How much matter can you put in space and time?

zeromodz

How much matter or energy density can be allowed in space and time before it closes in on itself?

Phrak

As most things GR, this depends on a coordinate chart. One such chart is the Schwarzschild solution for an uncharged black hole without angular momentum. This relates the radius of a ball--from which a volume can be implied, to an enclosed mass. See the Wikipedia for the Schwarzschild solution relating mass to radius.

This would put an upper bound on the amount of mass a volume could contain, given the various conditions given above and within the context of classical physics.

zeromodz

I researched the whole page and I found no direct answer. Shouldn't the actually quantity be invariant? Why can't you just tell me instead of telling me to go look it up?

yuiop

A lot depends on what you mean by time and space closing in on itself. If you mean what is the maximum density (p) you can have in a spherical region of space before it collapses into a black hole then the answer is:

[tex]p < \frac{3}{32} \, \frac{c^6}{\pi G^3M^2} [/tex]

It can be seen that from the inequality that the larger the mass, the lower the required density to form a black hole. The converse is that the smaller the radius of the enclosing volume, the greater the density required to to form a black hole and the question becomes how small can the radius of the event horizon of a viable black hole be? Some would conjecture the maximum density would be very roughly the Planck density (a Planck mass contained within a sphere of Planck radius) but there is no real proof of this. It can be also be noted that such a small black hole would evaporate very quickly due to Hawking radiation and so it could not be called stable.

From another point of view, the general consensus is that GR allows all the mass of a fully formed black hole to be contained within a point of zero volume and therefore GR allows the density of the singularity at the centre of a black hole to be infinite, but it is also acknowledged that the laws of physics (as we know them) break down at the black hole singularity and so we do not really know what happens there.

atyy

A precise answer is not known. Here's the latest: http://arxiv.org/abs/0912.4001

Also relevant is the Bekenstein bound, discussed in sections 2 & 3 of http://arxiv.org/abs/gr-qc/9508064 (the rest of the paper is speculative).