ds2 = −g(r)dt 2 + 2rd
⎛⎝⎜
⎞⎠⎟2
f (r) dψ + cosi=1
k∑ (θi )dφi
⎡⎣⎢
⎤⎦⎥
2
+ dr2
g(r) f (r)+ r
2
dd
i=1
k∑ Σ2(i )
2
!! f r( ) =1− a
d
rd!!!!!!!!!g r( ) =1− r
2
ℓ2
Clarkson/MannPRL 96 (2006) 051104
• Geometry depends on relative size of• No black hole horizon! • Can now have a cosmological horizon surrounding soliton
a / ℓr ≥ a
!! Λ = + (D−2)(D−2)2ℓ2
compact space
!!!Δψ=4π
p= 8πr2 ′f r( )
r=a
=2πRegularity!!
d =2k+2= D−1dΣ2(i )
2 = dθi2 + sin2(θi )dφi2
Mbarek/MannPLB 765 (2017) 352
ℓ2
Min/out = Ka
d
4πℓ2 +md2
ℓd−2+−
Vin/out = − 2K
d −1ad + 8(d − 2)
d(d −1)πmd
2
+ 4Kd
⎛⎝⎜
⎞⎠⎟ℓd+
−
R
d
Below the dashed line the Reverse Isoperimetric Inequality is violated
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/MannCQG (2018) to appear
!! !I = − 1
16π M∫ d4x g(R−2Λ+F 2)+ 18π ∂M∫ d3x k(K −K0)
!!
∂Ir(β ,r+ ,Rc ,q,Λ)∂r+
=0 → β(r+ ,Rc ,q,Λ)⇒T(r+ ,Rc ,q,Λ)
E =∂Ir∂β
, S = β∂Ir∂β
⎛
⎝⎜⎞
⎠⎟− Ir
Carlip/VaidyaCQG 20 (2018) 3827
!!
E =2(TS −λAc +PV )+ΦQdE =TdS −λdAc −VdP +φdQ
1) Check 1st law and Smarr
2) Analyze Free energy for phase transitions
Cavity area
De Sitter Black Holes in a Cavity
!! !I = − 1
16π M∫ d4x g(R−2Λ+F 2)+ 18π ∂M∫ d3x k(K −K0)
!!
!X(Λ)≡ 1− r+Rc
⎛
⎝⎜⎞
⎠⎟(1− q2
r+Rc− Λ3(Rc2 +Rcr+ + r+2))
!!!!!!!!!!!!!!!!!!!!!!! !Y(Λ)≡ 1− ΛRc2
3
!! I = βRc!Y(Λ)− !X(Λ)⎡⎣ ⎤⎦−πr+
2
!! E = ∂I
∂β= Rc !Y(Λ)− !X(Λ)⎡⎣ ⎤⎦
!!S = β ∂I ∂β − I =πr+2
!! T(r+ ,Rc ,P)=
1−Λr+2 −q2 r+24πr+ !X
!!
E =2(TS −λAc +PV )+ΦQdE =TdS −λdAc −VdP +φdQ
Swallowtube
!!
λ =4ΛRc3 −6Rc( )( !X − !Y)+ r+(Λr+2 −3) !Y − 3q
2 !Yr+
48πRc2 !X !Y
V = 4π3Rc3( !Y − !X )− r+3 !Y
!X !Y
Φ =(Rc − r+ )qr+Rc !X
Phase transitions for Charged dSBlack Holes in a Cavity
!! F(r+ ,Rc ,P)=
Λr+3
!X−Rc !X +
RcB
3−r+4 !X
!P
!T
!F
!T!P
De Sitter Black Holes in Equilibrium
• Generic dS Black holes have 2 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/MannJHEP (2018) to appear
!!I = 1
16πG dd∫ x −g L(k )k=0
kmax
∑ −4πGFµνF µν⎛
⎝⎜
⎞
⎠⎟!!L(k ) = ! 12k δ
(k ) ak Rµrνr
αrβr
r
k
∏⎛⎝⎜
+bkφd−4k Sµrνr
αrβr
r
k
∏ ⎞⎠⎟
!!Sµνγδ ! =φ2Rµν
γδ −2δ[µ[γδν ]δ ]∇ρφ∇
ρφ −4φδ[µ[γ ∇ν ]∇δ ]φ +8δ[µ[γ ∇ν ]φ∇
δ ]φ
!! δ(k ) =δ µ1ν1!µkνk
α1β1!αkβk
!!αk
k=0
kmax
∑ σ − fr2
⎛⎝⎜
⎞⎠⎟
k
! = 16πGM(d −2)ω
d−2( )(σ ) rd−1
+ Hrd
− 8πG(d −2)(d −3)
Q2
r2d−4
!!φ = N
rH = (d −3)!
