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Re: Re: Spacetime "continuations"



Date: Tue, 14 May 1996 12:39:28 EST
From: "Paul Camp" <pjcamp@coastal.edu>

The motivations for analytic contiunation are mostly mathematical --
analyticity is a necessary and sufficient condition for geodesic
completeness.

Unless there's too much confusion about the word "analytic," this is
false, so let's check. I meant by " analytic" that a function "is
locally representable in a power series" just as for complex analytic
functions, but for real coordinates we need real analytic instead.
(Just in case, the power series must have a strictly positive radius
of convergence. That's supposed to be covered by "representable.")
Thus Minkowski space minus a point has (real) analytic everything,
but is geodesically incomplete.

The reason you need the continuation to be analytic is
because otherwise you cannot write down the geodesic equation, a
second order differential equation.
No, I need only C^2 (twice continuously differentiable) to make good
standard sense of the geodesic equation, and less if I am willing to
accept weak (distributional or generalized function) solutions.

For a throoughly confusing
account of this, see Hawking and Ellis Large Scale Structure of
Spacetime.

I don't doubt their confusion here, but please supply a precise
reference and I'll go slog through it anyway. I'm used to them
(more or less), having studied their mathematical parts a bit.
(In fact, I wrote my PhD thesis on representing spacetime
singularities via distributional metric tensors etc.)

[snip---I'll leave the middle part for now, but may return later,
after we get some of these other points more hashed out]

Which coordinates do you choose to extend? Whichever look like they
may be extendable. For a black hole, only maximal extension of the
Kruskal coordinates provides geodesic completeness.
OK, let's assume for now that the objective is to extend to a
geodesically complete spacetime, and I'll come back to why later.

Minkowski spacetime is already geodesically complete and
so cannot be further extended. The proof of this is in the Hawking
and Ellis book but you really don't want to see it. Trust me.

I don't remember their proof, but using the embedding definition of
extension, it's easy to write one down. The problem here is choosing
a good notion of extensibility. The embedding definition says that
a spacetime is inextensible iff it cannot be (smoothly) embedded
into another spacetime which properly contains it.
There are two problems here. One is that a spacetime may be
inextensible but geodesically incomplete. The other is that an
extensible spacetime may have many different smooth extensions,
but they may not all be (real) analytic extensions. It appears to
me that the physicial hope is that requiring (real) analyticity
*might* provide uniqueness, allowing one to unambiguously specify
one extension as the only relevant one. (This would be similar to
the way that requiring the spinor group to be simply connected
selects exactly one of the double covering groups of the rotation
group.) Now complex analytic continuation is unique, but I am not
so sure about real analytic continuation in several variables.
I'll check that.
In any case, what we have here so far may at least provide a
plausible rationale for wanting analytic continuation of coordinates.
I think part of my problem was that it was never clear to me which
notions were supposed to be primary (especially given some of the
misleading statements in H&E!).
Thanks for helping me so far, and here's hoping we can do a
reasonable job of nailing this down a bit.

*************************
Phil Parker Internet: pparker@twsuvm.uc.twsu.edu
Math. Dept., Wichita St. Univ. Bitnet: pparker@twsuvm
I find [in mathematics] a wonderful beauty. This is no science, this is
art, where equations fall away to elements like resolving chords, and
where always prevails a symmetry either explicit or multiplex, but always
of a crystalline serenity.---Turjan of Miir (Jack Vance)