# Hawking singularity theorem - what if not all geodesics incomplete?

#### bcrowell

Staff Emeritus
Gold Member
Hawking singularity theorem -- what if not all geodesics incomplete?

The Penrose singularity theorem tells us that once you get a trapped surface, at least one geodesic is guaranteed to be incomplete, going forward in time. But this doesn't mean that 100% of the mass of a collapsing star has to go into the resulting singularity. It would be consistent with the Penrose singularity theorem if a collapsing star formed a microscopic black hole, blowing off the other 99+% of its mass. To set a lower bound on the mass of the resulting black hole, we need some other ingredient in the argument. For example, it must be at least equal to the Tolman–Oppenheimer–Volkoff limit, or else the star would have stabilized as a neutron star.

In the case of the Hawking singularity theorem, all we are guaranteed is that at least one geodesic is incomplete going backward in time. It's tempting to use the theorem as an ironclad argument that the Big Bang had to be the beginning of time, and therefore can't be interpreted as an explosion that occurred in a preexisting vacuum. Now I'm not proposing that the BB really was an explosion in a preexisting vacuum, but I would like to understand how to close the loophole in this argument that arises because it only proves geodesic incompleteness for a single geodesic, not all geodesics. It seems to me that we need some other ingredient in the argument.

Suppose for the sake of argument that our universe has some set of geodesics I that are incomplete, all of them springing out of the same BB singularity, but it has some other set C that are complete going backward in time. If geodesics from I never intersect geodesics from C, then we have two separate universes, each undetectable by the other; and then we'd know we lived in I, not C, since we do see the cosmic microwave background. Therefore the only really interesting case is the one in which some geodesics from C do intersect some geodesics from I. Observers whose world-lines were in C might go along minding their own business for a long time, and then one day they'd get their house knocked down by a piece of shrapnel whose world-line was in I. I suppose this is incompatible with isotropy, but isotropy is only approximate anyway. Is there any more fundamental way that we can rule out a case like this?

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#### PeterDonis

Mentor
It would be consistent with the Penrose singularity theorem if a collapsing star formed a microscopic black hole, blowing off the other 99+% of its mass.
Yes, but it would also require that at least a portion of the trapped region did not stay trapped indefinitely into the future. (At least, if what you're envisioning is that a portion of the star's mass gets blown off after the star has collapsed within a trapped surface.) It seems to me that that would require the null energy condition to be violated (basically because light rays would have to stop being trapped after they already were), which is a condition of the theorem. Or at any rate, for it to happen without violating the null energy condition would seem to me to require a highly unlikely, fine-tuned configuration.

Is there any more fundamental way that we can rule out a case like this?
For the universe as a whole, I would think homogeneity and isotropy would be the best ways to show that all geodesics into the past must be incomplete. The argument would be simple: all geodesics are identical by symmetry, so if any one is incomplete, they all must be.

"Hawking singularity theorem - what if not all geodesics incomplete?"

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