# I wish I was taught Einstein's Special Relativity this way!

- Channel: [FloatHeadPhysics](https://www.youtube.com/channel/UCGfFUc6eWxfbtjYeun6r9xg)
- Published: 2024-02-23
- Source video: [Watch on YouTube](https://youtube.com/watch?v=TJmgKdc7H34)
- Read on ReadTube: https://www.read.tube/p/videos/TJmgKdc7H34
- Language: English

## Summary

**Relativity Through Shadows**

This video explains Einstein’s special relativity by using a shadow-world analogy that makes the speed of light feel like a geometric limit rather than a mystery. By treating motion through space and time as different projections of one constant four-dimensional motion, it connects time dilation, length contraction, relativity of simultaneity, and the twin paradox in a single picture. The speaker also uses the idea to suggest why we experience three dimensions of space but only one of time.

The video argues that special relativity becomes easier to understand if we imagine our universe as a projection of motion in a higher-dimensional space-time. In the opening analogy, a moving object looks like it changes speed when viewed only as a shadow, but the object itself may be moving at a constant speed in the larger space. This is used to motivate the idea that everything in our universe may always be traveling through space-time at the speed of light, while we only perceive a projection of that motion.

To support the idea, the video introduces a second shadow and shows how measuring two visible components of motion can reveal a hidden constant speed through Pythagorean relationships. That same structure is then mapped onto physics: motion through space and motion through time behave like two perpendicular components of one invariant speed. The result explains why faster motion through space corresponds to slower passage of time, and why photon-clock thought experiments produce time dilation.

From there, the video shows how the same geometric turning of a four-dimensional velocity vector can explain more of special relativity at once. As an object moves faster through space, its projection shrinks along the direction of motion, producing length contraction. Because different parts of the moving object lie at different positions along the time axis, clocks that are synchronized in one frame no longer agree in another, which gives relativity of simultaneity. The twin paradox is then presented as a natural consequence: the traveling twin “spends” some of their constant space-time motion on space, so they age less than the twin who remains at rest.

The closing idea is more speculative but ambitious: the reason there are three spatial dimensions and one time dimension may be that the direction of motion at light speed gets collapsed into what we experience as time, leaving only the perpendicular directions available for ordinary spatial movement. Overall, the video presents special relativity as a single geometric story about projection, turning, and invariance rather than a set of separate effects.
- Constant speed in higher-dimensional space-time is the core idea.
- Shadows and projections explain why motion can look different from different viewpoints.
- Pythagorean geometry is used to link space speed and time speed.
- Time dilation follows from diverting motion into space.
- Length contraction follows from the “turn” of the motion vector.
- Relativity of simultaneity comes from different clocks projecting to different time positions.
- The twin paradox is framed as an age difference caused by different paths through space-time.
- The video ends by linking our experience of three space dimensions and one time dimension to this geometry.

## Article

### Why this idea about space-time blew my mind

This video is sponsored by Squarespace, the all-in-one website building tool.

I had read about this idea that **we are all traveling through space-time at the speed of light**. The problem with most resources I found was that they just told me that. And I do not like that. I want to discover things myself. I like to piece things together and then come up with a conclusion myself. I think that is one of the richest ways to learn science.

More importantly, I did not see a broader significance of that idea until a few weeks ago, when I finally started reading this book called *Relativity Visualized* by L. Epstein. Thank you, because a few of you actually recommended that for me. I think that has started to blow my mind away.

For one, it does exactly that: it helped me discover, by piecing things together, why we say that things are traveling through space-time at the speed of light, and what that really means. But more importantly, it showed me some truly incredible explanatory power. I was able to see **time dilation**, **length contraction**, **relativity of simultaneity**, even the **twin paradox**, all of it, in a brand-new perspective.

And that is not all. It was also able to help me ponder some of the deepest questions of physics, like:

- Why do we have **three dimensions of space** and just **one dimension of time**?
- Why is it that I can move freely back and forth through space, but I am always a prisoner of the current present moment when it comes to time?

So I am now super excited to share this with you. If you are as well, let’s get started.

### The airplane shadow analogy: how a higher-dimensional motion can look like changing speed

So, Louise, where do we start?

