Observer-‐Dependent  Entropy  in  Black  Hole  Thermodynamics?

Robert  Mann

Area  Law

What  happens  if  you  pour  a  cup  of  tea  into  a  black  hole?

• Tea  is  hot  – has  entropy• Black  hole  absorbs  everything  and  has  no  structure• Where  does  the  entropy  go?Bekenstein:  Tea  has  mass  à will  increase  mass  of  black  hole

à area  must  increase

A(area) S (entropy)

Bekenstein PRD  7 (1973)  2333    

Energy E↔ M Mass

Black  Holes  as  Chemical  Systems?

Thermodynamics Gravity

Entropy S↔ A4!

Horizon Area Temperature T ↔ !κ

2π Surface gravity

dE = TdS +VdP + work terms ↔ dM = κ8π

dA +ΩdJ +ΦdQ

First Law First Law

Dolan  CQG  28 (2011)  125020;  235017    

Kubiznak/Mann/Teo CQG  34  (2017)  063001

Scaling  Arguments

f α px,α qy( ) =α r f x, y( )Suppose

rf x, y( ) = p ∂ f∂x

x + q ∂ f∂y

y

M = M A,Λ( )AdS Black Holes M ∝ LD−3 A ∝ LD−2 Λ ∝ L−2

D − 3( )M = D − 2( ) ∂M∂A

A − 2 ∂M∂Λ

Λ

M = D − 2( )D − 3( )TS −

2D − 3( )VP

S = A4G

T = κ2π

= 4G ∂M∂A

P = − Λ8π

=D − 2( ) D −1( )16π l2

V = −8π ∂M∂Λ

Caldarelli/Cognola/Klemm,  CQG  17 (2000)  399

Creighton/Mann,  PRD53 (1995)  4569Padmanabhan,  CQG  19 (2002)  5387    

Kastor/Ray/Traschen CQG  26 (2009)  195011          Dolan  CQG  28 (2011)  125020;  235017    

Enthalpy H ↔ M Mass

Pressure  from  the  Cosmic  VacuumThermodynamics Gravity

Entropy S↔ A

4! Horizon Area

Temperature T ↔ κ

2π Surface gravity

dH = TdS +VdP + !↔ dM = κ8π

dA +VdP + !

First Law First Law

H = E + PV + ↔ M = E − ρVMass

= Total Energy- Vacuum

Contribution (infinite)

Pressure P↔− Λ8πG

Cosmological Constant

C.  Teitelboim PLB  158  (1984)  293    J.  Creighton  and  R.B.  Mann,  PRD  52 (1995)  4569  Caldarelli/Cognola/Klemm CQG  17 (2000)  399;          T.  Padmanabhan,  CQG  19 (2002)  5387    

Kastor/Ray/TraschenCQG  26 (2009)  195001

Cosmological  constant:  Einstein’s  biggest  blunder?  In  order  to  obtain  static  Universe  Einstein  introduced:

Dark Energy: Λ > 0 (cosmic tension)String Theory: Λ < 0 (cosmic pressure)

Λ

R = (D −1)VωD−2

⎛⎝⎜

⎞⎠⎟

1 D−1ωD−2A

⎛⎝⎜

⎞⎠⎟1 D−2

≤1V

A

Q:  What  is  the  smallest  area  that  encloses  a  given  Euclidean  volume  V?

A:  A  spherical  surface

Cvetic/Gibbons/Kubiznak/Pope  PRD84 (2011)  024037  RBH ≥1Conjecture:  All  black  holes  obey  theReverse  Isoperimetric  Inequality  

ωD−2 =2π

D−12

Γ D −12

⎛⎝⎜

⎞⎠⎟

à For  a  given  thermodynamic  volume,  the  entropy  of  a  black  hole  is  maximized  by  the  Schwarzschild-‐AdS sol’n

Meaning  of  Thermodynamic  Volume?

