Download Black Holes Jan Gutowski King’s College London

Black Holes
Jan Gutowski
King’s College London
A Very Brief History
John Michell and Pierre Simon de Laplace
calculated (1784, 1796) that light emitted radially
from a sphere of radius R and mass M would
eventually fall back to the sphere if
2GM
R<
c2
where G is Newton’s gravitational constant and
c is the speed of light.
A Laplace/Michell “black
hole” is therefore not
completely black - light
can escape from it and be
observed by a sufficiently
close observer. However,
an observer far enough
away will not see it.
The “critical radius”
2GM
R=
c2
is now known as the
Schwarzschild radius
Distance in Flat Space
In one dimension
ds2 = dx2
dx
In two dimensions
(Pythagoras)
2
2
ds = dx + dy
ds
dx
dy
2
2
2
2
In three dimensions ds = dx + dy + dz
2
dz
ds
dy
dx
This space is flat because the shortest path
between any two points is a line.
In four dimensions
(special relativity: Einstein 1905)
2
ds =
2
2
2
2
c dt + dx + dy + dz
2
where t denotes time.
2
ds
Here
can be negative, zero or positive.
The unification of time and space in this way is
called Minkowski space.
General Relativity
Problems with Newtonian Gravity:
In Newtonian theory, the orbit of Mercury
around the sun should be a perfect ellipse.
Taking into account the perturbations to this
solution due to the (Newtonian) gravity of the
other planets, the axis of the ellipse rotates by a
500
small angle - approx. 3600 of a degree per century.
Experiment has shown that the actual orbit
rotates by a bit more than this - in fact by an
43
additional 3600 of a degree per century.
Einstein (1907):
“...busy working on relativity
theory in connection with the
law of gravitation, with which
I hope to account for the still
unexplained secular changes
in the perihelion motion of the
planet Mercury -- so far it doesn’t seem to work”
Eight years later he got it to work!
In general relativity, the 4-dimensional distance
is given by
2
ds =
3
X
µ
gµ dx dx
µ, =0
xµ are spacetime co-ordinates (e.g (t,x,y,z)).
The choice of co-ordinates is not unique.
The metric gµ is symmetric in µ, : gµ = g
In general, the gµ are not constant, but are
functions of the co-ordinates x
µ
E.g. the surface of a sphere of radius 1
✓
Points on the sphere are labelled by the two
angles , ⇥
A point on the sphere has (x,y,z) co-ordinates
(x = sin cos ⇥, y = sin sin ⇥, z = cos )
So for small displacements d , d⇥ in the angles
between two points close together on the sphere
surface, the corresponding displacements in
(x,y,z) give
ds2 = dx2 + dy 2 + dz 2 = d
2
+ sin
2
d⇥2
This is the metric on the surface of the sphere.
The sphere is the simplest example of a curved
space.
The shortest distance between two points on the
sphere’s surface would in 3 dimensions be a
straight line. But this would involve digging a
tunnel through the sphere...
If, given two fixed points on the sphere, one tries
to find the curve of shortest distance between
the two, which lies on the surface of the sphere, then
the curve must be a great circle.
This is called a geodesic of the sphere.
Black Holes
In general relativity, physically interesting
metrics are required to solve the Einstein
equations:
G µ = 8 GN T µ
Here GN is Newton’s constant.
Gµ is the Einstein tensor. This is determined (in
a rather complicated way) by the metric, and
encodes geometric information.
is the stress-energy tensor and describes the
local distribution of matter (e.g. density)
Tµ
In 1915, while in the German
army on the Russian front,
Karl Schwarzschild found the
first exact solution.
The solution describes the
gravitational field outside a
single non-rotating spherical
body (with Tµ = 0 ). It gives
a good approximation to
the gravitational field outside the sun.
Later (1923) Birkhoff proved this solution is
unique.
The Schwarzschild metric is
2
ds =
2
c (1
rS 2
dr2
2
)dt +
rS + r d
r
(1
r )
2
+ sin
2
d⇥
2
Here t is a time co-ordinate, r is a radial
co-ordinate, and (θ, ϕ) are the angular
co-ordinates on the surface of the sphere.
rS is the Schwarzschild radius
For the sun, this is about 3km.
2GM
rS =
c2
Far away from the body, as r ! 1 the
geometry looks like flat (Minkowski) space.
A key principle of general relativity is that
freely falling physical objects, including light,
follow paths which correspond to geodesics.
