The Relativistic Submarine Paradox（原声文稿）

Imagine you have a submarine in the ocean able to reach very high velocities.

This submarine has the exact same density as the water around, so the buoyancy force equals the gravity and the submarine is floating at the same level.

Now according to special relativity, if the submarine moves, it lengths, contracts, making its volume smaller and therefore density the higher.

This means that the buoyancy force should decrease and the submarine should sink.

On the other hand, from the frame of the submarine, it's the water that contracts, making its density higher and therefore buoyancy stronger, making the submarine rise.

And as we know, only one option can be correct, if you don't want to break physics. So which one is it?

This puzzle was first studied by the physicist named James Supley in 1989 who published an article in the American Journal of Physics together with a solution.

But this solution used assumptions that were not obvious that they should hold.

I am going to present them here anyway so you can make your own opinion about it.

But another 14 years of this paradox and nobody cared until one day a curious student approached George Matzas and asked how to resolve this paradox and Matzas could answer on the spot he said it was embarrassing I thought it was a scandal that there was no answer to this paradox so I decided to waste time on it and then he provided a full general relativistic solution with surprisingly or not did But later in 2016, Vera makes a claim that's basically a coincidence caused by chosen geometry.

So I'm gonna present to you a solution, and you have to make your opinion about it. Are those assumptions unjustified?

And do we really need a general relativity to say with certainty that this paradox is solved?

To solve the paradox, we first need to understand what it really means when we say paradox is solved.

Now let's assume we are observers at rest relative to the ocean.

Imagine there is a small hill on the bottom of the ocean and the submarine is a certain distance away from it.

When the submarine starts moving, it shrinks and the gravity wins over buoyancy.

Now we set the initial velocity and distances in such a way that the submarine will eventually hit the hill on the bottom.

So now we consider the paradox to be solved when the calculations in the rest frame of the submarine show us that we'll also hit the hill there.

So simply the submarine must hit the rock in every frame in the universe.

So imagine the submarine is moving at a speed that equals certain gamma factor.

The horizontal distance to the hill is s and the vertical distance to the floor is h.

What acceleration the submarine has to have to hit the rock?

So we have accelerated motion in vertical direction and uniform motion in horizontal direction.

From this we can extract t and plug it here and find the a.

This is a very classic problem from kinematics.

In the frame of the submarine, though, the horizontal distance to the rock is gamma times smaller, while the vertical distance remains the same.

So now, we replace the S by S over gamma and plug it into the same equation, and we find out that the acceleration in the frame of the submarine has to be gamma squared times larger towards the bottom, so the paradox can be considered as soft.

This is not only about whether the submarine will sink or not, But it has to sink exactly at prescribed rate, to say that this paradox is solved.

So now let's try to solve this paradox.

We are going to denote all the non -moving quantities as unprimed and all moving quantities as primed. Let's start first in the frame of the ocean.

The gravitational force acting on the submarine is m times g.

Since the submarine is moving, the mass is supposed to be a relativistic mass. So the final force is gamma mg.

The buoyancy force is determined by the volume of the submarine which is moving, the stationary density of the water and gravitational acceleration.

The volume of the submarine scales linearly with gamma, so we can rewrite the expression in terms of the rest volume of the submarine.

But the rest volume times the density of the water is equal to the rest mass of the submarine.

Now we can calculate the total force acting on the submarine and we get this result.

Now if we want acceleration, we should divide the force by mass, but this has to be the relativistic mass, so the acceleration looks like this.

Now to solve the paradox, we need to calculate that the acceleration in the submarine frame is equal to this.

Now the frame of the submarine is a lot more tricky. First, let's consider the buoyancy.

The force equals to the volume of the non -moving submarine, the density of the moving water, and also the gravitational acceleration of the moving surface.

For the observer on the ship is not only the water that contracts, but the surface as well.

If we assume an approximation of an infinite plane surface, the gravitational acceleration scales as gamma times g.

so overall the buoyancy scales as gamma squared and this is the force pushing the submarine up so this is not heading in the right direction right but let's look at the force pushing the submarine down f is equal to mg and here m is the rest mass of the submarine and as we know g scales as gamma g the volume times density of the water at rest is equal to the rest mass of the submarine.

So we can substitute it in and subtract the two forces.

What we get is that the total force is equal to gamma mg1 minus gamma.

Since gamma is always larger than one for non -moving submarine, then we see that this force has the opposite sign compared to the one with the right for the observer at rest relative to the water. And this is the paradox.

In one frame the force is pointing down and in other frame it points up.

But there is one other effect we haven't taken into account.

It's an effect that most people know, but only attributed to electromagnetism.

If you have a current carrying wire, then you have probably learned that it creates a magnetic field around it.

If you placed an electron in the direction of the moving electrons inside a wire, then this electron will be attracted by the wire that the magnetic part of the Lorentz force.

But magnetism is not some kind of special independent force of nature.

The reason this happens is that in the frame of the electron the protons are moving and are Lorentz contracted, while the already contracted electrons are now spread out.

This creates a net positive charge density in the wire pulling the electron down.

So magnetism is present every time you have a force between bodies and special relativity.

It is a motion effect of attractive force.

Similar effect you have also in gravity, although you only have a single charge, if you have a mass current and a particle moving along, then from the point of view of this particle the mass density is weaker as the particles are more spread out. So But this makes the attractive force beaker. This is called gravel magnetism. And this is the effect we have to take into account.

The submarine sees a mass current below as the surface moves.

So it detects gravel magnetic field, which is given by this expression.

The velocity of the submarine is zero, so it has no effect on the submarine, but the water is moving inside this gravel magnetic field with the velocity minus v, and therefore it feels gravel magnetic force.

