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If it's true that

  1. every accelerating charge emits radiation
  2. every uniformly-accelerating object is inertial (with respect to its point of view)

Then every charge should emit radiation in every situation, either if it is accelerated (for example by a constant potential difference) in free space or it is stationary in a gravitational field. But in this last example, the particle has no kinetic energy, and we can set its potential energy to be zero at ground level. So where does the radiation energy come from? Where does the particle get it from? Its rest energy? No, for it should lose mass, which is absurd. I don't know.

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    $\begingroup$ An accelerating frame (the frame attached to the particle) is not inertial by definition (it's accelerating). Inertial frames move at a constant velocity with respect to all observers. $\endgroup$ Commented Feb 2, 2021 at 21:50
  • $\begingroup$ If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will continue to move among themselves, after the same manner as if they had been urged by no such forces. — Isaac Newton: Principia Corollary VI, p. 89, in Andrew Motte translation $\endgroup$
    – ric.san
    Commented Feb 2, 2021 at 21:57
  • $\begingroup$ Quoting Wikipedia: "This principle generalizes the notion of an inertial frame. For example, an observer confined in a free-falling lift will assert that he himself is a valid inertial frame, even if he is accelerating under gravity, so long as he has no knowledge about anything outside the lift. So, strictly speaking, inertial frame is a relative concept." $\endgroup$
    – ric.san
    Commented Feb 2, 2021 at 21:58
  • $\begingroup$ 1) The objects remain moving relative to each other the same, yes, but they are still in a non-inertial accelerating frame (even in classical mechanics, Newton's Second Law must be immediately edited, as each body feels a fictitious force $ -m\mathbf{A} $). 2) In general relativity, which that quote is discussing, there are no global inertial frames. See this question: physics.stackexchange.com/q/3193 $\endgroup$ Commented Feb 2, 2021 at 22:04
  • $\begingroup$ @ ric.san, The second assumption in your question clearly is wrong and needs more attention. And note that nonrotating freely falling reference fames can be regarded approximately as inertial reference frames (they are not exactly inertial). $\endgroup$
    – SG8
    Commented Mar 4, 2021 at 17:15

2 Answers 2

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Your assumption 2. is wrong. A uniformly accelerated observer is non-inertial.

Your question about charges sitting in gravitational fields is however a good one, and was addressed by DeWitt and Brehme in the 1960 paper Radiation damping in a gravitational field. It has been addressed several times on this site, for example here.

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  • $\begingroup$ But... what about Einstein's equivalence principle? Isn't any free-falling (uniformly accelerating) frame of reference inertial? $\endgroup$
    – ric.san
    Commented Feb 2, 2021 at 22:01
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    $\begingroup$ I agree with this answer. Point 2 is wrong. It is a misunderstanding of the equivalence principle. A free falling frame is inertial, but not all uniformly accelerating frames are free falling. $\endgroup$
    – Dale
    Commented Feb 2, 2021 at 22:25
  • $\begingroup$ Isn't free fall just a (locally) uniform acceleration? $\endgroup$
    – ric.san
    Commented Feb 2, 2021 at 23:22
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    $\begingroup$ @ric.san: the free-fall condition is only valid in the case where it happens over a small enough time that local spacetime curvature can be neglected. A charged particle falling over a macroscopic time interval, and over regions where the curvature varies beyond whatever "inertial" tolerance you've set up has already violated the "free fall" condition. The radiation term cited in this answer, in fact, depends on an integral of the particle's path, which implies that it is, in fact, a non-local effect, in the GR sense. $\endgroup$ Commented Feb 3, 2021 at 14:47
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We have to clear up the concept of "body radiates/body does not radiate".

Accelerated charged bodies do radiate, or "have electromagnetic field that had radiation-like characteristics" but not in a way a hot body does.

Hot body sends out chaotic electromagnetic field whose intensity falls off as $1/r$ in all directions, with random spatial fluctuations on the scale of IR wavelengths. Due to randomness of this EM field, it is detectable by observers at rest in all frames of reference, the radiation cannot be "transformed away everywhere" for an observer having appropriate acceleration (except when the observer is far enough from the body, then the radiating body can be behind the Rindler horizon, but let's ignore that for now).

Uniformly accelerated charge has very orderly EM field that is very different from EM field of a hot body. It does not have fluctuations, oscillations, it changes only slowly on the time scale it takes for the body to approach and leave the region of the observer. Hypothetically, such smooth orderly field can be transformed away by change of reference frame. This is what most probably happens when observer gets in the same reference frame as the accelerated charge. So yes accelerated charge "radiates" but only in the frame where it is accelerated.

Now let's look at the last example, a charged body stationary with respect to Earth's ground. We do not expect this body to have radiation field in Earth's frame, for the simple reason, the charged body has zero acceleration in this frame. Radiation field could be present in other frames, for example in the frame of free-falling observer that passes close to the charged body.

Suppose that is so. Where does the energy of radiation come from?

Well, consider energy of any body in this frame - kinetic energy of the charged body, kinetic energy of the Earth. All are increasing in time as the observer falls. This is immense amount of energy being gained in time from no apparent source.

This violation of conservation of energy is traditionally understood to be due to observing things from the "wrong frame" of accelerated observer.

The problem does not go away if the acceleration is due to gravity, like for free-falling observer.

In such frames we can introduce fictitious forces acting on the Earth body and the charged body. But there is no apparent source of these fictitious forces and no apparent source of corresponding energy gained by those bodies. So the problem with energy conservation in such frames is already with kinetic energies of solid bodies and already in non-relativistic theory.

In relativistic theory including gravitation this problem is even worse. There can be things such as bodies accelerating and EM energy increasing without apparent source of energy. There are also things such as Rindler/black hole horizons where things disappear or freeze in time. There are difficult problems and energy conservation is one of them.

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