1
$\begingroup$

Is there a way to understand test field regime as some limit of backreaction in general relativity?

Consider the Einstein-Hilbert action augmented with the standard electromagnetic field coupled minimally to gravity. The action is just sum of both, \begin{equation} S = S_{EH} + S_{EM} \sim \int d^nx\sqrt{-g} \frac{1}{16\pi G}R-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\,. \end{equation} Variation with respect to the metric will give us the Einstein field equation sourced by the stress-energy tensor of the EM field. In particular, the chared Reissner-Nordstrom black holes are obtained by suitably choosing a particular EM stress-energy configuration. This would be the situation where the EM field properly accounted for with dynamics of spacetime.

However, if I vary with respect to the Maxwell field $A_\mu$, what I get instead is Maxwell's equations: \begin{equation} \nabla_\mu F^{\mu\nu} = J^\nu \end{equation} for some current $J^\nu$ (I absorbed the constant prefactor for convenience). But this is the equation that we use in the test field limit, i.e., when the background metric could have been anything. But the RN metric must have been produced by suitable choice of $F_{\mu\nu}$ that solves the Maxwell's equations, so in general one should not have a test field limit at all.

I always thought of the test field regime as some sort of limit where the field is "weak enough" that we can ignore the backreaction. I guess in QFT in curved spacetimes, it makes sense since free quantized EM field has no charge so we never needed to worry about the background being Reissner-Nordstrom, but in this case I seem to have a situation where ignoring backreaction requires some sort of "weakness" (e.g., perhaps in the weak field regime). However, this would have run counter to the spirit of standard calculations of Hawking radiation (or really, anything where the classical/quantum field is a test field), where the background metric is freely specifiable and it only influences covariant derivatives of the test field.

In short, my question is whether there is a mathematically precise way I am missing (perhaps stupidly) in which test field regime is a "negligible backreaction" regime. Right now it just looks to me that we can study wave equation in curved spacetimes (fixed background metric) consistently without ever knowing Einstein equations (or any gravitational dynamics), so perhaps this consistency alone is why we can work with test fields?

Update: The problem I have is that when other contributions "outweigh" the EM test contribution, it should restrict the kind of solutions of the Maxwell's equation I can take; but at the same time, I am fairly sure if I just solve the (covariant) Maxwell's equation, there are "plenty" which are not "negligible". But it seems that there is no easy way to say, "ok this particular EM configuration is practically negligible". At the moment I will accept the answer provided below because I think it is probably the only (real) way to do this, but surprisingly I have not seen this being done explicitly anywhere in the literature.

$\endgroup$
1
  • $\begingroup$ I think the formal idea should be to modify action $S$ by introducing a constraint. The constraint should be such that extremizing S wrt metric will return the vacuum Einstein's equation, while for EM potential we will get the Maxwell's equations, but now with an additional gauge fixing condition. This gauge fixing condition will restrict to a particular EM configuration which is practically negligible. However, I'm not sure if this is feasible. $\endgroup$
    – KP99
    Dec 31, 2021 at 8:54

1 Answer 1

0
$\begingroup$

I think the problem is arising when taking the equations of motion out of the action. For the action to be minimized, the conditions one must satisfy are $$\left\lbrace \begin{aligned} &G_{\mu\nu} = 8 \pi T_{\mu\nu}, \\ &\nabla_{\mu}F^{\mu\nu} = J^\nu \end{aligned} \right.$$ at the same time. If you are accounting for backreaction of the electromagnetic field, you can't first solve the Einstein Equations "for some suitable electromagnetic field configuration" and only after solve for the electromagnetic field configuration. The second equations is not the test field limit, it is just the generic expression for a Maxwell field on a curved spacetime. The tricky thing is that to solve for (classical) Electromagnetism in curved spacetime while taking backreaction effects into account we must solve the entire system at once, which is particularly difficult. Nevertheless, this is what is done for Reissner–Nordström: by imposing the solution is an electrovac solution and imposing spherical symmetry and staticity on both the metric and the strength tensor, one can nail down the possible forms of the solutions and completely solve the system of equations (this is Problem 3 of Chap. 6 on Wald's General Relativity). $\nabla_{\mu}F^{\mu\nu} = J^\nu$ is just gonna be a test field limit if we impose the assumption that the metric was given beforehand (perhaps because there is something else sourcing the Einstein Field Equations and its contribution outweights the electromagnetic contribution).

Nevertheless, your final intuition is correct: if we are not taking backreaction into account, then we aren't exactly solving the Einstein Equations, we just have an approximate solution. One way of working around this perturbatively would be to compute the stress-energy tensor for the field configuration in the background we are considering, inserting it on the EFE, solving them again and repeating.

I'd say this procedure also allows to test for consistency of the test field approximation. By computing the stress-energy tensor of the test field on the fixed background, one can then see whether its contributions are indeed much smaller than those of the matter sourcing the background curvature or not.

$\endgroup$
3
  • $\begingroup$ The problem I have is that when other contributions "outweigh" the EM test contribution, it should restrict the kind of solutions of the Maxwell's equation I can take; but at the same time, I am fairly sure if I just solve the (covariant) Maxwell's equation, there are "plenty" which are not "negligible". But it seems that there is no easy way to say ok this particular EM configuration is "practically negligible". $\endgroup$
    – Everiana
    Dec 30, 2021 at 23:54
  • $\begingroup$ @Everiana I added an extra paragraph, but I think you are looking for a way to judge it before making the computations. I suggest adding your comment to the end of your question, since it clarifies what you are looking for. $\endgroup$ Dec 31, 2021 at 1:00
  • $\begingroup$ I will accept your answer; now that I think about it, there is probably no other way to do this, but I have not seen anyone do this explicitly. $\endgroup$
    – Everiana
    Dec 31, 2021 at 8:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy