🚀 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. 

(Stuff deleted...)

What you say is true in a sense, but it's not at all clear that it is an answer
to your original question, "How much mass is in the Schwarzschild solution?"
In order to answer that question, you'd need to define precisely what is meant
by the mass inside of a spacetime.  This is notoriously tricky to do.  Since
mass and energy are equivalent, and since there is something like "gravitational
potential energy" in a curved spacetime, you might like to define some quantity
that behaves like the total potential energy, integrated in some appropriate
way, as the mass.  You can do this in a self-consistent way for spacetimes that
are asymptotically flat, and not otherwise.  Fortunately the Schwarzschild solution
is asymptotically flat, and if you do this kind of calculation of the "mass"
contained in it, you get the usual answer, which reduces to the Keplerian answer
at distances much larger than the Schwarzschild radius.

Whether you want to call this computed quantity the total mass or not is
up to you, but if you don't want to, I think you ought to either supply
an alternate definition, or agree not to use the term "mass of a spacetime."
(The latter option seems to me to be the most sensible choice.)

-Ted