This is nothing to do with the Abraham-Lorentz force you referred me to in comment 14:

“Since momentum is conserved, the charge is pushed in the direction opposite the direction of the emitted radiation. The Abraham-Lorentz force is the average force on an accelerating charge due to the emission of radiation.”

- http://en.wikipedia.org/wiki/Abraham-Lorentz_force

From what you now say in comment 18, you’re not concerned with such a recoil at all (there isn’t any such recoil, as shown in comment 17). Instead, you are now making quite a different point: the electron experiences a decelerative force due to losing forward kinetic energy while accelerating, due to the transverse emission of radiation.

Your problem with conservation of momentum is that the electron is losing momentum (by being decelerated by kinetic energy loss due to emission of radiation), but that radiation isn’t going in the right direction to conserve momentum. But the electron is gaining momentum from the gravitational field which is causing it to accelerate in the first place. Whenever the electron accelerates, it flattens due to Lorentz contraction.

While it is at constant velocity, the electric isofield strength lines around the electron form an oblate spheroid shape, with propagation along the axis of symmetry. But in order to get into that shape for a constant velocity, the electron must first, during its acceleration, be distorted so that the field at the front moves slower than the field at the rear: this allows the field at the rear of the electron to catch up, squashing or flattening the electron in shape.

Hence, the electric isofield strength lines are not a perfect oblate spheroid while the electron is accelerating, and this distortion is physically necessary to explain the Lorentz contraction effect. This front-rear distortion during accelerations means that the emitted radiation is not emitted perpendicularly to the acceleration, but at a slight angle (hopefully satisfying conservation of momentum for the electron’s momentum loss which results from the deceleration of the electron due to loss of forward kinetic energy via radiation).

The nature of a transverse electromagnetic wave like light is that it can’t propagate outward in spherical symmetry. You need a charge acceleration in a specific direction in order to emit radiation, so a spherical radio antenna emits nothing: you need some asymmetry in the direction of charge acceleration in order to transmit radiation. An electron doesn’t radiate a radially-symmetrical photon: such a thing can’t be detected, so it’s not a physical concept, really.

Try looking up radiative cooling in gravitational collapse, particularly references to the evolution of white dwarfs and neutron stars, although the process is important for nearly all forms of stellar and astronomical evolution.

The classical phenomology to relate Abraham-Lorentz radition to Black Body radition is to consider the collision of two atoms in a hot gas of differing velocities as an acceleration of dipoles. Unfortunately this classical approach leads to a UV divergence. Hence Plancks need to invoke quantization.

Even with the tunnel through the earth example, we would still get integer steps in the radiation of the electron, because it is effectively bound by an inverse square field, to which we know the solutions very well.

Now the really tricky question is to ask what the gravitational force is between to photons (and if that is the same question as asking if space is hyperbolic, flat, or parabolic)

You have confused the recoil reaction to the momentum of the emitted radiation (which does depend on direction of radiation) with the acceleration-resistance force which normally does indeed act parallel to acceleration and velocity of the charges. That force acts for example in ordinary omni-directional antennas as an impedance, an antenna resistance. Otherwise, you could violate conservation of energy, gravity or not, by just running charges back and force without effort (like the energy recovery of classical masses). and collecting the radiation energy. Look up radiation resistance, Lorentz-Abraham force etc. in Wikipedia or etc. and see the difference.

However, your point is interesting in quantum mechanics: Well, atoms supposedly emit photons in a radially-symmetric wave (maybe not as strong along a given bidirectional axis as another, but with no net directional vector.) How then can absorption later of a photon somewhere (with its momentum pushing the absorber in a given direction) work back to give conservation of linear momentum to that atom and its environment? Does it depend on which one is measured first? etc.

Surely, there isn’t any such force in this case because the radiation is emitted perpendicular to the direction of acceleration of charge. The charge accelerates along a radial line to the earth’s core. Hence the radiation in the transverse direction to that line. Think about the radiation from electrons accelerating along the surface of a radio transmitter antenna. The radio waves come off with in all directions perpendicular to the direction of acceleration of the electrons. Ie, the radio beam goes out in a horizontal direction is the aerial is vertically orientated. There’s no recoil force because there is no preferred transverse direction for the radiated waves: they are emitted in all directions on the horizontal plane so the recoil forces cancel.

No, I am not talking about the relativistic mass increase, but rather the special Abraham-Lorentz force that specifically compensates for radiated power. The body falling through the earth is indeed making the classic sine wave SHM. BTW, the fact that the power would be so weak from an elementary charge is beside the point of the principle of the thing; also note that we could easily use a highly charged macroscopic body, and that the radiated power is proportional to q^2. (Also, what if we used a neutron type star instead! Billions of g’s…. Well, coring it is problematical….)

(BTW, could we please get sub and superscripts for this posting, it would be so helpful?)

Unless there is a cyclical acceleration, you don’t get actual electromagnetic waves being produced. Although your electron is oscillating through the earth, the frequency will be very low, because the maximum acceleration the electron experiences is that at the Earth’s surface, a = 9.8 ms^{-2}, so the maximum radiating power is merely

P = (e^2)(a^2)/(6*Pi*Permittivity*c^3) = 5*10^{-52} W.

Thanks for getting me started about charges in gravitational fields. I still wonder: does the charge oscillating in the tunnel show the effect of the self-force as given above, or not? That would actually change its rate of acceleration (slowing it down), and is independent of how you are considering distant fields.