The Speed of Light Limit Makes Space Travel EASIER not Harder（原声文稿）

Many people believe that special relativity is the major hurdle when it comes to interstellar travel, as the speed of light is the upper bound of how fast can you move.

Even though the speed of light is really fast, roughly 300 ,000 km a second, the distances in the universe are often measured in thousands or millions of light years.

But in my last video I showed you how the universe would look like if you kept accelerating at 1g forever.

And in the video we reached the closest star Proxima Centauri in just 2 years and 105 days, despite its distance from Earth being 4 .25 light years away.

So supposedly, speed of light was not a problem for us, moreover, it seems like relativity makes this kind of travel even easier.

As then, we crossed the entire Milky Way galaxy in just 12 years and reached the Andromeda galaxy in 15 years, despite it being 2 .5 million light years away.

In a video, I argue that this is due to length contraction.

For the observer on ship, the distance at the proxima center is smaller due to length contraction and therefore he doesn't need to travel as far.

So this upper bound on speed seems to be compensated by the length contraction.

But how does this compare to classical mechanics without any speed limit?

Or in other words, using classical mechanics, how long would it take to reach the Andromeda Galaxy?

In classical physics, things are very simple.

This is Earth, this is Andromeda Galaxy and this is our rocket.

An observer on Earth measures Andromeda Galaxy to be 2 .5 million light years away.

One lighter is this number of meters, so Andromeda is this much away.

Ship turns on the engine and the observer inside the ship experiences 1g of acceleration.

The observer on Earth measures that the ship's velocity is changing at a rate of 9 .81 meters per second squared, so they both agree on what the acceleration is. Their clock is perfectly synchronized during the whole trip.

Now you want a relation between distance, time and acceleration.

And this is one of the first things you'll learn at high school, at physics classes.

Distance is equal to one half times acceleration multiplied by time squared.

Luckily enough, both observers agree on the measured values, so it doesn't matter if you use the measurements from the Earth observer or the observer onboard the ship.

Or you mix them. You can use any of these equations.

You can understand classical physics as a system where each observer gets identical ruler and a clock.

Now, when one measures a certain value of distance and time, then other observers will measure the same values, since they have identical equipment.

Now, if you solve for time, you will find out that the time to reach the Andromeda galaxy is 2200 years.

This is significantly more than 15 years according to special relativity.

And this is the part where special relativity becomes so complicated, because mixing quantities like this is no longer possible.

And therefore we want to ask exactly what measurements do we really care about? So the same scenario again.

Earth, Andromeda and a ship.

The 2 .5 million light years of distance is what we as observers on Earth measured.

And this is the measurement we care about because all the distances we have ever measured so far were done here on Earth in this reference frame.

So I look at the Andromeda, I measure it to be 2 .5 million light years away.

And then I jump into the rocket and travel there, and I no longer care about the distance I would measure during the travel.

Then what I care about is how much of my biological age it will cost me, so I care about the ship's time, not the Earth's time.

And finally, I suppose that my engine will generate 1g of acceleration that me as a travelling observer feel as 1g.

So I don't care what change of velocity with time an observer on earth measures.

So I want to find a relation between these three quantities.

Looking at the classical relations, this is what we are interested in.

But as I said, in classical physics this doesn't matter.

So how does this relation look like in special relativity?

Would it be the case that by combining these three in special relativity, you will get something like this?

And obviously not, as I have already demonstrated, it would take 2200 years in classical physics to reach the Andromeda galaxy, and just 15 years in relativity.

So the relation has to be different.

I'm not going to show you the derivation, but you would get this expression.

It's a lot different than the classical one.

We can understand this in a way that in relativity it's no longer true that everybody has the same ruler and a clock.

Now everybody has its own equipment and the observers will no longer agree on measured values, because if you remember, time and space are relative.

But there is one thing they agree on, it's the speed of light.

All observers using their ruler and a clock always measure the same value, no matter how fast they move.

This is not the case in classical physics, where the speed would depend on the velocity of the observer.

So the special relativity doesn't make all quantities relative, but it makes some quantities we thought to be absolute, relative and vice versa.

But let's analyze this equation a little bit more.

Here in the argument of cosine hyperbolic, we have proper acceleration times the proper time divided by the speed of light.

At the start of the journey, this number in the numerator is small compared to the speed of light.

Hyperbolic cosine can be written as an infinite series like this.

And if the alpha is very small, then these higher orders of alpha are negligible compared to the first one.

So this is the cosine hyperbolic approximation for small alpha.

If we use this approximation in our relativistic expression and simplify, we get the equation known from classical physics.

So if you compare these two functions, they're almost identical on the start of the journey.

But at later times, this number grows much larger than c, and therefore this coefficient is much larger than 1.

We can rewrite cosine hyperbolic in terms of exponential functions.

