( ) = 1 r 2. ( r) r=a. Ka d. M in/out. 8(d 2) d(d 1) πm d. + 4K d. = 2K d 1 ad + V in/out

Transcription

1 ds = g(r)dt + f ( r) = 1 ad Regularity r d r d g r Δψ = 4π p = f (r) dψ + ( ) = 1 r r f l k cos(θ i )dφ i i=1 8π = π ( r) r=a a / l + Λ = +(D )(D ) l dr g(r) f (r) + r d Geometry depends on relative size of No black hole horizon r a Can now have a cosmological horizon surrounding soliton M in/out = + V in/out = K d 1 ad + Ka d 4πl + m dl d + 8(d ) d(d 1) πm d l + 4K d ld Mbarek/Mann PLB 765 (017) 35 R k dσ (i) i=1 compact space d = k += D 1 dσ (i) = dθ i + sin (θ i )dφ i Clarkson/Mann PRL 96 (006) Below the dashed line the Reverse Isoperimetric Inequality is violated d

2 De Sitter Black Holes in a Cavity Basic idea: put a cavity between the black hole and cosmological horizon Fix temperature of cavity to control (charged) BH temperature Study effects of cosmic tension Simovic/Mann CQG (018) to appear I = 1 16π d 4 x g(r Λ + F )+ 1 M 8π d 3 x k(k K M 0 ) I r (β,r +,R c,q,λ) r + = 0 β(r +,R c,q,λ) T(r +,R c,q,λ) E = I r β, S = β I r β I r 1) Check 1 st law and Smarr Carlip/Vaidya CQG 0 (018) 387 E = (TS λa c + PV )+ ΦQ de = TdS λda c VdP +φdq ) Analyze Free energy for phase transitions Cavity area

3 De Sitter Black Holes in a Cavity I = 1 16π d 4 x g(r Λ + F )+ 1 M 8π d 3 x k(k K M 0 ) X(Λ) 1 r + (1 q Λ r + R c 3 (R c R c + R c r + + r + )) Y(Λ) 1 ΛR c 3 I = βr c E = I β = R c Y(Λ) X(Λ) S = β I β I = πr + πr + Y(Λ) X(Λ) T(r +,R c,p)= 1 Λr q r + + 4πr + X E = (TS λa c + PV )+ ΦQ de = TdS λda c VdP +φdq

4 Phase transitions for Charged ds Black Holes in a Cavity λ = V = 4π 3 P Y ( 4ΛR 3 c 6R c )( X Y)+ r + (Λr + 3) Y 3q r + 48πR c X Y Φ = (R r )q c + r + R c X R c 3 ( Y X) r + 3 Y X Y T Swallowtube F F(r +,R c,p)= Λr 3 + X R c X + R c B 3 r + 4 X P T

5 De Sitter Black Holes in Equilibrium Generic ds Black holes have temperatures A variety of approaches have been employed Treat horizons independently (justification?) Average temperature Cavity (artificial structure) One possibility: equilibrate the temperatures Can do this with conformal scalar hair Provides a new control parameter Mbarek/Mann JHEP (018) to appear

6 Hairy ds Black Holes α 1 β 1 α k β k δ (k) = δ µ1 ν 1 µ k ν k I = 1 k 16πG d max d x g L (k) 4πGF µν F µν k=0 Oliva/Ray CQG 9 (01) k max k=0 α k φ = N r σ f r k = k max L (k) = 1 k δ (k) a k k R µr ν r r α r β r α r β r +b k φ d 4k S µr ν r S γδ = φ R γδ µν µν δ [γ δ [µ δ ] ν ] ρ φ ρ φ 4φδ [γ [µ ν ] δ ] φ +8δ [µ ds = fdt + f 1 dr + r dσ σ ( d ) 16πGM (σ (d )ω ( ) d ) r + H d 1 r 8πG d (d )(d 3) (d 3) H = b k=0 (d (k +1)) k σ k N d k k max Q r d 4 k b (d 1)(d(d 1)+ 4k ) k k=0 (d k 1) k max kb k k=1 r [γ ν ] φ δ ] φ σ k N k = 0 (d 1) (d k 1) σ k 1 N k = 0 Solution to the polynomial is a solution to the field equations Check 1 st law and Smarr at both horizons δm ± = T ± δ S ± +V ± δp ± + Φ ± δq +κδh (d 3)M ± = (d )T ± S ± V ± P ± +(d 3)Φ ± Q +(d )κ H

