What Density of Radiation Triggers Gravitational Clumping and Black Holes?

[moving finger]

Not being very mathematically inclined, I'm stumped with this one. Can anyone help?

Imagine a toy “closed” universe in the form of a very large box (say the size of our universe) with perfectly rigid and reflecting walls. The box is static in size (neither expanding nor contracting) and is uniformly filled with electromagnetic radiation, with an average energy density Rho, and nothing else.

The electromagnetic radiation will exert mutual gravitational attraction (vie E=mc^2).

At low values of Rho, we would expect to see no gravitational clumping of this radiation, in other words the toy universe would be stable with a uniform distribution of radiation.

Q1) What value of Rho is required before we observe gravitational clumping of the radiation (if at all)?

Q2) What value of Rho is required before we observe black holes forming?

Q3) If it is possible to spontaneously form black holes at high values of Rho, what would be the equilibrium mix of black holes vs background radiation as a function of Rho (taking into account the fact that black holes emit Hawking radiation)?

Best Regards

MF

 

[pervect]

Not being very mathematically inclined, I'm stumped with this one. Can anyone help?

Imagine a toy “closed” universe in the form of a very large box (say the size of our universe) with perfectly rigid and reflecting walls. The box is static in size (neither expanding nor contracting) and is uniformly filled with electromagnetic radiation, with an average energy density Rho, and nothing else.

The electromagnetic radiation will exert mutual gravitational attraction (vie E=mc^2).

At low values of Rho, we would expect to see no gravitational clumping of this radiation, in other words the toy universe would be stable with a uniform distribution of radiation.

Q1) What value of Rho is required before we observe gravitational clumping of the radiation (if at all)?

Q2) What value of Rho is required before we observe black holes forming?

The last time I thought about this I convinced myself that we would only observe "clumping" when the density was sufficient to form a black hole.

I don't have a really solid argument for this, though - my thinking is that if we draw a sphere around a clump of radiation of a density less than that required to form a black hole, that the radiation should be expanding on the average. Thus on the average there would be no tendency to "clump", there would be a tendency to "expand".

The argument isn't airtight, esp. if the radiation isn't really random.

Q3) If it is possible to spontaneously form black holes at high values of Rho, what would be the equilibrium mix of black holes vs background radiation as a function of Rho (taking into account the fact that black holes emit Hawking radiation)?

I don't think you'd have multiple black holes in the box. The larger the box, the lower the critical density. When you exceed the critical density, the whole box should become a black hole.