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From http://jila.colorado.edu/~ajsh/bh/schwp.html The amount by which radial distance is "squished" is:

$ \dfrac{1}{\sqrt{ 1 - r_s/r}}$

where the Schwarzschild radial distance (or "circumferential radius") r is the point at which measured circumference is $2 \pi r$ and the Schwarzschild radius is

$r_s=2 G M/c^2$

is the Scharzchild radius at which we get a black hole for that amount of mass ($M$). The amount by which time is dilated is the inverse of this (so pretty similar to the case for moving objects in flat space). Usually, this equation is integrated over dr to calculate the total distance between 2 points near a dense gravitational body.

Lets assume we have a thin spherical shell that is dense enough to make 50% time dilation immediately above the surface, which I believe comes out to

$r=(4/3)r_s$

The radial distance will be squished by 50% as well (though some people refer to this as "expansion" since you can fit more into the same space relative to the reference frame outside this gravitational well) and the circumference of the outside of the sphere will be

$2\pi(4/3)r_s$

According to Birchoff's theorem, the apparent gravitational field should be zero inside (just as it is for Newtonian gravity).

According to an answer to the question at Does a massive spherical shell expand the time inside itself?, "gravitational time dilation depends on the gravitational potential" so should be the same inside (which makes since if you think of photon red-shift as an indication of time difference).

It seems like you would still have the same length contraction inside as well (if it is just as much an effect of gravitational potential as time dilation) but it would be the same in all directions, which implies that the measured inner circumference of the sphere would be twice the outer circumference (so $2 (2\pi(4/3)r_s)$ ) and the inner (measured) radius would be $2 ((4/3)r_s)$.

Qmechanic
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Dustin Soodak
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1 Answers1

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As stated in On a common misunderstanding of the Birkhoff theorem:

In Newtonian gravity (NG) it is known that the gravitational field anywhere inside a spherically symmetric distribution of mass is determined only by the enclosed mass. This is also widely believed to be true in general relativity (GR), and the Birkhoff theorem is often invoked to support this analogy between NG and GR. Here we show that such an understanding of the Birkhoff theorem is incorrect [...] The correct metric, matching continuously to the location of an external observer, is determined both by the enclosed mass and mass distribution outside. The effect of the outside mass is to make the interior clock run slower [...]

Further in the section 2.0 the author shows that "the time term of the metric is always maintained continuous, but the space term is not". There is no length contraction or expansion inside an empty shell, only time dilation.

safesphere
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