If you have a sphere of gas of constant density, would something in the center of this gas experience time dilation relative to someone in empty space (not just someone outside the gas, but someone in the absence of any gravitational field)? What would happen if the density of the gas was finite, but the gas had infinite volume?
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Outside the sphere the spacetime geometry is described by the usual Schwarzschild metric. If we assume the pressure of the gas is negligable comapred to its energy density then inside the sphere the geometry is described by the Schwarzschild interior metric:
$$ d\tau^2 = \left[\frac{3}{2}\sqrt{1-\frac{2M}{R}} - \frac{1}{2}\sqrt{1-\frac{2Mr^2}{R^3}}\right]^2dt^2 - \frac{dr^2}{\left(1-\frac{2Mr^2}{R^3}\right)} - r^2 (d\theta^2 + sin^2\theta d\phi^2) $$
where $M$ is the total mass of the sphere and $R$ is the radius of the sphere.
To get the time dilation for a stationary observer inside the sphere relative to an observer at infinity we simply set $dr = d\theta = d\phi = 0$ to give:
$$ \frac{d\tau}{dt} = \frac{3}{2}\sqrt{1-\frac{2M}{R}} - \frac{1}{2}\sqrt{1-\frac{2Mr^2}{R^3}} $$
And at the centre of the sphere $r=0$ so this simplifies to:
$$ \frac{d\tau}{dt} = \frac{3}{2}\sqrt{1-\frac{2M}{R}} $$
If the sphere is infinite then you get a completely different geometry. For an infinite uniform distribution of matter Einstein's equations give the geometry as the FLRW metric i.e. the geometry of an isotropic and homogenous universe like (approximately) the one we live in. You imply the density is constant, in which case this can only be the moment of maximum expansion of a closed universe i.e. halfway between the Big Bang and Big Crunch. In an FLRW universe there is no time dilation - all points in the universe are the same as all other points and share the same proper time.
John Rennie
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