(d −2(k+1))!k=0
kmax
∑ bkσkNd−2k
!!
bkk=0
kmax
∑ (d −1)!(d(d −1)+4k2)(d −2k−1)! σ kN −2k =0
!!!!!!!!!!!!!! kk=1
kmax
∑ bk(d −1)!
(d −2k−1)!σk−1N2−2k =0
Hairy dS Black Holes
!!ds2 = − fdt2 + f −1dr2 + r2dΣ
σ d−2( )2
Solution to the polynomial is a solution to the field equations
Oliva/RayCQG 29(2012) 205008
!!
δM± =T±δS± +V±δP± +Φ±δQ+κδH(d −3)M± = (d −2)T±S± −2V±P± +(d −3)Φ±Q+(d −2)κH
Check 1stlaw and Smarr at both
horizons
!!2)!Equilbrate!the!temperatures:!!!T+ =
′f (rc )4π =
′f (rbh)4π =Tbh =T⇒ rc = rc Q,P ,rbh( )
!!1)!Solve!f (rc )=0!and!f (rbh)=0!for!!H !and!M
!!
4)!Compute!the!free!energy!(Grand!canonical!ensemble):!!!!!!!!!!cosmo!!!!G+ =M −TS+ −φ+Q!black!hole!!!!G− =M −TS− −φ−Q
!!total!!!!!Gtot ! =M −T S+ + S−( )+(φ+ −φ− )Q
!T
!rbh
!G !T
smalllarge
Reverse HP phase
transition!
!!3)!Require!postive!mass!and!entropy:!!Sh =
Σd−2σ
4G(d −2)kσ k−1αkrh
d−2k
d −2kk=1
kmax
∑ − dH2σ (d −4)
⎡
⎣⎢⎢
⎤
⎦⎥⎥>0
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
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 PRD2 (1970) 1359
Plebanski/DemianskiAnn. Phys. 98 (1976) 98
Dowker/Gauntlett/Kastor/Traschen
PRD49 (1994) 2909Gregory/HindmarshPRD52 (1995) 5598
Mann/Ross PRD52 (1995) 2254
Thermodynamics of ABHs• Basic features– 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/RangamaniCQG 27 (2010) 025001
Appels/Gregory/ Kubiznak PRL 117 (2016) 131303;
JHEP 1705 (2017)116 Astorino PRD95 (2017) 064007
ds2 = 1Ω2 [− fdt 2 + dr
2
f+ r2(dθ
2
Σ+ Σsin2θ dφ
2
K 2 )]
Ω = 1+ Ar cosθ Σ = 1+ 2mAcosθ
f (r) = (1− A2r2 )(1− 2mr)+ r
2
ℓ2
Metric for ABH in AdS
Parameterizes deficit angle
θφ
Hong/TeoCQG 20 (2003) 3269
µ± =δ ±
8π= 141− Σ(θ± )
K⎛⎝⎜
⎞⎠⎟ =
141− 1± 2mA
K⎛⎝⎜
⎞⎠⎟
StringTension
ds2 = 1Ω2 [− fdt 2 + dr
2
f+ r2(dθ
2
Σ+ Σsin2θ dφ
2
K 2 )]
f (r) = (1− A2r2 )(1− 2m
r)+ r
2
ℓ2
Metric for ABH in AdS
Ω = 1+ Ar cosθΣ = 1+ 2mAcosθ
m = 0Rindler-‐AdS
Boundary is beyond r = ∞
Slowly Accelerating Black Hole m ≠ 0
Distortion ofPoincare disk
Aℓ < 3 6
8
ds2 = 1Ω2 [− fdt 2 + dr
2
f+ r2(dθ
2
Σ+ Σsin2θ dφ
2
K 2 )]
