🚀 go-pugleaf

In article <1992Aug8.010716.16818@nuscc.nus.sg> matmcinn@nuscc.nus.sg (Mcinnes B T (Dr)) writes:
>OK, now that I have your attention: let me be more precise. Certainly
>there is a parameter in the S solution which tells you what the mass of
>this thing would be, calculated a la Kepler, if it were a Newtonian
>source...which it isn't. For certain purposes this is no doubt useful.
>But I am thinking of mass as a measure of "quantity of matter". So let
>me ask again:
>How much matter is there in the whatever-it-is that the Schwarzschild
>solution describes? [People have objected, quite rightly, to my use of
>the word "object"...another piece of bait, I must confess.] After all,
>there must be some matter there, right? I mean, "matter tells spacetime
>how to curve", right? And so there must be some matter there, "causing"
>spacetime to curve, otherwise it would not know how to curve...right?
>All right, to quit fooling, what I am driving at is this: The amount of
>matter in this spacetime is zero. Exactly zero.

That makes more sense than saying the mass is zero.  The mass is a
global concept in general relativity, and the Schwarzschild solution
does have mass according to the usual definition for asymptotically flat
solutions.  (Namely, the Kepler bit you refer to.)  However the
stress-energy tensor (which tells the density of energy among other
things) is identically zero.  So one can say that it is a vacuum
solution to Einstein's equations.

>There is no matter "in
>the singularity", because this statement does not make sense. 

That's true.  The singularity is not even a subset of spacetime in this
case.

>It's not
>that I am averse to delta functions etc: by all means, tell me that a
>Newtonian point source can be represented by a delta function, so that
>the point source has a non-zero mass. Just don't tell me that this
>situation is in any way comparable to the S solution.
>I ask because I am trying to understand why people consider the presence
>[actually, absence!] of singularities in classical GR to be some kind of
>blemish on the theory. 

It's a blemish for other reasons.  It wouldn't be too bad of a blemish
at the classical level if the Cosmic Censorship Hypothesis held.  (I
leave it to a friendly relativist to explain this.)  At the quantum
level it's more of a nuisance since it leads to questions like "What
happens to the singularity in the black hole after it evaporates
completely by Hawking radiation" -- a topic there are many papers about
these days.