L. Epstein says: let’s take a concrete analogy. We have an airplane, and let’s imagine that in this world the airplane will always move at a constant speed of **1 m/s**.

Okay, so in the first second, let’s say the airplane moves straight up, so that is **1 m**. Then in the next second, let’s say the airplane moves that way, another **1 m**. And in the last second, let’s say the airplane moves horizontally to the right.

But now, what if you were to look at the same thing one more time, but by putting some lights and focusing on the **shadow** that the airplane formed? What would be the speed of that shadow in each of these seconds?

Let’s see.

- In the first second, that shadow is at rest.
- In the next second, it starts moving, so it speeds up a little bit.
- And then in the last second, it speeds up even more. It moves even faster.

Now here is a big question. Imagine there were a couple of shadow people. You can imagine them to be two-dimensional beings who thought that their world was two-dimensional, that is it. They could not comprehend the third dimension. So for all they know, that shadow is an actual object.

So what would they be seeing, and how would they be perceiving things?

Well, from their perspective, they would say:

- In that first second, that shadow was at rest. It did not move at all. That object is not moving.
- In the next second, they would say, “Hey, this object started moving a little bit.”
- And then they would say, “Oh, it is moving even faster.”

So they see that object accelerating.

You and I know what is really going on. We understand that really that object is traveling at the same speed in three dimensions, but since that shadow only represents a two-dimensional projection, that is why that shadow will have variable speed.

But they do not know that. In fact, if they could see that shadow long enough and the plane kept on moving and twisting and turning, but all the while maintaining that constant speed of **1 m/s**, they would eventually realize that the shadow keeps moving, but it will have a maximum speed of **1 m/s**. And they would be wondering why that would be the case.

In fact, they would realize that in this world everything has a speed limit. Their universe has a speed limit of **1 m/s**. Nobody can break through it. They will postulate it, but they will have absolutely no idea why that is happening.

### What this is foreshadowing about the speed of light

Do you see what Louise is foreshadowing over here?

Just like those two-dimensional shadow people, we think that there is a speed limit in our universe, the speed of light, and nothing can go faster than light. But maybe that is because we are seeing a three-dimensional shadow of what is really happening in four-dimensional space-time.

Maybe everything in space-time is moving just like that airplane at a constant speed, always at the speed of light. But because we are seeing a projection of it in our universe, we perceive it as a speed limit. Maybe it is not the case. Maybe everything is moving at the speed of light.

So we can now begin to see where this idea stems from.

But we are far away from accepting this idea as of now, because remember, this is science. Unlike religion, we do not just accept ideas. In fact, one of the best ways to deeply understand science is to be a skeptic and always try to figure out what if — try to counter it, try to refute it from different angles.

So that is what we will do.

We should at this point look at Louise and say, “Cool story, bro, but how can we validate this? I mean, is there a way that these shadow people can actually somehow validate that from their perspective, indeed, there is a bigger world out there and objects may be traveling at a constant speed? Because if these shadow people cannot validate it, then we have no idea of validity, and it will just remain a story with no reason to have any confidence in it.”

And so this is where Lou says, maybe they can.

I am like: how? How could these shadow people ever hope to understand that there could be a bigger world out there? How would they do that?

### A second shadow on a perpendicular wall

The key here is to imagine a second shadow on a wall perpendicular to this first shadow.

Now I know what you are thinking at this point. I am also having the same thought: “Louise, what is the point of having to look at this shadow? Because these folks are stuck in their two dimensions, right? They will not be able to perceive the shadow or look at that shadow. So what is the point in looking at that?”

And Lou says, well, you are right. But remember, just like us, who are stuck in our three dimensions and cannot see time, but can still measure the flow of time with our clocks, similarly what if they cannot see this shadow, but somehow they can measure how quickly that shadow is moving? They can measure the speed of that shadow somehow.

Then would they be able to use the speed of this shadow and then use the speed of this shadow, which they can easily see, and then piece it all together and see there is something deeper going on over here?

Let’s find out.

### The right-triangle relationship between the two shadow speeds

Now let’s relook at this whole thing in its whole glory. Now by looking at a couple of shadows, let’s keep track of both shadows, the speed of both shadows.

