Behavior of a large spherical void in an infinite universe of uniform density matter

Question

I realise how unrealistic this scenario is, but nevertheless, it does not seem unphysical and it does raise interesting questions.
Imagine a flat stationary infinite isotropic homogeneous universe uniformly filled with an incompressible transparent fluid of mass-density, $\rho$. One can argue that such a universe could have zero gravitational field and field-gradient everywhere and that we could arbitrarily say that its gravitational potential is also zero.
A spherical void of radius, $R$, would have a mass deficit of $-\rho(4/3)\pi R^3$ compared with the rest of the universe. With Newtonian gravity, it would generate a field of $+G\rho(4/3)\pi R^3/r^2$ outside the void and a field of $+G\rho(4/3)\pi r$ inside the void (note the '$+$' sign). The corresponding potentials would be $+G\rho(4/3)\pi R^3/r$ outside the void and $+(3/2)G\rho(4/3)\pi R^2-(1/2)G\rho(4/3)\pi r^2$ inside the void.
For a void with large enough radius, R, the potential at the center of the void will exceed $c^2/2$. For even larger radii, the potential will exceed $c^2/2$ over a spherical region with radius R_s>R. This would correspond to the Schwartzchild radius for a black hole, but in this case the polarity is exactly opposite.
Admittedly, it's an implausible scenario but it does give rise to a myriad questions: what are the properties of such a void when the effects of general relativity become important? Is there the equivalent of an event horizon? Does it have a temperature? What does a person sitting at the center of the void see? What do they see if they start falling out of the center? What happens if two such voids meet? Can they orbit each other? etc., etc.

Answer 1

Imagine a flat stationary infinite isotropic homogeneous universe uniformly filled with an incompressible transparent fluid of mass-density, ρ.

I think this list of properties for the (non-holey) universe needs to be pared down somewhat. It's not consistent to demand that it be stationary; that would violate the Einstein field equations. The fluid can't be incompressible, because that would violate the dominant energy condition. I don't see why it's relevant whether it's transparent.

Since it's not consistent to say it's stationary, let's start from a simple example that is consistent, which is a flat FLRW universe. A calculation then shows that, ignoring unitless factors of order unity, $R=c/H$. This is the Hubble radius, i.e., the radius of a cosmological event horizon.

So although GR isn't linear, your heuristic argument about a negative-mass Newtonian black hole is sort of right. It suggests the existence of a horizon, and we do have a horizon. However, the horizon is less exciting than we might have imagined, since we get one even if there is no hole.

I think the interpretation here is that if homogeneity were to fail on very large scales, comparable to the cosmological horizon, we would not be able to detect that fact.