlety here about the rotation of the earth that we have to account for. But there's also an easy way to think about it which is that if you imagine the twin paradox in this experiment imagine that the earth twin is at the center of the earth and over the course of a day or so the center of the earth is going to move in pretty much a straight line. So we can imagine an earth twin that lives right at the core of the earth. Now then this experiment involves three space twins so to speak because we have the two flights around the world. One going east, the other going west and then we also have the clocks on the ground at the naval observatory. And you got to remember that because the earth is rotating a clock on the ground is actually going to be moving hundreds of miles an hour relative to the center of the earth. Of course it depends on the latitude. So when you think about that, you realize that when you fly around the world to the east, your motion is adding on to the rotation of the Earth. And so you're circling around the center of the Earth extra fast. So the eastward flight is the fastest of the three space twins. The clock on the ground is going to be the medium speed space twin. And then the flight going to the west is going to be counteracting the rotation of the Earth. And so that flight is actually going to be the slowest of all three of these space twins. Now, of course, there aren't any clocks at the center of the Earth. So they used the ground clock, that is the medium space twin, as the reference point for the experiment. So relative to the ground clock, you would expect the eastern flight, that is the faster space twin, to age more slowly during the trip. And conversely, you would expect the western flight to age more than the clock on the ground. So let's take a look at the numbers that they predicted and then compare those numbers to what they actually measured. And you can read more about their theoretical predictions in the paper cited here, but we're just going to summarize this very briefly in this table. So first off, for the flight around the world in the eastern direction, which should age most slowly out of all the three space twins, they predicted that the eastern moving clock would have aged by about 184 nonds less than the clock on the ground with an uncertainty of plus or - 18 nonds. So about plus or - 10%. And likewise for the western trip because that clock is going to be the slowest space twin. They calculated that that should age more than the clock on the ground by 96 n plus or - 10 nconds uncertainty. So again about plus or - 10%. So these are very small numbers of course I mean order of 100 nonds on a many hour trip but this is within the measurement capabilities of atomic clocks. And so you see this is literally an experimental measurement of the twin paradox. Now there's another nuance which we have to account for and that is gravitational time dilation. As it turns out if you're up in the sky the gravity up there is ever so slightly weaker than the gravity on the surface of the earth. And that is going to have a very slight effect of making the clocks run faster just because they're farther out of the earth's gravitational well. So remember, if you've seen the movie Interstellar, where they go down into the gravity well and then the guy on the ship ages way more, well, it's exactly the same effect, but to a much less dramatic extent. So you have the clocks on the ground at the naval observatory, but then you also have these clocks up in the sky, a little bit farther outside of the gravity well, and so they're going to be running a little bit faster. And according to general relativity, that should cause the eastern flight to pick up 144 nonds plus or -4 nonds uncertainty. And the western flight should pick up 179 nonds plus or - 18. Notice that those two numbers for east and west are pretty similar. The only differences have to do with slight differences in the duration of each trip as well as the altitude profile, but in essence, it's the same concept for both east and west because this effect doesn't have to do with your speed. It just has to do with where you are in the gravitational field. So when we take both of these things into account, the kinematic time dilation that has to do with the Loren factor and the twin paradox as well as the gravitational time dilation, we just add these things together. And in the case of the eastern trip, you see that the expectation is for the clock to have aged more slowly by about 40 n. And for the western trip, we expect the clock to age more quickly by 275 nonds relative to the clock on the ground. So that column is just the sum of the previous two columns. And by the way, if you look at the uncertainties, think of those as the hypotenuse of a right triangle whose legs are the uncertainties of the previous two numbers. That's how you add up independent uncertainties. Sort of like measuring the diagonal of a TV screen if you know the width and the height. cuz you can imagine an uncertainty box having two independent dimensions. All right then. So what did they actually measure? Well, for the eastern trip, they found that the clock ran more slowly by 59 n plus or - 10 ncond uncertainty, which is maybe a little bit off, but it's