Black Holes
Above: according to General Relativity, space around a black hole becomes so greatly warped that not
even light can escape if it passes too close and crosses the event horizon.
According to Einstein's Theory of General Relativity (GR), energy (matter) causes space and time to warp.
The larger the amount of energy, or the greater its density, the greater is the resultant warping or curving of
space-time. It is, according to GR, this warping that causes planets, which would otherwise drift through
space in straight lines (in the absence of other forces to accelerate them), to orbit stars like the Sun. It also
causes the Moon to orbit the Earth, and your feet to stick to the Globe!

This warping is of three-dimensional (3D) space, or more exactly 4D space-time (three dimensions of space
and one of time) in the fifth dimension and so is hard to visualise. The diagram above gives a 2D analogy.
Imagine holding a rubber sheet horizontally and stretching it taught and keeping it flat. Then somebody
places a heavy ball-bearing in the middle of the sheet, then the sheet deforms, stretches and curves
downwards under the weight of the ball. Note that the surface of the sheet is 2D but curves in the third
dimension. This is comparable to what space (in 3D) does, accept that space curves in the fifth dimension.
GR does not assert that the fifth dimension is real as such, but that the shapes or curvature of 4D
space-time that result are the equivalent shadows, projections or cross-sections of 5D space-time shapes.
What, if anything, the space-time actually curves into is debatable. I get around this by referring to this 5th
dimension as a pseudo-dimension (at least until it is proven otherwise). For example, to draw a 3D cube on
paper (2D) you might draw a square inside a square and connect the corners. Similarly, a cube in 4D space
(or 5D space-time) would appear something like a cube inside a cube in 3D space (or 4D space-time).

Note that we talk about 4D space-time, since space and time become curved or warped! GR postulates the
existence of the 4D spacetime continuum, though we shall later that space and time may in fact be granular
and not actually continuous.

Curving of spacetime not only accounts for the motions of planets, but it accounts for gravity in general,
though it gives a more accurate and more widely applicable explanation of gravity than Newton's theory. In
particular, it is energy that gives rise to gravity rather than mass alone (mass can be thought of as a 'form'
or type of energy so also contributes along with all other energies).

What is a Black Hole?

Gravity attracts objects to the Earth. The Earth has a gravitational field by virtue of the mass and energy it
contains. This causes space and time to curve around the Earth, though since the Earth has a relatively
modest mass of modest density, the curvature is slight, though it does keep the Moon orbiting the Earth!

To launch satellites into space, rockets are used to achieve the necessary acceleration to overcome the
Earth's gravitational field and exceed the escape velocity. The stronger the gravitational field, the higher the
escape velocity.

What happens when the escape velocity exceeds the speed of light? Light simply fails to escape from the
'surface' and the object becomes a black hole by definition. It appears that no signal can travel faster than
light (according to
Special Relativity) at least not in normal space-time and so we expect no massive object
or communication signal (like a light pulse) to be able to escape from a black hole.

Types of Black Hole

Black holes can be divided into three principal classes:

1)
Supermassive black holes, with a mass of a million to a billion times that of the Sun! The objects that
occur in the centre of many galaxies, including that in the Milky Way Galaxy, are probably black holes.
Calculations suggest that there is enough matter there to form a black hole, though more accurate
measurements are still needed to be absolutely certain. These galactic nuclei (see
AGN) have a mass of
about one million to one billion times that of the Sun! (The Sun has a unit of mass of one Solar mass). With
so much mass their escape velocity exceeds the speed of light even when the density of that matter is
comparable to water! It is hard, therefore, to see how a black hole could not form under such
circumstances! Supermassive black holes can have a low density.

2)
Stellar black holes. When massive stars die, they may undergo a supernova and collapse into a
neutron star or pulsar. In these stellar remnants the matter is extremely compressed, causing it to enter all
sorts of very strange states. If the mass of a neutron star exceeds a certain limit (about ? Solar masses)
then no known form of matter can resist the gravitational force and the object is expected to shrink to a
mathematical point or singularity. As it shrinks, the energy density and hence the gravitational field
increase. Once the object shrinks to a critical radius, called the
at its surface exceeds the speed of light and it becomes a black hole. The object is expected to shrink
further (to a point or not as we shall see), however, since no signal or information can escape from it we can
not say for certain what happens to it. The Scwarzschild radius marks the radius at which information and
light are lost, and this limit is known as the
event horizon. It is as if the region of space and time inside the
event horizon are detached and in a separate universe or region of spacetime. We really do seem to have a
'hole' in space! Stellar black holes have very high or perhaps infinite density.

