I'm sure there's a good explanation for the issues leading to my question so please read on:

Classically, we can represent Electromagnetism using tensorial quantities such as the Faraday tensor $F^{\alpha\beta}$ from which (with the help of the metric $g^{\mu\nu}$) we can construct the Maxwell Stress energy tensor $T^{\alpha\beta}$.

This is all well and good and I'v never had an issue with the covariance of electromagnetism, until I was reading about the “paradox” of a charge in a gravitational field.

(the Wiki page is a decent introduction: https://en.wikipedia.org/wiki/Paradox_of_a_charge_in_a_gravitational_field)

Rorlich's resolution to this (referenced on the wiki page) was to calculate that in the free falling frame (but rest frame of the charge) there is no radiation emitted from the charge. Meanwhile on the supported observer's frame (say at rest on the Earth's surface), one would observe radiation being emitted from the charge. It is argued that the coordinate transformation between the two frames is NOT a Lorentz transformation and hence the radiation observed in one frame vanishes upon transformation to the inertial frame.

How does this mesh with general covariance if we can choose a frame (or class of inertial frames) in which the radiation is zero?

Via the same arguments used for the gravitational stress-energy pseudotensor (if a tensor vanishes in one frame it vanishes in all frames) the radiation cannot be a tensorial object I would think.

Note that this argument applies even more simply to the concept of Unruh radiation (i.e we can choose frames in which it is zero). This makes me think Electromagnetism only respects Lorentz covariance which indicates it's not represented by true tensors (but rather some pseudotensor-like object akin to gravitational energy)?

I'm a huge fan of GR and reading about this “paradox” immediately reminded me of the gravitational pseudotensor vanishing in an inertial frame and hence the question. Perhaps there's a simple answer I'm missing.

EDIT: While the question is flagged as a duplicate, My question is more general. I'm curious about how the radiation stress energy tensor can change in non inertial frames. the radiating charge in a gravitational field is just one example, where the radiation appears to exist in one frame and not the other. From quantum physics the unruh and hawking radiations are other examples of radiations appearing in one frame and not in another. I get that a tensorial object shouldn't be able to vanish in one frame. I'm just curious about what we can say about making an arbitrary electromagnetic wave vanish in some frame. since it can be done in some cases, in what other cases can this be done? While we can say that the quantum and classical cases are totally different, note that they both involve radiation vanishing under a noninertial (nonlorenztian) coordinate transformation.

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    $\begingroup$ As any tensor EM is generally covariant, although of course you have to use the covariant version of maxwell's equations. $\endgroup$ – Slereah Jul 3 '17 at 8:24
  • $\begingroup$ @Slereah I don't doubt it, but how can you transform electromagnetic radiation "away" (granted only in strange cases) and still maintain general covariance of the electromagnetic Stress energy tensor? $\endgroup$ – R. Rankin Jul 3 '17 at 8:34
  • $\begingroup$ You can't. The Unruh effect is a quantum effect without any analogy to the classical case. In the classical case you can't transform a $0$ EM field into one that isn't (except up to gauge). $\endgroup$ – Slereah Jul 3 '17 at 8:35
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    $\begingroup$ Possible duplicate of Does a charged particle accelerating in a gravitational field radiate? $\endgroup$ – tparker Mar 31 '18 at 3:23
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    $\begingroup$ The falling charge paradox has been discussed to death on this site. $\endgroup$ – tparker Mar 31 '18 at 3:24

General covariance means that the 'form' of the fundamental laws of physics is independent of the coordinate system chosen - it does not mean that observations of observers using different coordinate systems necessarily agree. This statement is quite obvious but important. For example, in the context of SR, two observers might measure different elapsed times and distances for a particular event (say, $t/T$ and $x/X$). Even though x does not equal X and t does not equal T, if both observers construct the invariant proper time $t^2-x^2$ and $T^2-X^2$, they will get the same answer (c=1 here). Thus, the 'law of physics' that is invariant wrt Lorentz transformations is that the proper time elapsed for a particular event will be the same for all inertial observers.

Your example is similar - in the covariant formulation of Maxwell's equations, it is not a 'fundamental law of physics' that an accelerating charge must radiate. Indeed, we cannot even define acceleration until we choose a coordinate system. Instead, the covariant law of physics gives a relationship between tensors which is independent of the coordinates chosen (see https://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime). It is this law of physics that will be invariant with respect to arbitrary coordinate transformations (this is obvious because the law is written using tensors). The statement 'an accelerating charge radiates' already assumes a system of coordinates, and so cannot be a 'covariant law of physics'. Thus, this 'paradox' has no bearing on the covariance of electromagnetism.

  • $\begingroup$ I like your answer; however it doesn't resolve The issue of a radiation stress energy tensor appearing in one frame and not in another. Even for the quantum case, one can write a stress energy tensor for radiation. In the case of unruh radiation for example, how can it vanish then under certain coordinate trasformations? $\endgroup$ – R. Rankin Apr 1 '18 at 10:43
  • $\begingroup$ In your example of an accelerating charge, we have a field source (the particle). It sounds like you may be mistakenly equating the appearance of a radiation component of this field (i.e. an EM 'wave') with the existence of the field itself. The EM tensor will contain information about the total field. We cannot get rid of all the field components with a coordinate transformation. But we may be able to get rid of the radiation component. However, this would not mean the tensor vanishes, because we still have a source, so there would still be static components of the field left over. $\endgroup$ – Cicle Apr 4 '18 at 1:05
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    $\begingroup$ In the case of Unruh radiation, it may help to remember that in quantum mechanics a vacuum does not mean 'empty space' or 'absence of fields'. It just means 'lowest energy state of the fields'. Therefore, any tensors you write which contain information about the fields would presumably be non-vanishing, even in a vacuum. Therefore, perhaps it is not such a surprise that we can make a radiation component appear out of 'the vacuum' by a coordinate transformation, because the tensor was never vanishing in the first place. $\endgroup$ – Cicle Apr 4 '18 at 1:08
  • $\begingroup$ I would still expect a quantum vacuum expectation value to be zero in one frame and then not in another. However; in the classical realm, I think what I'm getting is that radiation in one frame can only be made to vanish in another frame if it is "absorbed" (for lack of better word) into a source field. Would you say that is (roughly) correct? In this case, there is an interesting thing that the radiation component of the field is invariant under conformal transformations, unlike the source field. Thank you I'll accept your answer. $\endgroup$ – R. Rankin Apr 8 '18 at 0:49
  • $\begingroup$ I suppose we're getting a little philosophical when we ask what the right way to 'describe' the mathematics is - whether we think of it as absorption, or any other way, the mathematics doesn't change... $\endgroup$ – Cicle Apr 11 '18 at 11:32

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