still well within the range of the uncertainties. And then for the western trip, they found that the clock aged more by 273 nconds. And that is right on. Like that's pretty much exactly what they were expecting. Now, one of the things that's really cool about this experiment is that it tests the twin paradox as well as general relativity at the same time. Because what you got to look at is for the total nancond predictions as well as measured. Think about how the difference in those two numbers primarily depends on the twin paradox and then the average of those two numbers really has to do with the gravitational time dilation. So this experiment tests both effects at the same time because gravity affects both of the trips more or less to the same extent whereas the twin paradox has an opposite effect on both trips. Oh, and by the way, this experiment has been reenacted since the original experiment to an even higher accuracy. In summary, the reason I bring up this experiment is just to show you that the twin paradox is something that has been literally demonstrated with actual clocks moving around in different ways. And so that's like a real thing that has actually been done. Albeit in the non-relativistic limit where the effects are super subtle. But even though the effect is subtle at those speeds, it has been measured. And so the twin paradox really is genuine physics. All right. So for everything that follows in this video, we're going to imagine a specific example of the twin paradox. But of course, all of this reasoning applies to a more generic situation when it comes to speeds and distances and all that. All right. So, suppose that a spaceship is going to fly away from Earth at 60% the speed of light to a cool spot 3 lightyear away and then it's going to turn around and come right on back at the same speed. This is exactly what we talked about at the beginning of the video, but now with some specific numbers for the speed and for the distance. Now, for the sake of simplicity, we're going to suppose that when the space twin turns around, their acceleration is fast enough that we can imagine it as effectively instantaneous. But rest assured that this simplification is only to avoid messy integrals and it does not change anything essential about the twin paradox. But don't take my word for that. We'll see that later on when we get into the space-time diagrams because you'll see then that acceleration would just round the corner on the space-time diagram in a way that doesn't change the moral of the story of the twin paradox. It would just make the analysis more complicated. Oh, and then also towards the end of the video, we'll see a version of the twin paradox which doesn't even have any acceleration. Well, anyway, we'll get to that, but first let's go ahead and calculate the Loren factor for 60% the speed of light. Well, as we know by now, that's going to be 1.25 25, also known as 5/4s. And so we're going to have a cool 54 resonance between the clocks of the Earth Twin and the Space Twin. Now, from the Earth Twin's point of view, it takes the space twin 10 years to complete their trip because it's a 6 lightyear round trip. Three there, three back, and the Space Twin is traveling at 60% the speed of light. So 6 light years divided by 60% the speed of light is 10 years. So from the Earth twin's point of view, the space twin takes off and 10 years later they come back. But now for the space twin, the trip only takes 8 years. And there's a couple of different ways to think about why the space twin only ages 8 years. First, from the Earth twin's point of view, the 8 years is just the 10 years divided by the Loren factor of 1.25. So the Earth twin can say, "Hey, I know If the space twin is moving at 60% the speed of light, then their clock is going to slow down by a factor of 1.25." And so even though the Earth twin ages 10 years, the Space Twin is only going to age 8 years. And that begs the question of why doesn't the same reasoning apply to the Earth Twin from the Space Twins point of view? But we'll get into that in just a moment. But first, think about how the space twin would explain the fact that they only aged 8 years. Well, what the space twin would say is that they're not moving at 60% the speed of light. But no, the universe is moving past them at 60% the speed of light. And so from their point of view, that 3 lightyear distance between the Earth and the cool spot is actually going to shrink because of length contraction by the Loren factor. So instead of a 6 lightyear round trip, it becomes 6 lightyear divided by 1.25. And so the space twin only experiences a roundtrip distance of 4.8 light years. So the space twin would say, "Hey, my clock is not running slowly. It's running at the normal rate. I just didn't have to go the full 6 lightyear because of length contraction. I only had to go 4.8 light years." And what is 4.8 lightyear divided by 0.6 times the speed of light? Well, that's 8 years. So, we have two different ways of accounting for the fact that the space twin only aged 8 years. So, that makes sense when you think about it. That totally aligns with everything we've been talking about so far. But the real mystery is how can the space twin and the earth twin both agree on the fact that