To a good approximation, as the mass of the sun
is much greater than that of Mercury, one can
regard Mercury as a probe particle moving in the
Schwarzschild geometry.
In reality, the sun is rotating, so the solution is
not exactly Schwarzschild, and Mercury is
massive, which will also deform the geometry,
but these effects are quite small.
So to determine the trajectory of Mercury,
one must find the geodesics (i.e. curves of
minimal length) of the Schwarzschild solution.
After some analysis, one finds that massive
particles with sufficient angular momentum can
have bound orbits, which are approximately
elliptical.
The deviation from an ellipse due to the
differences between Newtonian gravity and
GR is small, but it is non-periodic in the angle ϕ,
and it accounts for the extra orbital precession.
The Deflection of Light
In flat space, light travels in straight lines.
In curved space, it travels along “null geodesics”
which are generally not straight lines.
Outside a massive body, an investigation of the
null geodesics of the Schwarzschild solution
shows a deflection by a small angle '
4GM
= 2
c L
M is the mass of the body, and L is the closest
approach of the light to the body.
The body does not have to be a black hole,
deflection of light rays by the sun is also
observed (Eddington, 1919), for this case theory
and experiment agree very closely
1.75
=
60
In more extreme cases, gravitation lensing is
produced. Abel 2218 is a massive cluster of
galaxies which produces this effect.
Image from NASA(HST)
Stellar Collapse to Black Holes
The sun shines because fusion of hydrogen to
helium releases energy.
The outward pressure produced by this balances
the force of gravity, producing a stable system.
When a star starts to run out of hydrogen, the
heavier elements may also undergo fusion, but it
takes more energy to do this, and the process
becomes less efficient.
There are a number of possible outcomes.
If the star is less than 1.4 times the mass of the
sun, then a quantum process called the Pauli
exclusion principle results in the production of
electron degeneracy pressure which gives rise to a
white dwarf star.
If it is more than 1.4 times the mass of the sun,
processes occur (such as a supernova explosion)
which boil off the electrons leaving an extremely
dense neutron star supported by neutron
degeneracy pressure.
But if the star is more than a few times more
massive than the sun, then such quantum
processes are overwhelmed by gravity.
All of the matter in the star lies within the
Schwarzschild radius, and a black hole forms.
Singularity
r=0
Event horizon
r = rS
Classical vs Quantum Black Holes
General relativity is very good at describing the
dynamics of massive systems over large
distances.
Quantum mechanics is very good at describing
the dynamics of “light” systems over very small
distances.
Over conventional scales, the domains of validity
of these two theories do not intersect, so one
chooses a theory depending on what sort of
system you want to investigate.
There are two places at which one has a
problem: the Big Bang and the singularity at the
centre of a black hole.
A quantum theory of gravity (yet to be found -perhaps string theory??) is needed to adequately
describe these systems.
Black holes have been shown to satisfy a set of
classical laws which are analogous to those of
thermodynamics.
It is tempting to compare thermodynamics laws
to the black hole laws - when one does this
the black hole area is proportional to the
entropy.
In 1975, Hawking considered the quantum
properties of the vacuum near to the event
horizon of a black hole.
“Pair creation” of particles occurs in this region.
One particle can fall into the hole, while the
other heads off to infinity.
This gives rise to an effective temperature for
6
the black hole. For a solar size BH, it is 10 K
Unusual Black Holes in Higher Dimensions
In four dimensions, there are uniqueness theorems
which constrain the types of black holes which
can arise.
After formation, black holes can “wobble” and
emit gravitational radiation; but after this phase,
they settle down to a particularly simple form.
In four dimensions, the event horizons are
essentially spherical.
In five dimensions, solutions like black strings
can exist.
Simulation by
Frans Pretorius
Black string
collapse in 5
dimensions.
The string collapses to a series of black holes.
Simulation by
Frans Pretorius
Black string
collapse in 5
dimensions.
The string collapses to a series of black holes.
Simulation by
Frans Pretorius
Black string
collapse in 5
dimensions.
The string collapses to a series of black holes.
The same sort of instability means that one
cannot construct “black rings” in four
dimensions:
It is unstable, and disintegrates into black holes.
But in five dimensions, one can construct such a
black ring (Reall, Emparan) -- there is an extra
dimension in which to spin the ring, and this
extra angular momentum stabilizes the solution.
In fact one can construct a “Black Saturn”
solution (Gutowski, Gauntlett)
Higher dimensional solutions arise in String
Theory.
In ten and eleven dimensions, even more
peculiar black holes probably exist....