In the first second:

- This shadow is at rest.
- But look, this shadow moves completely **1 m**. It is moving at the maximum speed.
- So this one is at rest, but this shadow, the vertical shadow, is moving at a maximum speed.

In the next second:

- This shadow starts moving, its speed increases.
- But this one, its speed decreases. This distance is now a little smaller than **1 m**.
- So you see, this speed increases, this speed decreases.

In the last second:

- The horizontal speed is maximum.
- But the vertical speed has become zero.

So by looking at this, they will be seeing that there is some connection between the vertical speed and the horizontal speed. When the horizontal speed increases, the vertical speed decreases. When the horizontal speed decreases, the vertical speed increases. There is some connection between them.

And what is that connection?

Well, the connection is: if this arrow represents the horizontal speed and this arrow represents the vertical speed, we can now see that they form a **right-angle triangle** with the total three-dimensional speed. Which means if we label them — let’s call **H** for horizontal and **V** for vertical — and we know this speed is always **1**, because that speed is always constant, that is constant **1 m/s**, then just by using **Pythagoras**, we know that the squares of these speeds should add up to give us one.

This means if the shadow people start measuring things carefully and then eventually, maybe by trial or error, once they figure out the connection between the horizontal shadow speed and the vertical shadow speed, then they will get it. Then they will realize, “Hey, why is this happening?” And from that they would be able to use their own analogy in their two-dimensional space to sort of think about it in abstract terms, that there is a third dimension out there, and this plane and all the objects in the third dimension must be traveling at a constant speed, and we are seeing only a two-dimensional projection of it, and all of that.

And by using Pythagoras — Pythagoras, that is it — they can piece things together and say that maybe there is a third dimension out there, there is a higher dimension out there, and this object, which again they cannot perceive properly, must be traveling at the same speed, and we are seeing a two-dimensional projection.

Isn’t that insane? That is insane if you ask me.

### Squarespace: building a website the same way you piece together physics

But of course, what is even more important for us is: look, we are slowly and steadily piecing things together.

Isn’t it just like how I pieced together my website, `floatedphysics.com`, using Squarespace, the sponsor of this video?

So how do we piece together a website? Well, just like physics, you use some fundamental tools, and Squarespace provides one of the most powerful ones.

For example, you can use custom templates. So let’s go ahead and build a one-pager for ourselves. We pick something that we like, and boom — just like that, I have a skeleton for my website.

But wait, I do not want a templatized website. Come on, I want to be unique. My website needs to be unique.

Well, Squarespace says, buddy, it is hyper-customizable. For example, I want to introduce my brand somewhere in between, so I just do that. I add a block that allows me to introduce my brand, change the photo, and boom.

I do not like the font over here, you say? Well, yeah, go ahead and change the font to whatever.

I want to add my video, you say? Well, go ahead, let’s add a video over here. Oh, this is a cool video of me doing a physics experiment a long time ago.

But wait, what if we get stuck? We do not have website creation experience, right?

Ah, turns out they have a dedicated on-demand support team. I was actually having a problem, but they solved my problem within minutes and actually got my website live.

So if you want to create a digital presence for yourself — and I think everyone should — then try Squarespace for free by going to `squarespace.com/floatheadphysics`, and of course use the promo code `fload physics` at checkout to save 10% on your first purchase. The link is also in the description.

Now let’s go back to the video.

### From the shadow puzzle to motion through space-time

Once these folks figure this out, they will have a brand-new perspective, a brand-new way of thinking about speeds.

So they will realize that initially, when that airplane appeared to be at rest in their dimension, it did so because all of its speed was vertical, which they cannot see, of course. However, later on, that airplane just turned. It did not change its speed; its speed is still the same. But because it turned, now some of its velocity got directed in our horizontal direction, and that is why we saw it moving. And of course, the vertical speed decreased.

And later on, it turned almost completely, so we have the maximum speed in our horizontal, in our space, and the vertical speed almost became zero.

So we now have a way to validate this, which means we can now apply it to our own universe.

### Applying the idea to our universe: space and time as perpendicular components

In our universe, the horizontal over here, our world, represents **3D space**, and the vertical dimension over here would represent **time**.

