In relativistic physics, Supplee's paradox (also submarine paradox) arises when considering the buoyancy forces exerted on a relativistic object (such as a bullet) moving through a dense fluid such as water in a gravitational field. Analysis is simplified by ignoring frictional effects such as drag and viscosity. The paradox is named for James M. Supplee .

If a reference frame in which the fluid is at rest is chosen and the bullet moves at speed v, the Lorenz contraction of the bullet, together with its relativistic mass increase will make its density γ²ρ where γ = (1 - v² / c²) ^{- 1 / 2}.

If, for example, the rest densities of fluid and bullet are equal, ρ_f = ρ_b, then because γ > 1 the buoyancy force will be negative.

Alternately, consider the problem from the reference frame of the bullet, in which the fluid is moving past. The density of the bullet will be its rest value, and the density of the fluid will be γ²ρ_f. If we again choose equal densities, the buoyancy force would be expected to be positive.

Supplee considered the problem in terms of two coordinate systems: an unprimed system, in which the fluid was at rest, and a primed system in which the projectile was at rest. In both systems, a transverse acceleration provided "gravity".

Supplee observed that the "seeming paradox can be resolved by noting that in the primed frame the lake bottom is no longer flat", and showed that in both frames the projectile sinks with acceleration g(γ² - 1).

The paradox was also considered by George E. A. Matsas , who used the methods of general relativity to analyse the problem. Matsas included the curvature of spacetime induced by the nonzero density of the fluid in his analysis, and pointed out that different observers would measure different shapes for the submarine; Matsas's analysis agreed with Supplee's for the case of weak gravitational fields/accelerations. He commented that

the apparently contradictory conclusion reached in the submarine rest frame by the mariners, who would witness a density increase of the liquid volume elements, is resolved by recalling that the gravitational field is not going to "appear" the same to them as to observers at rest with the fluid.

Matsas went on to use his analysis to shed light on the thermodynamics of black holes.