The negative exponential goes to 0 if the coefficient is too large and becomes negligible compared to the positive exponential.

Now we have exponential expression.

So in relativity, the distance grows exponentially when this coefficient becomes much larger than 1, whereas classical expression grows only quadratically.

Therefore, these two curves diverge massively later on.

Here is the distance 2 .5 million light years, which corresponds to 15 years on a relativistic curve.

You can imagine the difference on a hand of speedometer in a car.

Well, in classical car, when you press the pedal, the speed hand grows at a constant rate.

Not ever faster, not ever slower, but steady.

if you accelerate exactly at 1g. In relativistic car this is different.

If you talk about the effective velocity, at the start the growth is steady the same as in classical physics.

But as we move closer to the speed of light, the effective speed growth starts to accelerate as well. And the faster you go, the faster it grows.

This is the same as compound interest on the stock market and this is the difference between being able to travel almost anywhere in the universe and not being able to live our galaxy so special activities not a bug in the interstore travel it's a feature but now we also have to talk about the downside of it imagine you want to visit Andromeda but you also want to go back home.

In classical physics you can accelerate towards Andromeda in the first part of the journey, then you turn the engine back and decelerate, and you reach the Andromeda in roughly 3 ,000 years.

Then you can repeat the same thing on the way back and you are home and your friend on earth is just 6 ,000 years older, the same as you. So now you can spend the retirement together.

In special activity you reach the half of the journey in 14 years and then you turn back the engine and decelerate for another 14 years and you reach the Andromeda Galaxy in 28 years.

Now you repeat the process on the way home and you reach the Earth in 56 years.

But your friend on Earth is almost 5 million years old so it will be quite hard to relate with such a large age gap.

as your friend would be a fossil, so special activity gave us the opportunity to travel almost anywhere in our lifetimes, but there is a price to pay somewhere else.

But there is a huge problem with everything what I just said, let me just break the speed of light limit, so at rest the Andromeda is roughly 2 .5 million light years away, and when you turn your engines you reach it in 15 years, so you must have seen In the Andromeda moving faster than light.

And that is true, Andromeda would in fact be approaching you faster than light.

So are we breaking the relativity here?

Special relativity was built upon inertial frames, which are frames without acceleration.

The first postulate states that all inertial frames are equal, and therefore there is no way to tell who is at rest and who is in uniform motion.

And the second one states that the speed of light is always the same, which is C in such a frame.

But it says nothing about accelerated motion, so these two postulates must hold for inertial observers, but not for accelerated ones.

The spacetime of an inertial observer is described by Minkowski spacetime, where you have time on the vertical axis and space on the horizontal axis.

light on this diagram has a straight world line at 45 degree angle and every inertial observer has straight world line in this region as observer in this region will have to travel faster than light and now things start to be a little bit weird such a diagram for accelerated observer looks different here on this diagram we have our ship at 1g of acceleration in this yellow starting from the Earth, moving towards Proxima Centauri 4 .25 light years away.

I used Proxima because Andromeda is too far and I would have to use logarithmic scale.

During the trip, Earth hits the Rindler horizon and becomes frozen in time, whereas Proxima is approaching very fast and dips into a forbidden zone, moving faster than light.

Interestingly enough, after you cross the star it starts to slow down and end up in the Rindler horizon as well.

If you want to know what is the Rindler horizon I recommend this video, but it's not important here.

So the speed of light limit makes it easier to travel through the Universe, provided you don't plan to come back home.

The Universe is expanding at a rate of 70 km per second per megaparsec.

This means that everything within roughly 14 billion light years is in principle within your reach, as everything further away moves away faster than the speed of light.

And you don't need absolutely high accelerations, just simple 1G.

But what about the fuel?

After all, the observer on Earth sees you accelerate for 2 .5 million years.

Do you need to have enough fuel to accelerate this long?

And this is the magic of time dilation.

Because the speed of light is finite, the time on the ship slows down to almost zero. And this also happens in your fuel tank.

So with time, the observer on Earth sees you spend less and less fuel per unit of time.

Eventually, if an observer on Earth, you will be an object frozen in time, traveling at the speed of light.

And this is the direct consequence of the finite speed of light.

As I said, the light always travels with the same velocity.

So if you have a light clock like this, which takes 2 bounces every second, then if such clock moves almost at the speed of light in this direction, there is only very little velocity left in the vertical direction so the overall speed remains c.

This is why the timer runs slower the faster you go.

And this doesn't apply only to the light clock.

It applies to every clock, mechanical one or biological one.

Because it's always about electromagnetic communication between neighboring atoms.

So the speed of light limit not only protects you from aging and gives you the ability to reach vast distances, but you also need a lot less fuel.

So that will be it for this video, thank you for watching and if you really want to know how the universe would look like if you kept accelerating forever then watch this video here.