Metric signature not preserved
Acc
eler
atio
nH
oriz
ons
pres
ent
f (r) = (1− A2r2 )(1− 2m
r)+ r
2
ℓ2Ω = 1+ Ar cosθΣ = 1+ 2mAcosθ
Slowly acceleratingblack holes
f (−1/ Acosθ ) >1No Acceleration Horizons
Asymptotics
ds2 = 1Ω2 [− fdt 2 + dr
2
f+ r2(dθ
2
Σ+ Σsin2θ dφ
2
K 2 )]
Ω = 1+ Ar cosθΣ = 1+ 2mAcosθ
1+ R
2
ℓ2 = 1+ (1− A2ℓ2 )r2 / ℓ2
(1− A2ℓ2 )Ω2 Rsinϑ = r sinθΩ
m = 0
dsAdS2 = −(1+ R
2
ℓ2)α 2dt 2 + dR2
1+ R2
ℓ2
+ R2(dϑ 2 + sin2ϑ dφ 2
K 2 )
α = 1− A2ℓ2Correct time coordinate is τ =αt
Thermodynamic Quantities
• Thermodynamic– Via the Smarr Relation
• Conformal– Via Electric part of Weyl Tensor
• Holographic– Via AdS/CFT counterterms
T = ′f (r+ )
4πα=1+ 3 r+
2
ℓ2− A2r+
2 2 + r+2
ℓ2− A2r+
2⎛⎝⎜
⎞⎠⎟
4παr+ (1− A2r+2 )
Temperature
S = A
4= πr+
2
K(1− A2r+2 )
Entropy
Mass? f (r+ ) = (1− A
2r+2 )(1− 2m
r+)+ r+
2
ℓ2= 0
P = 3
8πℓ2
Interpret in Context of Black Hole Chemistry
M = αm
K= 1− A2ℓ2 m
K
V = 4
3πKα
r+3
(1− A2r+2 )2 +mA
2ℓ4⎡
⎣⎢
⎤
⎦⎥
λ± =
1α
r+1− A2r+
2 −m 1± 2Aℓ2
r+
⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥
Consistency
Thermodynamic Mass of ABHs
δM = TδS +VδP − λ+δµ+ − λ−δµ−
M = 2TS − 2PV
C-metric: dSµ = δ µ
τ ℓ2 (d cosθ )dφ
αK
Q(ξ ) = ℓ
8πlimΩ→0
ℓ2
Ω"∫ NαN βCναµβ g[ ]ξνdS µ
ds2 = 1Ω2 [− fdt 2 + dr
2
f+ r2(dθ
2
Σ+ Σsin2θ dφ
2
K 2 )]= gµνdxµdxν
gµν = Ω2gµν Nµ = ∂µΩ ξ = ∂τ
M =Q(∂τ ) =α
mK
= 1− A2ℓ2 mK Ω = ℓΩr−1
Conformal Mass of ABHsAshtekar/Das CQG 17 (2000) L17Das/Mann JHEP 08 (2000) 033
I[g]= 116π
d 4
M∫ x −g R + 6ℓ2
⎡⎣⎢
⎤⎦⎥+ 1
8πd 3
∂M∫ x −hK
− 18π
d 3
∂M∫ x −h 2ℓ+ ℓ
2R h( )⎡
⎣⎢⎤⎦⎥
8πTab = ℓGab h( )− 2
ℓhab −Kab + habK
δS = − 1
2Tab∂M∫ δhab −h
M = Tab U
aUb∫ −γ (0) Tab = lim
ρ→∞
ρl
Tab
Holographic Mass of ABHs
Balasubramanian/Kraus CMP 208(1999) 413
Das/Mann JHEP 08 (2000) 033
ds2 = ℓ
2
ρ 2 dρ2 + habdx
adxb = ℓ2
ρ 2 dρ2 + ρ 2 γ ab
(0) + 1ρ 2 γ ab
(2) + ...⎛⎝⎜
⎞⎠⎟dxadxb
Fefferman/Graham NHS (1985) 95
F1(x) = −
1− A2ℓ2X( )3/2Aω (x)α
M = ρE∫ ℓ3 −γ (0) dxdφ = αm
K= 1− A2ℓ2 m
K
X = (1− x2 ) 1+ 2mAx( )For the C-‐metric
ρE =
m(1− A2ℓ2X)3/2
8πℓ2α 3ω 3 (2 − 3A2ℓ2X)
