Jackson: justification of the Poynting vector by GR

[PeterDonis]

Surprising as it may be, I've heard about Birkhoff's Theorem before

Of course. Again, I think you misunderstood me. I was providing the link so that more of the equations that I was using would be seen there. I was not providing it because I thought Birkhoff's Theorem would be new to you.

 

[PeterDonis]

Note also that the minimal form of the SET was assumed and used in the second insight article

But I didn't make use of that part of the article in my derivation in this thread.

 

[PeterDonis]

Wald is aware that it deserves a discussion, and that alternative possibilities are not entirely ruled out.

Not entirely ruled out mathematically.

But, for example, any alternative possibility, if it's going to lead to a different solution from R-N for the spherically symmetric electrovacuum case, would have to imply that the energy density of an electric field $E$ is not $E^2$ (again, modulo constant factors that depend on your choice of units). But what else could it be? $E^2$ is what makes sense on dimensional grounds; any other function of $E$ wouldn't. Nor would it make sense for the energy density to depend on something other than $E$.

 

[JimWhoKnew]

The condition that the energy density of an electric field $E$ is $E^2$ (modulo some constant factors that depend on your choice of units).

If, for example, $~Q'=(1+\epsilon)Q~~$ (in the same choice of units), then $~T^t{}_t~$ is still proportional to $~E^2~$. But Q' is what we have in the metric while Q is used in Maxwell's equations. Insisting that $~\epsilon=0~$ necessarily, is the same as assuming the minimal SET. For $~\epsilon\neq 0~$ the metric will still have the RN form, but charged test particles will behave differently (a physically observable effect, in principle).

Edit:
If you meant that the RN form of the metric is unique, your derivation seems OK (to me, as far as I checked). If you meant to show that the RN solution to the Einstein-Maxwell equations is unique, without assuming the minimal SET (or anything equivalent), you didn't convince me.

 

[PeterDonis]

For $~\epsilon\neq 0~$ the metric will still have the RN form, but charged test particles will behave differently (a physically observable effect, in principle).

This would amount to a different prediction for how much energy density a given amount of electric field produces; there would be an extra factor in the energy density that doesn't arise from the charge producing the field. So what does it arise from?

 

[PeterDonis]

If you meant to show that the RN solution to the Einstein-Maxwell equations is unique

The possible departure from uniqueness is pretty small: all we have, by your argument, is one additional free parameter in the solution, that specifies how much energy density a given amount of electric field produces.

 

[JimWhoKnew]

The possible departure from uniqueness is pretty small: all we have, by your argument, is one additional free parameter in the solution, that specifies how much energy density a given amount of electric field produces.

Any departure from uniqueness is non-uniqueness. The amount is irrelevant as far as the principle is concerned.

In the RN coordinates, the SET is diagonal, traceless, tangentially isotropic, and satisfies the TOV equation (as in your derivation). So all freedom left to begin with is a single parameter (which is eliminated too, if a further assumption like minimal SET is added).

In a general spacetime, we may expect more degrees of freedom due to SET modification.

This would amount to a different prediction for how much energy density a given amount of electric field produces; there would be an extra factor in the energy density that doesn't arise from the charge producing the field. So what does it arise from?

I have a speculation:
Consider for example the SR Lagrangian density $~\mathcal{L}~$ of QED as a classical function (before quantization). If we multiply it by a constant positive factor, that factor drops out of the Euler-Lagrange equations (ie. Maxwell and Dirac equations), but not from the canonical SET. So we get a family of theories which satisfy "classical QED" but their SETs differ by a factor (all these SETs are derived, none is "pulled out of a hat"). This suggests that $~T^t{}_t~$ is not necessarily the energy density itself, only proportionality to it is guaranteed (likewise for the other SET components).
In SR all these theories are just as good - no observable difference. But only one of these SETs (after symmetrization, etc.) should be used in EFE. It seems that our little RN example can tell them apart. If that's really the case, the assumption of minimal SET implicitly fixes the factor to be 1 (so $~T^t{}_t~$ is exactly the EM energy density). In that spirit, when Wald discusses the compatibility of CMB observations, it implies that any deviation from 1, if possible at all, is below our measurement capabilities.

Again, it is only a speculation.

Possible modifications of SET, as speculated in this thread, may have similar implications.

 

[PeterDonis]

Any departure from uniqueness is non-uniqueness.

Technically, yes, but the kind of departure is important. See below.

In the RN coordinates, the SET is diagonal, traceless, tangentially isotropic, and satisfies the TOV equation (as in your derivation). So all freedom left to begin with is a single parameter

No, there are two free parameters in the RN metric, $M$ and $Q'$ (to use your notation). Both of them contribute to the stress-energy.

Your third free parameter is not a free parameter in the RN metric itself, it's a free parameter in the relationship between the RN metric and the spherically symmetric electrovacuum solution to Maxwell's Equations. As you correctly pointed out, the parameter $Q$ in that solution does not have to be exactly equal to the parameter $Q'$ in the RN metric; they could differ by a constant factor, which you wrote as $1 + \epsilon$. So that factor is an additional free parameter in the overall solution.

(which is eliminated too, if a further assumption like minimal SET is added).

"Minimal SET" as you're using the term means $\epsilon = 0$, i.e., $Q' = Q$. That does fix the value of the additional free parameter I described above. But it still leaves $M$ and $Q'$ as free parameters in the RN metric.

In a general spacetime, we may expect more degrees of freedom due to SET modification.

Such as?

 

[PeterDonis]

I have a speculation

It's not really a speculation by you personally, it's just restating what's already in the literature (for example, in the Wald excerpt that was quoted earlier).

I bring this up because personal speculation is off limits here, so if what you say here actually were personal speculation, I'd be warning and deleting it. I'm not because, as I said, it's not actually personal speculation. But I would ask that you not use the word "speculation" for something like this. Call it what it is: you're restating an argument that's made in the literature.

only one of these SETs (after symmetrization, etc.) should be used in EFE.

More precisely, we can distinguish these different SETs by their different gravitational effects--which we predict by plugging each one into the EFE and seeing what comes out in the solution.

It seems that our little RN example can tell them apart.

Yes. For example, if we had a supply of RN black holes to experiment with, we could compare the motions of neutral test particles and test particles with known charges in the RN spacetime, to try to fix the value of your $\epsilon$.