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?