7 1)Solvef (r )= 0andf (r )= 0forHandM c bh )Equilbratethetemperatures:T + = f (r ) c 4π = f (r ) bh 4π 3)Requirepostivemassandentropy:S h = Σ σ k (d )kσ k 1 d k max α d k r h 4G k=1 d k 4)Computethefreeenergy(Grandcanonicalensemble): cosmog + = M TS + φ + Q blackholeg = M TS φ Q = T bh = T r c = r ( c Q,P,r ) bh dh > 0 σ(d 4) totalg = M T ( S + S tot + )+(φ + φ )Q Reverse HP phase transition T G T large small r bh

8 Phase Transitions in Charged ds Black Holes Features notably differ from AdS case in situations where we have control What is the correct system to consider? BH in a cavity? BH in a cosmological heat bath? ds- BH as a complete thermodynamic system Need other examples Need a method to treat heat flow

9 Accelerating Black Holes C- metric: an exact solution to the Einstein Equations describing an accelerating black hole Conical deficit along (at least) one polar axis Can be replaced with cosmic string or magnetic flux tube to accelerate the black hole Pair Creation of black holes Constant electromagnetic field Splitting of cosmic string Cosmic acceleration of spacetime Kinnersley/Walker PRD (1970) 1359 Plebanski/Demianski Ann. Phys. 98 (1976) 98 Dowker/Gauntlett/ Kastor/Traschen PRD49 (1994) 909 Gregory/Hindmarsh PRD5 (1995) 5598 Mann/Ross PRD5 (1995) 54

10 Basic features Thermodynamics of ABHs both an event horizon and an acceleration horizon Thermodynamic equilibrium can t be maintained Asymptotically AdS Acceleration horizon removed for small acceleration Cosmic string suspends BH from the centre of AdS Conflicting Results for Thermodynamics Differing identifications of mass Role of conical deficits in first law not clear Free energy/action not compatible Hubeny/Marolf/Rangamani CQG 7 (010) Appels/Gregory/ Kubiznak PRL 117 (016) ; JHEP 1705 (017)116 Astorino PRD95 (017)

11 Metric for ABH in AdS ds = 1 Ω [ fdt + dr f + r ( dθ Σ + Σsin θ dφ K )] String Tension Ω = 1+ Ar cosθ Σ = 1+ macosθ f (r) = (1 A r )(1 m r )+ r µ ± = δ ± 8π = Σ(θ ) ± K = 1 4 l 1 1± ma K Hong/Teo CQG 0 (003) 369 φ θ Parameterizes deficit angle

12 Metric for ABH in AdS ds = 1 [ fdt + dr Ω f f (r) = (1 A r )(1 m r )+ r + r ( dθ Σ + Σsin θ dφ l K )] Ω = 1+ Ar cosθ Σ = 1+ macosθ Boundary is beyond r = Distortion of Poincare disk Rindler- AdS m = 0 Slowly Accelerating Black Hole m 0 Al < 3 6 8

13 No Acceleration Horizons f ( 1/ Acosθ) > 1 Metric signature not preserved Slowly accelerating black holes Acceleration Horizons present ds = 1 [ fdt + dr Ω f f (r) = (1 A r )(1 m r )+ r + r ( dθ Σ + Σsin θ dφ l K )] Ω = 1+ Ar cosθ Σ = 1+ macosθ

14 Asymptotics Ω = 1+ Ar cosθ Σ = 1+ macosθ ds = 1 Ω [ fdt + dr f + r ( dθ Σ + Σsin θ dφ K )] ds AdS 1+ R l = 1+ (1 A l )r / l Rsinϑ = rsinθ (1 A l )Ω Ω m = 0 = (1+ R l )α dt + dr 1+ R l + R (dϑ + sin ϑ dφ K ) α = 1 A l Correct time coordinate is τ = αt