the Earth twin aged 10 years? I mean, the Earth twin can say, "Hey, I just waited around 10 years. It's that simple." But from the space twin's point of view, the Earth Twin is moving. And so you'd think that the space twin would think that the Earth twin would age less than 8 years, right? Or something like that. So what makes the space twin think that the Earth twin aged 10 years? That's the question. And the answer to that question is the solution to the All right. So to make sense of what's going on with the twin paradox, first let's go ahead and look at an animation of what happens when the space twin goes out and comes back. Now in this animation, each twin is also going to emit a pulse once a year. So let's say the space twin takes off on their birthday. And then on every birthday, each twin is going to send out a birthday signal to the other twin. And the birthday signals travel at the speed of light. And these birthday signals are going to help us see what's going on as far as cause and effect and aging and all of Now, this is all going to make a lot more sense in just a moment when we extrude this animation along the time dimension and look at it in a space-time diagram because then we can see the history of what happened throughout this trip. For now, I just want to look at the animation itself as a way of priming our imagination before we draw the So, let's go ahead and watch this animation and keep an eye on the number of birthday signals that are sent and received by each twin. One thing I'd like to point out is that if you think about it from the Earth twins perspective, they're putting out birthday signals at a constant rate. Once per year, they send out the signal. But now on the way out, the space twin receives the signals pretty slowly, purely because of the Doppler effect before we even think about time dilation. The space twin is moving away from the Earth. And so each signal takes longer and longer to catch up with the space twin. But then on the way back, the space twin is seeing the Earth twins birthday signals at an increased rate because when the space twin is moving back towards the signals, then each one has less of a distance to go to get to the twin. So on the way back in, the space twin is receiving the signals at an increased rate. Again, purely due to the Doppler effect, independent of relativistic time dilation. Though of course, time dilation is still playing a role here as well. And to see exactly how this works, we're going to draw the space-time diagram in a moment. Likewise, if you look at how the space twin is emitting birthday signals, you see that because it's moving in the Earth twin's inertial frame, it's putting out signals at a slower rate because the Loren factor is causing its clock to tick more slowly. But then also there's the Doppler effect as well. Whereas the space twin is moving away from the Earth Twin, each subsequent birthday signal therefore has a longer way to travel to get back to Earth. And so that's going to stretch out the frequency with which the Earth twin sees the Space Twins birthday signals. So on the way out from the Earth Twin's point of view, it's going to look like the Space Twin is aging even more slowly than time dilation would suggest. But you see, part of that is because of the Doppler effect. And then of course on the way back in as the space twin is approaching the earth the Doppler effect works the other way because on the way back in each subsequent birthday signal has a less of a distance to travel to get back to earth and so that is going to increase the apparent frequency of these birthday signals relative to what the earth twin would expect purely from a time dilation analysis. Okay. So by now our imagination is dialed in as far as what's going on here. But as you can see, when we think about the Doppler effects and time dilation, there are some nuances here that require a careful parsing of what exactly is going on. And in order to carefully examine those nuances, let's go ahead and draw a space-time diagram of the situation. So all we do to make this diagram is we extrude the situation along the time dimension, which in this case will be the vertical dimension. And you can see that over time the space twin goes out and then turns around and comes back in. And throughout this journey, we'll track the light speeded birthday signals with yellow lines. Now, in this diagram, we're going to use natural units where the speed of light is 1. And so, a light speeded signal is always going to be at 45° in the And also because in relativity the speed of light is always the same for all inertial observers. No matter how we luren boost this space-time diagram, notice that these yellow lines are always going to remain at a 45° angle. That is the speed of light is always going to be constant from everyone's point of view no matter what kind of Loren boost we do as we'll see in just a moment. But before we do any loren boosts, let's go ahead and take a moment to carefully examine this example of the twin paradox from the Earth twins point of view. So the space-time diagram shown here is from the Earth twins point of view because you can see that their world line is pointing purely along time in that vertical direction. Meaning that in this