[Eternal Student (OP)]

Hi all.

   Does anyone have some knowledge or insight about this "paradox" in the theory of Relativity?

   Imagine a submarine underwater.
Initially:
The submarine is at rest relative to the fluid and has adjusted it's tanks so that it has equal density with the fluid and remains at a depth of 100 metres.  (No thrust required from the engines, it just has neutral buoyancy).
 
Subsequently
    The submarine accelerates rapidly to reach relativistic speeds (let's say 0.9 c) relative to the fluid and then sustains a constant velocity.  This is intended to be a horizontal motion, the fins, bow planes etc. were not set to drive the submarine up or down.

Finally
     As is usual for these sorts of paradoxes,  we have two observers in two different frames of reference.

The submarine commander is at rest inside the submarine.  She should observe length contraction for the fluid in her rest frame and a corresponding increase in density of the fluid.  The submarine retains it's rest characteristics, including density in her frame.

A mermaid is at rest on the ocean floor.  In her rest frame, the density of the fluid has not changed,  however the submarine has undergone length contraction in her frame and it's density has increased.

Question
   Will the submarine rise or sink due to buoyancy?

Background info:   You may like to read the Wikipedia article about Supplee's paradox.
https://en.wikipedia.org/wiki/Supplee%27s_paradox
There is also a similar discussion about a Helium balloon moving through air on another forum.  (I'm not sure I should put links to another forum).

   I do not know the answer.  I can see references to articles in that Wikipedia entry but they seem to demand some application of General Relativity and a complete re-write of the Archimedian principle.  I was wondering if there is a resolution based only on Special Relativity - but I'll take any insight or discussion I can get.

[alancalverd]

Archimedes beat you to it!

The submarine displaces its own volume of water.

Each appears to shrink from the point of view of the other, but the volume ratio remains constant.

The increase in density of the water, as seen from the submarine, is the same as the increase in density of the submarine as seen from the water.

So it stays where it is.

[Eternal Student (OP)]

Hi and thanks Alancalverd.
   I'll take a day to consider what you've said but, at first glance, I'm not convinced with it.

The increase in density of the water, as seen from the submarine, is the same as the increase in density of the submarine as seen from the water.

   So, let's just take the view from the submarine and the sub commanders rest frame.  Initially, the submarine has volume V and the density of the fluid is d.   So an amount of fluid (in Kilograms)   Vd  was displaced,   giving a buoyancy force  of size  Vdg,    which exactly matches the submarines  weight of   Mg.
    Finally (after the acceleration), the sub is still the same in the commanders frame but the Density of the water is now D > d.    Giving a displaced fluid mass of   VD  >   Vd.   The mass of the submarine is unchanged in the commanders frame and (we presume) so is g the vertical acceleration due to gravity.   So   VDg  >   Vdg = Mg   and the submarine should be buoyant and start rising upward.   
    You seem to be suggesting that the sub wouldn't displace  a volume   V   of fluid in the submarine commanders frame afer the acceleration and I can't see why.

[Bored chemist]

Archimedes beat you to it!

The submarine displaces its own volume of water.

Each appears to shrink from the point of view of the other, but the volume ratio remains constant.

The increase in density of the water, as seen from the submarine, is the same as the increase in density of the submarine as seen from the water.

So it stays where it is.

Would you like to read the question again?

[Halc]

My gut says the whole concept of buoyancy is out the window once the object surpasses the speed of sound in the medium. There's going to be this vacuum cavity all around the thing that collectively displaces far more volume than the sub regardless of the frame in which it is considered. Somehow the analysis needs to proceed in a way that ignores that.

[Eternal Student (OP)]

Hi.

My gut says the whole concept of buoyancy is out the window once the object surpasses the speed of sound in the medium.

   Thanks Halc and you may be right.

   I've had most of the day to think about it while shopping and cutting the grass.   I can't see an easy way to consider and resolve the paradox with Special relativity alone.  I think it demands that Arhimedes' principle is thrown out of the window and replaced with a relativistic version.  This still isn't enough, the rest frame of the fluid has more problems...

  In the rest frame of the fluid (the mermaid's frame) it isn't clear that the downward force on the submarine due to it's weight would remain constant before and after the acceleration of the sub.  It's almost as if the idea of relativistic mass is needed so that downward force  on the submarine  becomes more like  M_rel g  rather than just Mg.   Alternatively you need to consider that gravity may push harder on an object with kinetic energy than it pushes on the rest mass alone.

   I'm part way through reading the references that are cited at the bottom of the Wikipedia article.  It seems that if the full machinery of General relativity is applied, then almost everything goes out of the window:   For example, the ocean floor curves upwards in the rest frame of the submarine, so that it's still possible for the sub to get closer to the ocean floor despite being positively buoyant.  I'm increasingly less convinced that there is a satisfactory resolution using SR alone, so much has to be modified and adjusted that you are almost using a version of GR anyway.

   If anyone's interested,  the currently accepted resolution of the problem is that the submarine would actually start to sink (get closer to the ocean floor).  Like all "paradoxes", this must happen in both frames of reference so that the paradox is resolved.

   We aren't building any really fast submarines but apparently there is some application of this result in Astronomy.   Fast moving particles (in orbit around stars) and immersed in fluid (stellar fluid) would be less buoyant than anticipated.  Hence, such particles may sink further in toward the centre of the star than was originally thought.

[Halc]

I can't see an easy way to consider and resolve the paradox with Special relativity alone.

The problem is considered (on the wiki webpage) using Rindler coordinates in Minkowski spacetime, making it fall squarely in the realm of special relativity. There's no gravity then, and no force pushing anything down.
Not claiming to be personally up to the mathematics, but it can very much be handled by SR alone.
OK, the ocean becomes flat, not curved around a planet, but we're not trying to make the sub accelerate in a circle round the planet several times a second.