So if this is true, that everything is moving through space-time at the speed of light — the maximum speed that we perceive, the speed of light — then we should have a very similar equation. They should also obey a very similar equation:

> speed through space squared + speed through time squared = 1

One being the speed of light in our case. In this video, the speed of light is **1**.

So we now have a way to validate if this makes sense.

And so the big next question is: is this true? Let’s find out again.

### Photon clocks: why moving things tick slower

For this, we go back to our photon clock. Remember, just to recap, a photon clock is where a photon ping-pongs between a couple of mirrors. Every time it hits one of the mirrors, we get a tick.

So from this, we realize that because the speed of light is a constant, what we find is that moving things will have photon clocks that are ticking slower the faster they move.

Look, the slower it takes.

But now let’s try to see if that relationship holds between them.

### The photon clock also gives a right triangle

So for that, let’s look at this one more time. Here we go.

And if I concentrate on the speed of this photon, this photon is traveling this way, of course, with the speed of light, **1**. But the horizontal component of that photon represents the horizontal speed of this cart, and the vertical component of that photon represents the speed of time — like how fast time is passing in this cart.

So if you understand this, this is the speed through space, and this truly represents the speed through time, then look: we are indeed getting the same equation. Pythagoras makes a comeback.

Pythagoras is incredible because he transcends dimensions of space and time.

So this is incredible.

So now we can also use the same story and say, “Hey, look here: the cart is at rest in the spatial dimension, but all of that speed, the cart is moving almost at the speed of light through the time dimension at full speed. Therefore its clock is ticking at the maximum rate.”

However, later on, some of that speed is diverted to space. Therefore we see that cart moving through space, but now the speed through time decreases, and therefore this clock ticks slower. And the same effect happens over here.

And everything now starts making sense. This is incredible, isn’t it?

This idea beautifully and elegantly explains why, as you move faster through space, you move slower through time: **time dilation**.

### Length contraction, relativity of simultaneity, and the turning of the four-dimensional velocity

But folks, we are not done, because this is the climax. This is where L. Epstein truly starts blowing my mind away. Because now we will see its true explanatory power at explaining other consequences.

Are you ready for this? Okay, let’s look at it.

Again, let’s go back to that plane, but this time in full three-four dimensions.

Let’s say that plane is at rest in space, and this is our space right now. I am only showing two dimensions of space because I want to use the rest of the other dimension to represent time. And when I say time, I mean **proper time** — the time on the clock. You can imagine a clock attached, a clock inside the airplane. That is the time I am talking about, not the time that we would be looking at in our clock, because the flow of that time will always be the same. That will never get dilated, that will never change. I am talking about this clock, the time on the clock on the airplane.

Okay, that makes sense, right? That is the proper time.

Anyways, now if this airplane is at rest in space, that means it is moving at full speed through time. And therefore that means it is moving in this particular diagram vertically upwards. In other words, this velocity — which is not three-dimensional velocity, it is a velocity in four dimensions — is directed perpendicular to the spatial dimensions.

And I know it is hard to comprehend that, because how do you imagine perpendicular to three-dimensional space? We cannot comprehend that. But just like how these space folks could not comprehend that, that should not stop us from trying to understand the implications of this.

Okay, so what do we see? We see only a shadow of it. And because it is moving completely through time, we do not see it moving, but it is moving through time at full speed.

Now comes the question: what happens as the shadow starts moving?

If the shadow has to move, the plane has to turn. This velocity has to turn. Only then will the shadow start moving. Only then will it be able to give it some speed in the spatial direction, right?

But now look, if that arrow turns, look at how that plane turns. The plane turns like this. It has to turn like this, because from its own perspective the time axis should always be perpendicular to itself. From its own perspective — remember, it is not moving — so from its own perspective it should be in the spatial dimension, the time axis should always be perpendicular to it. It should be flowing only through time from its own perspective. Therefore that is what it looks like.

And therefore when we now look at its projection on space, look, it looks contracted. And that makes sense, right? Because the whole thing has turned like this, and so its projection becomes smaller. That is **length contraction**, folks.

This explains a lot of things. For example, what happens if it is moving faster through space? If it has to move faster through space, it has to turn even more, and now that shadow will become even more contracted. Length contraction depends on speed: more speed, more length contraction.

