Figure 1: The ergosphere of Kerr s black hole A Panoramic Tour in Black Holes Physics - A brief history of black holes The milestones of black holes physics Astronomical observations - Exact solutions of Einstein s equations The Schwarzschild solution The Kerr solution The Kerr-Newman solution - Some results on classical black holes - Thermodynamic properties of black holes Hawking s effect The laws which govern black hole physics The nature of black hole entropy The information puzzle The final state of a black hole This is part of a seminar given at CBPF (Centro Brasileiro de Pesquisas Físicas), Rio de Janeiro.
- A brief history of black holes The term black holes for completely collapsed stars is due to J.A. Wheeler in 1969, but such astronomical objects have a longer history. In fact, Michell (1783) and independently Laplace (1749-1827) stated that: If a star is sufficiently dense, then the escape velocity of a test particle on its surface can be larger than the speed of light. This means that, in the framework of Newtonian corpuscular theory of light, the light emitted from the star cannot reach a distant observer. This idea is abandoned after the appearance of the Maxwell s theory of Electromagnetism, which replace the corpuscular theory of Newton, but it appears again in a natural way in the framework of General Relativity (Einstein 1915), where, like the matter, also the light feels the gravitational field. If a star with mass M has a radius smaller that the radius of Schwarzschild (r h = 2MG/c 2, G and c are the Newton s constant and the velocity of light respectively), then the ligth emitted by the star cannot rich a distant observer. The sphere of radius r h around the star is called event horizon; such a (virtual) surface separate the space in two parts. The inner part cannot communicate with the exterior one and therefore it is invisible to a distant observer. A free falling observer can across the event horizon without notice anything strange (if M if large enough), but after the crossing, he will not be able to communicate with his home. He is fallen into a black hole and his fate is written. He will rich the central singularity, in which the curvature has an infinite value, in a finite amount of time. The milestones of black holes physics 1916 - Einstein publishes the complete version of the General Theory of Relativity. It permits to derive the geometry of the space-time for a given distribution of matter. 1916 - Schwarzschild finds the first exact solution of Einstein s equations. It corresponds to the gravitational field (or, what is the same, the geometry) created by a spherical body of mass M. 1928 - Chandrasekar and independently Landau find that if a body has a mass larger than a critical value M C 1.5M (M = 10 33 g is the mass of the sun) then its pressure is not enough strong to contrast the gravitational force. 1939 - Oppenheimer, Tolman, Volkoff find that if the mass of a star is greater than the critical value M C, then the star collapse indefinitely. The gravitational field on its surface becomes stronger and stronger and the light cones curves toward the inside of the star. When the body overcomes a critical radius (event horizon), the emitted light can never escape. 1963 - Kerr finds another exact solution of Einstein s equation. It describes the gravitational field created by a spinning star, with mass M and angular momentum J. 1974 - Hawking discovers that black holes radiates like a black body of temperature T BH = hκ/4π 2 k B (Hawking effect; κ is the gravity on the horizon, while h and k B are Planck s and Boltzmann s constants respectively). Astronomical observations Recently, by Hubble Space Telescope a lot of possible candidates to be black holes have been found (with 99% of confidence level). It is also believed that the invisible partners of some binary systems could be black holes.
- Exact solutions of Einstein s equations The Schwarzschild solution It describes the gravitational field outside a spherical body of mass M. It is static (there is a choice of coordinates in which the metric g ij, i, j = 0, 1, 2, 3 does not depend on time t). The metric is usually written in the form ( ds 2 = 1 r ) s r ( dt 2 + 1 r s r ) 1 dr 2 + r 2 (dϑ 2 + sin 2 ϑ dϕ 2 ), r s = 2GM c 2 = 3 M M Km (radius of Schwarzschild), r, ϑ, ϕ being polar coordinates, G the Newton s constant, c the speed of light and M the mass of the sun. You can see that such a solution is singular when r = 0 and when r = r s. The first is a true singularity of space-time (for example the square of the Riemann tensor R ijrs R ijrs diverges as r 6 for r 0, while the second is only a singularity of the metric due to a bad choice of coordinates (it disappears in the Kruskal extension). r h = r s (event horizon position), physical parameters: A = 4πr h = 16πG2 M 2 c 4 (area of the horizon), κ = c4 4MG 1.5 1013 M M m sec 2 (surface gravity).