diagram, the Earth twin is not moving. They're staying still. And by the way, if you want to factor in the motion of the Earth around the Sun, we can. But think about how compared to 60% the speed of light, the Earth going around the Sun is almost no speed whatsoever. So even if you factor that in, that would just be the most subtle of wiggling along this vertical line. So it's really something we can neglect. Now then, as we talked about before, the Earth Twin has the simplest explanation for why there's a difference in aging between them and the space twin. And that is they say, look, the space twin's going three light years and back at 60% the speed of light. You do the Loren factor calculation and you see that 10 years on Earth corresponds to 8 years for the space twin. And it's as simple as that. And you can see this in our space-time diagram. If you imagine that once a year the Earth twin meditates on the present moment, well then we can draw a bunch of horizontal lines in our diagram like so, corresponding to the points in spaceime that are simultaneous with the Earth Twin's birthday from the Earth Twins point of view. And then if you look at where those horizontal lines intersect the space twin's trajectory, you see that one year for the Earth Twin comes up short of a year for the Space Twin. And you see if the Earth Twin wanted to think about when is the Space Twin's birthday from the Earth Twin's point of view, well, the Earth Twin would have to celebrate once every 1.25 years. So, it's a longer interval because of time dilation. But if you've made it this far in the video, then time dilation isn't so weird. You know, more moving, less swirling. It's as simple as that. And so from the Earth twins point of view, the twin paradox is not so paradoxical. The Earth Twin can tell a very simple story about why the space twin aged less. It's just time dilation cuz the space twin was moving. And the reason the Earth twins point of view is so simple is that they remain in a constant inertial frame throughout this whole process. Now, in just a moment, we're going to look at the same situation from the space twins point of view, but that necessarily requires Loren boosting into their two different inertial frames. And so, the space twin is going to have a more complicated experience of what's happening. And in particular, when the space twin boosts from the outgoing trip to the incoming trip, there's going to be a tilting of their now slice that will fast forward what they think of as the present moment on Earth. And that effect is really the essential thing to understand about the twin paradox. It all comes down to the tilting of the now slice that happens when the space twin turns around. But while we're here, before we boost, let's think a little bit about the Doppler effect. So even if we're not dealing with relativity, the Doppler effect is still a thing because if you have a thing that's pulsing while moving away from you, then the pulses are going to come in more slowly. So think of like an ambulance siren. You know, when the ambulance is coming towards you, it's high pitched. And when the ambulance is going away, it's lower pitched. And that effect is entirely because of the motion of the pulsing thing and the distance that each pulse has to travel between the thing and you. If the thing is moving towards you, then each pulse has a shorter distance, and so it'll increase the frequency that you observe. Whereas, if the thing is going farther away, each pulse has farther and farther to go, and so you'll observe a lower frequency. Well, now when we're dealing with relativity, the same basic concept still applies. You're still going to have a Doppler effect, but now you also have to take into account the Loren factor and time dilation. Now, in the interest of time, I'm not going to derive the relativistic Doppler effect formula, but I will show it on the screen here so we can see how it works. So, let's go ahead and use tow prime to represent the space twin's one-year period. Now on the way out as the space twin is receding away from earth the earth twin is going to observe the space twin's birthday signals coming in with an observed period towo observed which is equal to tow prime. So in this case 1 year that the space twin is putting out these signals once a year times this factor of the square root of 1 + v / c over 1 minus v / c and that factor is greater than 1. So the observed period of the birthday pulses coming into earth is going to be longer than the period that the space twin is sending them out from the space twin's point of view. So in this specific example to prime is 1 year from the space twin's point of view. And this factor if you think about 1 + v over c well that's going to be 1.6 and then 1 minus v over c is going to be 0.4 and 1.6 / 0.4 4 is 4 and the square roo<unk> of 4 is 2. So therefore the observed period on earth of the birthday signals coming in from space is going to be 2 years. And in fact you can see that you see if you draw these lines at the moment when the space twins birthday signal reaches earth you can see that they're spaced out with a period of 2 years from the earth twins point of Now then on the way back in as the space twin is approaching the earth it's the