We have a submarine (sub) in a frictionless liquid and the average sub density is equal to the density of a liquid. Sub is fully submerged not touching the bottom. Now the sub moves at near speed of light. In the reference frame of liquid the sub undergoes relativistic contraction, has higher density now and must sink. In the reference frame of a sub the liquid undergoes relativistic contraction, is now more dense. Hence, the sub must resurface. Contradiction. How to resolve it? Idon't think we need GTR, Newtonian gravity should be enough.
$\begingroup$
$\endgroup$
4
-
2$\begingroup$ My first thought, not having thought through this much granted, is that special relativity might not apply here since in the event that buoyant/gravitational forces are not perfectly balanced the submarine will be accelerating and thus not in an inertial reference frame. Plus, there is a clear preferred frame here anyway: the rest frame of the liquid infinitely-far from the sub. $\endgroup$controlgroup– controlgroup2025-10-10 23:49:30 +00:00Commented Oct 10 at 23:49
-
5$\begingroup$ Moving a moderately large object so fast in a liquid that you need to consider SR, would be a catastrophic event. It will be a nuclear fusion explosion. $\endgroup$naturallyInconsistent– naturallyInconsistent2025-10-11 00:52:01 +00:00Commented Oct 11 at 0:52
-
$\begingroup$ +1 for introducing me to this interesting thought experiment. $\endgroup$cms– cms2025-10-11 01:10:57 +00:00Commented Oct 11 at 1:10
-
$\begingroup$ When the submarine is at rest, $$F_g - F_b = \frac{GMm}{r^2} - \frac{\rho V GM}{r^2} = 0$$ When the submarine is traveling fast horizontally, $m$ increases uncontrollably (due to relativistic mass / gravitational transverse Doppler blueshift), while $\rho$ likely decreases overall (higher in front of the submarine, lower behind the submarine, and lower on the sides of the submarine). So now, $$F_g - F_b > 0$$ $\endgroup$James– James2025-10-11 23:28:13 +00:00Commented 2 days ago
Add a comment
|
5 Answers
$\begingroup$
$\endgroup$
This is an apparent paradox (not actually a paradox in the sense of a logical contradiction) known as Supplee's paradox, first presented in 1989 in the paper:
Relativistic buoyancy, https://doi.org/10.1119/1.15875
This page provides a non-technical discussion, and a corresponding Wikipedia page is also available. A more technical approach and a convincing solution to the paradox was proposed in 2003 in:
Relativistic Archimedes law for fast moving bodies and the general-relativistic resolution of the "submarine paradox", https://doi.org/10.1103/PhysRevD.68.027701
The relativistic submarine (treated as an ideal projectile in a perfect fluid medium) does "go down" when the full energy-momentum tensor is considered in General Relativity. The fix is that ordinary Archimedes' law is not Lorentz-invariant. If you transform the full stress–energy (pressure + energy density) and gravity consistently, both frames agree: a neutrally buoyant projectile at rest will sink once it moves fast parallel to the surface.
Edit: I discovered now that there are already other PSE posts on the same paradox: Is there a simple derivation of the solution to the Submarine Paradox in terms of Special Relativity?, Explanation of Supplee's paradox, as well as this answer. Feel free to add links here if you find more PSE references.
$\begingroup$
$\endgroup$
3
As @controlgroup's comment pointed out, I believe that in this fluid there is a preferred frame of reference, which could lead to the answer that there is no need to consider SR at all (recall that when Maxwell's equations weren't compatible with Galilean relativity, before Einstein people thought that the EM waves were travelling in "Ether" and thus that there was no inconsistency).
This suggests that you probably need general relativity, in which the laws of physics are the same in all frames of reference. That said, it seems that solutions to this "paradox" have been considered in both frameworks: Solution of Supplee's submarine paradox through special and general relativity, and that certain assumptions allow us to solve it even in SR.
Also, I just saw Quillo's answer when I refreshed the page. Still posting mine because it has the link to an openly accessible paper on the matter (my institution doesn't have access to the ones in his post, and I assume OP might not have access either).