3)
Mini black holes. These have not been observed with certainty, though some report to have created or
simulated them in laboratory experiments. Due to Hawking radiation (see below) mini black holes are hot
and so radiate energy at an increasing rate as their mass diminishes and so are likely to be short-lived.
However, they may have existed in abundance in the early Universe.

Do Black Holes Really Exist?

The existence of black holes has yet to be proven. However, there is incontrovertible data demonstrating
the existence of objects that are probably supermassive and stellar black holes. That is they have the
necessary characteristics, according to current measurements, for their escape velocitiies to exceed the
speed of light. They also exhibit other behaviours we would expect from matter falling into black holes.

Sceptics should also bear in mind that if we argue that black holes do not exist, then equally bizarre physics
must be imagined to prevent the escape velocity exceeding the speed of light! I think of it as this: bizarre
objects that we call black holes do indeed exist and we expect that for at least some of these objects the
escape velocity should prevent light from escaping, so black hole is an apt label. What we can not be so
sure about is the exact nature of these objects, or if you will, what they really are exactly.

Mini black holes remain hypothetical, though it is thought they may have existed in abundance in the very
early Universe, since they are expected to be unstable and short-lived due to hypothetical Hawking

Spacetime Metrics and More Black Hole Types

Scwarzschild Black holes

In GR the curvature of spacetime varies from one region of space to another, according to the energy
present in the local vicinity. (In addition the Universe as a whole may have curvature on the 'global' or
universal scale). The curvature of spacetime in a given region of space is described by the
metric
equations
or simply the metric (a tensor or matrix which appears in these equations). The metric tells us
how to calculate the distance between two points. In the case of a spacetime metric they describe the
distance between points in space and between points in time, or specifically the separation (temporal and
spatial) between two events in spacetime. These events could be the positions of two space-ships at a
given time according to a distant observer (remember that spacetime is relative so different observers can
see things differently).

Flat spacetime is described by the Minkowski metric, which for 2D space reduces to Pythagoras' Theorem
which is the metric for a fat and uniform 2D space. A spherical non-rotating object, like a stationary star,
planet or black hole (non-rotating relative to a distant observer) is described by the
Scwarzschild metric
(given in the article on General Relativity).

The Scwarzschild metric describes space-time outside the event horizon (though solutions can be extended
to cover the region of spacetime inside the event horizon, a so-called Scwarzschild solution). We assume
that for such a black hole collapses to a singularity (mathematical point) since no form of matter can resist
such immense gravitational fields (and in any case unless there is some strange form of incompressible
matter a singularity must form if the gravity is strong enough, at least according to GR).

Falling into a Schwarzschild Black Hole

The changes in gravitational field strength that occur as one approaches the black hole (the gradient of the
curvature) gives us
tidal forces. Tidal forces are due to variations in gravitational field strength over
distance. For a low-density supermassive black hole the tidal forces are calculated to be negligible for a
small object like a human being (although they have been observed to distort stars and to strip off some of
the star's atmosphere from the end closest to the black hole) and an astronaut crossing the event horizon
would notice nothing catastrophic. For a stellar black hole, however, with its smaller radius (event horizon)
and higher energy density, the tidal forces can be enormous according to some calculations. In this case
our astronanut, if he feel feet or head first, would be stretched lengthwise and ripped apart!

To an outside observer, remote from the black hole, any light beamed out from near the event horizon (but
still outside it) becomes
red-shifted and dimmer. The wavelength increases (rather like a stretched spring)
as the light struggles to escape the increasingly strong gravitational field - the light reddens. Also due to
time dilation (see Special Relativity) the astronaut would appear to slow as they neared the event horizon
and would never seem to get there. The light would redden and grow dimmer and the astronaut would
appear to slow down, grow redder and then slowly fade from view. From the astronaut's perspective things
happen much more quickly. If he survived crossing the event horizon then they would reach the central
singularity in a fraction of a second. Calculations show that inside the event horizon, space and time swap
places, so space becomes timelike (and time spacelike) meaning that space acquires only one dimension -
the astronaut or spaceship could not manoeuvre or steer away or change course, no matter how strong its
thrusters, because space only goes one way - toward the singularity! Once at the singularity he would be
compressed to a methematical point! {I am not certain what it means for time to become spacelike, however.]

In all familiar situations in physics where a singularity occurs, some new phenomenon intervenes to prevent
it. Nature seems to abhor singularities! It just makes no sense for an object to shrink to a mathematical point
of zero width, height and breadth with infinite energy density! For example, in fluid flowing under pressure,
singularities can occur in the equations of fluid-flow (the Navier-Stokes equations) (for example in water
driven by a propeller) but instead of an actual singularity occurring the fluid cavitates and air bubbles form!
We shall look later at what might prevent a black hole singularity from forming.

