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Nonextremal Black Holes, Subtracted Geometry and Holography

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Nonextremal Black Holes, Subtracted Geometry and Holography Nonextremal Black Holes, Subtracted Geometry and Holography Mirjam Cvetič Einstein’s theory of gravity predicts Black Holes Due to it’s high mass density the space-time curved so much that objects traveling toward it reach a point of no return à Horizon (& eventually reaches space-time singularity) Black holes `behave’ as thermodynamic objects w/ Bekenstein-Hawking entropy: S=¼ Ahorizon Ahorizon= area of the black hole horizon (w/ ħ=c=GN=1) Horizon-point of no return Space-time singularity Key Issue in Black Hole Physics: How to relate Bekenstein-Hawking - thermodynamic entropy: Sthermo=¼ Ahor (Ahor= area of the black hole horizon; c=ħ=1;GN=1) to Statistical entropy: Sstat = log Ni ? Where do black hole microscopic degrees Ni come from? Horizon Space-time singularity Black Holes in String Theory The role of D-branes D(irichlet)-branes Polchinski’96 boundaries of open strings with charges at their ends closed strings I. Implications for particle physics (charged excitations)-no time II. Implications for Black Holes Dual D-brane interpretation: extended massive gravitational objects D-branes in four-dimensions: part of their world-volume on compactified space & part in internal compactified space Cartoon of (toroidal) compactification; D-branes as gravitational objects Thermodynamic BH Entropy & wrap cycles in internal space: Statistical field theory interpretation intersecting D-branes in compact dimensions & charged black holes in four dim. space-time (w/ each D-brane sourcing charge Q ) i D-branes as a boundary of strings: microscopic degrees Ni are string excitations on intersecting D-branes w/ S = log Ni Strominger & Vafa ’96 the same! Prototype: four-charge black hole w/ S= π√Q1Q2P3P4 M.C. & Youm 9507090 Microscopic origin of entropy for extremal (BPS), multi-charged black holes with M = Σi |Qi| + Σi |Pi | (schematic) M-mass, Qi - el., Pi- magn. charges Systematic study of microscopic degrees quantified via: AdS/CFT (Gravity/Field Theory) correspondence [A string theory on a specific Curved Space-Time (in D-dimensions) related to specific Field Theory (in (D-1)- dimensions) on its boundary à Holographic Approach] Maldacena’97 For multicharged (near)-BPS black holes: AdS3/CFT2 correspondence The rest of the talk: Highlight recent progress on studies of Internal Structure of Non-Extremal Black Holes Outline: I. General asymptotically flat black holes in string theory [in four (&five) dimensions – prototype STU black holes] thermodynamics, suggestive of conformal symmetry II. Subtracted Geometry: non-extremal black holes in asymptotically conical box manifest conformal symmetry III. Variational Principle and Subtracted Geometry conserved charges and thermodynamics IV. Holography via 2D Einstein-Maxwell-Dilaton gravity full holographic dictionary V. Outlook Black holes with U(1) charges in 4-dimensions a) Asymptotically Minkowski space-time (zero cosmological constant) - in maximally supersymmetric theories b) General rotating solutions: mass, angular momentum and four-charges (special cases: importance of extremal solutions) c) Thermodynamic properties à indication of microscopic structure associated with dual 2D CFT Background: Initial work on subtracted geometry M.C., Finn Larsen 1106.3341, 1112.4846, 1406.4536 M.C., Gary Gibbons 1201.0601 M.C., Monica Guica, Zain Saleem 1301.7032 … Recent: variational principle, conserved charges and thermodynamics of subtracted geometry Ok Song An, M.C., Ioannis Papadimitriou, 1602.0150 Most recent: subtracted geometry and AdS2 holography M.C., Ioannis Papadimtiriou,1608.07018 I. 4D general non-extremal black holes in string theory, asymptotically flat (zero cosmological constant Λ=0) M - mass, Qi, Pi - multi-charges, J - angular momentum w/ M > Σi |Qi| + Σi| Pi | Prototype solutions of a sector of maximally supersymmetric D=4 Supergravity [sector of toroidally compactified effective string theory] à so-called STU model 2TheSTUmodelanddualityframes 2TheSTUmodelanddualityframes In this section we review the bosonic sector of the 2-charge truncation of the STU model that is In this section we review the bosonic sector of the 2-charge truncation of the STU model that is relevant for describing the subtracted geometries. We will do so in the duality frame discussed in relevant for describing the subtracted geometries. We will do so in the2TheSTUmodelanddualityframes duality frame discussed in [14], where both charges are electric, as well as in the one used in [17],[14], where where there both is one charges electric are and electric, as well as in the one used in [17], where there is one electric and one magnetic charge. We will refer to these frames as ‘electric’ andone ‘magnetic’In magnetic this section respectively. charge. we