Fourth Meeting on Constrained Dynamics and Quantum Gravity
Sardinia — September 2005
Mass and entropy of the exact string black hole
speaker: Daniel Grumiller (DG) affiliation: Institute for Theoretical Physics, University of Leipzig, Augustusplatz 10-11, D-04109 Leipzig, Germany
supported by an Erwin-Schr¨odinger fellowship, project J-2330-N08 of the Austrian Science Foundation (FWF)
Based upon: DG, hep-th/0501208, hep-th/0506175 Outline
1. Motivation/Actions
2. The exact string black hole (ESBH)
3. Action for the ESBH
4. Mass and entropy of the ESBH
5. Loose ends
1 1. Motivation
Why 2D?
2 1. Motivation
Why 2D?
dimensionally reduced models (spherical • symmetry)
strings (2D target space) •
integrable models (PSM) •
models for BH physics (information loss) • study important conceptual problems with- out encountering insurmountable technical ones 2D gravity: useful toy model(s) for classi- → cal and quantum gravity most prominent member not just a toy model: Schwarzschild Black Hole (“Hydrogen atom of General Relativity”)
2-a Actions (2D dilaton gravity)
EH: S = dDx√ gR + surface Z − R: Ricci scalar g: determinant of metric gµν Physical degrees of freedom: D(D 3)/2 − JBD/scalar-tensor theories/low energy strings:
S(2) = dDx√ g XR U(X)( X)2 + 2V (X) − − ∇ Z X: “dilaton field” U, V : (arbitrary) potentials “dilaton gravity in D dimensions”
Often exponential representation: 2φ X = e−
Extremely useful: First order action! In 2D:
(1) a a S = XaT +XR+ (XaX U(X) + V (X)) , Z Review: DG, W. Kummer, D. Vassilevich, hep-th/0204253
3 3. The exact string BH
µ NLSM (target space metric: gµν, coordinates: x )
σ 2 ij µ ν S d ξ√ h gµνh ∂ x ∂ x + α φ + tachyon ∝ − i j 0 R Z set B-field zero. neglect tachyon for the time being. conformal invariance: ! 2πT i = βφ + βg hij∂ xµ∂ xν = 0 i R µν i j thus, β-functions must vanish. LO: 16π2 βφ = 4b2 4( φ)2 + 4φ + R α − − ∇ 0 g β = Rµν + 2 µ νφ µν ∇ ∇ φ g conditions β = 0 = βµν follow from
S = dDx√ ge 2φ R + 4( φ)2 4b2 − − ∇ − Z for D = 2: “Witten BH” and CGHS model: 2D dilaton gravity (U = 1/X, V = 2b2X) − − Note: b2 = (26 D)/(6α ) − 0 In 2D: non-perturbative generalization to all orders in α0 – but no action, “just” geometry: R. Dijkgraaf, H. Verlinde, and E. Verlinde, Nucl. Phys. B371 (1992) 269–314.
4 X=0 I+
i0 B _ I
0 + i , I , I− and B de- + note spatial infinity, I future light-like
X=0 infinity, past light- i0 like infinity and the bifurcation point, respectively; the _ I Killing horizon is denoted by the dashed line
5 Why an action?
Needed for mass definition (“Gibbons-Hawking • term”) – clarify which, if any, of previous mass definitions is correct
Needed for entropy – get insight into ther- • modynamics of a non-perturbative BH so- lution of string theory
Not needed for surface gravity and Hawk- • ing temperature (still, various papers get different results here)
Supersymmetrization •
Quantization •
Get a new theory in this way which is • interesting on its own and which gen- eralizes the CGHS model – may couple geometric action to some matter action, study critical collapse, etc.
6 3. Action for the ESBH nogo result DG, D. Vassilevich, hep-th/0210060
R
7 3. Action for the ESBH nogo result DG, D. Vassilevich, hep-th/0210060 circumvent it by allowing matter – but: don’t want propagating physical degrees of free- dom!
R
7-a 3. Action for the ESBH nogo result DG, D. Vassilevich, hep-th/0210060 circumvent it by allowing matter – but: don’t want propagating physical degrees of free- dom! suggestive: consider (abelian) gauge field
R
7-b 3. Action for the ESBH nogo result DG, D. Vassilevich, hep-th/0210060 circumvent it by allowing matter – but: don’t want propagating physical degrees of free- dom! suggestive: consider (abelian) gauge field result: it works! DG, hep-th/0501208
a SESBH = XaT + ΦR + BF ZM2 a + (XaX U(Φ) + V (Φ)) , with Φ = γ arcsinh γ and γ = X/B +: ESBH, : ESNS − The potentials read 1 V = 2b2γ , U = , − −γN (γ) with an irrelevant scale parameter b R+ and ∈ 2 1 1 N (γ) = 1 + 1 + . γ γ s γ2! Note that N+N = 1. − 7-c 2
1.75
1.5
1.25
1
0.75
0.5
0.25
1 2 3 4 5 Plot of U as a function of γ Red: ESBH, Blue: ESNS, Black: Witten BH
Asymptotic (“weak coupling”) limit (γ ): → ∞ Witten BH: U = 1/Φ, V = 2b2Φ − − valid for both branches (ESBH, ESNS)
Strong coupling limit (γ 0): → ESBH branch: JT model (U = 0, V = b2Φ) − ESNS branch: 5D Schwarzschild! (U = 2/(3Φ), V = 2b2(6Φ)1/3) − −
8 4. Mass and entropy of the ESBH
Constants of motion: U(1)-charge: value of DVV dilaton at origin mass: determined by level k! (MADM = bk) conservation law in presence of matter: d (g) + W (m) = 0 geometryC matter Thus, matter |has{z to} “defo| {zrm”} the level k/the central charge – consistent!
Similar interpretation in V.A. Kazakov and A.A. Tseytlin, hep-th/0104138:
b 2 TH = 1 2πs − k
For mass knowledge of action pivotal! The same holds for entropy! S = 2πΦ = 2π (x + arcsinh x) |horizon with x := 2 M(M 1) and M = k/2 − Limit of largeq mass (k ) → ∞ S M 1 = SLO+2π ln SLO + (1) | O with SLO = 4πM 9 6
5
4
3
2
1
1.2 1.4 1.6 1.8 2 2.2 2.4 Entropy as function of the Mass: ESBH, Witten BH, Fluctuations
0.012
0.2 0.4 0.6 0.8 1 0.01
-0.2 0.008
-0.4 0.006
-0.6 0.004
-0.8 0.002
-1 0.02 0.04 0.06 0.08 0.1 Entropy difference to (C T 2) 1 as func- Witten BH as func- V − 1 1 tion of M − tion of M −
10 5. Loose ends
Summary: The ESBH action is constructed! DG, hep-th/0501208
To-do list:
Further thermodynamical considerations •
Supersymmetrization •
S-matrix calculations •
Application to 2D type 0A/0B strings •
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