RBH ≡(D −1)VBH

ωD−2

⎛⎝⎜

⎞⎠⎟

1 D−1ωD−2ABH

⎛⎝⎜

⎞⎠⎟

1 D−2

Black  Holes:

4D Kerr-AdS BH

RBH ≡3V4π

⎛⎝⎜

⎞⎠⎟1 3 4π

A⎛⎝⎜

⎞⎠⎟1 2

= r+lr+2 + a2

l2 − a21+

a2 r+2 + l2( )

2r+2 (l2 − a2 )

⎛

⎝⎜

⎞

⎠⎟3

l2 − a2

r+2 + a2

≥1 →1a = 0

The  Chemistry  of  AdS Black  Holes

D − 3D − 2

M = ThSh + (Ωhi −Ω∞

i )i∑ J i + D − 3D − 2ΦhQ −

2D − 2

PVh

δM = ThδSh + (Ωhi −Ω∞

i )i∑ δ J i +ΦhδQ +VhδP

First Law

Smarr Relation

Thermodynamic Potential: Gibbs Free Energy

G = M −TS = G(T ,P, Ji ,Q)• Equilibrium: Global minimum of Gibbs Free Energy• Local Stability: Positivity of the Specific Heat

CP = T∂S∂T

⎛⎝⎜

⎞⎠⎟ P,Ji ,Q

> 0

Results  from  Black  Hole  Chemistry• Hawking  Page  Transition

– solid/liquid  phase  transition  with  infinite  coexistence  line• Black  Holes  as  Van  der  Waals  Fluids

– Complete  correspondence  between  intrinsic  and  extrinsic  variables  

• Reentrant  Phase  Transitions– Change  from  one  phase  to  another  and  back  again  as  one  parameter  (eg.  temperature)  monotonically  changes

• Black  Hole  Triple  Points  ßà Solid/Liquid/Gas

N.  Altimirano,  D.  Kubiznak,  Z.  Sherkatgnad,  R.B.  Mann  

Galaxies 2 (2014)  89

N. Altimirano, D. Kubiznak, Z. Sherkatgnad, R.B. Mann

CQG 31 (2104) 042001

Kubiznak/Mann CJP 93 999 (2016)

Kubiznak/MannJHEP  1207  (2012)  033  

• Multiple  re-‐entrant  phase  transitions– Higher  curvature  gravity

• Reverse  VdW phase  transitions– Exhibit  similar  phenomena  but  in  lower  dimensions

• Isolated  Critical  Points– Isotherms  cross  at  a  particular  value  of  the  volume  V– Some  black  holes  are  like  polymers:  they  do  not  have  standard  critical  exponents

• Heat  Engines– Can  compare  black  hole  efficiencies  through  various  engine  cycles

Frassino/Kubiznak/Mann/SimovicJHEP  1409 (2014)  080

Brenna/Hennigar/MannJHEP  1507 (2015)  077  

B.  Dolan,  A.  Kostouki,  D.Kubiznak,  R.B.  Mann,  CQG  

31 (2014)  242001

Hennigar JHEP  1710 (2017)  082

Johnson  CQG  31 (2014)  205002    Henningar/McCarthy/Ballon/Mann

CQG  34 (2017)  175005

• Van  der  Waals  black  holes– Black  holes  +  exotic  matter  yield  exact  VdW equation

• Lifshitz Black  Holes– Standard  Smarr Formula  holds  in  all  cases– Can  use  this  to  define  mass  (otherwise  intractable)

• Holographic  Smarr Relation– Higher  curvature  è non-‐planar  loops

• Superfluid  Black  Holes– Black  holes  with  scalar  “hair”                                                                              can  exhibit  a  superfluid  phase                                                            transition  analogous  to  4He

A.Rajagopal,  D.  Kubinzank,  R.B.  Mann,  PLB  B737  (2014)  277  

T.  Delsate,  ,  R.B.  Mann,    JHEP  1502(2015)  070  

W.G.  Brenna,  M.  Park,  R.B.  Mann,  PRD  92 (2015)  044015  

E.  Tjoa,  R.  Hennigar,  R.B.  Mann PRL  118 (2017)  021301  E.  Tjoa,  R.  Hennigar,  R.B.  Mann    JHEP  1702 (2017)  040H.  Dykaar,  R.  Hennigar,  R.B.  Mann  JHEP  1705 (2017  )  045      

Karch/Robinson JHEP  1512 (2015)  073  Sinamuli/Mann      PRD  96 (2017)  068008

Van  der Waals  fluid

Parameter  a measures  the  attraction between  particles  (a>0)  and  b  corresponds  to  “volume  of  fluid  particles”.  