2P = ρEEqn of State: thermal gas
of massless particles
π xx = − 3mA
3XF116πα 2ω 2 = −πφ
φ
Shear tensor: anisotropic dual fluid
Tab = P + ρE( )UaUb + Pl2γ ab +π ab
1Ar
= −x − Fn∑ x( )ρ−n
cosθ = x + Gn∑ x( )ρ−n
Convert to FG coords
ds(0)
2 = γ ab(0)dxadxb = −ω
2dτ 2
ℓ2+ Xω 2α 2dφ 2
K 2 (1− A2ℓ2X)+ ω 2α 2dx2
X(1− A2ℓ2X)2
x = Al ∈(0,1)y = r+ / l > 0
Reverse Isoperimetric Inequality for ABHs
R =2 + x2 (1+1/ y2 )− 2x4 (1+ y2 / 2)+ x6y2( )2
4((1− x2y2 )(1− x2 ))⎡
⎣⎢⎢
⎤
⎦⎥⎥
1/6
≥1
x = Al
y = r+l
R321
R = 3V
ω 2
⎛⎝⎜
⎞⎠⎟
13 ω 2
A⎛⎝⎜
⎞⎠⎟
12
I = β
2αKm − 2mA2ℓ2 − r+
3
ℓ2 (1− A2r+2 )2
⎛⎝⎜
⎞⎠⎟
Action
●●
µ± =δ ±
8π= 141− 1± 2mA
K⎛⎝⎜
⎞⎠⎟
µ− = 0 : Hawking-Page transitionµ− > 0 : No clear interpretation
F = I / β = M −TS
β = 1T
Conformal Boundary
γ ab(0)dxadxb = −ω
2dτ 2
ℓ2+ Xω 2α 2dφ 2
K 2 (1− A2ℓ2X)+ ω 2α 2dx2
X(1− A2ℓ2X)2
ds2 = 1Ω2 [− fdt 2 + dr
2
f+ r2(dθ
2
Σ+ Σsin2θ dφ
2
K 2 )]
→ ℓ2
ρ 2 dρ2 + ρ 2 γ ab
(0) + 1ρ 2 γ ab
(2) + ...⎛⎝⎜
⎞⎠⎟dxadxb
ω 2 = (1− A
2ℓ2X)α 2 1− x( )
AdS2 × S1
asymptoticregion
ω 2 = (1− A
2ℓ2X)α 2 1− x2( )
Two AdS2 × S1 regions
connected via wormhole
ω2 = (1− A2ℓ2X)α −2
A2ℓ2X >1⇒ Black droplet
No equilibrium T
Hubeny/Marolf/RangamaniCQG 27 (2010)
025001
Charge and Rotation
ds2 = 1Ω2 − f (r)
Ξ θ( )dtα
− asin2θ dφK
⎡⎣⎢
⎤⎦⎥
2
+Ξ θ( )f (r)
dr2 +Ξ θ( )r2
Σ(θ )dθ 2⎧
⎨⎪
⎩⎪
+ Σ(θ )sin2θΞ θ( )r2
adtα
− (r2 + a2 ) dφK
⎡⎣⎢
⎤⎦⎥
2 ⎫⎬⎪
⎭⎪
Ω = 1+ Ar cosθ Σ = 1+ 2mAcosθ +[A2 (a2 + e2 )− a2
ℓ2 ]cos2θ
f (r) = (1− A2r2 )[1− 2mr
+ a2 + e2
r2 ]+ r2 + a2
ℓ2 Ξ = 1+ a2
r2 cos2θ
F = dB B = − eΞ θ( )r[
dtα
− asin2θ dφK]
Anabalon/Gregory/Gray/Kubiznak/Man
n 1811.04936
µ± =
141− Ξ(θ± )
K⎡⎣⎢
⎤⎦⎥= 141−1± 2mA + A2 (a2 + e2 )− a2 / ℓ2( )
K⎡
⎣⎢⎢
⎤
⎦⎥⎥
StringTension
Ω =ω h −ω∞
ω h =Ka
α (r+2 + a2 )
ω∞ = Ka(1− A4l2a2 − A4l2e2 + a2A2 − A2l2 )(a2 − l2 − A2l2a2 − A2l2e2 )α (1+ a2A2 )
Angular Velocity
φ = er+(a2 + r+
2 )α Q = e
K
Gauge Potential and Charge
M =Q(∂t )−ω∞J J =Q(∂φ )
Mass and Angular Momentum
Rotating ABH Thermodynamics
T = ′f (r+ )r+2
4πα (r+2 + a2 )
Temperature
S = A
4= πr+
2
K(1− A2r+2 )
Entropy
f (r+ ) = (1− A
