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I'm looking for ways to find the stable stationary solutions of a modified form of the Einstein-Maxwell equations. "Stationary" would mean that the solution is time-independent; "stable" would mean that if an arbitrary small perturbation were added to the solution its time evolution would return it to the stationary solution. In the standard form of the Einstein-Maxwell equations, the energy-momentum density of the electromagnetic field acts as a source for the gravitational field, but the sources of the electromagnetic field are specified at the outset. In the modified form I'm considering, coupling between the EM and gravitational fields produces an effective charge/current distribution as well as an effective source for the gravitational field; and the fields are otherwise source-free. I expect that any stationary solutions will be axially symmetric. Tips from anyone who has experience with stationary solutions of the standard Einstein-Maxwell equations would be very useful.

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  • $\begingroup$ I expect the question is too vague to answer. Do you have a Lagrangian/field equations? Do you know of an analogue of the solution you want to obtain in Einstein-Maxwell theory (or other models)? $\endgroup$ – A.V.S. Mar 3 '18 at 6:06
  • $\begingroup$ yes, I've got a Lagrangian and field equations. There are a host of stationary solutions to the Einstein-Maxwell equations described in the literature, which would probably provide a good starting point for finding solutions to the equations I'm working with. However, my single year of Calculus of Variations a few decades ago didn't give me the skills that seem necessary for this problem. $\endgroup$ – S. McGrew Mar 3 '18 at 15:46
  • $\begingroup$ If you are constructing vector current to couple with EM field from gravitational degrees of freedom you are breaking general covariance, aren't you? $\endgroup$ – A.V.S. Mar 4 '18 at 7:25
  • $\begingroup$ I think my equations are covariant, but am not completely sure. If you have the time and interest to discuss in more detail, I'll start a chat room for the purpose. $\endgroup$ – S. McGrew Mar 4 '18 at 13:03
  • $\begingroup$ Sure, but the 'latency' of my responses could be rather large. $\endgroup$ – A.V.S. Mar 4 '18 at 15:28

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