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8/16/2019 Paradox of a Charge in a Gravitational Field - Wikipedia, The Free Encyclopedia

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4/14/2016 Paradox of a charge in a gravitational field - Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Paradox_of_a_charge_in_a_gravitational_field

Paradox of a charge in a gravitational fieldFrom Wikipedia, the free encyclopedia

The special theory of relativity is known for its paradoxes: the twin paradox and the ladder-in-barn paradox

for example. Neither are true paradoxes; they merely expose flaws in our understanding, and point the way

toward deeper understanding of nature. The ladder paradox exposes the breakdown of simultaneity, while

the twin paradox highlights the distinctions of accelerated frames of reference.

So it is with the paradox of a charged particle at rest in a gravitational field: it is a paradox between the

theories of electrodynamics and general relativity.

Contents

1 Reca p of K ey Points of Gravitation and Electrodynamics2 Statement of the Paradox

3 Resolution of the Paradox4 References

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 in addition to its rest-frame

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 cance

out when equated, with the result discovered by Galileo that all bodies fall at the same rate in a gravitation

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

(https://upload.wikimedia.org/wikipedia/commons/3/3c/Apollo_15_feather_and_hammer_drop.ogg) 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. One can find the weightlessness

https://upload.wikimedia.org/wikipedia/commons/3/3c/Apollo_15_feather_and_hammer_drop.ogghttps://upload.wikimedia.org/wikipedia/commons/3/3c/Apollo_15_feather_and_hammer_drop.ogghttps://en.wikipedia.org/wiki/Photonhttps://en.wikipedia.org/wiki/Special_theory_of_relativityhttps://en.wikipedia.org/wiki/Twin_paradoxhttps://en.wikipedia.org/wiki/Ladder_paradoxhttps://upload.wikimedia.org/wikipedia/commons/3/3c/Apollo_15_feather_and_hammer_drop.ogghttps://en.wikipedia.org/wiki/Mass#Galilean_free_fallhttps://en.wikipedia.org/wiki/Mass#Inertial_masshttps://en.wikipedia.org/wiki/Mass#Universal_gravitational_masshttps://en.wikipedia.org/wiki/Equivalence_principlehttps://en.wikipedia.org/wiki/Photonhttps://en.wikipedia.org/wiki/Classical_electrodynamicshttps://en.wikipedia.org/wiki/Maxwell_equationshttps://en.wikipedia.org/wiki/General_relativityhttps://en.wikipedia.org/wiki/Electrodynamicshttps://en.wikipedia.org/wiki/Ladder_paradoxhttps://en.wikipedia.org/wiki/Twin_paradoxhttps://en.wikipedia.org/wiki/Special_theory_of_relativity


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of outer space right here on earth: just jump out of an airplane. It is a lynchpin 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 t

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. Since

we know the earth's gravitational field of 1 g is equivalent to being accelerated constantly upward at 1 g,

and we know 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 gravitationalfield and acceleration, because an electric charge apparently only radiates when it is being accelerated

through motion, but not through gravitation.

Resolution of the Paradox

The resolution of this paradox, like the twin paradox and ladder paradox, comes through appropriate care i

distinguishing frames of reference. We follow the excellent development 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 spac

away from any forces. The surface of the earth is not an inertial frame. It is being constantly accelerated.

We know 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 o

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 fallingcharge 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 th

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. Let us see if any contradictions arise.

https://en.wikipedia.org/wiki/Inertial_framehttps://en.wikipedia.org/wiki/Frames_of_reference


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Now we are in a position to establish what an observer at rest in a gravitational field, the supported

observer, will see. Given the electric and magnetic fields in the falling frame, we merely 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 we must bring to bear the machinery of general relativity.

In this case our gravitational field is fictitious because it can be transformed away in an accelerating 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, cylindricalmetric and line element can be written:

where is the speed of light, is proper time, are the usual coordinates of space and time, is

the acceleration of the gravitational field, and is an arbitrary function of the coordinate but must

approach the observed Newtonian value of . 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:

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

When this coordinate transformation is applied to the rest frame electric and magnetic fields of the charge,

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 supportedcharge, then? Does it not radiate due to the equivalence principle? To answer this question, start again in th

falling frame.

In the falling frame, the supported charge appears to be accelerated uniformly upward. The case of constan

acceleration of a charge is treated by Rohrlich [1] in section 5-3. He finds a charge uniformly accelerated

at rate has a radiation rate given by the Lorentz invariant:

https://en.wikipedia.org/wiki/Minkowski_spacehttps://en.wikipedia.org/wiki/Metric_tensorhttps://en.wikipedia.org/wiki/Riemann_curvature_tensorhttps://en.wikipedia.org/wiki/General_relativity


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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 the field of such a charge is not precisely .

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.

References

1. Rohrlich, F. (1965). Classical Charged Particles. Reading, Mass.: Addison-Wesley.

2. Boulware, David G. "Radiation from a Uniformly Accelerated Charge". Ann. Phys. 124: 169–188.doi:10.1016/0003-4916(80)90360-7.

3. 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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Categories: Physical paradoxes Special relativity

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