Paradox of a charge in a gravitational field

"Paradoxes" in relativity theory are thought experiments used to expose flaws in conventional ways of understanding physics when applied to relativistic speeds; their resolution leads to correct ways of viewing the problem. Examples in the special theory of relativity (SR) are the twin paradox and the ladder-in-barn paradox. Neither are true paradoxes; they merely illustrate that the results of measurement differ in different reference frames, so in relativity the state of motion of the observer must be taken into account.

A similar paradox in general relativity (GR) is the paradox of a charge in a gravitational field. 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 space 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. The explanation of the apparent paradox is that Maxwell's equations hold only in an inertial reference frame, whereas a particle at rest in a gravitational field is not in an inertial reference frame.

Recap of key points of gravitation and electrodynamics

It is a standard result from the Maxwell equations of classical electrodynamics that an accelerated charge radiates. That is, it produces an electric field that falls off as $1/r$ in addition to its rest-frame $1/r^{2}$ 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. We understand a photon to be the quantum of the electromagnetic radiation field, but the radiation field is a classical concept.

The theory of general relativity is built on the principle of the equivalence of gravitation and inertia. This means 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. This can also be understood in terms of the equivalence of so-called gravitational mass and inertial mass. The mass in Newton's law of gravity (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 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, of course, 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 true absence of forces, because a charged particle in a free-falling laboratory would begin to be pulled 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 to balance the Earth's downward gravitational field of 1 g. This is equivalent to being accelerated constantly upward at 1 g, and we know that a charged particle accelerated upward at 1 g would radiate, why don't we 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 of the paradox by Rohrlich

The resolution of this paradox, like the twin paradox and ladder paradox, comes through appropriate care in distinguishing frames of reference. This section follows the analysis of Rohrlich (1965),^[1] section 8-3, who shows that a charged particle and a neutral particle fall equally fast in a gravitational field, despite the fact that the charged one loses energy by radiation. Likewise, a charged particle at rest in a gravitational field does not radiate in its rest frame. The equivalence principle is preserved for charged particles.

The key is to realize that the laws of electrodynamics, the Maxwell equations, hold only in an inertial frame. That is, in a frame in which no forces act locally. This could be free fall under gravity, or far in space away from any forces. The surface of the Earth is not an inertial frame. It is being constantly accelerated. We know that the surface of the Earth is not an inertial frame because an object at rest there may not remain at rest—objects at rest fall to the ground when released. So we cannot naively formulate expectations based on the Maxwell equations in this frame. It is remarkable that we now understand the special-relativistic Maxwell equations do not hold, strictly speaking, on the surface of the Earth, even though they were, of course, discovered in electrical and magnetic experiments conducted in laboratories on the surface of the Earth. Nevertheless, in this case we cannot apply the Maxwell equations to the description of a falling charge relative to a "supported", non-inertial observer.

The Maxwell equations can be applied relative to an observer in free fall, because free-fall is an inertial frame. So the starting point of considerations is to work in the free-fall frame in a gravitational field—a "falling" observer. In the free-fall frame the Maxwell equations have their usual, flat-spacetime form for the falling observer. In this frame, the electric and magnetic fields of the charge are simple: the falling electric field is just the Coulomb field of a charge at rest, and the magnetic field is zero. As an aside, note that we are building in the equivalence principle from the start, including the assumption that a charged particle falls equally as fast as a neutral particle.

The fields measured by an observer supported on the surface of the Earth are different. Given the electric and magnetic fields in the falling frame, we have to transform those fields into the frame of the supported observer. This is not a Lorentz transformation, because the two frames have a relative acceleration. Instead the machinery of general relativity must be used.

In this case the gravitational field is fictitious because it can be "transformed away" by appropriate choice of coordinate system in the falling frame. Unlike the total gravitational field of the Earth, here we are assuming that spacetime is locally flat, so that the curvature tensor vanishes. Equivalently, the lines of gravitational acceleration are everywhere parallel, with no convergences measurable in the laboratory. Then the most general static, flat-space, cylindrical metric and line element can be written:

c^{2}d\tau ^{2}=u^{2}(z)c^{2}dt^{2}-\left({\frac {c^{2}}{g}}{\frac {du}{dz}}\right)^{2}dz^{2}-dx^{2}-dy^{2},

where $c$ is the speed of light, $\tau$ is proper time, $x, y, z, t$ are the usual coordinates of space and time, $g$ is the acceleration of the gravitational field, and $u(z)$ is an arbitrary function of the coordinate but must approach the observed Newtonian value of $1+gz/c^{2}$ . This is the metric for the gravitational field measured by the supported observer.

Meanwhile, the metric in the frame of the falling observer is simply the Minkowski metric:

c^{2}d\tau ^{2}=c^{2}dt'^{2}-dz'^{2}-dx'^{2}-dy'^{2}.

From these two metrics Rohrlich constructs the coordinate transformation between them:

{\begin{aligned}x'&=x,\\y'&=y,\\{\frac {g}{c^{2}}}(z'-z_{0}')&=u(z)\cosh g(t-t_{0})-1,\\{\frac {g}{c}}(t'-t_{0}')&=u(z)\sinh g(t-t_{0}).\end{aligned}}

When this coordinate transformation is applied to the electric and magnetic fields of the charge in the rest frame, it is found to be radiating, as expected for a charge falling away from a supported observer. Rohrlich emphasizes that this charge remains at rest in its free-fall frame, just as a neutral particle would. Furthermore, the radiation rate for this situation is Lorentz-invariant, but it is not invariant under the coordinate transformation above, because it is not a Lorentz transformation.

