Is it correct using Larmor Formula for identifying a non-inertial reference frame?

Question

As is well known Newton's laws:

$$m\frac{\text{d}^2 x_i}{\text{d}t^2} = F_i$$

whose relativistic generalization is:

$$m \left(\frac{\text{d}^2 x^\mu}{\text{d}\tau^2} + \sum_{\sigma,\nu} \Gamma_{\sigma \nu}^{\mu} \frac{\text{d} x^\sigma}{\text{d}\tau}\frac{\text{d} x^\nu}{\text{d}\tau}\right) = F^\mu$$

only work in an inertial reference system, in a non-inertial system additional fictitious forces appear, which in the relativistic case translate into the Christoffel symbols are not to be identically null $\Gamma_{\sigma \nu}^{\mu} \neq 0$.

In principle, Larmor's formula for the radiated energy:

$$ \frac{\text{d}E}{\text{d}t} = \frac{q^2 a^2}{6 \pi \varepsilon_0 c^3} $$

provides a method for an inertial observer to detect relative accelerations: an observer could notice that his system is not inertial if by placing a sufficiently large charge he is able to detect the electromagnetic radiation emitted by it (in the quantum context this leads to the Unruh effect).

My doubt is the following, in relativity theory it is said that an observer is inertial if its Christoffel symbols cancel out, but on the other hand an electric charge falling in a gravitational field describes a coordinate system with null Christoffel symbols, but still that charge emits radiation because the curvature of spacetime deforms its electric field and as a consequence that charge emits radiation, even being in an "inertial system". Therefore, we have two non-equivalent definitions of inertial system in relativity:

1. An observer in whose coordinate system the Christoffel symbols cancel out.
2. An observer who sees an electric charge at rest with respect to him does not emit radiation.

Which of these two definitions seems more appropriate to identify a non-inertial system?

Answer 1

In principle, Larmor's formula for the radiated energy:

$$ \frac{\text{d}E}{\text{d}t} = \frac{q^2 a^2}{6 \pi \varepsilon_0 c^3} $$

provides a method for an inertial observer to detect relative accelerations: an observer could notice that his system is not inertial if by placing a sufficiently large charge he is able to detect the electromagnetic radiation emitted by it

Larmor's formula is a needlessly specific result for this purpose (derived under very special assumptions about the EM field).

It seems better to work with the concept of EM radiation being present or not being present. EM radiation of accelerated charge being present in a specific frame should mean that in that frame, the EM field has a component falling off with distance $r$ as $1/r$, so that this field component is much stronger than the Coulomb field at large enough distances.

Your idea seems to assume two things:

1. radiation presence is absolute, i.e. either radiation is present in all frames, or it is not present in all frames;
2. a charge that accelerates in an absolute sense gives off radiation (assuming EM field is retarded field of the charge), while the charge that accelerates only apparently, due to observer's reference frame being non-inertial, does not give off radiation.

So the idea is if we detect such radiation in the observer frame, we know 1) there is radiation present in the absolute sense 2) there had to be a charge accelerating in absolute sense and a frame connected to it could not be inertial.

All of this seems experimentally unjustified, because we do not have much experience with detecting presence of EM radiation of charges static in freely falling frames, or radiation due to freely falling charges in Earth's frame.

It is quite possible that assumption 1. is not true, i.e. that radiation presence is frame-dependent.

an electric charge falling in a gravitational field describes a coordinate system with null Christoffel symbols, but still that charge emits radiation because the curvature of spacetime deforms its electric field and as a consequence that charge emits radiation, even being in an "inertial system".

Maybe the freely falling charge radiates in the non-inertial frame of the Earth, but it does not radiate in the inertial frame of the charge.