# Gravitational field (metric tensor) and speed of light between two massive plates

Suppose I have two massive plates of size $$l\times h\times w$$ mounted parallel to each other with a distance of $$d$$ and with a mass density of $$\rho$$. I send a light beam in the middle between them along the length $$l$$ and in parallel to them.

How long does it take the light beam, in coordinate time, to pass through the empty space between the plates?

I assume the speed between the plates is basically determined by the gravitational field between the plates and how this then determines the speed of light.

Maybe the better question is: what is the metric tensor between the plates and how does it determine the speed of light?

Given the highly regular setup, I would hope that at least for larger $$l$$ and $$w$$ there is a nearly homogeneous (read, constant in a plane or all over) metric tensor "near the middle" of the setup.

• You might consider the same question for a massive sphere with a small hole drilled through it. Then you could use the Schwarzchild metric to answer it. – mmesser314 Jun 16 at 15:09
• @mmesser314 A good thought and should be doable, but the metric inside a planet is not Schwarzschild. The time dilation at each radius is defined by two factors. One is indeed the Schwarzschild metric of the inner "smaller planet" (discounting the outer shell). The other is the outer shell. For example, the time dilation just outside a thin massive shell is Schwarzschild while the time dilation everywhere inside is the same as just outside. See: arxiv.org/abs/1203.4428 - While gravity at the center is zero, a bit counter intuitively the time dilation there is maximal. – safesphere Jun 16 at 18:45