So a falling charge will appear to radiate to a supported observer, as expected. What about a supported charge, then? Does it not radiate due to the equivalence principle? To answer this question, start again in the falling frame.

As observed from the falling frame, the supported charge appears to be accelerated uniformly upward. The case of constant acceleration of a charge is treated by Rohrlich^[1] in section 5-3. He finds a charge $e$ uniformly accelerated at rate $g$ has a radiation rate given by the Lorentz invariant:

R={\frac {2}{3}}{\frac {e^{2}}{c^{3}}}g^{2}.

The corresponding electric and magnetic fields of an accelerated charge are also given in Rohrlich section 5-3. To find the fields of the charge in the supported frame, the fields of the uniformly accelerated charge are transformed according to the coordinate transformation previously given. When that is done, one finds no radiation in the supported frame from a supported charge, because the magnetic field is zero in this frame. Rohrlich does note that the gravitational field slightly distorts the Coulomb field of the supported charge, but too small to be observable. So although the Coulomb law was of course discovered in a supported frame, relativity tells us that the field of such a charge is not precisely $1/r^{2}$ .

The radiation from the supported charge is something of a curiosity: where does it go? Boulware (1980)^[2] finds that the radiation goes into a region of spacetime inaccessible to the co-accelerating, supported observer. In effect, a uniformly accelerated observer has an event horizon, and there are regions of spacetime inaccessible to this observer. De Almeida and Saa (2006)^[3] have a more accessible treatment of the event horizon of the accelerated observer.

Notice that this solution contradicts the bremsstrahlung effect. In any case, this solution could be considered a classic approach to a problem that cannot be handled in classic electrodynamics.

Proposal of Feynman

Classic (non-quantum, general relativistic) radiation equation is obtained assuming the Maxwell's equations in curved spacetime. These are four tensorial differential equations and their solution is the radiation field.

To integrate the equations several assumptions must be made. The classic one is the principle that "a charge does not interact with its own field". Feynman modified this hypothesis while he was working in the Wheeler–Feynman absorber theory. Assuming that a charge does in fact interact with itself, he obtained a different expression for the radiation Feynman radiation equation yields the same value that classic equations in circular acceleration cases, but the big difference is that it yields no radiation for an uniformly accelerating charge seen from an inertial frame. Should Feynman be right, a charge uniformly accelerated in an inertial frame would not radiate in any frame. And a charge hold static in a gravitational field would not radiate in any frame. Not in its own non-falling frame, and not in any other frame.

Though maybe it is surprising, there is no experiment that measures directly if an uniformly accelerated charge radiates in an inertial frame. Classic relativistic electromagnetic solution says it should, and Feynman solution says it shouldn't, but this divergence has never been confirmed by an experiment, because experiments use circular accelerating charges to produce the radiation, or use small particles like electrons that obey quantum laws (see again bremstrahlung)

Notice also that in reality there is not such a thing as a classic relativistic point charge. Real charges behave as quantum objects and they obey quantum electrodynamic theory, not classic GR. Therefore the problem of the classic relativistic accelerated charge is more a theoretical problem than a real one. The question should be reformulated as "how should we modify the concept of a classic relativistic point charge so that it becomes consistent with GR?"

New Proposal

In order to solve this paradox, one has to notice the difference between gravity force and Lorentz force. Lorentz force only acts on charge. Gravity force uniformly acts on mass. If a charged particle such as electron is viewed as its electromagnetic field, then its mass is distributed in its field. As it free falls under gravity, every part of its electromagnetic field is accelerated at the same rate. Thus there is no ripple and distortion in its electromagnetic field and there is no radiation. However Lorentz force only acts on charge which is at the center of its electromagnetic field. The acceleration is non-uniform and propagated from its center to its entire field. Thus there is ripple and distortion in its electromagnetic field. It is the ripple which contributes to the radiation.

As a similar example, when a spring free falls under gravity, its length doesn't change because every part of the spring has the same acceleration. However if you drag one end of the spring and accelerate, the spring will not only accelerate as a whole, but its length will oscillate as well because the acceleration applied to the spring is non-uniform and propagated from one end to the other. The field lines of the electromagnetic field of a charged particle can be viewed as springs governed by the Maxwell equations instead of a mechanical equation.

Notice that though this explains perfectly the lack of radiation for a free-falling charge, this says nothing about charges hold "static" (non free falling) in the same field, and does not explain why charges on the earth surface do not radiate.

References

1 2 Rohrlich, F. (1965). Classical Charged Particles. Reading, Mass.: Addison-Wesley.
↑ Boulware, David G. "Radiation from a Uniformly Accelerated Charge". Ann. Phys. 124: 169–188. doi:10.1016/0003-4916(80)90360-7.
↑ de Almeida, Camila; Saa, Alberto (2006). "The radiation of a uniformly accelerated charge is beyond the horizon: A simple derivation". Am. J. Phys. 74: 154. doi:10.1119/1.2162548.

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