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, contradictions and paradoxes are generally assumed to be artifacts of error and incompleteness because reality is assumed to be completely consistent, although this is itself a philosophical assumption. When, as in fields such as quantum physics and relativity theory, existing assumptions about reality have been shown to break down, this has usually been dealt with by changing our understanding of reality to a new one which remains self-consistent in the presence of the new evidence.
Paradoxes relating to false assumptions
Certain physical paradoxes defy common sense predictions about physical situations. In some cases, this is the result of modern physics correctly describing the natural world in circumstances which are far outside of everyday experience. For example, special relativity has traditionally yielded two common paradoxes: the twin paradox and the ladder paradox. Both of these paradoxes involve thought experiments which defy traditional common sense assumptions about time and space. In particular, the effects of time dilation and length contraction are used in both of these paradoxes to create situations which seemingly contradict each other. It turns out that the fundamental postulate of special relativity that the speed of light is invariant in all frames of reference requires that concepts such as simultaneity and absolute time are not applicable when comparing radically different frames of reference. Another paradox associated with relativity is Supplee's paradox which seems to describe two reference frames that are irreconcilable. In this case, the problem is assumed to be well-posed in special relativity, but because the effect is dependent on objects and fluids with mass, the effects of general relativity need to be taken into account. Taking the correct assumptions, the resolution is actually a way of restating the equivalence principle. Babinet's paradox is that contrary to naïve expectations, the amount of radiation removed from a beam in the diffraction limit is equal to twice the cross-sectional area. This is because there are two separate processes which remove radiation from the beam in equal amounts: absorption and diffraction. Similarly, there exists a set of physical paradoxes that directly rely on one or more assumptions that are incorrect. The Gibbs paradox of statistical mechanics yields an apparent contradiction when calculating the entropy of mixing. If the assumption that the particles in an ideal gas are indistinguishable is not appropriately taken into account, the calculated entropy is not an extensive variable as it should be. Olbers' paradox shows that an infinite universe with a uniform distribution of stars necessarily leads to a sky that is as bright as a star. The observed dark night sky can be alternatively resolvable by stating that one of the two assumptions is incorrect. This paradox was sometimes used to argue that a homogeneous and isotropic universe as required by the cosmological principle was necessarily finite in extent, but it turns out that there are ways to relax the assumptions in other ways that admit alternative resolutions. The Mpemba paradox is that under certain conditions, hot water will freeze faster than cold water even though it must pass through the same temperature as the cold water during the freezing process. This is a seeming violation of Newton's law of cooling but in reality it is due to non-linear effects that influence the freezing process. The assumption that only the temperature of the water will affect freezing is not correct.
Paradoxes relating to unphysical mathematical idealizations
Mathematical idealizations used in approximations lead to paradoxes if pushed beyond their range of validity. For example, point sources describe physical phenomena well at distant or global scales but break down at the point itself. These paradoxes are sometimes seen as relating to Zeno's paradoxes which all deal with the physical manifestations of mathematical properties of continuity, infinitesimals, and infinities often associated with space and time. For example, the electric field associated with a point charge is not defined at the location of the point charge. This reflects the limits of the classical field theory approximation. A consistent theory of quantum electrodynamics removes the need for point charges altogether. Similarly, general relativity, another classical field theory, has a gravitational singularity associated with the Schwarzschild solution that describes the geometry of a black hole. The curvature of spacetime at the singularity is undefined, but the approximate theory predicts that a falling particle must reach it within a finite amount of proper time. It is hoped that the solution to this paradox will be found with a consistent theory of quantum gravity. The presence of relativistic singularities also poses an issue in Big Bang theory. Before a theoretical extrapolation of a singularity can occur, quantum mechanical effects become important during the Planck era. Without a consistent theory, there can be no meaningful statement about the physical conditions associated with the universe as it approaches this point. Another paradox due to mathematical idealization is D'Alembert's paradox of fluid mechanics. When the forces associated with two-dimensional, incompressible, irrotational, inviscid steady flow across a body are calculated, there is no drag. These conditions, however, do not hold in most physical systems, with the notable exception of superfluids moving below their Landau critical velocity. More general mathematical models work without some or all of these assumptions, thus giving way to more accurate solutions involving boundary layers.
Quantum mechanical paradoxes A significant set of physical paradoxes are associated with the privileged position of the observer in quantum mechanics. Two of these are:
the EPR paradox and the Schrödinger's cat paradox, These thought experiments supposedly to use principles from quantum mechanics to derive conclusions that are seemingly contradictory. In the case of Schrödinger's cat this takes the form of a seeming absurdity.
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