Beyond the horizon

Schwarzschild’s solution of Einstein’s equations states that when
a massive enough body collapses under its own weight it will
reach a critical size beyond which there is nothing to stop it from
collapsing further, no matter how squashed the body is. This is
the size which, according to Newtonian gravity and as calculated
by Michell, the star must be for the escape velocity to equal the
speed of light. But there is a major difference in relativity.
If we use only the rules of Newtonian gravity then, provided
the internal pressure of the collapsed star is strong enough, there is
no reason why it could not stop collapsing at, or just beyond, this
critical size. It just depends on the stage at which the molecules,
atoms or even the subatomic particles say enough is enough, we
will not tolerate any further compression.
The force of gravity according to Newton grows in what is
known mathematically as an inverse square relation with distance.
This means that if a star collapses to a size that has a radius that is
half its original value then the force of gravity on its surface will be four times as strong. If it collapses to one third of its original
radius then gravity will be nine times as strong, one quarter and
it is sixteen times as strong, and so on. As the radius gets smaller
the force of gravity gets bigger. If it were possible for all the mass
of the star to be squeezed to a pinpoint of zero size (it now has a
radius of zero) then the force of gravity would be infinite.
General relativity says something dramatically different. If a
star were to collapse down to some critical size, such that its escape
velocity equals the speed of light, then the gravitational force on its
surface would be infinite! By this I mean the force that would be
required to stop it from collapsing further would be infinite. The
radius of this critical size is called the Schwarzschild radius and
marks the boundary of a black hole. Now we see that the collapse
must continue beyond this radius. If you are mathematically
inclined2 you may be wondering3 how it is possible for this force
to be infinite at the Schwarzschild radius and grow even stronger
after the star has collapsed further. How can anything get bigger
than infinite?! The answer lies in the discussion (in Chapter 2) of
objects in free fall. Recall that when you are in free fall, say at the
end of a bungy jump rope (and before you reach the bottom) your
acceleration cancels out the effect of gravity and you do not feel
any gravitational pull. In the same way, as the surface of the star
collapses through its critical radius it is in free fall, and its surface
does not feel the gravitational pull of the interior of the star. This is
why the star cannot stop at the critical radius since it is impossible
now to stop it from collapsing further.
Within the Schwarzschild radius, nothing—not just light—
can escape. Imagine a sphere that has a radius equal to the
Schwarzschild radius and which surrounds the collapsed star.
Such an imaginary spherical surface is known as the event horizon
and is an artificial boundary in space which marks the point of no
return. Outside the horizon gravity is strong but finite and it is
possible for objects to escape its pull. But once within the horizon,
an object would need to travel faster than light to escape, and this is not allowed. Thus the event horizon is a rather unpleasant concept
in that it allows one-way traffic only.
The event horizon is an appropriate name since it can be
compared (loosely) with the common meaning of ‘horizon’ on
Earth. This is the artificial line that marks the furthest distance
we can see and appears as the place where the ground meets the
sky. We understand that this boundary is due to the curvature
of the Earth and, since light travels more or less in a straight line
near the surface of the Earth, we are unable to see beyond it. In
the same way, a black hole’s horizon marks the boundary beyond
which we cannot see any ‘event’. But unlike the horizon on Earth
which continually moves back as we approach it, an event horizon
is fixed and we can get as close to it as we like, and even pass
beyond it if we were foolish enough.
All bodies have their own potential event horizon with its
own Schwarzschild radius. Even the Earth could be made into a
black hole, but since it does not have enough mass to collapse by
itself it would have to be squeezed from the outside. Don’t ask me
how, I am just saying that if it could be squeezed hard enough
then it would eventually pass through its own event horizon,
by which time its collapse would be self-sustaining. The Earth’s
Schwarzschild radius is less than half a centimetre, which means
that any black hole with as much matter in it as our planet would
be the size of a pea.
Once a collapsing star has contracted through its event
horizon, nothing can stop it from continuing to collapse further
until its entire mass is crusheddownto a single point. This is called
the singularity and is a very strange entity indeed. It is so strange
in fact, that the laws of physics that work—as far as we know—
perfectly well everywhere else, describing the behaviour of the
tiniest subatomic particle to the properties of the whole Universe,
break down at a black hole’s singularity. It is therefore quite a
relief for the outside Universe that the event horizon shields us
from such a monstrosity.
Without an event horizon, who knows how a singularity
would corrupt the laws of physics outside the black hole. In
fact, the horizon is so important that physicists have invented the grand-sounding law of cosmic censorship which, they
hope, applies everywhere in nature. They act like the Mary
Whitehouses4 of cosmology guarding the Universe against the
chaos, unpredictability and infinities of the singularity. So what
does this law state? Quite simply that: “Thou shalt not have naked
singularities”. You have to bear in mind that this rather tongue-incheek
statement is really only a hypothesis and may well turn out,
in certain theoretical scenarios at least, not to hold. For instance it
is claimed that tiny black holes, smaller than atoms, may have been
created just after the Big Bang and slowly evaporate away through
a process known as Hawking radiation (which we will meet later
in the chapter). Some calculations have shown that what might be
left behind at the end of this evaporation are naked singularities.
However, this is by no means clear.
According to the equations of general relativity the singularity
is the place where matter has an infinite density, space is
infinitely curved and time comes to an end. There is a common
misconception that time comes to an end at the event horizon. This
is because of what distant observers see as they watch something
falling into a black hole. I will deal with this later; for now I want
to return to the singularity marking the end of time.
Ring any bells? It should do. This is precisely how I described
the Big Bang itself. Only then the Big Bang marked the beginning
rather than the end of time. Apart from that the two cases are
remarkably similar with the Big Bang being the mother of all
singularities; a naked one to boot.
Back to black holes and their interior which, as defined by
the event horizon, is completely empty space apart from the
singularity in its centre (and apart from any matter that has been
captured by the hole and is falling in). The reason why the
singularity has infinite density can be seen if we consider how
we calculate the density of an object. It is the ratio of its mass
to its size. Thus if an object of any mass has zero size then to obtain its density we must divide a number that is non-zero by
zero. And this, believe me, is a highly undesirable thing to do in
mathematics. Try it for yourself. Divide any number by zero on a
pocket calculator. Mine gives me the symbol ‘-E-’ which stands for
‘error’ since a humble calculator cannot cope with infinity. Come
to think of it neither can my powerful workstation computer I
use for research at university. If it encounters a division by zero
the program it is running simply crashes. At least it does me the
courtesy of telling me where the problem is in the code. It turns
out, however, that the singularity is not quite as nasty as this.
When we apply the rules of quantum mechanics, as we must do
at this level, we discover that the singularity has an extremely
tiny (much smaller than an atom) but non-zero size. Many of the
details of the physics have yet to be ironed out, since applying
the rules of quantum mechanics at the same time as the rules of
general relativity is something no one knows how to do properly
yet.
A black hole is therefore very simple in its structure. It has a
centre (the singularity) and a surface (the event horizon). All else
is gravity. Of course what makes black holes so fascinating is the
way their tremendous gravity affects space (and time5) nearby.