Critical  point:

Chamblin/Emparan/Johnson/Myers  Phys.Rev.  D60 (1999)  064018Cvetic/Gubser-‐JHEP  9904 (1999)  024  

Kubiznak/Mann  JHEP  1207 (2012)  033Gunasekaran/Kubiznak/Mann  JHEP  1112  (2012)  110  

Charged  AdS black  holes  as  Van  der  Waals  fluids  

Charged  AdS Black  Hole

Critical  point:

Kubiznak/Mann  JHEP  1207 (2012)  033Gunasekaran/Kubiznak/Mann  JHEP  1112  (2012)  110  

Charged  AdS black  holes  as  Van  der  Waals  fluids  

Phase  diagrams

Coexistence  line  

• MFT  critical  exponents

govern  specific  heat,  volume,  compressibility  and  pressure  at  the  vicinity  of  critical  point.

• Clausius-Clapeyron and Ehrenfest equations  are  satisfied

Kubiznak/Mann  JHEP  1207 (2012)  033Gunasekaran/Kubiznak/Mann  JHEP  1112  (2012)  110  

Phase  diagrams

Coexistence  line  

• MFT  critical  exponents

govern  specific  heat,  volume,  compressibility  and  pressure  at  the  vicinity  of  critical  point.

• Clausius-Clapeyron and Ehrenfest equations  are  satisfied

Small/large  black  hole  phase  transition

A  system  undergoes  an  RPT  if  a  monotonic variation  of  any  thermodynamic  quantity  results  in  two  (or  more)  phase  transitions  such  that  the  final  state  is  macroscopically  similar  to  the  initial  state. C. Hudson

Z. Phys. Chem. 47 (1904) 113.First  observed  in  nicotine/water  

T. Narayanan and A. Kumar Physics Reports 249 (1994) 135

• multicomponent  fluid  systems

• gels• ferroelectrics• liquid  crystals• binary  gases

And  later  in  many  other  systems:

And    recently  in  Black  Holes!

Reentrant  Phase  Transitions

Coexistence Lines

N.  AltimiranoD.  KubiznakR.B.  Mann  

PRD  88 (2013)  101502  

Example:  Re-‐entrant  Phase  Transition  in  Rotating  AdS Black  Holes

P

T

Pz

Tz

Pt

Tt

INTERMEDIATE BH

SMALL BH

LARGE BH

0.054

0.056

0.058

0.06

0.23 0.235 0.24

Low T Medium T High T mixed ⇒ water/nicotine ⇒ mixedIntermediate BH ⇒ Small BH ⇒ Large BH

P

T

large/small/large  black  hole  phase  transition

Altimirano/Kubiznak/Sherkatgnad/Mann  Galaxies 2 (2014)  89

Takes  place  in  many  examples

Surprises  in  Black  Hole  Entropy

• Super-‐entropic  Black  holes– Entropy  exceeding  the  reverse  isoperimetric  inequality

• Entropy  of  de  Sitter  Black  Holes–With  and  without  a  cavity

• Accelerating  Black  Holes– Thermodynamics  with  Radiation?

