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 where G is Newton s gravitational constant and c is the speed of light. R< 2GM c 2
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 R = 2GM c 2 is now known as the Schwarzschild radius
Distance in Flat Space In one dimension ds 2 = dx 2 dx In two dimensions (Pythagoras) ds ds 2 = dx 2 + dy 2 dy dx
In three dimensions ds 2 = dx 2 + dy 2 + dz 2 ds dz dy dx This space is flat because the shortest path between any two points is a line.
In four dimensions (special relativity: Einstein 1905) ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 where t denotes time. Here ds 2 can be negative, zero or positive. The unification of time and space in this way is called Minkowski space.
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 small angle - approx. of a degree per century. Experiment has shown that the actual orbit rotates by a bit more than this - in fact by an additional 43 3600 General Relativity 500 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 ds 2 = 3X µ, =0 g µ dx µ dx x µ are spacetime co-ordinates (e.g (t,x,y,z)). The choice of co-ordinates is not unique. The metric is symmetric in : In general, the g µ µ, g µ = g µ g µ functions of the co-ordinates are not constant, but are x
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 ds 2 = dx 2 + 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 G N T µ Here G N is Newton s constant. G µ is the Einstein tensor. This is determined (in a rather complicated way) by the metric, and encodes geometric information. T µ is the stress-energy tensor and describes the local distribution of matter (e.g. density)
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 T µ =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 ds 2 = c 2 (1 r S r )dt2 + (1 dr2 r Sr ) + r2 d 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. r S is the Schwarzschild radius For the sun, this is about 3km. r S = 2GM c 2 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 c 2 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 = r S
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 the black hole. For a solar size BH, it is 10 6 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...