Fast moving particles (in orbit around stars) and immersed in fluid (stellar fluid) would be less buoyant than anticipated.

Orbiting stuff has a concept of buoyancy???  I.E. if it's immersed in a stellar fluid, how is that an orbit?

[Eternal Student (OP)]

Orbiting stuff has a concept of buoyancy???  I.E. if it's immersed in a stellar fluid, how is that an orbit?

    "Buoyancy" seems like a fair description.  An effect of differing pressure on two surfaces of an object, that causes a force on the body that is opposed to the pressure gradient in the fluid.  A pressure gradient in a fluid is always likely when there is a gravitational field.
     I would have thought that even the moon experiences buoyancy in the low density fluid that is thin atmosphere spread out from earth.  However, the effect is so slight we wouldn't notice it.  The moon is already receeding due to tidal effects at about 4cm per year  (if I recall correctly).  Objects can be in orbit (not necessarily a perfectly stable orbit) and experience buoyancy.
     The example about stars was a bit arbitrary.  It's just something I was thinking of, not something I know an Astronomer is looking at now.  There is an example concerning black holes but I wasn't going to get into that because it's more complicated.  That last section should not have been written in a way that looked like there is a concrete example being studied now.  Sorry everyone.  It's gone midnight and I just wanted to close the post.

[Janus]

Hi all.

   Does anyone have some knowledge or insight about this "paradox" in the theory of Relativity?

   Imagine a submarine underwater.
Initially:
The submarine is at rest relative to the fluid and has adjusted it's tanks so that it has equal density with the fluid and remains at a depth of 100 metres.  (No thrust required from the engines, it just has neutral buoyancy).
 
Subsequently
    The submarine accelerates rapidly to reach relativistic speeds (let's say 0.9 c) relative to the fluid and then sustains a constant velocity.  This is intended to be a horizontal motion, the fins, bow planes etc. were not set to drive the submarine up or down.

Finally
     As is usual for these sorts of paradoxes,  we have two observers in two different frames of reference.

The submarine commander is at rest inside the submarine.  She should observe length contraction for the fluid in her rest frame and a corresponding increase in density of the fluid.  The submarine retains it's rest characteristics, including density in her frame.

A mermaid is at rest on the ocean floor.  In her rest frame, the density of the fluid has not changed,  however the submarine has undergone length contraction in her frame and it's density has increased.

Question
   Will the submarine rise or sink due to buoyancy?

Background info:   You may like to read the Wikipedia article about Supplee's paradox.
https://en.wikipedia.org/wiki/Supplee%27s_paradox
There is also a similar discussion about a Helium balloon moving through air on another forum.  (I'm not sure I should put links to another forum).

   I do not know the answer.  I can see references to articles in that Wikipedia entry but they seem to demand some application of General Relativity and a complete re-write of the Archimedian principle.  I was wondering if there is a resolution based only on Special Relativity - but I'll take any insight or discussion I can get.

As with most apparent "paradoxes" in Relativity, this one is likely due to only focusing on one aspect of the theory and ignoring the rest.
For example, here, you say that the Water appears more dense due to length contraction from the Sub's frame of reference, and thus should be more buoyant. However, what causes buoyancy?  First you need gravity ( this in of itself means you need GR to properly address the issue).   This in turn causes a difference in the water pressure between the top of bottom of the sub.  It is this difference in pressure ( water pushing up on the bottom more than it pushes down on the top) that produces the net upwards force.
Now you need to consider the effect that water flowing past the sub has on the static pressure of the water. Bernoulli's principle states that a moving fluid's static pressure decreases with fluid velocity.   That suggests that if the sub starts with neutral buoyancy at rest, it will not remain so once it begins to move relative to the water.  Thus if it wants to stay at constant depth, it will need to adjust its bow planes to a new angle.* 
But now, because of length contraction, the Mermaid will measure that angle as being different, changing the effect they produce...

The point is that just considering how length contraction acts on the water or sub as seen by sub or mermaid doesn't give you the full picture as to what is going on.  SR has been proven to be entirely self-consistent. Which means that no such thought experiments can never produce a real paradox, as long as you properly apply it.**

*and this doesn't even go into how Relativity would factor in to calculating the resultant effect.
** Now this, in of itself, does not prove the correctness of SR, just that thought experiment alone is not enough to disprove it.  You need a real life experiment or observation that doesn't match what Relativity predicts for that.

[Eternal Student (OP)]

Hi Janus.  I hope you are well.
   Thanks for your time.  I don't think there is a lot of disagreement with anything you said.

First you need gravity ( this in of itself means you need GR to properly address the issue).

   Yes,   or you can create the effect using an accelearted reference frame and SR.   For example, Halc mentioned using Rindler co-ordinates earlier.   I tend to agree with you, why go half way to re-creating the theory of General Relativity from SR,   someone (mainly Einstein) has already done that for us when they produced GR.  However, Halc is technicallty correct using Rindler Co-ordinates is considered an acceptable trick that keeps us in the terriotory of Special Relativity rather than GR.  The full power of GR is only required if you were trying to re-create a gravitational field that wasn't assumed to be uniform everywhere. 

Bernoulli's principle states that a moving fluid's static pressure decreases with fluid velocity.

   Yes.  Although we can model the submarine as being perfectly symmetric top and bottom and the fluid is often modelled as having a uniform density.  So the net forces applying to the top and bottom of the sub can change equally when the fluid is flowing.

SR has been proven to be entirely self-consistent.

  Just to make it clear,  I don't think anyone was trying to disprove SR.   I was trying to understand how the paradox is resolved and what the easiest way to do this may be.