Consider the following thought experiment:

Imagine an object of a certain mass density which allows it to float in water.

Now if this object is viewed from a moving frame with high speed, it will look Lorentz contracted hence its density will look greater than the rest density.

So from the rest frame it floats but from the moving frame it sinks!

What am I missing?

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    $\begingroup$ If the fluid was at rest relative to the object moving at relativistic velocity, we would be in an extremely high Reynold's number flow regime and buoyancy would become meaningless well before any effect would be felt. I don't think it's unreasonable that some sort of shift in buoyancy would exist as a high order correction to a relativistic theory of fluid flow. $\endgroup$ – Stephen McAteer Jan 8 '12 at 22:30

Trivially the density of the water increases identically to the object and so bouyancy is maintained...

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  • $\begingroup$ What happens when it is moving through the water? Sure, in practice the effects of deflecting the water will dominate, but if that could be eliminated, this would actually be a somewhat interesting thought experiment. $\endgroup$ – wnoise Jan 8 '12 at 19:36
  • $\begingroup$ then the answer depends on what effect the questioner is claiming will occur, $\endgroup$ – Nic Jan 8 '12 at 22:24
  • $\begingroup$ @wnoise, an object moving through water at relativistic speeds? I don't know about water. how about air? what-if.xkcd.com/1 $\endgroup$ – Solomon Slow Mar 11 '16 at 22:41

There is an answer that is--in some sense--even more trivial than nic's argument: the whole point of relativity is that the result of the experiment (does it float or sink?) is not dependent on the observer's frame of reference.

I know, I know: your first introduction to relativity is always about how things do depend on the frame, but it is always about how things you thought were invariant (because there are under Galilean transforms, really aren't)

  • How long are the cavities?
  • How long does the beam take to get from the hall entrance to the target?
  • Is the speeding spaceship ever full closed into the dock (i.e. does the rear door close before or after the front door opens)?

These kinds of questions ask about only part of a 4-vector (the space component and the time component respectively) between two events. Those question are not Lorentz invariant, so the answers depend on the frame you choose.

But consider questions about

  • Does the electron beam have enough energy to produce charmonium on a hydrogen target?
  • What was the mass[*] of the neutral particle that decayed to produce the two charged particles we see in the detector?
  • Does the spaceship hit the front door?
  • Does that stuff sink or float?

These kind of questions must have the same answer in every frame.

Please, take away the idea that physics does not depend on the choice of frame. This is very important.

[*] The invariant mass, guys. This is why particle physicists don't talk about "relativistic mass": because the real mass is the Lorentz invariant mass.

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    $\begingroup$ Actually I think Revo knows that "Does that stuff sink or float?" must have the same answer in every frame. That's why he asks "What am I missing" as he found an ostensible paradox. $\endgroup$ – Siyuan Ren Jan 8 '12 at 2:00
  • $\begingroup$ @KarsusRen and dmckee: I think my problem is that I am not sure which questions will have the same answers in all inertial frames and which questions will not. For me I do not know how to know that intuitively in advance by just reading the question. If you guys know how please share. $\endgroup$ – Revo Jan 8 '12 at 12:39
  • $\begingroup$ You have to be careful about this idea that the "same" event exists for all frames. A trivial example is the event of a particle having some velocity in one frame whereas it has a different velocity for the "same" event in another frame. I would say the only invariant event is the labeling of some space-time point by all possilble coordinate systems. All reference frames will agree on how the other frames labelled this space-time point. $\endgroup$ – John McVirgooo Jan 9 '12 at 2:07

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