-
1$\begingroup$ The paper by Vieira seems to assume that in the frame of the seabed, the force due to liquid on the submarine is as if the liquid was still and only the submarine has decreased volume (since it applies the Archimedes law to a contracted volume). But the liquid around the submarine is in relativistic flow, and the liquid around the submarine gets contracted and denser too. $\endgroup$Ján Lalinský– Ján Lalinský2025-10-11 17:20:05 +00:00Commented 2 days ago
-
$\begingroup$ @JánLalinský Agree, to be honest I believe (as I kind of hinted at at the beginning of the post), that SR/GR does not tell the full story. In my opinion, the most important part of the discussion is about how the fluid behaves at relativistic speeds, which is a topic that I am unfamiliar with and which, at first glance, looks relatively underdeveloped. Relativistic Fluid Dynamics is one of the only resources I've found, and I am unsure about its application to the problem (I don't really have the time to check the details currently). $\endgroup$hecate– hecate2025-10-12 10:53:06 +00:00Commented yesterday
-
$\begingroup$ I think the paradox does not have to be interpreted too much in a literal sense, as it is clearly intended not to go into details of fluid dynamics: it is more about a projectile in a frictionless fluid once a steady state is reached. See it as a neutrally buoyant spherical impurity in a relativistic superfluid that has a Landau velocity greater than that of the impurity. Of course, it is extremely far from anything realisable in nature. $\endgroup$Quillo– Quillo2025-10-12 14:00:47 +00:00Commented yesterday
$\begingroup$
$\endgroup$
1
Density of a submerged body is obviously determinative of the direction of its motion only when the sub is initially at rest with respect to the water.
If the sub already moves horizontally even a bit, lots of water of the same volume moves in the opposite direction in the frame of the still water far from the sub, with the opposite velocity (the water moves and fills in the space the sub was in previously). This motion of water could be directed by the sub control surfaces (diving planes) to produce thrust in any direction, up or down. Then, density of the sub and the water alone is not enough to determine the direction of motion; the shape of the sub enters the analysis too.
We can eliminate this effect - if the sub and the flow are very symmetric (topside vs bottomside), there may be zero thrust in vertical direction on the sub of the above kind. Then, if the sub is fast enough to undergo substantial relativistic contraction, its density increases, so the usual simple argument is that in the frame of the water, the sub should move down, due to greater density of the sub. However, this argument assumes, erroneously, that whether the sub goes up or down is determined by densities of the sub and still water far from the sub alone.
In fact, the pressure forces on the sub are not due to hydrostatic forces of still water, but they are forces due to moving water. Water is moving near and around the sub, and this motion is relativistic too (moving in the opposite direction to fill in the space the sub was in). Thus that water undergoes contraction as well, and its density increases as well. Not all water gets denser, only that moving fast, near the sub. This denser water itself is not hydrostatically stable and experiences net force downwards. A complicated flow of dense water mixing with slower, less dense water, may occur. Net result of this on the sub is not clear; this complicated flow can send the submarine in any direction.
answered Oct 10 at 23:55
-
$\begingroup$ We ignore viscosity, structural considerations, control surfaces, shock waves. As well as the fact that if we rub hands at half light speed, we get energy yield of a thermonuclear bomb. $\endgroup$sixtytrees– sixtytrees2025-10-12 10:22:54 +00:00Commented yesterday
$\begingroup$
$\endgroup$
4
We can take gravity (and GR) out of the picture by supposing a mass of liquid being accelerated by its container with $a = g$. The mass of the liquid would be very small compared to the mass of the Earth, and all pressures would be consequence of the acceleration.
Simplifying the sub geometry to a prismatic shape, the requirements for the equilibrium $F = (p_{bottom} - p_{top})A = m_{sub}g$ would not be modified by its horizontal velocity.
The length contraction results from desynchronizing of the clocks in the RF's of the liquid and the sub. I don't see how they could have any effect on the pressures and the mass of the sub.
Or, we can say that the Archimedes principle is valid for densities (liquid and object) measured in the same frame.