2r+2 )[1− 2m
r++ e
2
r+2 ]+ r+
2
ℓ2= 0
Mass M = m(1− A2l2 )(1+ a2A2 )
Kα 1− a2
l2+ a2A2⎛
⎝⎜⎞⎠⎟
α = (1− A2l2 )
δM = TδS +VδP +Ωδ J − λ+δµ+ − λ−δµ−
M = 2TS − 2PV + 2ΩJ
1st LawSmarr
Action F = M −TS −ΩJ
Angular Momentum J = ma
K 2Ω =ω h −ω∞
V = 4πr+3
3α− 4πma2
3Kα 1− a2
l2 + a2A2⎛⎝⎜
⎞⎠⎟
+4πmA2l 4 1− a
2
l2 + a4
l 4 α2 1+α 2( )⎡
⎣⎢⎤⎦⎥
3Kα 1− a2
l2 − a2A2⎛⎝⎜
⎞⎠⎟
2⎛
⎝⎜
⎞
⎠⎟
Thermodynamic Volume
Conjugates to tension
λ± =r+
α 1± Ar+( ) −m 1+ a2
l2 − a2A2⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
α 1− a2
l2 + a2A2⎛⎝⎜
⎞⎠⎟
2 ∓
Al2 1− a2
l2 + a2A2⎛⎝⎜
⎞⎠⎟
2
+ a4
l 4 α2
⎡
⎣⎢⎢
⎤
⎦⎥⎥
α 1− a2
l2 + a2A2⎛⎝⎜
⎞⎠⎟
2
α = (1− A2l2 )
Charged ABH Thermodynamics
T = ′f (r+ )4πα
= 1+ 3r+2 / l2 − A2r+
2 (2 + r+2 / l2 − A2r+
2 )4πr+ (1− A
2r+2 )α
− e2 (1− A2r+
2 )4παr+
3
Temperature
S = π (r+2 + a2 )
K(1− A2r+2 )
Entropy
f (r+ ) = (1− A
2r+2 )[1− 2m
r++ a
2
r+2 ]+ r+
2 + a2
ℓ2= 0
MassM = m
K1− A2l2 − A4e2l2
αα = (1− A2l2 − e2l2A4 )(1+ e2A2 )
δM = TδS +VδP +φδQ − λ+δµ+ − λ−δµ−
M = 2TS − 2PV +φQ1st LawSmarr
Action F = M −TS −φQGauge Field does NOT vanish at
infinity!
Rotating Charged ABH Thermodynamics
MassM = m
Kαl2 (1+ a2A2 + A2e2 )(1− a2A4l2 − A4l2e2 + a2A2 − l2A2 )
(a2A2 +1)(l2A2a2 + l2A2e2 + l2 − a2 )
δM = TδS +VδP +Ωδ J +φδQ − λ+δµ+ − λ−δµ−
M = 2TS + 2ΩJ − 2PV +φQ1st LawSmarr
Action F = M −TS −ΩJ −φQ
T = ′f (r+ )r+2
4πα (r+2 + a2 )
Temperature
S = A
4= πr+
2
K(1− A2r+2 )
Entropy
α = (1+ A2e2 + a2A2 )(1− a2A4l2 − A4l2e2 − l2A2 + a2A2 )1+ a2A2 Gauge Field does
NOT vanish at infinity!Angular Momentum
J = ma K 2
Ω =ω h −ω∞
Snapping Swallowtails
F
TT
FCharged AdS Black Hole Accelerating Charged AdS Black Hole
Abbasvandi/Cong/ Kubiznak/Mann
1812.00384
Critical pressure below which small black holes don’t exist
Mini-‐Entropic Black Holes
mAmA
Aℓ Aℓ
eA = 0 eA = 0.22
extremal
signaturesignature
α = 0 α = 0
• Black holes near X are `mini-entropic'• volume divereges whilst area remains finite
Abbasvandi/Cong/ Kubiznak/Mann
1812.00384
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?
Top Related