I mean, we can go very close, but we can never ever reach the speed of light.

Why is that?

The most common explanation I got is if you take the graph of the kinetic energy versus the speed, look, as the speed approaches C, kinetic energy goes to infinity, so it takes infinite energy.

But my problem is that that's not an explanation.

I don't have an intuition behind why that is true.

Another explanation was mass goes to infinity, but again, there's no evidence for that.

So what's really going on?

What's the physically happening that's limiting that ship from going towards the speed of light?

That's what we're gonna try and answer in this video.

And so if you're ready for it, let's begin.

So Einstein, where do we start?

Einstein says, imagine you are inside a very fast-moving ship and you have a photon clock with you.

Now at this point, I say, wait a second, Einstein.

I don't wanna deal with hypothetical clocks.

I wanna talk about real clocks.

Einstein says, well, buddy, patience.

If you understand photon clocks, you'll be able to understand everything else.

I'm like, okay, cool, let's trust him.

So what's a photon clock?

It's a clock where you have two mirrors and photon bounces between the two.

And we can say that, hey, when the photon hits the bottom mirror, it hits a tick.

So we get tick, tick, tick, tick, tick, and so on and so forth.

Now Einstein asks, what would the same clock look like if you are seeing the whole thing from outside the ship?

Well, let's see.

At first, I'm like, hey, it looks the same to me.

The photon is bouncing between the two mirrors.

But Einstein says, if you look carefully, you see the photon is now traveling a diagonal path.

This means it's gonna travel a longer distance.

But remember, the photon always travels with the speed of light.

See, the speed of light is the same in all reference frames, which means the photon now is gonna take a longer time between the ticks.

In other words, we will see that photon clock ticks slower.

So again, let's compare.

Here's what the photon clock would look like from inside the ship.

Look at the ticks.

Tick, tick, tick, tick.

Now from outside.

Tick, tick, tick.

You can see it has become slower.

And now what happens if the ship moves even faster?

Well, it will travel even longer path, as you will see.

And therefore, the ticks will become even slower.

So the photon clock shows that clocks slow down when they are moving.

And at this point, I say, cool, photon clocks tick slower.

That's great.

Now what about real physical clocks, Einstein?

Well, Einstein says, if you look at this clock, and if you zoom in to the second hand, what would the animation look like from the atomic scale?

And I say, well, if the second hand is just ticking like this, then the animation would look something like this.

But Einstein says, that can't happen.

You can't have something like this.

And the reason is when, say, this atom accelerates, this atom in particular accelerates, all the other atoms cannot accelerate at the same time.

Information takes time to travel.

These atoms are bonded electromagnetically.

So when one atom accelerates, we need to wait for an electromagnetic wave to go from one atom to another, and only then the other atom can accelerate.

An electromagnetic wave is basically a photon.

So the better way to think about this is, when this atom accelerates, it actually sends a photon, and only then the next atom accelerates, and then the next one, and the next one, and so on, and so forth.

So you see, the whole thing is not instantly going, but it's so fast, it feels like it's instantly going.

But now Einstein asks,

Mahesh, what if now this set of atoms were moving?

What would the animation look like now?

Wait a second.

This also is very similar to the photon clock, which means if this was moving, the photons would end up taking a diagonal path again, and therefore the whole thing slows down.

Well, let's look at it.

Let's look at it, here we go.

Yes, the photons are slowing down.

The photon is not slowing down, the transfer is slowing down, because the photons are taking a diagonal path just like before, and therefore, real physical clocks actually slow down.

Look at this, this one is ticking slower than this one.

Real physical clocks slow down when they're moving.

Whoa.

But at this point, we could ask Einstein, is there evidence for this?

This is just a theory, right?

So Einstein says, well, probably not in his time, but today we do.

You see, we've actually taken atomic clocks, which are synced on Earth, and then we put them on planes, and then we flew around the world, and then we brought them back, they went out of sync.

And we did this multiple times, and every time we checked, we saw that the amount they went out of sync perfectly predicts, is perfectly predicted by special relativity.

Now, of course, gravity also affects it, which is dealt in general relativity, so there is this additional effect that's happening, but when you put all of that together, there is evidence that suggests that time dilation is real.

So with this, we know that time itself is slowing down, or is it?

You see, clocks are slowing down, but how does it tell us that the time itself is slowing down, Einstein?

Einstein says, for that, let's look at radioactivity.

Radioactive decay is where you have an unstable atom that spontaneously splits.

The cool thing about radioactivity is that they have something called the half-life.

For example, if you take a particular radium isotope, it has a half-life of about 1,600 years, which means that if you take a billion radium isotopes, wait for 1,600 years, half of them would have decayed, and the rest half would have stayed.

Wait for another 1,600 years, another half would have decayed, and so on and so forth.

Now, it turns out that if you take elementary particles like muons, they have a half-life of about 1 1⁄2 microseconds.

We did that in lab, we have tested this in labs, but we also get muons in our atmosphere due to the cosmic rays colliding with our atmosphere.

Turns out that these muons, when we looked at them, they have a half-life of 10 times more.

At first, I'm like, Einstein, why is that a problem?

Like, these are maybe different muons, and Einstein says, no.

Just like how the charge of an electron stays the same, regardless of where the electrons come from, or when you do the experiment, or where you do it, the half-life of radioactive sample of a particular radioactive isotope would be the same, regardless of where that isotope comes from.

So, how is it possible that these muons would have 10 times more half-life?

Well, it turns out that's because they're traveling close to speed of light.

