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Fourth Meeting on Constrained Dynamics and Quantum Gravity 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 = dDxp 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) = dDxp g XR U(X)( X)2 + 2V (X) − − r 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 ξp 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 − − r 0 g β = Rµν + 2 µ νφ µν r r φ g conditions β = 0 = βµν follow from S = dDxp ge 2φ R + 4( φ)2 4b2 − − r − 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 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 ): ! 1 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) jhorizon with x := 2 M(M 1) and M = k=2 − Limit of largeq mass (k ) ! 1 S M 1 = SLO+2π ln SLO + (1) j 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 • 11.
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