The Kerr solution It describes the gravitational field outside a rotating body with axial symmetry, of mass M and angular momentum J and is believed to be the unique solution for the description of all (uncharged) rotating black holes formed by collapse. ds 2 = ρ2 Σ 2 ( ρ2 dt+ dr2 + Σ2 ρ 2 dϕ 2aMr ) 2 Σ 2 dt sin 2 ϑ, = r 2 2Mr + a 2, Σ 2 = (r 2 + a 2 ) 2 a 2 sin 2 ϑ, ρ 2 = r 2 + a 2 cos 2 ϑ, J = Ma (angular momentum). For this case, the irremovable singularity of space-time is given by ρ 2 = 0 and has the structure of a ring, while r ± = M ± M 2 a 2 are singularities of the metric only. r h = r + = M + M 2 a 2 (event horizon position), physical parameters: A = 4π(rh 2 + a2 ) = 8πMr h (area of the horizon), κ = r+ r 2Mr + (surface gravity), (in natural units) G = c = h = k B = 1. Another important region is the ergosphere where g 00 > 0 (see picture 1 in the first page). It is given by r h R < r < R +, R ± = M ± M 2 a 2 cos 2 ϑ. In such a region, a test particle at a fixed (coordinate) distance r from the body must rotate (with respect to the inertial observer at infinity) with angular velocity and ω(r) = 2aMr Σ 2 Ω = ω(r h ) = 2aMr h Σ 2 = a 2Mr h is identified with the angular velocity of the event horizon. The Kerr-Newman solution It describes the gravitational field outside a charged, rotating body with axial symmetry, of mass M, angular momentum J and change Q and is believed to be the unique (and more general) solution for the description of all black holes formed by collapse. The metric has the same form as above, with the replacement + Q 2.
- Some results on classical black holes 1965-1970 - Penrose, Hawking, Lifshitz, Khalatnikov, Belinsky formulate the singularity theorems: General Relativity predicts the existence of singularities of space-time. In particular, the complete collapse of a body produces a singularity, in which the physical predictability is lost. - Penrose proposes the hypothesis of cosmic censorship: the singularity is always hidden by an event horizon (weak version); the singularity is entirely in the past (big bang) or entirely in the future (black hole) of the observer (strong version). - Israel shows that (non spinning) black holes must have spherical symmetry and must depend only by a parameter (the mass M). A real star, which is not perfectly spherical at the beginning, during the collapse emits gravitational radiation and becomes more and more symmetric. The final state must have spherical symmetry and so it becomes a black hole. The alternative could be that it completely collapses to form a nude singularity, but this is forbidden by cosmic censorship. 1970-1973 - Carter, Hawking, Robinson extend the result of Israel to a spinning black hole. A stationary-spinning black hole is described by the Kerr solution, which depends only on two parameters (the mass M and the angular momentum J). The no hair theorem states that independently of the original star, which collapsed to form a black hole, the final state (the black hole) is described by the Schwarzschild solution (one parameter M) or the Kerr solution (two parameters M and J) or the Kerr-Newman solution (three parameters M, J and Q, the charge) 1971 - Hawking proves that in any process involving black holes (matter falling into a black hole, black holes collisions, etc.) the sum of all horizon areas after the process is equal or greater than the sum of all horizon areas before the process. In General Relativity, the horizon area never decrease (Hawking s theorem). 1973 - Bardeen, Carter and Hawking prove that the surface gravity of a stationary black hole must be constant over the event horizon. It has to be also recalled the Penrose s process (superradiance phenomenon), which permits to extract (classically) energy from a spinning black hole. An incident wave can be reflected from the Kerr black hole with an amplitude larger that the original one. This is possible for the presence of the ergosphere (this is the origin of the name). The energy is subtracted to the rotational energy of the black hole, which loses its angular momentum until the process stops. Quantum-mechanically this phenomenon can be seen as particle creation (in the superradiant modes) by the black hole itself (stimulated emission). From this point of view one also expects spontaneous emission of particles always in the superradiant modes (Starobinsky (1973), Churilov, Unruh).