15 Thermodynamic Quantities Temperature Entropy Mass? T = f (r ) + 4πα = 1+ 3 r + l A r + S = A 4 = πr + K(1 A r + ) + r + l A r + Thermodynamic Via the Smarr Relation Conformal Via Electric part of Weyl Tensor Holographic Via AdS/CFT counterterms 4παr + (1 A r + ) f (r + ) = (1 A r + )(1 m )+ r + r + l = 0

16 Interpret in Context of Black Hole Chemistry P = 3 8πl Thermodynamic Mass of ABHs Consistency δ M = Tδ S + Vδ P λ + δµ + λ δµ M = TS PV λ ± = 1 r + α 1 A r + V = 4 3 Al m 1± r + 3 π r + Kα (1 A r + ) + ma l 4 M = αm K = 1 A l m K

17 Conformal Mass of ABHs ds = 1 Ω [ fdt + dr f + r ( dθ Σ + Σsin θ dφ Ashtekar/Das CQG 17 (000) L17 Das/Mann JHEP 08 (000) 033 K )]= g µν dx µ dx ν g µν = Ω g µν N µ = µ Ω ξ = τ Q(ξ) = l 8π lim Ω 0 l " N α N β C ν [ αµβ g]ξ ν ds µ Ω C-metric: ds µ = δ µ τ l (d cosθ)dφ αk Ω = lωr 1 M = Q( τ ) = α m K = 1 A l m K

18 Holographic Mass of ABHs I[g] = 1 d 4 x g R π M l 1 8π M d π x h l + l R h M ( ) ds = l ρ dρ + h ab dx a dx b = l ρ dρ + ρ d 3 x hk Balasubramanian/Kraus CMP 08 (1999) 413 Das/Mann JHEP 08 (000) 033 γ (0) ab + 1 ρ γ ab () +... dxa dx b δ S = 1 T ab δ h ab h M Fefferman/Graham NHS (1985) 95 8πT ab = lg ab ( h) l h K + h K ab ab ab T ab ρ = lim ρ l T ab M = T ab U a U b γ (0)

19 For the C- metric Convert to FG coords ds (0) 1 Ar = x cosθ = x + F n G n ( x)ρ n ( x)ρ n ( ) X = (1 x ) 1+ max ( ) 3/ F 1 (x) = 1 A l X Aω(x)α = γ ab (0) dx a dx b = ω dτ l + Xω α dφ K (1 A l X) + ω α dx X(1 A l X) T ab = ( P + ρ E )U a U b + Pl γ ab + π ab P = ρ E Eqn of State: thermal gas of massless particles ρ E = m(1 A l X) 3/ 8πl α 3 ω 3 ( 3A l X) M = ρ E l 3 γ (0) dxdφ = αm K = π x x = 3mA3 XF 1 16πα ω = π φ φ 1 A l m K Shear tensor: anisotropic dual fluid

20 Reverse Isoperimetric Inequality for ABHs R = 3V ω 1 3 ω A 1 R 3 1 x = Al y = r + l R = ( + x (1+1/ y ) x 4 (1+ y / ) + x 6 y ) 4((1 x y )(1 x )) 1/6 1 x = Al (0,1) y = r + / l > 0

21 Action I = β αk m ma l r + 3 l (1 A r + ) β = 1 T F = I / β = M TS µ ± = δ ± 8π = ± ma K µ = 0 : Hawking-Page transition µ > 0 : No clear interpretation

22 ds = 1 Ω [ fdt + dr l ρ dρ + ρ Conformal Boundary f + r ( dθ Σ + Σsin θ dφ K )] γ (0) ab + 1 ρ γ () ab +... dxa dx b γ (0) ab dx a dx b = ω dτ + Xω α dφ l K (1 A l X) + ω α dx X(1 A l X) A l X > 1 ω = (1 A l X)α No equilibrium T Black droplet ω = (1 A l X) α 1 x ( ) AdS S 1 asymptotic region Hubeny/Marolf/Rangamani CQG 7 (010) ω = (1 A l X) α 1 x ( ) Two AdS S 1 regions connected via wormhole

