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Re: Spacetime "continuations"

Good questions, all. Unfortunately, the answers are not all so good.

The motivations for analytic contiunation are mostly mathematical --
analyticity is a necessary and sufficient condition for geodesic
completeness. This means that all geodesics are either infinitely
long or terminate only on physical singularities (as opposed to
coordinate singularities arising from a bad choice of coordinate
system). The reason you need the continuation to be analytic is
because otherwise you cannot write down the geodesic equation, a
second order differential equation. For a throoughly confusing
account of this, see Hawking and Ellis Large Scale Structure of
Spacetime.

This doesn't say very much to me personally since I don't see the
physical requirement for geodesic completeness. It seems to me
analogous to driving down the freeway until you reach the end of your
road map and saying "Well, I need some more map. I'll just tape some
paper to the side and draw me some more." Even if you know the rules
necessary to continue the map, that doesn't mean there will be more
road when you get there. In the relativistic continuation, when you
change coordinate systems (e.g. for a black hole, from Schwarzchild
to Kruskal-Szekeres coordinates) you find that the old coordinate
system only covers part of the geometry covered by the new coordinate
system. It is sort of analogous to starting with a coordinate system
that only covers the first quadrant, changing to Cartesian
coordinates, and noticing that allowing x and y to assume negative
values gives you a bigger geometry. That does not, however, insure
physical meaning for those extra values.

Analytic continuation also raises global issues that relativity does
not address. For example, analytic contiunation of a static
spherical black hole (which can only be maximally done in Kruskal
coordinates) yields a white hole solution as well. Leaving aside the
fact that none of these have ever been observed, there is a question
of whether the link is between widely separated points in the same
universe (a wormhole solution) or between objects in different
universes. Since GR doesn't address global issues, both are
allowable.Misner, Thorne and Wheeler has a good account of this
in chapter 31, section 6.

Essentially, by analytically continuing the coordinate system, you
have abstracted it away from all sources. However, the sources are
still there and only half of the Kruskal coordinate system is
relevant for black holes. The other half is fully replaced by the
interior of the star which collapsed to form the bh.

Also, if you allow the black hole to rotate, then the only way to
analytically continue the spacetime to the point of geodesic
completeness, as with the Schwarzchild geometry which yielded the
wormhole solution you end up needing an infinite number of extra
coordinate regions. Every time you pass through the rotating
singularity, you end up in an entirely different universe (or point
in our own universe). You don't ever get to go back to your starting
point.

Of course, the extended solution is a perfectly acceptable solution
of the Einstein field equations and this is another reason for
analyticity since the field equation also impose differentiability
requirements on the geometry. However, it is hard to imagine how to
construct one. A real physical source which might create a black
hole, like a collapsing star, cannot be present since it would block
off the extended part, rendering it simply a mathematical artifact.
To make it real, you have to find some way of connecting widely
separated parts of the universe, or of different universe, without
the aid of a gravitationally collapsing object. Doing this requires
violation of some of the energy conditions (e.g. that it always be
nonnegative). It is possible that such a thing could happen in
quantum gravity, where the energy density of the vacuum can assume
nonnegative values, but nobody really knows. This is the origin of
Wheelers idea of a multiply connected quantum geometry, the
"spacetime foam."

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. The
Eddington-Finklestein coordinates my be extended but in so doing only
part of the Kruskal extension is covered and geodesic completeness is
not insured. 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.


From: Bowman_David/tiger_mpc@tiger.gtc.georgetown.ky.us

[snip]
possible inside the event horizon. If a more complicated type of BH is being
considered (i.e. one with sufficient angular momentum and/or charge) then
there are analytic extensions to the home spacetime (region) of the observer
[snip]

This incidental comment prompts me to ask some questions about one of the
things mentioned. I address them to the list in general, but especially
to our resident GR experts.

Q1: What is the physical motivation for wanting or requiring (or whatever)
the extension to be analytic (as opposed to any lesser notion of smoothness,
such as C^\infty)?

Q2: What is the physical motivation for considering only extensions by
(analytic or otherwise) continuation of coordinates? Many other extension
procedures are known in differential geometry/topology, some of which
are independent of coordinate choices.

Q3: Even in the restricted context of analytic continuation of coordinates,
I do not really understand how the choice of which coordinates are to be
continued is made. For example, one can choose polar coordinates on flat
Minkowski spacetime and apply the analytic continuation procedure to obtain
a "helical" spacetime. Why is this not physically viable? (I assume it is not,
because I've never seen this example discussed in print.)

*************************
Phil Parker

Paul J. Camp "The Beauty of the Universe
Assistant Professor of Physics consists not only of unity
Coastal Carolina University in variety but also of
Conway, SC 29526 variety in unity.
pjcamp@csd1.coastal.edu --Umberto Eco
(803)349-2227 The Name of the Rose
fax: (803)349-2926