Kerr Black Holes

If the black hole is rotating appreciably, as most probably are, then spacetime curvature around it is given
by the Kerr metric. In this instance the event horizon is smaller than in the non-rotating Schwarzschild case.
Indeed if the rotation is fast enough then the event horizon disappears altogether and then a raw singularity
is exposed to the outside universe, especially at the poles where another boundary called the static limit
meets the event horizon (this seems unsatisfactory and nobody really knows what this would mean). The
singularity is also now not a point, but a ring. Outside the event horizon is an ellipsoidal region of spacetime
called the
ergosphere, whose outermost limit forms a surface called the static limit (or stationary limit).
Above: a section through the regions around a Kerr black hole. AR, axis of rotation (direction
of rotation indicated by the arrow); E, ergosphere; EH, event horizon, and SL, static limit.
Falling into a Kerr Black Hole

Inside the ergosphere (i.e. within the static limit) rotation is compulsory! A space ship in this region could
still manoeuvre with thrusters and even escape the black hole, however, they would rotate with the black
hole, since in this region
spacetime itself is dragged around by the black hole! (That is this region of
spacetime rotates compared to a distant observer). Approaching closer to the event horizon, it would
require ever increasing engine power to pull away from the black hole. Once the spaceship or anything
else crosses the event horizon, however, escape is simply impossible. Some solutions suggest that it may
be possible to avoid the singularity, however, and emerge into a white hole. A white hole is the
mathematical converse of a black hole, it spews out everything instead of sucking out everything. It is
thought by some that such a white hole could open into another universe or a different region of spacetime
in the same universe. However, to an outside observer the white hall requires an infinite time to form (just
as it appears to take infinity for matter to finally fall into a black hole so it appears to take infinitely long for
matter to emerge from a white hall). To the astronaut, however, things would happen quickly and maybe
they would emerge somewhere else, although some physicists state that there is no known natural
mechanism for producing a white hole. To be honest i don't think anybody really knows what will happen to
our astronaut! Indeed, perhaps nobody accept our astronaut ever can!

Kerr-Neuwman Black Hole

This describes spacetime around a black hole with both rotation (angular momentum) and electric charge.
Interestingly, although photons carry the force between electric charges and photons can not escape from
inside the event horizon, electric field lines can. This means that a black hole can generate an external
electric field. The effect of a net electric charge on the blackhole causes minor changes to the metric but
apparently no new major phenomena. For a black hole with no net electric charge (i.e. a neutral black hole
with positive charges exactly balanced by negative charges) the metric reduces to the Kerr metric.

Properties of a Black Hole and Information Loss

So far we have seen that the only observable properties a black hole can posses are its mass, its rotation
(angular momentum) and its net electric charge. All the details of what went in to make up the black hole,
such as the abundance of different elements, appears completely lost to the outside universe. The loss of
information is a serious business. Information relates to entropy, order and chaos, and thermodynamics.
According to the second law of thermodynamics, the entropy (loosely a measure of the chaotic nature of a
system or the number of states or the amount of 'information' it possesses; it takes more information to
describe the many components of a disordered system; see entropy) of a closed system, such as the
Universe, can never decrease - the Universe is gradually collapsing into a chaotic or disordered state of
high entropy. If matter falling into a black hole loses its information then it loses its entropy and black holes
then allow the entropy of the Universe to decrease. This will not happen if the black hole does indeed
retain the entropy of in-falling matter, that is if black holes have entropy. GR predicts that a Scwarzschild
black hole singularity has zero entropy - a vanishing point has few properties with which to describe it! (For
entropy_1 and entropy_2).

Quantum Gravity

String Theory is one of the candidate theories being developed to explain gravity on a quantum scale. The
singularity of a Scwarzschid black hole is infinitely small, infinitely dense and so has an immense
gravitational field - a rather unsatisfactory state of affairs. Such a singularity, as predicted by GR, might not
exist due to the effects of quantum gravity at these extreme scales. GR has never been tested on these
scales and might break down. Furthermore, GR predicts the existence of
gravity waves, and we should
therefore expect these waves to be quantised, according to quantum mechanics, that is we expect there to
be a particle for gravity, called the
graviton. GR, however, is not compatible with quantum mechanics and
one or both theories must be inexact under the extreme conditions in the centre of a black hole. Since
gravity waves are ripples in spacetime, what actually needs to be quantised is spacetime itself! That is we
expect space and time to be grainy on a minute scale and not smooth and continuous, i.e. there is a
smallest fundamental unit of space and of time. It would make no sense to consider smaller quantities of
space and time than these spacetime 'particles' as anything smaller loses precise meaning. This smallest
unit of space can be calculated from theory and it is called the
Planck length (about 1.6 x 10^-35 m -
extremely minute!). (There is also a smallest unit of time, called the Planck time, 5.31 x 10^-44 s). If the
'singularity' cannot meaningfully be smaller than the Planck length then it might contract down to the Planck
length and then stop - rather than a singularity, the nucleus of a black hole might have a diameter equal to
the Planck length.

