B Helium Balloon: Explaining Special Relativity Effects

[Freaky Fred]

Hi, guys o/
Suppose a person sees a helium balloon moving horizontally at close to the speed of light. From this perspective, the observer will see the helium balloon contract in accordance with special relativity. If helium contracts to the point where it is denser than air, will it fall? how to explain it to the person? That's curious because, for the balloon, the air will be denser.

 

[Vanadium 50]

Such a balloon would be obliterated by the collisions with air molecules.

 

[hutchphd]

Balloonists usually avoid launching in wind speeds above 0.1 c.

 

[Will Learn]

You're assuming the balloon floats beacuse of pressure considerations and conventional fluid dynamics. Is that all that is causing you concern? For real fluid dynamics, the air around the balloon would be dragged along with the balloon and therefore also show length contraction and density changes.

 

[Orodruin]

For real fluid dynamics, the air around the balloon would be dragged along with the balloon and therefore also show length contraction and density changes.

For real fluid dynamics the balloon would quickly be oblitterated.

To the OP’s actual question: Two observers will always agree on the existence of actual events. Their explanations may differ, but anything that occurs in a frame must occur in all frames.

 

[pervect]

I thought I'd mention some of the available literature on this problem.

The paradox has a name, known as the submarine paradox, or Suplee's paradox, after the person who first published a formulation of it. Wiki has a short entry on it, see https://en.wikipedia.org/w/index.php?title=Supplee's_paradox&oldid=1020483779

From the wiki bibliography, there's an an arxiv paper, https://arxiv.org/abs/gr-qc/0305106.

I'd have to read the papers more carefully to give a considered answer - but I thought I'd mention there was literature written on the topic.

 

[DaveC426913]

This question needs some clarification before it can be answered. (Posters have been attempting to address these but the OP needs to do the clarifying)

Is the balloon in a medium such as air? (It would have to be to presume it's floating, so let's assume yes)

Is the balloon moving relative to the air? This will lead to very different outcomes depending on your answer.

 

[Freaky Fred]

Let's assume this balloon is indestructible. I'm assuming a scenario where the air is denser than the balloon at rest, but the balloon at relativistic velocity is denser than air to the observer.

 

[Ibix]

Let's assume this balloon is indestructible. I'm assuming a scenario where the air is denser than the balloon at rest, but the balloon at relativistic velocity is denser than air to the observer.

In that case, read @pervect's references.

 

[Freaky Fred]

I thought I'd mention some of the available literature on this problem.

The paradox has a name, known as the submarine paradox, or Suplee's paradox, after the person who first published a formulation of it. Wiki has a short entry on it, see https://en.wikipedia.org/w/index.php?title=Supplee's_paradox&oldid=1020483779

From the wiki bibliography, there's an an arxiv paper, https://arxiv.org/abs/gr-qc/0305106.

I'd have to read the papers more carefully to give a considered answer - but I thought I'd mention there was literature written on the topic.

This is very interesting, basically the ''same situation''.

 

[DaveC426913]

Let's assume this balloon is indestructible.

Alas, relativity explicitly forbids infinitely rigid objects. Pretty sure indestructible objects are the same deal. The motion, structure strength and rigidity of any object, real or theoretical, is limited.

If you try to assume such infinities, relativity will give you nonsensical answers*.

*(for example: an infinitely rigid stick 240,000 miles long would allow you to send messages to and from the Moon infinitely fast, creating a paradox).

These aren't merely pendantic details getting in the way of your thought experiment; they are fundamental in the relationship between spacetime, mass and energy.

 

[Ibix]

Pretty sure indestructible objects are the same deal.

Probably, but he doesn't need indestructible, only ridiculously strong. I think the problem is well enough defined that pervect's references on Supplee's paradox are an answer.

 

[Bandersnatch]

I've got to say, the solution is enjoyably counter-intuitive.

 

[Office_Shredder]

I've got to say, the solution is enjoyably counter-intuitive.

The solution also appears intuitive in some way, in that the observer who is always at rest ends up being right, and the moving object that is potentially accelerating and hence not in an inertial frame is the one that is "wrong" (i.e. not properly analyzed)

 

[Vanadium 50]

First, have you actually looked at a helium balloon in a windstorm? It is not gently buoyed up - it's zipping around hither and yon. And those are air velocities of less than one millionth the speed of light.

Next, even in Newtonian mechanics, Archimedes Principle is only an approximation.

Technically, you need GR to solve this, because it involves gravity. I think we can use an analogy where the forces are due to magic, but again, this will be an approximation.

It is probably simplest to solve this by using the Cauchy stress tensor and recognizing that the fluid is non-uniform. An alternative would be to look at a microscopic picture where pressure and buoyancy arise from molecular collisions and Lorentz transform the collisions.

 

[PeterDonis]

Technically, you need GR to solve this, because it involves gravity.

One could set the scenario inside an accelerating rocket in flat spacetime, which can be analyzed using SR. Tidal gravity, which is what would require GR, is not really necessary for the scenario.

 

[PeterDonis]

One could set the scenario inside an accelerating rocket in flat spacetime

This appears to be what the Matsas paper referenced in the Wikipedia article on Supplee's paradox is doing. It uses Rindler spacetime, which is just flat Minkowski spacetime in non-inertial coordinates. (Note that the line element given in the paper is not the usual Rindler coordinates, but it is easily verified that its Riemann tensor vanishes, so it is indeed flat Minkowski spacetime in disguise.) This means that the paper's claim to be giving a "general relativistic" resolution is not really correct.

 

[pervect]

If we consider a "submarine" whose top is at the surface of the water, the pressure on the surface is zero, and the submarine is entirely supported by the pressure on the bottom.

The magnitude of the pressure needed is the same as that of the relativistic sliding block. This happens to be a problem I'm somewhat familiar with.

The short version is that the pressure under the block scales as $\gamma^2$, increasing, the contact area scales as $1/\gamma$, decreasing, and the total force, pressure * area, scales as $\gamma$.

In the sliding block case, we talk about "weighing" the block by measuring the pressure under it and multiplying by the contact area. The fact that the weight of the block increases by a factor of $\gamma$ while the contact area shrinks by the same factor is what gives the $\gamma^2$ pressure dependence.

In the submarine case, we replace "weighing" the block with the question of whether the submarine is neutrally buoyant or not.

Because the downward force on the submarine increases, and the buoyant force decreases due to the shrinking contact area, the neutrally buoyant submarine sinks when it's in motion.