Dynamic solutions in time-independent spacetimes

[TrickyDicky]

Hi, I would like to clarify this probably trivial little issue that is bugging me:
How should dynamical solutions be understood in the context of a static spacetime?
To exemplify what I mean I'll use a well known case, the source-free Maxwell eq. in their explicitly covariant form set in Minkowskian static spacetime, reduce to a EM wave eq. in the EM tensor Fab, and you can obtain solutions like the monochromatic plane wave.
My confusion arises from not seeing how such dynamical solution can happen in a static spacetime (Minkowski) that is not just stationary, which would allow time symmetry, but static so time evolution cannot even show up from crossed (dtdr..) terms.
Is the wave solution time-dependency introduced thru boundary conditions? Or am I missing anything important?
Thanks.

 

[Ben Niehoff]

You can take any static spacetime and solve Maxwell's equations on it, sure. You'll get wave solutions that evolve dynamically.

But these will not be solutions of the full Einstein-Maxwell theory, because we haven't taken into account the stress-energy of the dynamic Maxwell field. Essentially, we're ignoring the effects of the EM waves on the background geometry. In my field, we'd say we're ignoring the "back-reaction" of the EM waves.

So what we have is an approximation, that holds where the geometrical effects of EM radiation are negligible.

 

[TrickyDicky]

You can take any static spacetime and solve Maxwell's equations on it, sure. You'll get wave solutions that evolve dynamically.

Right, this is the starting point of my post and what I'd like to understand better.

But these will not be solutions of the full Einstein-Maxwell theory, because we haven't taken into account the stress-energy of the dynamic Maxwell field.

Yes, when we do that we get the electrovacuum solutions(null for the radiation case and non-null for the rest like for instance the Reissner-Nordstrom one). But these unlike Minkowski spacetime are singular spacetimes.

Essentially, we're ignoring the effects of the EM waves on the background geometry. In my field, we'd say we're ignoring the "back-reaction" of the EM waves.

So what we have is an approximation, that holds where the geometrical effects of EM radiation are negligible.

Yes, we assume the field strength is small enough not to dramatically affect the geometry, this assumption is understood but it refers to the slight cheat of considering it a vacuum (here I'm paraphrasing Carroll in his exercise 6 in chapter 4 of his GR book).
My concern above is not about the solution not being strictly vacuum but with obtaining dynamical solutions in a static background, so I'm looking for a justification analogous to the one just commented for calling them vacuum but for introducing time changing fields in a static setting. Note that I'm specifically referring here to the Maxwell equations in their Minkowskian explicitly covariant form, so this is basically a formal question.

 

[PAllen]

Yes, when we do that we get the electrovacuum solutions(null for the radiation case and non-null for the rest like for instance the Reissner-Nordstrom one). But these unlike Minkowski spacetime are singular spacetimes.

It may be true that a complete manifold for an electrovac solution is always singular (true of the cases I know); however it is certainly true that there are exact solutions with no singularities that have a charged fluid ball optionally rotating, and electrovac outside of the ball.

 

[TrickyDicky]

It may be true that a complete manifold for an electrovac solution is always singular (true of the cases I know); however it is certainly true that there are exact solutions with no singularities that have a charged fluid ball optionally rotating, and electrovac outside of the ball.

Interesting, can you name one?

 

[TrickyDicky]

Ok so I guess I might be either misunderstanding what a static background entails, or failing to see that the approximation in the same way it ignores the geometric effects of the EM field on the background so that it can be considered a vacuum, it ignores the time-varying nature of the EM far field so that the static Minkowskian background isn't altered, nevertheless deriving dynamic consequences like wave solutions.
I'm inclined to this last posibility, maybe this is such a usual assumption in classical field theory nobody pays much atention to it.

 

[PAllen]

Ok so I guess I might be either misunderstanding what a static background entails, or failing to see that the approximation in the same way it ignores the geometric effects of the EM field on the background so that it can be considered a vacuum, it ignores the time-varying nature of the EM far field so that the static Minkowskian background isn't altered, nevertheless deriving dynamic consequences like wave solutions.
I'm inclined to this last posibility, maybe this is such a usual assumption in classical field theory nobody pays much atention to it.

This is correct. I'm sure some mathematical physicist somewhere has justified and bounded the degree of validity of this, but I would not be able to point to any references for this.

 

[PAllen]

Interesting, can you name one?

I found one online: http://link.springer.com/content/pdf/10.1007/s10509-010-0521-9#page-1 (In this link, you can look inside for the first couple of pages).

Synge, 1960 has chapter on them (leading into pure electrovac solutions), but that is reference that is very hard for people to locate.

 

[TrickyDicky]

I found one online: http://link.springer.com/content/pdf/10.1007/s10509-010-0521-9#page-1 (In this link, you can look inside for the first couple of pages).

Synge, 1960 has chapter on them (leading into pure electrovac solutions), but that is reference that is very hard for people to locate.

Ok, thanks.