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 )

引力塌缩和致密星 天体 质量 半径 表面引力 太阳 ^(-6) 白矮星 < ^(-2) 10^(-4) 中子星 ^(-5) 10^(-1) 黑洞 > 3 2GM/c^2 1

Schwarzschild Black Holes: the first exact solution of Einstein equations (K. Schwarzschild, ) 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。

Singularity Horizon I Minimal black hole ? Black hole horizon 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

Non-extremal BHExtremal BH Euclidean Time periodarbitrary Horizon topology S^1 x S^2 Horizon

Kerr-Newman Black Holes : where 1)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 Horizons:

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)

视界 Kerr-Newman 黑洞 M, J, Q 无毛定理（ No Hair Theorem ）

Cosmological Constant: asymptotically (anti-)de Sitter BHs Cosmological constant Schwarzschild-(A)dS Black Holes:

When M=0, de Sitter (anti-de Sitter) space: Cosmological horizon I Other solutions? Reissner-Nordstrom-(A)dS BHs Kerr-(A)dS BHs Kerr-Newman-(A)dS BHs

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:

S Consider an integration for over a hypersurface S and transfer the volume on the left to an integral over a 2-surface bounding S. Note the Komar Integrals: measured at infinity

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 where Integral mass formula The Differential Formula: first law

The 0 th lawk =const. The 1 st law d M=k dA/8 πG + J d Ω+Φd Q The 2 nd lawd A >0 The 3 rd lawk ->0 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,

The 0 th law T=Const. on the horizon The 1 st law d M= T d S + J d Ω+Φ d Q The 2 nd law d (S BH +S matter )>=0 The 3 rd law T->0 黑洞热力学四定律：

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,

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_1 S_2

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 a boundary with a fixed radius r_0, one has For a flat metric, one has

Complex spacetime and Euclidean action (1)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 also write the amplitude as

Put and 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 box of 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 Hawking is, 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 > , there are no turning points and the maximal value of f is at x=0. (2) When =y_c < y < , there are two turning points: a local minimum for x 4/5. However, a global maximum of f is still at x=0. (3)when y x_c=

Black holes in AdS space: embedded BHs to AdS space i) AdS space  Anti de Sitter space (1)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? (1)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)

ii)AdS BHs and Hawking-Page phase transition Ref: S. Hawking and D. Page, CMP 87, 577 (1983)) E. Witten, Adv. Theor.Math.Phys.2: ,1998 Consider the action has solutions

(1) AdS space the time coordinate has period Euclidean AdS space Thermal AdS space with temperature T=1/beta: Local temperature

Thermal radiation gas in AdS Where The energy density The total energy is finite.

(2) AdS BH solution One horizon is located at f(r)=0. The inverse temperature The maximum value

The Euclidean action (1) (2) The reference background: thermal AdS space R boundary a)Take a finite boundary at R, b) R->infinity AdS BH thermal AdS space

For larger BHs: The density of states: The partition of function: converges; the CE is well-defined. For Schwarzschild BHs: The partition function diverges; the CE is pathological!

If r_+ < l Small BHs If r_+ > l Large BHs The thermal radiation in AdS: C > 0 C < 0 HP transition

iii) Topological BHs in AdS space Consider the following action: The metric ansatz : One has the solution: Without loss of generality, One can take I) k=1; ii) k=0; iii) k=-1.

ii) The case of k=0 horizon: temperature: radius beta C >0 AdS soliton? If has a period,the solution is regular everywhere. The solution has a minimal mass (negative)! (Horowitz and Myers, 1998)

iii) The case of k=-1: (1)Even when M=0, a horizon exists, r_+=l “massless black hole”. (2) Hawking temperature T > 0 “negative mass BHs” (3)when, the BH has two horizons when M>0, the BH has only one horizon.

Euclidean action The background: The heat capacity:

iv) BTZ black hole and its higher dimensional generalization 3-dim.gravity: Einstein tensor ~ Riemann tensor since Weyle tensor=0 (1) The Einstein equations still hold (2) GR is dynamically trivial (3) The vacuum solution is flat (4) The localized source has effect only on the globally geometry (5) A static point source forms a cone (Deser, Jackiw and ‘t Hooft, Ann. Phys(NY), 1983)

When a cosmological constant is present for a static point source at the origin, one has the solution (Deser and Jackiw, Ann. Phys. (NY), 1984) where If deficit angle:

BTZ Black Hole (Banados, Teitelboim and Zanelli, PRL, 1992) Where The BTZ BH has two horizons provided M>0 and J< Ml.

If J=0: (1)AdS space, if –8M=1, in global coordinates (2) AdS space, if M=0, in Poincare coordinates (massless BTZ black hole since r=0 is a null singularity) Therefore there is a gap between the BTZ BH and AdS space!

Some Remarks: (1) BTZ BH is locally equivalent to a 3-dim. Ads Space, but not globally. (2) BTZ BH appears naturally in SUGRA (4-dim. BH with 4 charges and 5-dim. With 3 charges) (3) The statistical degrees of freedom of the BTZ BH i) Carlip’s method (1994, gr-qc/ , ) ii) Strominger’s method (1997, hep-th/ )

The higher dim. generalization of BTZ BHs: Constant curvature case: k = -1 and M=0. This is not the counterparts of the BTZ BH in higher dims. What is the counterparts? As an example, consider the 5-dim. Case There are 15 Killing vectors: 7 rotations and 8 boosts

Consider a boost vector: its norm There are two special cases: (1), it is a null surface

(2) When, it is a hyperloloid The killing vector is spacelike in the Region contained-in-between S_f and S_p; is null at S_f and S_p; is timelike in the causal future of S_f and in the causal past of S_p;

Remarks: (1) Identifying the points along the orbit of \xi, another one-dim. manifold becomes compact and isomorphic to S^1. (2) The region \xi^2 <0 has a pathological chronological structure and therefore must be cut off from the physical spacetime. (3) Identifying the points along a boosting Killing vector to construct a black hole; if along a rotating Killing vector, results in a conical singularity. (4) topological structure: M_4 x S^1 usual case (Sch. BH): R_2 x S^3 (5) This is a constant curvature BH since we start from an AdS.

The Penrose diagram:

In the region, introducing six diemnsionless local coordinates Here with the restriction –1 <y^2 <1.

Introducing local “spherical” coordinates : It does not cover the full outer region of black holes!

The coordinates covering the entire exterior of BH:

Dual Field Theory: Stress-energy tensor Starting from the action: The surface counterterm Resulting in the quasi-local stress-energy tensor of the gravitational field

Take a surface with a fixed $r$ as the boundary, one has The boundary metric, on which dual CFT resides

Using the relation One has (1)Independent of r_+, (2)Vanishing trace, (3) No Casimir energy (For details, see for example, R.G.Cai, PLB 544, 176 (2002); PLB552, 66 (2003))

v) AdS Black Hole in string/M theory Type II SUGRA: Self-dual NS sector RR sector 11-dim. SUGRA:

Dp-brane solution: Take the near-horizon limit (decoupling limit)

D3-brane case: AdS_5 S^5 where Entropy S ~N^2 SU(N) SYM theory IIB theory on AdS_5 x S^5 N=4 SU(N) SYM (J. Maldacena, 1997) AdS/CFT Correspondence

M2-brane: Near horizon geometry: AdS_4 x S^7 M5-brane: Near horizon geometry: AdS_7 x S^4

BTZ BH and 5-dim. BH in string theory Consider the D1+D5+W system: Near-horizon geometry:

Thanks!