Maxwell's equations in general relativity

[Gavroy]

Hey

there is something that I don't really understand

In Landau Lifgarbagez"Classical Theory of Fields" it is said that one of the Maxwell's equations in the presence of a gravitational field is:

div E= \frac{\rho}{\epsilon_0}\sqrt{g_{00}}

So I thought that if you have a hydrogen atom in the gravitational field of a black hole, than this would cause the Electric field strenght between the proton and the electron to become smaller, which would be caused by the g(00), which is value less than 1 in a gravitational field.

But now an extremely smart person told me, that this is not so, because this formula is only valid, if the body that causes the gravitational field is the same as the one that causes the electric field. so in my case this would be not so, as the black hole that produces the curved spacetime and the proton that causes the electric field in the hydrogen atom are two different bodies?

can somebody explain this to me? i don't really understand, why one has to make a difference between the curvature of the charged body itself or the curvature of a body and the electric field that is in the curved space?

 

[bcrowell]

Hi,

To make your math show up properly, you need to surround it in itex tags. Here's your equation done that way: div E= \frac{\rho}{\epsilon_0}\sqrt{g_{00}}. To see how I did that, click on the QUOTE button under my post.

The first thing you need to realize is that this equation isn't predicting any new or interesting physical phenomenon. One of the basic ideas of GR is the equivalence principle, which says that spacetime is locally compatible with coordinates in which it's the same as Minkowski space. These coordinates are the coordinates measured by a free-falling observer. So if you want to know Maxwell's equations in the region near a black hole, you can always use the ordinary form of Maxwell's equations, and they will agree with experiment if your lab is in free fall.

That means that the equation you're quoting from LL can't really have any special physical significance without its context, and you'd really need to tell us more about the context if you want us to help you interpret it. The value of g₀₀ is not a measurable property of a region of spacetime; it's coordinate-dependent. Therefore the significance of the equation can't be evaluated without knowing something about the coordinate system that LL are assuming.

What this smells like is something to do with tensor densities. GR has two types of quantities with nice transformation properties: tensors and tensor densities. Here http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html#Section4.6 is my attempt to explain those. (Some of the equations are not quite formatted right in this html version; there is a pdf version here: http://www.lightandmatter.com/genrel/ ) When you're working with tensor densities, your equations will tend to have factors of \sqrt{-g} in them, where g is the determinant of the metric (not just its 00 component, although LL may be talking about an example where the determinant happens to equal its 00 component).

This WP article http://en.wikipedia.org/wiki/Maxwell's_equations_in_curved_spacetime shows Maxwell's equations in curved spacetime, in a form that is independent of the coordinate system chosen. Notice the factors of \sqrt{-g}.

-Ben