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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 ). 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:

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3344063

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)


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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


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Einstein’s Equations (1915):

{Geometry matter (energy-momentum)}


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Thermodynamics of black holes :

Schwarzschild Black Hole: Mass M

horizon

More general:

Kerr-Newmann Black Holes

M, J, Q

No Hair Theorem


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Four Laws of Black Hole mechanics:

k: surface gravity,

J. Bardeen,B. Carter, S. Hawking, CMP,1973


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Four Laws of Black Hole Thermodynamics:

Key Points: T = k/2π S= A/4G

J. Bekenstein, 1973; S. Hawking, 1974, 1975


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Black hole is a window to quantum gravity

Thermodynamics of black hole:

dM = T dS

(S.Hawking, 1974, J. Bekenstein, 1973)


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Holography of Gravity

Entropy in a system with surface area A: S<A/4G

(‘t Hooft)

(L. Susskind)

The world is a hologram?


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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)


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Scaling symmetry:

Lifshitz theory:

Gravity dual?

(S. Kachru, arXiv: 0808.1725)







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Gauss-Bonnet Black Holes

Equations of motion:

metric ansatz:


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The solution:

[D. Boulwareand S. Deser, PRL 55, 2656 (1985)

J. T. Wheeler, NPB 268, 737 (1986)

R.G. Cai, PRD65, 084014 (2002) ]


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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)]



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Now consider the spacetime:

Equations of motion:


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Some examples:

[H. Maeda and N. Dadhich,

arXiv:hep-th/0605031;

arXiv:hep-th/0611188 ]



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Wald formula and euclidean action:

1) when m is odd,

2) When m is even,


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An example:

Euclidean action:

M=0


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(3) Black holes in gravity with conformal anomaly

and logarithmic term in black hole entropy

(M. Duff, hep-th/9308075)

In four dimensions:


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(3) Additional assumption

i) Two dimensions; ii) FRW universe


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The meanings of Q:

Soften the singularity at r=0:



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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…..



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