same formula but different you see because now the observed period of signals on earth is going to be the space twins's 1-year period t prime but now it's times a factor which is actually the reciprocal of the previous factor. So whereas before we had a factor of two, now we're going to have a factor of 1/2. And so the birthday signals that the space twin is sending on the way back in are going to arrive at Earth every half a year. So think about what that means. If you're the Earth twin and you're looking out into space and you're saying, "Hey, every 6 months I'm getting a birthday signal from my space twin. They must be aging faster than I'm aging here on Earth." But then you remember, uh uh uh nope, because they're coming in. And so there's a Doppler effect. And when you factor in the Doppler effect, you realize that their clock is still running slower than yours. It's just that they're coming in and so there's a frequency adjustment because of that. Well, anyway, that's a few words about the Doppler effect. And with that in mind, I hope this diagram gives you a pretty good perspective of how things look from the Earth Twins point of view. Now, let's go ahead and loren into the space twins outgoing reference frame. Are you ready for this? Here we go. Here goes the coast into the space twins. >> All right. So, this is what the space twin sees as they go out from the Earth to the cool spot way out in space. And you know that this is the space twin's reference frame because in this diagram the space twin is traveling in a straight vertical line purely along the time dimension on their way out. Now then accordingly the earth twin is moving to the left from this point of view at 60% the speed of light. And what do we know about 60% the speed of light? Well we know that's going to be a loren And so the space twin is going to see the Earth twin's clock ticking more slowly, just like the Earth twin saw the space twin's clock ticking more slowly. That is to say, so far before the space twin turns around and comes on back, there's actually complete symmetry between the Earth twin and the space twin's point of view as far as time specifically when the space twin reaches the cool spot in space and four years have passed for them before they turn around and come back only 3.2 years have passed on Earth because 4 divided by the Loren factor of 1.25 equals 3.2. But now, as we'll see in a moment, the crux of the twin paradox is the observation that when we hop into the space twins's incoming reference frame, there's going to be a tilting of the now slice. And that tilting is going to fast forward from 3.2 years on Earth all the way up to 6.8 years on Earth. And that tilting of the now slice is where all of the asymmetry in aging comes in. We'll talk about that in just a moment. But first, I want to point out that because in the space twins point of view, the universe is moving at 60% the speed of light, that 3 lightyear distance between the Earth and the cool spot is going to contract to only 2.4 4 lightyear. And that contraction shows up in our space-time diagram as we do the Loren's boost. Because if you look carefully at what happens as we boost from the Earth twins point of view to the space twins outgoing point of view, you'll notice that that distance is actually going to shrink along the x-axis as we do our Loren boost. And that aligns with what we've explored earlier as far as length Now, I guess just to be totally clear here, when I say the universe is moving at 60% the speed of light relative to the space twin, it would be more precise to say that the Earth and the cool spot are moving at 60% the speed of light relative to the space twin. In reality, as far as we can tell, the universe does not have a preferred reference frame. And so, really, it is specifically the Earth cool spot system that is moving relative to the space twin. Well, anyway, that length contraction of 3 light years down to 2.4 lightyear is why the space twin is allowed to say, "Hey, I've only aged 4 years, and yet the Earth is now 2.4 light years away." But you see how that works, right? That's coherent. Everything makes sense because you see space and time transform together as we do our Loren boost. So, the distance shrinks to 2.4 light years. And as we just talked about, after four years for the space twin, the Earth twin has only aged 3.2 years. Also notice that in this diagram, all of these yellow lines, that is the light speeded birthday signals, remain at 45° before and after the Loren boost. Because as we're doing our Loren boost, all of these points in spaceime are sliding along hyperbolic curves. And that hyperbolic sliding is the unique transformation that ensures that these 45° angle light speeded signals remain at 45° no matter whose inertial reference frame we hop into. And that is the connection between the speed of light having to be the same for everyone and the specific details of the Loren boost as we talked about earlier. Hey, speaking of those light speeded birthday signals, let's go ahead and talk about the Doppler effect. Well, if you think about the rate at which the space twin is receiving the Earth twins's birthday signals, then you see that just as before, the space twin is going to receive birthday signals from