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， ）

2. (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:
(2) RGC, L.M. Cao and N. Ohta, PRD 81, (2010), arXiv:
(3) RGC, L.M. Cao and N. Ohta, JHEP 1004, 082 (2010), arXiv:

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: 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

6. 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

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

8. 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？

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: )

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

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

17. The solution: [D. Boulware and S. Deser, PRL 55, 2656 (1985)
J. T. Wheeler, NPB 268, 737 (1986)
R.G. Cai, PRD65, (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: arXiv:hep-th/0611188 ] [H. Maeda and N. Dadhich,

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/ )
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:
* 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！