answered Oct 11 at 2:59
-
$\begingroup$ The general consensus is that the sub sinks if it moves relative to the sourse of gravity. I would not pretend that I understand the gravimagnetic correction in this case, but it makes the liquid undergo relativistic compression and even strunger effect on the submarine gravitational pull (due to the gravimagnetic component). $\endgroup$sixtytrees– sixtytrees2025-10-12 10:19:29 +00:00Commented yesterday
-
$\begingroup$ @sixtytrees that is why I used the principle of equivalence to avoid the complications of curved spacetime. For a flat spacetime approximation I don't see any reason for the sub sinks of floats. $\endgroup$Claudio Saspinski– Claudio Saspinski2025-10-12 13:55:21 +00:00Commented yesterday
-
$\begingroup$ The $A$ in that equation is the projected horizontal surface area of the submarine. That surface area changes due to relativistic length contraction. That's why it is reasoned that the equilibrium should change. As side note, the $p_{bottom}$ and $p_{top}$ also change due to Bernoulli's principle, but I don't see anybody talking about that. $\endgroup$fishinear– fishinear2025-10-12 14:29:05 +00:00Commented yesterday
-
$\begingroup$ @fishinear The length contraction is a side effect of the desynch of clocks. From the perspective of the crew, its length doesn't change, and there is liquid outside everywhere doing pressure. Bernoulli principle is a consequence of Newton second law for fluid flow inside pipes of varying sections. It doesn't applies here. Of course there would be big dragging effects for a real liquid. But it was supposed frictionless. $\endgroup$Claudio Saspinski– Claudio Saspinski2025-10-12 18:17:40 +00:00Commented yesterday
$\begingroup$
$\endgroup$
The motion of the sub is not allowed to have a relativistic effect on buoyancy. Extra math working provided, two ways, for fun.
"Visual" Special Relativity:
Ignoring the effect of the intervening water on the visual appearance of the submarine, the raw photographable appearance of the moving sub is still affected by differential timelag effects.
If the submarine recedes (or approaches) at v, SR predicts it being seen to elongate (or contract) by a total factor of length'/length = SQRT [(c-v) / (c+v)] . So if it recedes or approaches at 0.8c , we see it squashed by a ratio of 1:3 (or stretched by a factor of 3:1), depending on where we stand. This is analogous to the SR Doppler shift for frequency (Penrose: "spatial analogue of the Doppler effect").
If we place observers before and behind the sub, it would be strange if the same submarine was seen to sink and scrape into the seabed and be covered in mud and crabs for one observer, and to instead have risen and broken the water's surface and be spattered with seagull poop for the other, depending on the observer's position.
"Interpreted" Special Relativity
SR's total, physical prediction (for visual lengths and frequencies) can be broken down into two components, the "light-propagation" Doppler and differential-timelag effects, and the Lorentz effects.
- If we decide that the speed of light is fixed in our own frame, then the Doppler effect is E'/E = length'/length = c/(c+v) = 1/1.8 = 5/9 , and the Lorentz redshift/contraction is sqrt[1 - v^2/c^2] = sqrt[1- 0.64] = sqrt[0.36] = 0.6 = 3/5
Total length-change and frequency change is then E'/E = length'/length = propagation * Lorentz effects = 5/9 * 3/5 = 3/9 = 1/3
- If we decide that the speed of light is fixed in the sub's frame, then the Doppler and differential-timelag length effects are different, they are now E'/E = length'/length = (c-v)/c = 0.2/1 = 1/5 , and since WE are moving, and time-dilated and contracted, we see a Lorentz blueshift/elongation of 1.666' = 5/3
Total length-change and frequency change is now E'/E = length'/length = propagation * Lorentz effects = 1/5 * 5/3 = 1/3
... exactly as before
So in any given symmetrical situation, it is physically impossible to isolate and verify the Lorentz component in anything, as a matter of principle. A single SR-compatible photograph can be interpreted as showing either a Lorentz contraction and redshift when propagation effects are taken into consideration, or a Lorentz elongation and blueshift.
It depends which propagation effects we believe to be correct, and since the choice has zero effect on the final experimental outcome, whether the Lorentz factor is nominally red or blue, or whether it corresponds to a nominal lengthening or a shortening, is purely interpretative, and has no consequences for anything in the real world, outside the experimenter's head (unless they choose to use that belief in calibrating their equipment).
Since the experimenter is free to believe that the Lorentz effect is either lengthening or shortening the sub, that internal metaphysical belief cannot have physical real-world consequences. One cannot sink or float a submarine just by thinking about it.
Conclusion
If the sub floated to the top, everyone would know that the Lorentz effect was "really" an elongation, and that we were "really" moving. If it sank, everyone would know that it was the sub that was "really" moving.
Since both of these outcomes would allow us to identify a physically-preferred global frame for the propagation of light, violating the relativity principle, both are illegal, and relativistic effects cannot be allowed to change the sub's buoyancy.