Furthermore, according to some of these quantum theories, like String Theory, black holes do indeed have
entropy (and also a temperature) so that the information might not be lost after all! In String Theory, black
holes can be modeled as one or more strongly interacting strings in high-energy states of vibration.

Hawking calculated that, according to particle physics theory, black holes should not be black at all!
According to this theory they can actually radiate energy due to a quantum mechanical effect. The vacuum
of space is not actually empty, even if we removed all matter from it, then particle/antiparticle pairs would
still spontaneously form all the time. According to the
uncertainty principle, these particle/antiparticle
pairs can only exist for a short time before one particle bumps into its equivalent antiparticle and then both
particles disappear again! Otherwise, energy conservation would be violated. However, what happens if
such a pair form near the event horizon and one of them falls in? Clearly once inside the event horizon it
can not get out! Its partner is left outside as a spare particle.

The conceptual details of what happens exactly are described differently by different sources, but imagine
a photon/antiphoton pair being produced. To avoid energy conservation violation, and in accordance with
the uncertainty principle of quantum mechanics, these particles live on borrowed time and must disappear
again very quickly, by meeting again and annihilating each other (as matter and antimatter do). The photon
or antiphoton falling in ends up reducing the mass or energy of the black hole. Picture it annihilating with a
photon or antiphoton inside the black hole instead of its original partner, then the black hole loses energy.
However, so as not to violate energy conservation, the partner left outside now becomes a normal photon,
it is no longer ephemeral - it contains an equivalent amount of energy to that lost by the black hole and
carries away this energy as
Hawking radiation. It is as if the black hole radiated away energy!

It can be shown that the profile (the spread in energies or frequencies) of Hawking radiation should be that
of thermal energy - again we find that black holes have a temperature! For a stellar black hole, this
temperature is extremely cold, lower than the 2.73K temperature of space (due to the Cosmic microwave
background radiation) and so the black holes absorbs more heat energy than it radiates. However, for a
mini black hole, the Hawking temperature exceeds that of space and it suffers a net loss of energy - it
radiates away and evaporates! The smaller the black hole, the hotter it becomes and the faster it loses
energy and mass, so the final stages may see a tiny black hole radiating as much energy as the Sun
before ending in a brilliant flash.

Hawking radiation from stellar black holes is too weak to detect by current methods, but perhaps min black
holes in the early Universe could be observed? (As we look across space then we look back in time).

The Final State of an Evaporating Black Hole

Now we have a dilemma. Does the black hole evaporate away to nothing, or does a Planck size remnant
remain? I think there is no easy answer to this question. When particle/antiparticle pairs spontaneously
form from the vacuum of space, they are entangled. Quantum entanglement means that the two particles
are in a common state. It is a mysterious process, because whatever happens to one partner has an
instant effect on the other partner (this has been called 'ghostly action at a distance'). If the black hole
evaporates to nothing, then the partner is entangled with nothing! This is rather nonsensical, but means
that the radiation emitted will be in a 'mixed state' (which necessitates changes to the theory of quantum
mechanics) and will contain no information about the initial state of the matter in the black hole, so again
information has been lost! There are also concerns that such a state of affairs may violate energy
conservation in certain processes or reactions involving the particles of this radiation.

If, on the other hand, a Planck size remnant remains, then this remnant remains entangles with the
radiation. As the radiation is carried away to infinity, the number of states becomes infinite and so the
Planck remnant must have an infinite number of possible states. This can lead once again to problems with
energy conservation.

What do we mean by a Planck size remnant? As we have seen the black hole nucleus may already have a
diameter equal to the Planck length. However, there is also a fundamental unit called the Planck mass
(2.18 x 10^-8 kg), or equivalently (since E = Mc^2) the Planck energy (1.22 x 10^19 GeV). Our black hole
might end-up as a remnant of Planck length and Planck mass.

Bibliography / References

• The Open University course textbooks for S357, 2001: Space, Time and Cosmology.

• Kenyon, I.R., 1995. General Relativity. Oxford Science Publications.

• Stephani, H., 2004. Relativity: An Introduction to Special and general relativity, 3rd ed. Cambridge
University Press.

• Zwiebach, B., 2004. A First Course in String Theory. Cambridge University Press.

• Das, S.R. and Mathur, S.D., 2000. The Quantum Physics of Black Holes: Results from String Theory.
Annu. Rev. Nucl. Part. Sci. 50: 153-206.
Accreting black holes in binary star systems.