But you can also see length contraction only happens in the direction of the spatial motion. In the perpendicular direction there is no length contraction, and you can see why, because it does not turn like that. It only turns in the direction of the spatial motion. That is why we only see length contraction this way.

Isn’t that beautiful, folks?

### Why moving clocks desynchronize: relativity of simultaneity

Okay, but we are not done.

If we go back over here, when it was at rest with respect to space, we see that the whole airplane maps to one specific point in time over here on our diagram, and therefore all the clocks attached to that particular airplane will read the same time. The clock on the front and the clock on the back, they will all be synchronized. They can all be synchronized, and they will read the same time.

But now what happens when that plane starts moving through our spatial dimension? What happens when it turns in the four-dimensional space-time?

Well, let’s look at it.

Here we go. If you look at it this time, you see that the front of that plane is slightly lower in the time axis compared to the back of the plane. The back of the plane is elevated in the time axis. In other words, the front of the plane will be behind in time compared to the back of the plane.

Can you see that?

This explains **relativity of simultaneity**. From the plane’s perspective, these two clocks are in sync, like we saw in the previous slide. But from our perspective, the two clocks are no longer in sync. The clock behind is actually ahead in time.

And it makes perfect sense. And if you add again what we saw earlier — because it is now moving, it has diverted some of its velocity through space, the velocity through time has decreased, therefore it is also time dilated — the time is also dilated.

Now we see all three consequences: **time dilation**, **length contraction**, and **relativity of simultaneity**, the clock desynchronization, all in one picture.

I promised you that, and that is what we got. Just look at it. Isn’t that beautiful?

Oh my God, that is just insane.

### The twin paradox in the same picture

But we are not done. There is more.

I want to show you this time we have two twins who are at rest with respect to each other, and therefore they are moving full speed through time. But let’s say the twin sister — this one is the sister — decides to venture off into space.

To do that, she will have to turn her velocity vector, because remember, this is the velocity in four-dimensional space-time. That will always be one, the speed of light. That cannot change. The only thing she can do is turn.

When she turns, somehow her speed gets diverted into space, and so she can now start moving away from her twin brother. But in doing so, she is now moving slower through time.

And because her brother is moving full speed through time, you can immediately see that when she returns, she will not be able to catch up to him in time. When she returns, she would be behind him.

Doesn’t this make perfect sense now?

The reason she will be behind him is because she diverted some of her speed to space. The moment she did that, she started lagging behind, and there is no way she can catch up because the total speed is the same. He is using his total speed to go only through time, so of course she will be behind him in time, and therefore she is the one that is younger than him when they meet up.

Look at that. Amazing, isn’t it?

### Why there are three dimensions of space and only one of time

This now brings us to the last question: why do we have three dimensions of space and we can move through it, but we have only one dimension of time and we are locked in it?

Louise, let’s take it home.

So Lou says: imagine when the universe was born, we had all four spatial dimensions, which were all four dimensions of space-time. They were identical.

Then we put a particle in it, and that particle always has to travel at the same space speed in four-dimensional space-time. Remember, everything is traveling at the speed of light in some direction.

Now the direction in which it is traveling at the speed of light, that particular direction will be length contracted to completely zero, because remember, length contraction becomes zero if you are traveling at the speed of light, right?

Therefore, in that particular dimension, you will not have any access. You will only see a ghost of it. But you only have access to all the dimensions perpendicular to it, and you can freely move in those dimensions only. That is what ends up becoming our spatial dimensions.

And the ghost of that dimension that is left over is what becomes our time dimension.

This is why, from our own perspective, we will find that whichever direction we are traveling through space-time at the speed of light, that dimension is length contracted and ends up becoming our time dimension. That is why we are moving — from our own perspective, we are always moving through time at the maximum speed. Our clocks are always running at the maximum speed, **1 second per second**.

But in the perpendicular direction, there is no length contraction, and so I have full access to it to move around. That ends up now becoming my spatial dimension.

I will be honest with you: I still have not fully wrapped my head around this, and I am sure there is more to it. But that is such a profound way of thinking about some of these incredibly deep things in physics, isn’t it?

So you and I, we both are indeed traveling through space-time at the speed of light.