几个有趣的黑洞解

1. 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）

2. 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

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

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

5. Four Laws of Black Hole mechanics: k: surface gravity, J. Bardeen,B. Carter, S. Hawking, CMP,1973

6. Four Laws of Black Hole Thermodynamics: Key Points: T = k/2π S= A/4G J. Bekenstein, 1973; S. Hawking, 1974, 1975

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

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

9. 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)

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

11. Consider the action:

12. The Lifshitz spacetime

13. Non-extremal black holes:

14. Thermodynamics: =0! =0!

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

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

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

18. 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)]

19. Thermodynamic quantities

20. Now consider the spacetime: Equations of motion:

23. Some examples: [H. Maeda and N. Dadhich, arXiv:hep-th/0605031; arXiv:hep-th/0611188 ]

24. Thermodynamics:

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

27. An example: Euclidean action: M=0

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

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

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

33. Thermodynamics:

34. 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…..

35. 谢谢！