R = (D −1)VωD−2

⎛⎝⎜

⎞⎠⎟

1 D−1ωD−2A

⎛⎝⎜

⎞⎠⎟1 D−2

Cvetic/Gibbons/Kubiznak/Pope  PRD84 (2011)  024037  R ≥1

Conjecture:  All  black  holes  obey  theReverse  Isoperimetric  Inequality  

ωD−2 =2π

D−12

Γ D −12

⎛⎝⎜

⎞⎠⎟

Implication:  For  a  given  thermodynamic  volume,  the  entropy  of  a  black  hole  is  maximized  by  the  Schwarzschild-‐AdS sol’n

Utility?    Works  for  most  (charged,  rotating)  black  holes

Counterexample!    Super-‐entropic  Black  Holes

Black  Hole  Volume  and  EntropyRecall:  Isoperimetric  ratio

Super-‐Entropic  Black  Holes• New  ultraspinning limit  to  the  class  of  Kerr  black  hole  metrics• Non-‐compact  horizons  with  finite  area  • Asymptotically  AdS,  but  with  boundary  rotating  at  the  speed  of  light• Obtained  in  the  context  of  Black  Hole  Chemistry• First  counterexamples  to  the  Reverse  Isoperimetric  Inequality

à Super-‐entropic!• Many  other  examples  exist

ds2 = − ΔΣ

dt − l sin2θdψ⎡⎣ ⎤⎦2+ ΣΔdr2 + Σ

sin2θdθ 2 + sin

4θΣ

ldt − (r2 + l2 )dψ⎡⎣ ⎤⎦2

A − qrΣ

dt − l sin2θdψ( )Σ = r2 + l2 cos2θ

Δ = (l + r2

l)2 − 2mr + q2

D.  Klemm PRD  89(2014)    048007  

Hennigar/Kubiznak/MannPhys Rev  Lett 115

(2015)  031101

Hennigar/Kubiznak,   /Muskoe/Mann  JHEP  1506 (2015)  096  

Entropy Volume

S = µω d−24

(l2 + r+2 )r+

d−4 = A4

V = r+Ad −1

R =d −1( )Vµω d−2

⎛⎝⎜

⎞⎠⎟

1d−1 µω d−2

A⎛⎝⎜

⎞⎠⎟

1d−2

= r+2

l2 + r+2

⎛⎝⎜

⎞⎠⎟

1(d−1)(d−2)

Kerr-‐CFT  Correspondence

Kerr-‐AdS Black  Hole

Super-‐Entropic  Black  Hole

Kerr-‐CFT  Limit

Super-‐Entropic  Kerr-‐CFT

1) ψ = φ /Ξ 2) a→ l3) Compactify ψ ~ψ + µ

1) ψ = φ /Ξ 2) a→ l3) Compactify ψ ~ψ + µ

t = r0tε

r = r+ + εr0r

φ = φ +Ξω hr0t / ε

t = r0tε

r = r+ + εr0r

φ =ψ

C.M.  Sinamuli,  R.B.  Mann  JHEP  1608 (2016)  148  arXiv:1512.07597  

ds2 = − ΔaΣa

dt − asin2θ

Ξdφ⎡

⎣⎢

⎤

⎦⎥

2

+ ΣaΔa

dr2 + ΣaSdθ 2 + S sin

2θΣa

adt − r2 + a2

Ξdφ⎡

⎣⎢

⎤

⎦⎥

2

A = − qrΣa

dt − asin2θ

Ξdφ⎛

⎝⎜⎞⎠⎟

Σa = r2 + a2 cos2θ , Ξ = 1− a

2

l2,

S = 1− a2

l2cos2θ Δa = (r

2 + a2 )(1+ r2

l2)− 2mr + q2

Kerr-Newman AdS Black Hole

Γ(θ ) = l2

2x2 + cos2θ

1+ 3x2 α (θ ) = 2

sin2θ(1+ 3x2 ) γ (θ ) = l2 sin4θ (1+ x

2 )2

x2 + cos2θ

r02 = l

2

21+ x2

1+ 3x2 k = x

(1+ x2 )(1+ 3x2 ) f (θ ) = q (1+ x

2 )x

x2 − cos2θx2 + cos2θ

x = r+l

ds2 = Γ(θ )[−r2dt 2 + dr2

r2+α (θ )dθ 2 + γ (θ )

Γ(θ )(dψ + krdt)2 ]

Super-‐Entropic  Kerr-‐CFT  Limit

A = f (θ )(dψ + krdt)

c = 3kµ2π

Γ(θ )γ (θ )α (θ )∫ dθ =3kµπ

l2 (1+ x2 )Central  Charge

CardyFormula SCFT =

π 2

3cLTL S=

µπ2(l2 + r+

2 )

TR = 0

TL =1

2πk

CFT  TemperaturesWorks  in  any  

dimension  and  for  Gauged-‐SUGRA

• What  is  the  significance  of  the  Reverse  Isoperimetric  Inequality  Under  what  conditions  does  it  hold?

