Black Holes, Event Horizons, and the Universe

Is it conceivable that the entire universe is a "black hole"?  To 
even begin to answer this question, we need clear definitions of 
"black hole" and "universe", neither of which are as easy to define 
as one might think.  The formal definition of a black hole is a region
of spacetime that is not in the causal past of the infinite future.
If we combine this with any Robertson-Walker type universe, it's clear
that neither an "open" infinite universe nor a "closed" finite universe
can be considered a black hole, because the definition explicitly 
requires a black hole to be a proper SUB-set of the overall universe,
i.e., a black hole is defined as being EXCLUDED from some portion of
the universe, so the universe itself cannot be a black hole according
to this definition and with this class of "universes".

On the other hand, we're free to imagine "our universe" as a "closed" 
(though not necessarily finite) subset of some meta-universe that can
be said to posses an "infinite future" of which our universe is not in
the causal past, in which case by definition our universe IS a black 
hole.  Of course, this putative embedding may not be of the sort that
is usually contemplated in the context of relativistic cosmologies, 
but this illustrates how much the answer to our original question 
depends on the precise definitions that we choose, as well as on 
whether we choose to imagine "our universe" as just a subset of some
meta-universe.

It's also worth noting that (if we confine ourselves to just "our 
universe") not only does the formal definition of a black hole rule 
out regarding a closed Friedmann universe as a black hole, it implies
that a closed finite universe cannot even CONTAIN a black hole, 
because it has no infinite future.  In other words, the formal 
definition of a black hole has no applicability at all in the context
of a closed finite universe.  Nevertheless, there are some conceivable
configurations of matter and spacetime in a finite universe that could,
with some justification, be called black holes, provided we are 
willing to adopt a more provisional definition.

For example, one rough definition of a "black hole" is a mass whose 
Schwarzchild radius is outside of itself.  If we can define all the
terms adequately in the context of a finite universe, then we can
agree to call such configurations "black holes", even though they
do not (and cannot) satisfy the more formal definition.  

If the average density of a region is rho, then very roughly we could
say the Schwarzchild radius of a spherical region in asymptotically
flat spacetime (and neglecting gravitational binding energy, etc) of 
radius r is about (8 pi rho)(r^3)/3.  Therefore, regardless of the 
density, if we consider a spherical region (in flat space) of radius
r greater than sqrt[3/(8 pi rho)] then the region would be called a 
black hole.  However, this definition still doesn't apply to a closed 
Friedman universe because of two related effects: (1) the overall
space has positive curvature on a cosmological scale (rather than
being asymptotically flat), meaning that volume (and quantity of 
enclosed mass) increases more slowly with increasing radius, and (2) 
the entire space closes up on itself before reaching the radius that 
would constitute an event horizon.  In a sense, this is almost the 
definition of a closed universe.

On the other hand, in an open infinite universe with overall negative
curvature on a cosmological scale, volume increases with increasing 
radius faster than in flat spacetime, and there is no limit to the 
size of region we can consider.  Therefore, we might think that for 
any fixed density there must exist regions large enough to be within 
their own Schwarzchild radii, so there would be actual cosmological 
event horizons, making vast regions of the universe inaccessible to 
each other.  Inflationary cosmologies involve structures such as this,
but such regions cannot really be regarded as "black holes" relative 
to each other.  For one thing, the overall negative curvature of the 
universe in these models is inconsistent with the idea of the universe
as a black hole.  More fundamentally, we need to distinguish between 
the general notions of an event horizon and the boundary of a black 
hole.  The definitions given in Wald may be paraphrased as follows

    An asymptotically flat [and strongly asymptotically
    predictable] spacetime M is said to contain a black
    hole if not every point of M is contained in the causal
    past of future null infinity.

    The black hole region, B, of such a spacetime is
    defined to be the points of M not contained in the 
    causal past of future null infinity.  The boundary
    of B in M is called the event horizon.

These are often cited as the kind of precise and rigorous definitions 
needed to actually prove theorems about black holes, in contrast with
more prosaic definitions such as

   A black hole is a mass inside its own Schwarzschild radius.

It has sometimes been claimed that this definition is not actually used 
by any general relativist, because it depends on the quantity of mass, 
which is not well-defined in general relativity unless we make some
artificial assumption such as considering the mass to reside in an
asymptotically flat spacetime.  However, notice that Wald's definition
is explicitly restricted to the very same class of spacetimes.  As he
says,

    ...we have defined the notion of a black hole only for
    strongly asymptotically predictable spacetimes [which are
    asymptotically flat by definition]...