the Earth twin every 2 years. So here again, we have perfect symmetry between the two twins points of view. As they move apart, each twin receives the other twins birthday signals once every 2 years. But now our twins are experts in relativity. So they realize that hey the frequency that you see the signal is not the time dilation factor because you also have to account for the Doppler effect and the fact that the source of the signal is moving away. So the twins can work out the math and say okay I'm seeing a birthday signal every 2 years but from what I know of the Doppler effect that corresponds to an actual time dilation factor of 1.25. And then you couple that with the fact that the motion is going away from each other and that gives an apparent receive frequency of birthday signals of once every 2 years. Okay. So if we just want to sum up everything that's shown here, the gist of this is that as the space twin is going out at a constant speed, we have total symmetry between the space twin and the earth twin as far as time dilation and the Doppler effect. On the way out, both twins know that the other one is aging 1.25 times more slowly, and both twins see the birthday signals of the other one coming in at a rate of once every 2 years. So, if you followed along with everything so far, you'll notice that we haven't addressed the core of the twin paradox, which is where does this asymmetry come from? Why is it that the space twin ends up aging less total along the whole journey? Well, now it's time to address that question. So, we are about to hop into the space twins incoming reference frame. So, we're going to luren boost such that the returning trip becomes a vertical line in the space-time diagram. And there's a couple of things we'll talk about about that transformation. But the most important thing to notice is the tilting of the now slice. So, as we talked about just now, when the space twin arrives at the cool spot 3 lightyears from Earth or 2.4 lightyears from Earth from the space twin's point of view, then right when they get there, the present moment on Earth is 3.2 years from the departure time. And just to be clear, when the space twin gets there, they receive the Earth twins year 2 birthday signal, but they know, hey, that signal had to travel across a lot of space to get here. And in reality, right now, it's 3.2 2 years on Earth. But as we know from relativity, the present moment is a tricky thing. It tilts depending on your speed. It has to be that way. This is a consequence of everyone. That's just how reality works. It's counterintuitive, but that's how it is. And so just by reversing their velocity in that moment, in that 4year for the space twin moment, the now slice is going to tilt such that the present moment on Earth becomes 6.8. and 8 So all of a sudden by turning around the space twins now slice is tilting such that the present moment on earth from their point of view fast forwards from 3.2 years to 6.8 years and that is when all of this time traveling into the future happens. Now here I should point out that for simplicity we're imagining instantaneous acceleration and so we just have two straight lines in the space-time diagram. But as you can see, if we had some real acceleration that wasn't instantaneous, well, that would just round the corner. And so, if we want to be really precise, we'd have to solve an integral around that rounded corner. And that would adjust the number slightly because instead of a clean corner, now you have a rounded thing you have to integrate over. But notice that that would not change anything fundamental because you can just draw a circle around that corner and say, "Don't worry about that. Let's think about what happens when we go from the straight line to the straight line." And you see that when you hop from the one straight line into the other, the now slice is going to end up tilted. So when you have a real finite acceleration, that's basically just going to spread the tilting of the now slice across some finite amount of space and time. But for the purpose of understanding the twin paradox, that really doesn't change anything at all. And you may as well imagine that tilting is happening all at once just for the sake of getting the gist of what's going on without getting lost in all kinds of messy calculations. And also, if you're not satisfied with that explanation, if you think that I'm sweeping something under the rug and maybe the details of the acceleration matter, well, later on we're going to see an example of the twin paradox where there is no acceleration whatsoever. And then you'll really see that it's all good to imagine a situation where the acceleration is instantaneous and that the details don't matter. Well, all right then. With all that in mind, let's go ahead and loren boost into the space twins incoming reference frame again into the returner frame. All right, here we are in the space twins incoming reference frame. So the first thing I want to point out about this is that after turning around, the space twin is coming back to Earth at the same speed it was going out. That is in this example 60% the speed of light. And so on the return journey, the space twin sees the Earth twin's clock ticking 1.25 times more slowly. The