So this effect directly affects the effective force pulling water down, which creates the buoyancy force on the submarine.

So let's compute the buoyancy once again, but this time properly.

So the buoyancy force on the submarine is given by the rest volume of the submarine times Fw, which is force acting on the water per unit volume.

This force has now two parts, the gravel electric part and gravel magnetic part, which depends on the velocity of the water and the velocity of the mass current.

Now if you plug in the numbers and you can pause the video if you want details, we get this expression, where this term is inverse gamma squared and therefore all the gamma dependencies cancel out.

And eventually, we find out that the buoyancy force is independent of the velocity of the water and the velocity of the mass current.

So we have managed to cancel off this nasty gamma square term from the buoyancy.

So this is going in the right direction, right? Now what about the force of gravity? So here nothing really changed.

The gravitational force is just amplified by gamma due to more dense floor, producing the gravity.

So now, if we add these two forces, we get this.

Now we can compare it with the result we got from the reference frame of the surface.

If we want to calculate the acceleration, we have to divide this by relativistic mass in the surface frame, and by the rest mass in the submarine frame.

And we said that if we want to claim the paradox is solved, we need this relation between accelerations, which we clearly didn't get. it.

The reason is that there is some asymmetry in our thinking.

If we assume infinite plane surface, the gravitational force acting on the body is given by this expression.

Here we took the relativistic mass of the object in the rest frame of the surface and this is the surface density of the floor.

But when we look at it from the rest frame of the submarine, we get more dense floor in one direction which gives us one gamma contribution, but also each chunk of matter now has kinetic energy since it's moving and therefore more mass.

More mass gives more gravity and therefore we end up with gamma squared scaling.

Exactly what we wanted, because now if you had gamma squared in the gravitational force, the math checks out perfectly.

But we have treated the relativistic mass as gravitational mass, But how can we be so sure that we can do this?

Also in the submarine frame, this relativistic mass density increase might seem to also affect buoyancy as it pulls more water down, which completely breaks our gravel magnetism.

But the water is in the surface rest frame, so it shouldn't feel it.

It only sees the rest mass of the surface the same way as it only sees the rest density.

So, this will be the paradox solved using standard theory of gravity, coupled with special relativity somehow, using some weekly justified assumptions.

As Vera says in his article about the topic, it should be mentioned however that the argument presented by Supley in his first explanation of the paradox concerning the gravitational force between moving bodies cannot be justified.

In fact, Supley assumed that Newton's law is valid even when the interacting bodies are in motion.

However, he also assumed, as many others do, that the gravitational mass should be determined by the Einstein's formula E is equal mc squared, which leads to the speed -dependent mass m equal gamma m0, where m0 is so -called rest mass, and u is the velocity of the body.

This, by its turn, leads to the same speed -dependent force as deduced by us in the previous section.

Notice however that the relation between mass and energy expressed by Einstein's formula about only holds when the momentum of the body is no.

In fact, the correct expression is this.

But anyway, we are in this article provided a solution using general overall activity and got exactly the same solution.

But he calls it a coincidence, the two arguments mentioned previously.

But this Vera argument seems kinda off, because this expression holds for any momentum of the body if m is taken to be a relativistic mass.

In fact, these expressions are equal, but I think there is a better argument, where you shouldn't always trust this analysis we just did previously.

In 1911, Einstein calculated how much a light ray would be deflected moving nearby the sun.

He simply used a particle with a finite relativistic mass moving at the speed of light. Which is exactly what photon is.

But then, when he discovered a general relativity, he did this calculation again and got deflection that is twice as strong.

And therefore Einstein thought that he must have done a mistake, because he trusted his original solution.

But then he realized that general relativity really produces genuinely different results to the same problem.

And this turned out to be a blessing, because now you had an exact experiment that you could do that would ultimately determine which one of those theories is the true law of nature.

And later it was proven many times that light really bends twice as strong compared to the classical calculation of a particle with a mass moving at the speed of light.

I think the coincidence we got the same result in the submarine paradox by using this shaky argument with general relativity is because in general relativity, in this kind of situation, we always only consider the time curvature, which is responsible for the gravitational attraction we feel right now.

But space is completely flat, and this is exactly what Biradit in his GR calculation.

But when you consider some full general relativistic scenarios of this paradox, then it can become even more interesting.

For example, in 2006, Johnson studied this paradox on a spherical ocean, so you have to take the centrifugal force into account.

And you know that every object has its own Schwarzschild radius, which is a radius at which this object would become a black hole.

So we can make the Schwarzschild radius comparable with the size of the planet.

A black hole has the so -called photon ring that lies exactly at 1 .5 times the Schwarzschild radius.

What Johnson found is that if the submarine is below this photon ring, it will sink. If it's exactly at the photon ring, it will stay the same depth.

And if it's above, it will float upwards relative to the ground.

And this is the kind of calculation you wouldn't be able to do without full GR.

But when you think about it, this can give you a significant insight into what happens with a matter in the vicinity of the black hole.

When you approach the horizon of a black hole, you experience stronger and stronger Hawking radiation, which creates force upward, while the black hole is pulling you down.

When the spaceship is moving, it contracts, experiencing less force from the Hawking radiation. Exactly as our submarine in the water.

Another thing that one might wonder is what if we removed the gravity completely and made the surface just accelerating infinite plane.

And this is what Suplee has been studying and he found out that in the frame of the submarine the surface wouldn't be straight but it would be curved upwards.

But this could be a topic for another video maybe I don't want to dig into it here that would be it for this one thank you for watching and I see you next time