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The dipole formula for the power loss emitted by a time varying electric dipole is (in natural units) $P = \frac{\dot d_i^2}{6 \pi}$. This is clearly even under time reversal symmetry $T$, but a power should be odd under $T$.

The same puzzle appears in the power loss in gravitational radiation as described in the quadrupole formula $P = \frac{G}{5} {\dddot I}_{ij}^2$, which also appears even under $T$. What am I missing?

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It is not even under time reversal in a general case: $\dot d^2(T)\ne \dot d^2 (-T)$. Consider, for example a fading oscillator and chose the point t=0 in the "middle" of its lifetime. – Vladimir Kalitvianski Oct 18 '12 at 15:35

This is the old puzzle of the electromagnetic arrow of time, and it is resolved in quantum mechanics, when quantizing the electromagnetic field. The classical field starts out in a non-radiative state for the problem of dipole emission, and this is thermodyanmically infinitely improbable (what's the chance that there is no radiation at infinity?) and so the dipole emission is not balanced by absorption, and classically, it can't be, because there is a Rayleigh Jeans ultraviolet catastrophe and no thermal equilibrium state for radiation.

This makes it that if you look at a radiating dipole, the behavior is not time-reversal invariant. The rate of radiation emission is even under time reversal--- an oscillating dipole emits radiation whether you reverse it's motion or not, and this is because the field boundary conditions are not time reversal invariant.

But if you have a quantum field in thermal equilibrium at the beginning, and a radiating dipole, this dipole can also absorb quanta from the field, and you can have an equilbrium of absorption and emission. In this case, you have no net average radiation, any emission is due to the fact that the dipole is dissipating energy to the colder environment and increasing its entropy. The entropy of a field only makes sense in quantum mechanics, so all the classical radiative formulas are wrong with respect to time-reversal, as you noticed.

This problem has a long history, it was understood to be solved by quantum mechanics more or less by Einstein and Planck in the early years of the 20th century, but it didn't really die, it just faded away.

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I do not think quantum theory is necessary to answer this. The power of energy radiation is given by the above formula only under special assumptions on the charges (have to be extended) and initial conditions on the field. More concretely, whether you choose retarded, advanced, or some other special solution of the wave equation. There are good reasons to assume retarded fields, and the above formula is valid only for them. If you do time reversal, the retarded waves change into advanced waves and the power will become negative (oscillator absorbing energy).

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