review We As will it the refer bosonic to these sector frames of the as 2-charge ‘electric’ truncation and ‘magnetic’ of the STU respectively. model that As is it will become clear from the subsequent analysis, in order to comparewill the becomerelevant thermodynamics clearfor describing from in the the twothe subsequent subtracted analysis, geometries. in order We will to comparedo so in the the thermodynamics duality frame discussed in the in two frames, it is necessary to keep track of boundary terms introducedframes, by[14], the duality it where is necessary transformations. both charges to keep are electric, track of as boundary well as in terms the one intro usedduced in [17], by where the duality there is one transformations. electric and one magnetic charge. We will refer to these frames as ‘electric’ and ‘magnetic’ respectively. As it 2.1 Magnetic frame 2.1will Magnetic become clear frame from the subsequent analysis, in order to compare the thermodynamics in the two frames, it is necessary to keep track of boundary terms introduced by the duality transformations. The bosonic Lagrangian of the STU model in the duality frame uThesed in bosonic [17] is given Lagrangian by of the STU model in the duality frame used in [17] is given by 1 1 2 2ηa a a 2.1 Magnetic frame 1 1 2κ4 4 = R ⋆ 1 ⋆ dηa dηa e ⋆ dχ dχ 2 κ 2STU= R Model⋆ 1 Lagrangian⋆ dη dη e 2ηa ⋆ dχa dχa L − 2 ∧ − 2 ∧ 4L4 − 2 a ∧ a − 2 ∧ 1 η0 0 0 1 2ηa η0 a a 0 Thea bosonica 0 [A Lagrangian sector of toroidally of the STU compactified model in the effective duality string frame theory] used in [17] is given by e− ⋆ F F e − ⋆ (F + χ F ) (F + χ F ) 1 η0 0 0 1 2ηa η0 a a 0 a a 0 − 2 ∧ − 2 ∧ e− ⋆ F F e − ⋆ (F + χ F ) (F + χ F ) 2 − 2 1 ∧ − 2 1 2ηa a a ∧ 1 a b c 1 a b 0 c 1 a b c 0 2κ0 4 4 = R ⋆ 1 ⋆ dηa dηa e ⋆ dχ dχ + Cabcχ F F + Cabcχ χ F F + Cabcχ χ χ F F , L 1(2.1)− a2 b ∧c 1− 2 a b 0 ∧ c 1 a b c 0 0 2 ∧ 2 ∧ 6 ∧ + Cabcχ F F + Cabcχ χ F F + Cabcχ χ χ F F , (2.1) 1 η0 0 0 1 2ηa η0 a a 0 a a 0 3 2 e− ⋆ F∧ F 2 e − ⋆ (F∧+ χ F 6) (F + χ F )∧ where ηa (a =1, 2, 3) are dilaton fields and η0 = a=1 ηa.ThesymbolCabc is pairwise symmetric− 2 ∧ − 2 3 ∧ with C =1andzerootherwise.TheKaluza-Kleinansatzforobtaininwhere ηa gthisactionfromthe(a =1, 2, 3) are dilaton1 fields and η10 = a=1 ηa.Thesymbol1 Cabc is pairwise symmetric 123 ! + C χaF b F c + C χaχbF 0 F c + C χaχbχcF 0 F 0, (2.1) 6-dimensional action (1.1) is given explicitly in [17]. Thisframepossessesanexplicittrialitysym-with C123 =1andzerootherwise.TheKaluza-Kleinansatzforobtaininabc abc abc gthisactionfromthe 2 ∧ 2 ! ∧ 6 ∧ metry, exchanging the three gauge fields Aa,thethreedilatonsηa6-dimensionaland the three axions actionχa (1.1).Inthis is given explicitly in [17]. Thisframepossessesanexplicittrialitysym-(a=1,2,3; C -anti-symmetric tensor) where η (a =1, 2, 3) are dilaton fields and η = 3 η .ThesymbolabcC is pairwise symmetric frame, the subtracted geometries source all three gauge fields Aa metry,magnetically, exchanginga whilew/ A the0 &is threeelectri- gauge fields Aa,thethreedilatons0 a=1 a ηa and theabc three axions χa.Inthis with C =1andzerootherwise.TheKaluza-Kleinansatzforobtainingthisactionfromthe cally sourced. Moreover, holographic renormalization turns out toframe, be much the more123 subtracted straightforward geometries source all three gauge! fields Aa magnetically, while A0 is electri- 6-dimensional action (1.1) is given explicitly in [17]. Thisframepossessesanexplicittrialitysym- in this frame compared with the electric frame. cally sourced. Moreover,Black holes: holographic explicit solutions renormalization of equations turns of out motion to be for much the moreabove straightforward metry, exchanging Lagrangian the three w/ metric, gauge fields fourA gaugea,thethreedilatons potentials andηa andthree the axio-dilatons three axions χ a.Inthis In order to describe the subtracted geometries it suffices toin consider this frame a truncation compared of with (2.1), the electric frame. a 0 frame,1 the2 subtracted3 geometries source all three gauge fields A magnetically, while A is electri- corresponding to setting η1 = η2 = η3 η, χ1 = χ2 = χ3 χ,andA = A = A A.The ≡ ≡ Incally order sourced. to Prototype, describe Moreover,≡ the four-charge holographic subtracted renormalization geometriesrotating black it tur su holeffinsces out, tooriginally to c beonsider much obtained a more truncation straightforward via of (2.1), resulting action can be written in the σ-model form 1 2 3 correspondingin this frame tosolution compared setting generatingη with1 = η the2 = electrictechniquesη3 frame.η, χ1 = χ 2 = χ 3 χ ,and M.C.,A =YoumA =9603147A A.The 1 1 ≡ Chong,≡ M.C., Lü, Pope 0411045≡ S = d4x√ g R[g] ∂ ϕI ∂µϕJ F Λ F Σµν resultingϵµνρσIn actionF orderΛ F Σ canto+ describeS be, written(2.2) the in subtracted the σ-model geometries form it suffices to consider a truncation of (2.1), 4 2 IJ µ ΛΣ µν ΛΣ µν ρσ Four- GHSO(1,1) transfs.
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