Some Topics on Black Hole Physics Rong-Gen Cai（蔡荣根） Institute of Theoretical Physics Chinese Academy of Sciences
1. Black Holes in GR Einstein equations (1915) (This equations can be derived from the following action)
Schwarzschild Black Holes: the first exact solution of Einstein equations 真空的爱因斯坦场方程的精确解， 描写一个天体的引力场。当这一 天体的半径<2GM,它就是一个黑洞。 (K. Schwarzschild,1873-1916)
Singularity Horizon Black hole horizon I Minimal black hole ? Singularities ?
The geometry near horizon: where Making a Wick rotation on w, to remove the conical singularity at ,the Euclidean time w has a period In the time coordinate t, the period is
Reissner-Nordstrom Black Hole: gravity coupled to a U(1) gauge field where horizons:
The geometry near outer horizon: where after a Wick rotation on w, w has the period In the time coordinate t, the Euclidean time it has the period
Extremal black Holes: where In this coordinate, double horizon is at r=0. The geometry near the horizon has the form AdS_2 X S^2
Kerr-Newman Black Holes: where Horizons: • When a=0, Reissner-Nordstrom black hole solution • 2) When Q=0. Kerr black hole solution • 3) When a=Q=0, Schwarzschild black hole solution
No-hair theorem of black holes (uniqueness theorem): The most general, asymptotically flat stationary solution of Einstein-Maxwell equations is the Kerr-Newman solution! Ref: M. Heusler Black Hole Uniqueness Theorem Cambridge University Press, 1996 (W. Israel)
Kerr-Newman 黑洞 M, J, Q 无毛定理（No Hair Theorem ） 视界
Cosmological Constant: asymptotically (anti-)de Sitter BHs Cosmological constant Schwarzschild-(A)dS Black Holes:
When M=0, de Sitter (anti-de Sitter) space: Other solutions? Reissner-Nordstrom-(A)dS BHs Kerr-(A)dS BHs Kerr-Newman-(A)dS BHs I Cosmological horizon
2. Four Laws of Black Hole Mechanics (Ref: J. Bardeen, B. Carter and S. Hawking, CMP 31, 161 (1973)) Kerr Solution: where There are two Killing vectors:
These two Killing vectors obey equations: with conventions:
Consider an integration for S over a hypersurface Sand transfer the volume on the left to an integral over a 2-surface bounding S. measured at infinity Note the Komar Integrals:
Then we have where Similarly we have
For a stationary black hole, is not normal to the black hole horizon, instead the Killing vector does, where is the angular velocity. where Angular momentum of the black hole
Further, one can express where is the other null vector orthogonal to , normalized so that and dA is the surface area element of . where is constant over the horizon
For Kerr Black Holes: Smarr Formula Integral mass formula where The Differential Formula: first law
经典黑洞的性质：黑洞力学四定律 （k, 表面引力，类似于引力加速度） (J.M. Bardeen,B. Carter, S. Hawking, CMP,1973)
Wheeler 问： 假如某个热力学体系掉入黑洞，将导致什么？ 热力学第二定律将违背吗？ J. Bekenstein（1973）：黑洞有热力学熵！ S ~ A, 视界面积
Hawking不以为然, 大力反对Bekenstein 的观点！ 可是考虑了黑洞周围的量子力学后， Hawking （1974，1975）发现黑洞不黑，有热辐射！ For Schwarzschild black hole,
Information Loss Paradox (信息丢失佯谬） S. Hawking, PRD, 1976; a bet established in 1997
3. Black Hole Entropy: Area Formula Refs:a) The path-Integral Approach to Quantum Gravity by S. Hawking, In General Relativity: An Einstein Centenary Survey, eds. S. Hawking and W. Israel, (Cambridge University Press, 1979). b) Euclidean Quantum Gravity by S.W. Hawking, in Recent Developments in Gravitation Cargese Lectures, eds. M. Levy and S. Deser, (Plenum, 1978) c) Action Integrals and Partition Functions in Quantum Gravity, by G.W.Gibbons and S. Hawking, PRD 15, 2752-2756
The path-integral approach: The starting point is Feynman’s idea that one can express the amplitude to go from a state with a metric $g_1$ and matter $\phi_1$ on a surface $S_1$ to a state with a metric $g_2$ and matter $\phi_2$ on a surface $S_2$, as a sum over all field configurations $g$ and $\phi$ which takes the given values on the surface $S_1$ and $S_2$. S_2 S_1
The action in GR: Equations of motion where is the energy-momentum tensor of matter fields.
