Hi all.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.
Does anyone have some knowledge or insight about this "paradox" in the theory of Relativity?
Imagine a submarine underwater.
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).
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.
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.
Will the submarine rise or sink due to buoyancy?
Background info: You may like to read the Wikipedia article about Supplee's 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.
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.
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