same Loren factor is on the way out and also the same Loren factor as seen from the Earth Twins point of view. So all that's to say the same relative constant speed is going to yield the same Loren factor for both twins. That's true on the way out. It's true on the way back in. It's true for the Earth twin and it's true for the space twin. And also go ahead and think about the Doppler effect and you can see that while it's coming in, the space twin receives birthday signals from the Earth twin every half a year. that is twice a year which is exactly what we saw before from the earth twins point of view when the space twin was coming in. So not only is the lorren factor the same for the earth twin and the space twin during the incoming trip but so is the rate at which each twin observes the birthday signals from the other twin. And so there really is perfect symmetry across most aspects of the twin paradox which is what our intuition would expect, right? you go into thinking about the twin paradox and you're thinking, hey, shouldn't there be symmetry here and that is actually the right instinct to have because for the most part, yeah, there is. And both there and back on both parts of the space twin's journey, the space twin sees the Earth Twin aging more slowly. So, the asymmetry really all comes down to the bending of the space twin's path through spaceime as it turns around. That is the one part of this whole story where there's an asymmetry in perspective because that's where you have a tilting of the now slice. And that tilting of the now slice, which fast forwards the apparent time on Earth fast forwards by so much that even though the space twin sees the Earth twin aging more slowly on both halves of its trip, that tilting of the now slice more than compensates for that and still leaves the space twin in a situation where they end up aging less throughout the trip. Now, if this is your first time seeing this concept in these diagrams, it's going to feel a bit weird and it might take some getting used to. As a good exercise, if you want to go ahead and repeat this analysis using 80% the speed of light for the space twin speed, well, that's another pretty convenient number to use. And you can go ahead and calculate all these things and check the story for 80% the speed of light. And you'll find that in essence, everything's the same, but the numbers are a little bit different. So, that's a great exercise if you want these ideas to become more familiar. And in fact, if you want to go ahead and do that exercise, I'm going to put on the screen in just a moment all of the numbers for that scenario so you can check your work against these numbers. But if you're planning on doing the exercise and you don't want any spoilers, go ahead and close your eyes for the next few seconds because the numbers are going to flash on the screen right now. And now they're gone. So you can open your eyes again. It's all good. All right, then. So that is an example of the twin paradox. We got into the math. We saw where the asymmetry comes from and I hope that it shed some light on what's going on here and why the space twin is able to age more slowly even though throughout the journey each twin sees the other one's clock moving more slowly. But I can imagine maybe some of you are not satisfied with how we neglected the details of the acceleration. And so I want to show you now something I like to call the twin exchange program which is a version of the twin paradox where we don't have acceleration. And so we don't even have to worry about ignoring acceleration because there's no acceleration to ignore. So here's the thing. Because the asymmetry of the twin paradox comes from the space twins now slice tilting when they turn around. People often get the impression that the time travel magic of the twin paradox is an effect of acceleration per se, as if the gforces are what impart the fast forwarding effect. Now, there is a sense in which that's quantitatively right in that if you're traveling along a curved path in spaceime, then you're necessarily accelerating. So, that much is true. But it's not spiritually right that the G forces are what cause the fast forwarding into the future. But rather, it's all about thinking about different reference frames. And in general, it's about the proper time along nonlinear paths through spaceime. So, that's what the following thought experiment is going to illustrate. We're going to parse out the subtle difference between a curved path in spaceime and acceleration, per se. So, here's the deal. Suppose that your twin is more attractive than you, which is fine, you know, whatever. That's fine. You can have a more attractive twin. But then on second thought, you know what? Maybe that's not fine because maybe you're tired of living in their shadow as the second best version of yourself, you know, and you grow resentful and you begin to curse the creator of this universe for putting you in such a predicament in life. It's not fair. So, you concoct a devious scheme where you decide to send your twin off into deep space on an endless journey away from the Earth at a constant 60% of the speed Now, suppose you also somehow coordinate with an alien so that they're going to come in towards the Earth at 