And so, if you, again, plug in the numbers according to special relativity, you get the exact same result.

Time dilation.

Time dilation is literally making these muons age slower.

But what's physically going on?

Something very similar to what we saw in the photon clock.

The fact that these muons are moving, this means that photons are taking longer, or whatever that force carrier, photons are not the force carrier, this is a weak nuclear force, but whatever the force carriers are, they will now take a longer time, and therefore the radioactive process gets slowed down.

And that's why muons are literally aging slower because they're moving.

Time dilation.

And if muons can age slower, if radioactivity process can slow down, all biochemical processes can also slow down.

And that means living beings will age slower.

So, if you have two twins, one baby, sorry, two twins, one moving, then you will see the moving twin will age much slower than the twin that's at Earth.

This is a real process.

And the fact that muons age slower somewhat gives us an indirect evidence that this is a real, real phenomenon.

Whoa.

And we could still argue that Einstein, the processes are slowing down over here, and because the force carriers take a longer time because of the motion, but that doesn't mean that the time itself slows down.

And at this point, Einstein says, well, Mahesh, in science, you need some process to measure anything.

And so if you believe that time exists beyond the processes and measurements, then it's now beyond the realm of science, and now we're going into philosophical debate, and that's meaningless for science.

So science deals with the measurables.

And so as far as we can measure, everything that is affected by time is being slowed down, and so we say time itself slows down.

Okay.

Einstein, now, how does this whole time dilation thing explain why objects can never be accelerated to speed of light?

Einstein says, for that, let's first derive the expression for this time dilation, and at first, I'm like, no, no, no,

I want an intuitive explanation, and Einstein says, don't worry, it will only add to the intuition.

And it's also cool to actually derive it because we can do it logically as well.

So let's quickly do that.

So for that, we're gonna use distance equals speed over time, and if you look at our clock from inside the space, when the, inside the spaceship, when the clock is at rest, the distance traveled by the photon is just speed into time, so c into t.

So this is the time that we see from inside the ship.

Now, when we look at it from outside the ship, this is what it would look like.

And so if the time taken now is longer, and let's call that time as t dash, this distance would now again be the speed of light into this new time, the time that we see from outside the ship, t dash.

And now you can see, you have two sides of the triangle.

If you can figure out the third side, we're done.

But what is that third side?

The third side is the distance traveled by the mirror in the time t dash, or the distance traveled by the ship in the time t dash, and we know the speed of the ship is v, then it'll become v times t dash.

And now it's just the Pythagoras theorem.

We can use Pythagoras theorem to isolate t dash, and we can figure it out.

Great idea for you to pause and derive this historical expression yourself.

It's such a proud moment.

But if you've tried it, here it goes.

So we'll just use Pythagoras theorem.

And now we're just gonna do some algebra to isolate t dash.

And boom, this is the time dilation equation.

And let's pause here because this gives me goosebumps.

It's one of the cornerstones of physics, and what did we use?

Pythagoras theorem to derive it.

And yet, more than 150 years ago, even the smartest folks couldn't comprehend this.

They had no idea that universe behaved this way.

Oh my God.

Wow.

It can be confusing when you're first learning it, and so Einstein says, let's make sure that we're on the same page with the vocabulary.

This t is what we call the proper time.

This is the time that we see when the clock is at rest.

And since when things are at rest, we see time flowing normally for us, that's why it's called the proper time.

Anybody inside the spaceship will basically say nothing weird is happening.

The time is proper for me, so proper time.

And this t dash is what we call the dilated time.

This is the time that we see when we're looking at things from outside the spaceship.

And to make this equation slightly more intuitive, this is how we like to write it.

This is the usual notation as well, so let's familiarize ourselves with this.

Gamma is what we call the Lorentz factor.

And we like to talk in terms of gamma because gamma is a number that's bigger than one.

And again, it makes sense.

The dilated time must be bigger than the proper time by the factor gamma.

And now let's look at what are some values for gamma for different values of speed.

So let's do that.

And you can immediately see, even at 50% the speed of light, gamma is so small, 1.2.

And so for lower speeds, gamma is almost one.

And that's the reason why Newtonian physics works because if gamma is almost one, dilated time is the same as the proper time and we don't see any discrepancy, we just have one time.

But notice as you go closer and closer to the speed of light, gamma value increases.

So for example, at 87% the speed of light, gamma is two.

This means the dilated time is twice the proper time.

That means when I'm looking at the ship, in my clock when two seconds tick, in that ship I will see one second ticking.

It's dilated twice.

Similarly, I'd say 99.9% the speed of light, gamma is 22.

It means I have to wait 22 seconds for one second to tick in that moving clock.

Does that make sense?

All right, so we have the intuition for where the time dilation comes from because the photon ends up traveling a longer distance.

We have the numbers now.

Einstein, it's time to answer the question.

Why can't objects be accelerated to the speed of light?

So Einstein says, well, imagine we have a spaceship, very advanced spaceship that is accelerating at say 1,000 kilometers per second per second.

This means that every second that spaceship is gaining a velocity of 1,000 kilometers per second.

This will happen if there's a constant force acting on the spaceship, okay?

Now, initially at very low speeds, we will see nothing weird.

We will see every second the spaceship gain 1,000 kilometers per second.

So it'll start with zero, then it'll get 1,000, then it'll get 2,000, 3,000.

Every second it'll keep getting 1,000, 1,000, 1,000, 1,000.

But what happens when we go close to the speed of light?