23 ds = 1 Ω f (r) Ξ θ ( ) F = db Ω = 1+ Ar cosθ Charge and Rotation dt α asin θ dφ K + Ξ ( θ ) + Σ(θ)sin θ adt Ξ( θ )r α (r + a ) dφ K B = f (r) = (1 A r )[1 m r + a + e f (r) dr + Ξ ( θ )r Σ(θ) e Ξ( θ )r [dt α asin θ dφ K ] Anabalon/Gregory/ Gray/Kubiznak/Man n Σ = 1+ macosθ +[A (a + e ) a l ]cos θ dθ r ]+ r + a l Ξ = 1+ a r cos θ

24 String Tension ( ) µ ± = Ξ(θ ) ± K = ± ma + A (a + e ) a / l K Angular Velocity ω h = Ka α(r + + a ) Gauge Potential and Charge φ = Mass and Angular Momentum Ω = ω h ω ω = Ka(1 A4 l a A 4 l e + a A A l ) (a l A l a A l e )α(1+ a A ) er + (a + r + )α Q = e K M = Q( t ) ω J J = Q( φ )

25 Rotating ABH Thermodynamics Temperature T = f (r + )r + 4πα(r + + a ) Entropy Mass M = m(1 A l )(1+ a A ) Angular Momentum Action J = ma K Kα 1 a l + a A F = M TS ΩJ S = A 4 = πr + K(1 A r + ) f (r + ) = (1 A r + )[1 m r + ]+ r + r + l = 0 + e α = (1 A l ) Ω = ω h ω 1 st Law Smarr δ M = Tδ S + Vδ P + Ωδ J λ + δµ + λ δµ M = TS PV + ΩJ

26 Thermodynamic Volume V = 4πr 3 + 3α 4πma 3Kα 1 a l + a A Conjugates to tension + 4πmA l 4 3Kα 1 1 a l α = (1 A l ) ( ) + a4 l 4 α 1+ α a l a A λ ± = r + ( ) α 1± Ar + m 1+ a l a A α 1 a l + a A Al 1 a l + a A + a4 l α 4 α 1 a l + a A

27 Charged ABH Thermodynamics Temperature Entropy Mass T = f (r ) + 4πα = 1+ 3r + / l A r + ( + r + / l A r + ) 4πr + (1 A r + )α M = m K S = π(r + + a ) K(1 A r + ) 1 A l A 4 e l α Action F = M TS φq 1 st Law Smarr f (r + ) = (1 A r + )[1 m r + e (1 A r + ) 4παr a ]+ r + + a = 0 r + l α = (1 A l e l A 4 )(1+ e A ) Gauge Field does NOT vanish at infinity δ M = Tδ S + Vδ P + φδq λ + δµ + λ δµ M = TS PV + φq

28 Rotating Charged ABH Thermodynamics Temperature Mass Action T = f (r + )r + 4πα(r + + a ) Entropy M = m l (1+ a A + A e )(1 a A 4 l A 4 l e + a A l A ) Kα (a A +1)(l A a + l A e + l a ) F = M TS ΩJ φq S = A 4 = πr + K(1 A r + ) α = (1+ A e + a A )(1 a A 4 l A 4 l e l A + a A ) 1+ a A Gauge Field does Angular Momentum J = ma K NOT vanish at infinity Ω = ω h ω 1 st Lawδ M = Tδ S + Vδ P + Ωδ J + φδq λ + δµ + λ δµ Smarr M = TS + ΩJ PV + φq

29 Snapping Swallowtails Abbasvandi/Cong/ Kubiznak/Mann F Charged AdS Black Hole F Accelerating Charged AdS Black Hole T T Critical pressure below which small black holes don t exist

30 Mini- Entropic Black Holes Abbasvandi/Cong/ Kubiznak/Mann ma ea = 0 ma ea = 0. α = 0 α = 0 signature signature extremal Al Al Black holes near X are `mini-entropic' volume divereges whilst area remains finite

31 Summary Full and consistent description of Accelerating Black Hole thermodynamics obtained Computation is independent of the conformal frame Dual description is that of an anistropic relativistic fluid Charge and Rotation basics now understood Future work Inclusion of Scalars, Constant EM field Interpretation of Free Energy diagram(s) Weak coupling calculation stress tensors with conical deficits Phase transitions, other phenomena?