## When Must Black Holes Form?

#### John Baez

Oz re-appears in a kaleidoscope of naked women and strange goings on.

"Hey!" said Oz "There's a serious transdimensional time warping going on in the link to Wiz-time over in the United Plates. A whole lotta demons are using the lode-lines over there for their own nefarious purposes, and the lode-masters have gone on short time working. "

"Gosh" said erg "you had better be careful. You definitely wouldn't want to cross-link with some of those. You couldn't IMAGINE what you might drop into."

"It's really nice to talk to someone who doesn't think a quick ZAP! will make you think better." sighed Oz. "I guess we can chat over this in peace and quiet. What we gotta show Wiz is that d2V/dt2 heads negative as V-> something small no matter what the momentum flow is."

Now d2V/dt2 = -(1/2)[T00 + T11 + T22 + T33]

lets say T11 + T22 + T33 = Tss for simplicity.

and as you say there must be a relationship between T00 and Tss and for expansion Tss has gotta be negative. Now it seems to me that T00 is a volume term so if we keep the same amount of energy in the volume V the energy density will vary as 1/V or perhaps better 1/r3 if r is the characteristic radius of the volume V. If Tss is really a pressure term then you would expect it to vary as the surface area or as 1/r2. The trouble is that I don't know of this is right. Dimensionally it doesn't look right, but there are so many c's that have dissapeared off as 1, that it's hard to be sure.

Anyway if this is about right then as r gets very small the T00 term will always win out in the end over the Tss term. Then d2V/dt2 will become negative and the volume will start to decrease (as long as dV/dt=0 at some point) and as it decreases so it will decrease even faster, and eventually become (horror) zero.

Then there's this E2 = p2 + m2 thingy I came across years ago. I'm sure we ought to be able to work this in somewhere. What do you think?

Erg wiped his rusty sword, jumped up and swiped it around impressively. Oz dived at the floor as it swung round over his thinning hair.

"Well," interjected erg.

"Well...." He stopped, put the sword down, and scratched his head.

"Well, I don't know!"