    ... there appears to be no natural notion of a black hole 
    in a "closed" Robertson-Walker universe which recollapses 
    to a final singularity...

Thus, the range of possible universes for well-defined (in Wald's
sense) black holes seems to closely coincide with the universes in 
which the total mass-energy of a system can be rigorously defined.  
They both evidently require an asymptotically flat universe.  Of 
course, as Wald says,

   ... an approximate notion of a black hole still exists for
   any region of a closed Robertson-Walker universe that can be
   treated as an isolated system.

and in this same sense we can define the total mass-energy inside a
radius R for a spherically symmetrical configuration of matter by the
integral

                                   R
        Total Mass-Energy         /       2
        Inside Radius R     =     |  4pi r  rho dr
                                  /
                                 r=0

According to Birkhoff's theorem the Schwarzschild solution is 
the essentally unique solution of the field equations outside a 
spherically symmetrical configuration of mass, whether static or
not.  In particular, there are no such things as spherically 
symmetrical gravitational WAVES, so there is no ambiguity about
possible energy transfer across the boundary.  This is, for example,
the mass-energy that is used to establish the correspondence
between the relativistic and Newtonian versions of Kepler's third
law.  On this basis, in a spherically symmetrical context, we can 
apply the "prosaic" definition of a black hole, but notice that we're
now talking about "approximate notions of black holes", rather than 
precise and rigorous definitions.  (It's unclear to what extent 
all the celebrated theorems about BLACK HOLES are applicable to 
these "approximate black holes".)

Applying the prosaic definition in situations that do NOT possess
spherical symmetry is even more difficult.  We then have to give an 
invariant (i.e., coordinate-independent) definition of "inside the 
Schwarzschild RADIUS" for a region that is not even spherical.  We
must also decide if we should use proper distance, and, if so, proper
distance on what time slice?  And so on.  There's no doubt that giving
a definition of black holes that's applicable to all possible local 
and cosmological contexts is difficult.  For example, we can define 
a black hole as a region that's not in the causal past of future null
infinity, but then we're likely to have trouble in a closed universe,
since such a universe doesn't possess "infinity".  Moreover, this 
prescription for a black hole has a certain tautological quality:

  A black hole is that which is surrounded by an event horizon.  

  An event horizon is that which surrounds a region of no escape,
  i.e. a black hole.

In contrast, the "mass within its own Schwarzschild radius"
prescription is of a different and, in a sense, more ambitious 
nature.  It seeks to describe an actual physical circumstance that
would constitute a black hole.  Not surprisingly, people have only
succeeded in doing this in a rigorous way for certain simplified
cases, such as with spherical symmetry.  No known definition of a 
black hole gives meaningful and unambiguous results for all possible
contexts and, in particular, all possible cosmologies.  Fortunately,
stars, some galaxies, and even some cosmologies can be treated as 
spherically symmetric, so this special solution has a fairly wide 
range of applicability.

Another interesting point of definition concerns "event horizons".
As noted above, Wald defines an EVENT HORIZON as the boundary of 
the points of spacetime not contained in the causal past of future
null infinity.  However, Rindler defines an EVENT HORIZON of any
point P as the boundary between the region of points whose causal
future includes P and the region of points whose causal future does
not include P.  Roughly, points that are more than c/H (where H 
is Hubble's constant) away from the point P are outside P's event
horizon.  If you imagine a spherical universe whose "circumference"
exceeds c/H, then every point is on the event horizon of infinitely 
many other points.

So which is the "true" definition of an event horizon?  Weinberg uses
Rindler's definition.  In fact, he credits Rindler with having coined
the term "event horizon" [and "particle horizon"] in a 1956 paper.
On the other hand, Misner, Thorne, Wheeler distinguish between two 
kinds of "horizons": (1) horizons in cosmology, and (2) horizons in
black hole physics.  The first correspond to Rindler's definition,
and the second to Wald's.  However, MTW don't actually use the term 
"EVENT horizon" for either of these concepts.

As an aside, it's interesting to note that when naive people first 
hear about black holes they often wonder: "If it takes infinite 
'external time' for matter to collapse inside its Schwarzchild radius,
and if there is only a finite amount of time in the entire existence 
of a closed universe, how can a true black hole exist?"  The answer, 
of course, is that those naive people are right: a black hole (strictly
defined) IS inconsistent with a closed universe, precisely because of 
it's finiteness.