几个有趣的黑洞解 蔡 荣 根 中国科学院理论物理研究所 (中科大交叉中心，2010.5.20）

(1) A Lifshitz black hole in R^2 Gravity 一、有温度，没有质量和熵的黑洞 (1) A Lifshitz black hole in R^2 Gravity (2) Black holes in Lovelock gravity 二、考虑了共形反常的黑洞解 (3) Black holes in gravity with conformal anomaly and logarithmic term in black hole entropy References: (1) RGC, Y. Liu and Y.W. Sun, JHEP 0910, 080 (2009), arXiv: 0909.2807 (2) RGC, L.M. Cao and N. Ohta, PRD 81, 024018 (2010), arXiv:0911.0245 (3) RGC, L.M. Cao and N. Ohta, JHEP 1004, 082 (2010), arXiv: 0911.4379

Einstein’s Equations (1915): {Geometry matter (energy-momentum)}

Thermodynamics of black holes : Schwarzschild Black Hole: Mass M horizon More general: Kerr-Newmann Black Holes M, J, Q No Hair Theorem

Four Laws of Black Hole mechanics: The 0th law k =const. The 1st law d M=k dA/8πG + Ω d J +Φd Q The 2nd law d A >0 The 3rd law k ->0 k: surface gravity, J. Bardeen,B. Carter, S. Hawking, CMP,1973

Four Laws of Black Hole Thermodynamics: The 0th law T=const. on the horizon The 1st law d M= T d S + Ω dJ+Φ d Q The 2nd law d (SBH +Smatter)>=0 The 3rd law T->0 Key Points: T = k/2π S= A/4G J. Bekenstein, 1973; S. Hawking, 1974, 1975

Black hole is a window to quantum gravity Thermodynamics of black hole： dM = T dS （S.Hawking, 1974, J. Bekenstein, 1973)

Entropy in a system with surface area A: S<A/4G Holography of Gravity Entropy in a system with surface area A: S<A/4G             (‘t Hooft) (L. Susskind) The world is a hologram？

AdS/CFT correspondence (J. Maldacena, 1997) IIB superstring theory on AdS5 x S5 N=4 SYM Theory “Real conceptual change in our thinking about Gravity.” (E. Witten, Science 285 (1999) 512)

A Lifshitz black hole in R^2 gravity Scaling symmetry: Lifshitz theory: Gravity dual? (S. Kachru, arXiv: 0808.1725)

Consider the action:

The Lifshitz spacetime

Non-extremal black holes:

Thermodynamics: =0! =0!

(2) Black holes without mass and entropy in Lovelock gravity

Gauss-Bonnet Black Holes Equations of motion： metric ansatz:

The solution: [D. Boulware and S. Deser, PRL 55, 2656 (1985) J. T. Wheeler, NPB 268, 737 (1986) R.G. Cai, PRD65, 084014 (2002) ]

More general case: Lovelock black holes [J.T. Wheeler, NPB 273, 732 (1986); R. Myers and J. Simon, PRD 38, 2434 (1988); R. G. Cai, PLB 582, 237 (2003)]

Thermodynamic quantities

Now consider the spacetime: Equations of motion:

Some examples: arXiv:hep-th/0611188 ] [H. Maeda and N. Dadhich,

Thermodynamics:

Wald formula and euclidean action: 1) when m is odd, 2) When m is even,

An example: Euclidean action: M=0

(3) Black holes in gravity with conformal anomaly and logarithmic term in black hole entropy (M. Duff, hep-th/9308075) In four dimensions:

Two conditions: Its trace is given by it is covariant conserved (3) Additional assumption i) Two dimensions; ii) FRW universe

The meanings of Q: Soften the singularity at r=0:

Thermodynamics:

Entropy formula of interest: * S. Solodukhin, PRD 57, 2410 (1998) * J.E. Lidsey, arXiv: 0911.3286 * RGC, L.M. Cao and Y.P. Hu, JHEP 0808, 090 (2008) * S~ A + ln A +1/A +1/A^2+…. However, Wald formula…..

谢谢！