- Thermodynamic properties of black holes In 1972 Beckenstein proposes to interpret a multiple of the area of the event horizon as an entropy associated with the black hole itself. The fact that the area never decreases is consistent with second law of thermodynamics. The analogy between black holes dynamics and thermodynamics becomes more stringent and credible in 1974, when Hawking shows that quantum mechanical effects cause black holes (spinning or not) to create and emit particles like a black body of a given temperature (Hawking s effect: black holes are not really black). This also permits to exactly compute the constant of proportionality which relates entropy and area. As a consequence of the Hawking s effect, the area of event horizon can also decrease, but the area plus the entropy of the emitted radiation never decrease. By the end of 1970 s, this idea was generally accepted, at least for quasistatical and semiclassical black holes. Hawking s effect (particle creation by black holes-1974) Quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature T BH = hκ 10 6 M 2πk B M K, h (Planck s constant over 2π), k B (Boltzman s constant), κ (superficial gravity), M (mass of the sun). This thermal emission leads to a slow decrease in the mass of the black hole and eventually to its complete disappearance. The time necessary for the complete evaporation of a black hole assuming Hawking s result to be valid at any time (in deriving Hawking s effect a lot of approximations have been made) is about 10 28 M 3 sec. Primordial black holes (formed after the big-bang) of mass about 10 15 g would have evaporate by now by emitting X and γ rays. In the last 10 23 sec a primordial black hole explodes with the emission of 10 35 ergs of energy. Note that for an astronomical black hole the temperature is much smaller than the cosmological microwave background (which is about 3 K) and so such a black hole absorbs more than it emits. Also disregarding the background, its life its of the order of 10 71 years. This astonishing result (according to DeWitt) can be obtained in a semiclassical way (quantised fields in a curved space-time), by studying the asymptotic solutions of field equations. The initial vacuum state 0 > in (the vacuum for an observer at the past infinity) does not appear to be a vacuum state to an observer at the future infinity. Physically, this is due to the fact that the gravitational field can create particles (see picture 2 in the last page). If one computes the number of particles in a given mode, one obtains a Planckian spectrum. Thus, a black hole behaves like a black body at temperature T BH.
The laws which govern black hole physics dynamics laws of black holes 0) the surface gravity is constant over the event horizon; 1) dm = κ 8π da + Ω dj + Φ dq, M, J, Φ, Q are respectively mass, angular momentum, electric potential and charge of the black hole, while κ, A, Ω are gravity, area and angular velocity of the horizon; 2) S matter+radiation + A 4 0. thermodynamic laws 0) the temperature is uniform over a body in thermal equilibrium; 1) du = T ds + work terms, U, T, S are respectively internal energy, temperature and total entropy of the system; 2) in any isolated system S 0. In the previous diagram there is perfect correspondence between left and right if one identifies the black hole entropy S BH with A/4. In standard units S BH = A 4l 2 P, l P = hg c 3 1.6 x 10 33 cm (Planck s length). With such identification the dynamics laws of a black hole become equivalent to the laws of thermodynamics. Moreover, temperature and entropy are related by the thermodynamic formula T BH = M S BH, M being the energy (the mass) of the system. It is a remarkable fact that the black hole entropy is proportional to the area of the event horizon in contrast with ordinary systems, where the entropy is proportional to the volume.
The nature of black hole entropy One of the most exciting open question about black hole physics is the understanding of the deep origin of entropy. Has it a statistical origin? Some proposals: It is the logarithm of the number of ways in which the black hole may be formed, that is a measure of the information lost during the collapse; It is the logarithm of the number of internal states associated with a single black hole, that is a measure of internal black hole states; It is the logarithm of the number of horizon quantum states; It is the entropy of the fields outside the black hole (entanglement entropy); It is the entropy of thermal atmosphere near the horizon (quantum hair). The final answer would probably arrive from quantum gravity. At the moment, our understanding of quantum gravity is so rudimentary that we run into severe difficulties when we try to interpret black hole entropy according to previous proposals. The information puzzle It is strictly related to the understanding of black hole entropy. Probably, the black hole was formed by collapsing matter, which was initially described by a quantum mechanical wave function (a pure state). The phase correlations between all degrees of freedom would evolve in a unitary way, according to the law of quantum mechanics. If the black hole evaporates completely, then the final state of the system is simply thermal radiation, then the phase correlations are even in principle completely lost (mixed state, described by a density matrix). This is clearly a violation of unitarity (information paradox). The final state of a black hole black hole evaporates completely: the final state is not a pure state; quantum mechanics has to be modified; black hole obeys quantum mechanics: it does not evaporate completely; it decays into a stable, infinitely degenerate final state, with mass of the order of the Planck s mass, called remnant; black hole evaporates completely: the final state could be a pure state in an exact treatment of the problem.
Figure 2: The Hawking s effect