Electric field in curved spacetime

[Gavroy]

hi

i am looking for an equation that gives me the electric field( if possible in 3-dim form) of a point charge in a distance r in a schwarzschild spacetime, where the point charge and the the gravitating body should not be the same position.

 

[bcrowell]

Try the book by Stephani, Exact solutions of Einstein's field equations.

 

[Gavroy]

afais, they calculate only the metric, which is not what i am interested in.

i need an equation for the electric field of a charge at position r at a place x in the e.g. schwarzschild field of a gravitating body at position z.

 

[Mentz114]

Have a look at this

http://en.wikipedia.org/wiki/Maxwell's_equations_in_curved_spacetime

I don't know if it's all accurate.

 

[Gavroy]

thank you too, but actually , they do not calculate this stuff but just write down the maxwell equations in general relativity

 

[atyy]

Maybe http://arxiv.org/abs/physics/9910019? Their comments about the equivalence principle are just for fun, since it's well known that the EP doesn't apply to charged particles.

 

[pervect]

I'm a bit suspicious of that paper, the way it talks about "the radiation from a static charge" seems to me to be a bit of a danger sign. Though apparently it was published in genuine print :-).

Conceptually, the electromagnetic field is a differential form. So, you can imagine the electric field as an array of tubes , or field lines.

The mathematical expression of this is d*F = J. This works in both flat and curved space-times.

The pictorial expression of this is a bunch of field lines and or tubes (you can think of the lines as being at the center of the tubes) radiating out from the charge, as per MTW, or see also.

http://125.71.228.222/wlxt/ncourse/DCCYDCB/web/condition/9.pdf also has some pictures, esp fig 6.

Intuitively, I'd expect then that locally, the field lines of the charge would start out as the flat-space field lines one is familiar with, and that in curved space one would just extend the field lines as space-like geodesics.

One goes from the field lines to the "force on a unit charge" in the usual way, the magnitude being proportional to the density of the field lines and he direction the direction of the field lines.

This provides a nice visual picture, but, unfortunately, no hard numbers.

 

[Mentz114]

thank you too, but actually , they do not calculate this stuff but just write down the maxwell equations in general relativity

The EM field tensor contains the electric field ? What more do you want ?

 

[Gavroy]

well of course does the electromagnetic field tensor contain the electric field, but nowhere in this article is an equation that determines the electromagnetic field tensor on the basis of the information i have, which is that the metric is the schwarzschild metric and that there is a point charge in space.

i am extremely wondered that this problem is not more "popular", because i really need an equation.

 

[bcrowell]

Are we talking about a full electrovac solution, or just one in which the charge is small, so that it doesn't affect the metric much?

 

[Sam Gralla]

This was done in a classic paper of Cohen and Wald, J. Math. Phys. 12, 1845 (1971). I think the solution may have been worked out previously by copson or linet or something--I don't have a clear memory of the history. But I'm sure you'll find what you need in Cohen and Wald.

 

[Gavroy]

yes, this was really exactly what i was looking for...thank you