In order to be well-defined for the variation, a Gibbons and Hawking surface term has to be added to the action where C is a term which depends only on the boundary metric h and not on the values of g at the interior points. and
Consider a metric which is asymptotically flat in the three spatial directions but not in time If the metric satisfies the vacuum Einstein equations near infinity, then M_s=M_t. Consider aboundary with a fixed radius r_0, one has For a flat metric, one has
Complex spacetime and Euclidean action • For ordinary quantum field theory, make a Wick • rotation, Where is called Euclidean action, which is greater than or equal to zero for fields which are real on the Euclidean space defined by the real coordinates. Thus the integral over all such configurations of the field will be exponentially damped and should therefore converge.
(2) Quantum field theory at finite temperature To construct a canonical ensemble for a field which expresses the amplitude to propagate from a configuration $\phi_1$ on a surface at time $t_1$ to a configuration $\phi_2$ on a surface at time $t_2$ . Using the Schrodinger picture,one can alsowrite the amplitude as
Putand and sum over a complete orthonnormal basis of configurations . One has the partition function for the field at a temperature . One can also express Z as a Euclidean path integral where the integral is taken over all fields that are real on the Euclidean section and are periodic in the imaginary time coordinate with period.
(3) Apply to quantum gravity Introducing an imaginary time coordinate , the Euclidean action of gravitational field has the form (The problem is that the gravitational part of the action is not positive-definite.)
Canonical ensemble for gravitational fields: One can consider a canonical ensemble for the gravitational fields contained in a spherical boxof radius r_0 at a temperature T, by performing a path integral over all metrics which would fit inside a boundary consisting of a timelike tube of radius r_0 which was periodically identified in the imaginary time direction with period 1/T.
(4) The stationary-phase approximation * Neglecting the questions of convergence. *Expecting the dominant contribution to the path integral will come from metrics and fields which are near a metric g_0 and $\phi_0$, which are an extremum of the action, i.e. a solution of the classical field equations. *Can expand the action in a Taylor series about the background field where
Ignoring the higher order terms results in the stationary-phase approx. Zero-loop One-loop
Some Examples (1) Schwarzschild BH Introducing the Euclidean time Introducing the coordinate
The Euclidean section of the Schwarzschild solution is periodic in the Euclidean time coordinate tau, the boundary at radius r_0 has the topology S^1xS^2, and the metric will be the stationary phase point in the path integral for the partition function of a canonical ensemble at temperature T=1/beta. The action is (2) Reissner-Nordstrom BHs where
(3) Kerr Black Holes The Euclidean section of the Kerr metric provided that the mass M is real and the angular momentum J is imaginary. In this case the metric will be periodic in the frame that co-rotates with the horizon, i .e. the point is identified with
Black Hole Entropy Schwarzschild BHs According to the formulas
Kerr-Newman BHs According to where
4. Black Holes in anti-de Sitter Spaces • * For a Schwarzschild BH, its Hawkingis , • Therefore its heat capacity • It is thermodynamic unstable. • * The lifetime of a Schwarzschild BH, • * If
Put a BH in a cavity where set
(1) When y > 1.4266, there are no turning points and the maximal value of f is at x=0. (2) When 1.0144=y_c < y < 1.4266, there are two turning points: a local minimum for x < 4/5; and a local maximum for x >4/5. However, a global maximum of f is still at x=0. (3)when y< y_c, there is a global maximal with x > x_c=0.97702.
Black holes in AdS space: embedded BHs to AdS space i) AdS space Anti de Sitter space • Negative energy density • (2) Closed timelike curves • (3) Not globally hyperbolic • (4) Ground states of some gauged supergravities • (5) The positive mass theorem holds, the AdS is stable Why AdS?
AdS Space: SO(2,D-1) with topology
Why AdS BHs? • AdS BHs are quite different from their conterparts in • asymptotically flat spacetime or de Sitter spacetime. • (2) Horizon topology may not be S^2 (S. Hawking, 1972) • (3) AdS/CFT correspondence (J. Maldacena, 1997) • AdS BHs/thermal CFTs (E. Witten, 1998)