After some hesitation and having checked the Stack Exchange policy as indicated on Meta Stack
Exchange, I will attempt to answer my own question and hope to clarify
what I was asking. Putting this together did help my own appreciation
for the topic. Perhaps it might be of interest to others. It goes without
saying that if I have made an error then I trust that someone will point it out to me.
My answer does perhaps involve a re-framing: Does
General Relativity add any clarification to the dynamics of the electron,
in particular pertaining to the radiation reaction?
In the following, I have left out some details, including some physical constants.
The direct answer is in the negative. The analysis of the dynamics of
the electron does not particularly depend, other than on some technical
details, on whether the background mechanics is classical electrodynamics,
special relativity, or general relativity. More over, whether the path can
actually be determined depends on decisions about the properties of the
electron, decisions that are outside the scope of every one of these
theories as given.
In a space-mode classical approach, the power associated with charged particle
behaviour is determined by the power associated with the Coulomb $qE$ and
Lorentz $qv\times{}B$ forces as well as the radiation power $\nabla\cdot{}S$,
where $E$ is the electric field, $B$ is the magnetic field, $q$ is the charge
on the particle, $v$ is the particle velocity, and $S=E\times{}B$ is the
electromagnetic energy flux vector. The magnetic field vector $B$ can be
replaced by an anti symmetric matrix making the Maxwell equations and the
Lorentz force dimension agnostic. The practical difficulty in determining
the actual power absorbed or emitted by the electron lies in the difficulty
of solving the field equations.
The spacetime-modes of classical electromagnetic and special relativistic
mechanics are essentially the same as each other. The velocity and charge
of the particle are replaced by a 4-vector $J=(q,qv)$, and the electric and
magnetic fields are packed into a 4-tensor.
$F = \left[\matrix{0 & E \cr E & -B\times}\right]$
The product $FJ = (E\cdot qv,\ qE + qv\times B)$, a force-power vector, which
is the electromagnetic power and the Lorentz force. This is the rate of change
of energy with time, and with space. The stress-energy tensor packs in the
stress associated with the Lorentz force and the energy density, as well as
the flux:
$T = \left[\matrix{ E^2+B^2 & E\times B \cr
E \times B & E \otimes E + B \otimes B}\right]
- \frac{1}{2}(E^2+B^2)I $
It follows that $\nabla\cdot T = FJ$, that is, the gradient of the stress
tensor is a power-force. The temporal component of this is
clearly the Poynting theorem $\nabla\cdot S + E\cdot J - \frac{\partial}{\partial{}t}e$,
which is zero from the Maxwell equations on which this is all based.
So, the divergence of the stress-energy tensor is a force-power
tensor. Hence, it does state both how much force is on the electron
and how much power is being emitted - that is, the rate of change
of energy of the particle with time and the rate of change of
energy with position.
But, this is for virtual displacements. While the divergence of the
stress-energy tensor is a force-power vector, it is not immediate that
it is the force-power vector of the electron. The equating of the
divergence of the stress-energy tensor with the force-power vector of
the particle itself is a matter of mechanical assumption.
However, even if this is taken as axiomatic, and so the momentum-energy
exchange between the electron and the field is known. To turn this
into a path, which is kinematic, requires knowledge of the relation
between velocity and momentum. In classical mechanics this is taken
to be $p=mv$ and in special relativity $p=\gamma{}mv$, and there is
an identical electromagnetic expression for an effective momentum.
But, this contains assumptions about the structure of an electron
filled only by a further theory of matter, such as Quantum Mechanics,
or more radically, stochastic electrodynamics. If the electron has,
for the sake of argument, excited states, then the electron might
become excited, like a partially ionised atom, on the absorption of
a photon, and change its momentum and energy without changing its velocity.
In general relativity, the derivatives and connections required for
curved spacetime make the analysis more sophisticated, but does not
essentially change the structure of the discussion.
