In
relativistic physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, Supplee's paradox (also called the submarine paradox) is a
physical paradox
A physical paradox is an apparent contradiction in physical descriptions of the universe. While multiple physical paradoxes have accepted resolutions, others defy resolution and may indicate flaws in theory. In physics as in all of science, c ...
that arises when considering the
buoyant force exerted on a relativistic bullet (or in a submarine) immersed in a fluid subject to an ambient
gravitational field
In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
. If a bullet has
neutral buoyancy
Neutral buoyancy occurs when an object's average density is equal to the density of the fluid in which it is immersed, resulting in the buoyant force balancing the force of gravity that would otherwise cause the object to sink (if the body's de ...
when it is at rest in a
perfect fluid and then it is launched with a relativistic speed, observers at rest within the fluid would conclude that the bullet should sink, since its density will increase due to the
length contraction
Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald ...
effect. On the other hand, in the bullet's proper frame it is the moving fluid that becomes denser and hence the bullet would float. But the bullet cannot sink in one frame and float in another, so there is a paradox situation.
The paradox was first formulated by James M. Supplee (1989),
where a non-rigorous explanation was presented. George Matsas
has analysed this paradox in the scope of general relativity and also pointed out that these relativistic buoyancy effects could be important in some questions regarding the
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
of
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s. A comprehensive explanation of Supplee's paradox through both the special and the general theory of relativity was presented by Ricardo Soares Vieira.
Buoyancy
To simplify the analysis, it is customary to neglect
drag and
viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
, and even to assume that the fluid has constant
density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
.
A small object immersed in a container of fluid subjected to a uniform gravitational field will be subject to a net downward gravitational force, compared with the net downward gravitational force on an equal volume of the fluid. If the object is ''less dense'' than the fluid, the difference between these two vectors is an upward pointing vector, the buoyant force, and the object will rise. If things are the other way around, it will sink. If the object and the fluid have equal density, the object is said to have ''neutral buoyancy'' and it will neither rise nor sink.
Resolution
The resolution comes down to observing that the usual
Archimedes principle cannot be applied in the relativistic case. If the theory of relativity is correctly employed to analyse the forces involved, there will be no true paradox.
Supplee
himself concluded that the paradox can be resolved with a more careful analysis of the gravitational buoyancy forces acting on the bullet. Considering the reasonable (but not justified) assumption that the gravitational force depends on the kinetic energy content of the bodies, Supplee showed that the bullet ''sinks'' in the frame at rest with the fluid with the acceleration
, where
is the
gravitational acceleration
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag (physics), drag). This is the steady gain in speed caused exclusively by gravitational attraction. All bodi ...
and
is the
Lorentz factor
The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
. In the proper reference frame of the bullet, the same result is obtained by noting that this frame is not inertial, which implies that the shape of the container will no longer be flat, on the contrary, the seafloor becomes curved upwards, which results in the bullet getting far away from the sea surface, ''i.e.'', in the bullet relatively sinking.
The non-justified assumption considered by Supplee that the gravitational force on the bullet should depend on its energy content was eliminated by George Matsas,
who used the full mathematical methods of
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
in order to explain the Supplee paradox and agreed with Supplee's results. In particular, he modelled the situation using a
Rindler chart, where a submarine is accelerated from the rest to a given velocity ''v''. Matsas concluded that the paradox can be resolved by noting that in the frame of the fluid, the shape of the bullet is altered, and derived the same result which had been obtained by Supplee. Matsas has applied a similar analysis to shed light on certain questions involving the
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
of
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s.
Vieira
has recently analysed the submarine paradox through both special and general relativity. In the first case, he showed that gravitomagnetic effects should be taken into account in order to describe the forces acting in a moving submarine underwater. When these effects are considered, a ''relativistic Archimedes principle'' can be formulated, from which he showed that the submarine must sink in both frames. Vieira also considered the case of a curved
space-time
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional continuum (measurement), continu ...
in the proximity of the Earth. In this case, he assumed that the space-time can be approximately regarded as consisting of a flat space but a curved time. He showed that in this case the gravitational force between the Earth at rest and a moving body increases with the speed of the body in the same way as considered by Supplee (
), providing in this way a justification for his assumption. Analysing the paradox again with this ''speed-dependent gravitational force'', the Supplee paradox is explained and the results agree with those obtained by Supplee and Matsas.
See also
*
Bell's spaceship paradox
*
Ehrenfest paradox
*
Ladder paradox
*
Twin paradox
References
External links
Light Speed Submarine- article about the paradox in Physical Review Focus
{{DEFAULTSORT:Supplee'S Paradox
Physical paradoxes
Theory of relativity
Relativistic paradoxes