Daryl McCullough wrote:
> I'm a little bit confused about white holes. Many physics
> web pages, including http://casa.colorado.edu/~ajsh/schww.html
> and http://en.wikipedia.org/wiki/White_hole
> and http://cosmology.berkeley.edu/Education/BHfaq.html#q10
> all describe white holes as the time-reversal of black holes.
> It's hard for me to know what that could possibly mean, since
> the black hole metric is *symmetric* under time-reversal:
> ds^2 = -(1-2m/r) dt^2 + 1/(1-2m/r) dr^2 + r^2 dOmega^2
Yes, neither the Einstein field equation nor this particular solution of
it dictate in which direction the future lies. So you must make a
choice, and the usual one is that +d/dt in the exterior region is future
pointing. This is merely a convention, and calling -d/dt the future
would be equally valid (but will confuse your readers).
So pick +d/dt in the exterior region to be the future. In a Kruskal
diagram that makes the interior region at the bottom be a white hole,
and the interior at the top be a black hole. All that "white holes are
the time-reversal of black holes" means is that if you took -d/dt as the
direction of the future then the black and white holes would exchange
places in the Kruskal diagram.
To those who claim "time reversal is meaningless", let me point out that
in the real world this is true, but in a mathematical model of the world
time reversal can make sense, and in this particular MODEL it does. Time
reversal has become one of the important symmetries of MODELS in modern
physics. Nobody really knows why it is important, but it is a part of
all modern theories of physics.
> For an eternal black hole,
> there is no preferred direction in time, so I don't see how
> it makes sense to talk about the time-reversal being a white
You must select a direction to be the future. Until you do that you
don't know which region is the black or white hole; as soon as you
choose then their locations are determined.
This is normal -- you must select all the criteria necessary to make the
model conform as best as possible to the real world. In this case (and
in most cases of manifolds in GR) that includes selecting a direction in
the manifold to correspond to the future. Of course Schw. spacetime is
not a good model of the universe at all, but that's irrelevant to the
issue of selecting a future direction in the model.
[Schw. spacetime IS a good model for the geometry near a
spherically symmetric static object, such as an isolated
planet or star.]
This is no different from a roadmap -- you must select a direction on
the paper to correspond to North in the real world. Of course the map
maker does that for you, but the principle is the same.
> On the other hand, it seems to me that choosing the parameter
> m to be *negative* is a perfectly valid solution to Einstein's
> field equations (if unrealistic). That solution is equivalent
> to reversing *r*, not t. Is there a name for this exotic
It's still Schw. spacetime. Such values of M do not correspond to any
objects we observe in the real world. Indeed, requiring the energy
density to be everywhere non-negative is called the "energy condition"
of GR, and corresponds to this.
Real world black holes are expected to not be primordial (as in Schw.
spacetime), but rather due to the collapse of massive objects. Such
objects with M<0 would not collapse, but would "explode" (as you point
BTW this is related to the ambiguity in extending the exterior Schw.
region across the horizon, and which of the two Eddington-Finkelstein
coordinate charts one selects. One extension goes only backwards in
time, and the other only forwards in time; the first leads from the
white hole, and the latter leads to the black hole.