Black Holes 99An interesting quantity is the Schwarzschild radius of the Universe,푟c,U. Combining
Equations (5.50) and (5.70) we find
푟c,U= 3 푐푡max>12 Gpc. (5.77)Comparing this number with the much smaller Hubble radius 3ℎ−^1 Gpc in Equa-
tion (1.14) we might conclude that we live inside a black hole! However, the
Schwarzschild metric is static and takes the black hole to be surrounded by a nonex-
panding universe, whereas the Hubble radius recedes in expanding Friedmann models
with superluminal velocity, as was seen in Equation (2.52), so it will catch up with푟c,U
at some time. Actually, it makes more sense to describe the Big Bang singularity as a
white hole, which is a time-reversed black hole. A white hole only emits and does not
absorb. It has a horizon over which nothing gets in, but signals from inside do get out.
Black holes possessing either charge or angular momentum are calledReissner–
Nordström black holesandKerr–Newmann black holes, respectively, and they are
described by different metrics. It is natural to consider that matter attracted by a
hole has angular momentum. Matter can circulate a hole in stable orbits with radii
exceeding 3푟c, but if it comes any closer it starts to spiral in towards the horizon, and
is soon lost into the hole with no possibility to escape. Since angular momentum is
conserved, the infalling matter must speed up the rotation of the hole. However, cen-
trifugal forces set a limit on the angular momentum퐽that a rotating black hole can
possess:
퐽⩽퐺푀^2
푐
. (5.78)
This does not imply that the hole is ripped into pieces with one increment of rotating
matter, rather, that it could never have formed in the first place. Remember that angu-
lar momentum is energy, and energy is curvature, so incremental energy is modifying
the space-time geometry of the black hole, leading to a smaller event horizon. Thus
the angular momentum can never overcompensate the gravitational binding energy.
If it could, there would be no event horizon and we would have the case of a visi-
ble singularity, also called anaked singularity. Since nobody has conceived of what
such an object would look like,Stephen HawkingandRoger Penrosehave conjectured
that space-time singularities should always be shielded from inspection by an event
horizon. This is called the principle ofcosmic censorship—in Penrose’s words ‘Nature
abhors a naked singularity’. The reader might find further enjoyment reading the book
by Hawking and Penrose on this subject [5].
Event Horizons. Classically a black hole is a region from which nothing can escape,
not even light. Black holes are mathematically simple objects obeying general relativ-
ity, as seen from outside their event horizon, they have only the three properties: mass,
electric charge and angular momentum. Their size depends only on their mass so that
all holes with the same mass are identical and exactly spherical, unless they rotate.
All other properties possessed by stars, such as shape, solid surface, electric dipole
moment, magnetic moments, as well as any detailed outward structure, are absent.
This has led to John Wheeler’s famous statement ‘black holes have no hair’.