60% the speed of light and they're going to pass by your twin at a cool spot 3 lighty years from Earth. I don't know how you do this coordination with the alien. You know, this is a plan that's been a long time in the making. Okay, so let's go ahead and draw some diagrams here and see what's going on. In the classic twin paradox, it's really simple. Your twin goes out and then they come back in. So, the spac-time diagram looks like this. And as you see, because the twins path through spaceime bends at the cool spot, they were necessarily accelerating in order to change their trajectory and come on back in. Now, what we have in mind for the twin exchange program is that your twin is going to fly away forever, never to return. So, they go right on through the cool spot and they keep floating away until the end of time. And also at the same time, an alien is coming in from far away and they happen to pass by the cool spot on their way in. So neither your twin nor the alien are accelerating. They just pass right on by each other at the cool spot. And at that spot, no acceleration takes place. Oh, by the way, you might be wondering, well, doesn't the twin have to accelerate up to 60% the speed of light in the first place to leave Earth? Two ways to think about that. One is imagine that acceleration is instantaneous in which case the tilting of the now slice doesn't matter because the twin is at earth when that acceleration happens and so there's no shifting in time as a result of the tilting of the now slice. Another way to think about it which is totally fine is that the twin takes off into space does a warm-up lap around the sun accelerates to 60% the speed of light and then does a flyby right next to earth and then that's when you start the clock. So in any case, we can imagine this thought experiment starting with the twin moving at 60% the speed of light and there's no time warping, time traveling weirdness having to do with those initial conditions. And the same concept applies in reverse to the alien coming in. Two options. One is either they slow down instantaneously, in which case the tilting of the now slice doesn't matter because the Earth twin and the alien are in the same spot, or totally fine to imagine the alien as just doing a flyby of the Earth. In any case, the main thing to see here is that as the alien passes right next to your twin ship at the cool spot, suppose they look through the window and they see a clock on your twin's ship that shows that the elapsed time for the space twin has been 4 years. So, the alien says, "Oh, okay. They've been going 4 years along that path." And then the alien starts their clock and they're going to measure an additional 4 years until the alien reaches the Earth. And now the net result of all of this is that your attractive twin is gone, but you have a cool new alien friend to hang out with. Now, in reality, of course, you should never send your twin off into space forever. It's not cool. This is purely a hypothetical thought experiment. In any case, during this whole exchange program, 10 years passed for you on Earth, but only 8 years passed for your twin going out plus the alien coming in. And because there was no acceleration involved here, that tells us that G forces are not the causal agent of the fast forwarding through time that we see in the twin paradox. See, the way to think about it is that the bent path through spaceime has a shorter proper time. And that proper time that is the elapsed time of a clock traveling along that path is going to be shorter than the proper time along a straight path that starts and ends at the same points in spaceime. Now here if we want to be pedantic we can say yes but if you want to travel along a curved path in spaceime you necessarily have to accelerate. That is true. But you see, the purpose of the twin exchange program is to be second order pedantic and say you only need G-forces if you want the same thing to travel along a bent path in spaceime. But if you're thinking about multiple things and the space-time geometry as such, then you see that G forces are not a causal agent. They're just a requirement of having to put the same thing on that path. So anyway, it's a subtle point. It's a nuanced point, but I think it's one worth talking about. And it segus into probably the most elegant way of thinking about the twin paradox, which is that in uklitian space, a straight line minimizes the distance between two points because the shortest path between two points is a straight line. But on the other hand, in Manovsky spacetime, a straight line maximizes proper time. So between two events in spaceime, the inertial frame that goes from one event to another is going to maximize the proper time between those events. And any curvy path relative to that, any acceleration and going out and coming back and turning around, those alternative paths are all going to have a lower proper time than the straight path of the inertial frame. And if you see that, then you understand the twin paradox. Hey, thank you to everyone who has been supporting my channel on Patreon. Your support really means a lot and is the reason I'm able to devote so much time to these videos. I really couldn't do this without you. So, thank you all so much.