The paradox of a charge in a gravitational field is an apparent physical paradox in the context of general relativity. A charged particle at rest in a gravitational field, such as on the surface of the Earth, must be supported by a force to prevent it from falling. According to the equivalence principle, it should be indistinguishable from a particle in flat spacetime being accelerated by a force. Maxwell's equations say that an accelerated charge should radiate electromagnetic waves, yet such radiation is not observed for stationary particles in gravitational fields. One of the first to study this problem was Max Born in his 1909 paper about the consequences of a charge in uniformly accelerated frame. Earlier concerns and possible solutions were raised by Wolfgang Pauli (1918), Max von Laue (1919), and others, but the most recognized work on the subject is the resolution of Thomas Fulton and Fritz Rohrlich in 1960.
Background
It is a standard result from Maxwell's equations of classical electrodynamics that an accelerated charge radiates. That is, it produces an electric field that falls off as 1 / r {\displaystyle 1/r} in addition to its rest-frame 1 / r 2 {\displaystyle 1/r^{2}} Coulomb field. This radiation electric field has an accompanying magnetic field, and the whole oscillating electromagnetic radiation field propagates independently of the accelerated charge, carrying away momentum and energy. The energy in the radiation is provided by the work that accelerates the charge. The theory of general relativity is built on the equivalence principle of gravitation and inertia. This principle states that it is impossible to distinguish through any local measurement whether one is in a gravitational field or being accelerated. An elevator out in deep space, far from any planet, could mimic a gravitational field to its occupants if it could be accelerated continuously "upward". Whether the acceleration is from motion or from gravity makes no difference in the laws of physics. One can also understand it in terms of the equivalence of so-called gravitational mass and inertial mass. The mass in Newton's law of universal gravitation (gravitational mass) is the same as the mass in Newton's second law of motion (inertial mass). They cancel out when equated, with the result discovered by Galileo Galilei in 1638, that all bodies fall at the same rate in a gravitational field, independent of their mass. A famous demonstration of this principle was performed on the Moon during the Apollo 15 mission, when a hammer and a feather were dropped at the same time and struck the surface at the same time. Closely tied in with this equivalence is the fact that gravity vanishes in free fall. For objects falling in an elevator whose cable is cut, all gravitational forces vanish, and things begin to look like the free-floating absence of forces one sees in videos from the International Space Station. It is a linchpin of general relativity that everything must fall together in free fall. Just as with acceleration versus gravity, no experiment should be able to distinguish the effects of free fall in a gravitational field, and being out in deep space far from any forces.
Statement of the paradox Putting together these two basic facts of general relativity and electrodynamics, we seem to encounter a paradox. For if we dropped a neutral particle and a charged particle together in a gravitational field, the charged particle should begin to radiate as it is accelerated under gravity, thereby losing energy and slowing relative to the neutral particle. Then a free-falling observer could distinguish free fall from the true absence of forces, because a charged particle in a free-falling laboratory would begin to be pulled upward relative to the neutral parts of the laboratory, even though no obvious electric fields were present. Equivalently, we can think about a charged particle at rest in a laboratory on the surface of the Earth. In order to be at rest, it must be supported by something which exerts an upward force on it. This system is equivalent to being in outer space accelerated constantly upward at 1 g, and we know that a charged particle accelerated upward at 1 g would radiate. However, we do not see radiation from charged particles at rest in the laboratory. It would seem that we could distinguish between a gravitational field and acceleration, because an electric charge apparently only radiates when it is being accelerated through motion, but not through gravitation.
Resolution by Rohrlich
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