• What  are  the  underlying  degrees  of  freedom  of  super-‐entropic  black  holes?    Are  there  “more”  degrees  of  freedom  than  we  expect?

• Not  all  super-‐entropic  black  holes  violate  the  Reverse  Isoperimetric  Inequality  

à What  is  the  meaning  of  the  transition?

x = r+l

y = bl

R

NOT  Super-‐entropicReverse  Isoperimetric  Inequality  obeyed

Super-‐entropicExample:  Doubly-‐spinning  super-‐entropic  Black  Hole    

horizon  sizeorthogonalspin

Entropic  Questions

Rq=0

12 = 127

⎛⎝⎜

⎞⎠⎟

(3x2 + y2 − 2x2y2 )3

x2 (1− y2 )2 (x2 +1)(x2 + y2 )

Thermodynamics  of  de  Sitter  Black  Holes

P = − Λ

8π= − 3

8π1ℓ2

Negative Pressure (tension) in de Sitter Spacetime?

δM = ThδSh + (Ωhi −Ω∞

i )i∑ δ J i +VhδP

D − 3D − 2

M = ThSh + (Ωhi −Ω∞

i )i∑ J i − 2D − 2 PVh

SmarrRelation

First Law

Black Holes

δM = −TcδSc + (Ωci −Ω∞

i )i∑ δ J i +VcδP

D − 3D − 2

M = −TcSh + (Ωci −Ω∞

i )i∑ J i − 2D − 2 PVc

SmarrRelation

First Law

dS Horizon

Thermodynamics  of  Kerr  de  Sitter  Black  Holes

ds2 = −W (1− r2

ℓ2)dt 2 + 2m

UWdt − aiµi

2dϕiΞii=1

N∑

⎛⎝⎜

⎞⎠⎟

2

+ r2 + ai

2

Ξii=1N∑ µi

2dϕi2 + dµi

2( )

+ Udr2

X − 2m+ εr2dν 2 + 1

W (ℓ2 − r2 )r2+ai

2

Ξii=1N∑ µidµi+εr

2νdν⎛⎝⎜

⎞⎠⎟

2

W = µi

2

Ξii=1N∑ + ν 2

X = rε−2 (1− r

2

ℓ2) (r2 + ai

2 )i=1

N∏

2Λ = (D −1)(D − 2)

ℓ2

U = Zℓ

2

ℓ2 − r21− ai

2µi2

r2 + ai2

i=1

N∑

⎛⎝⎜

⎞⎠⎟

Ξi = 1+

ai2

ℓ2

= 1 D=even

0 D=odd ⎧⎨⎩

µi2

i=1

N∑ + ν 2 = 1

• Multiply-‐rotating  Kerr  de  Sitter  Black  hole  in  D  dimensions• 2  horizons  at  different  temperatures

Cosmological Horizon Black Hole Horizon

Even  Dim’l Kerr-‐dS Black  HolesM = mωD−2

4π Ξ jj∏

1Ξii

∑ , Ji =maiωD−24πΞi Ξ j

j∏

Sc =ωD−24

rc2 + ai

2

Ξii∏ =

Ac4

Sh =ωD−24

rh2 + ai

2

Ξii∏ =

Ah4

Tc = −

rc2πℓ2

(ℓ2 − rc2 )

rc2 + ai

2i∑ +

ℓ2 + rc2

4πrc ℓ2

Th =

rh2πℓ2

(ℓ2 − rh2 )

rh2 + ai

2i∑ −

ℓ2 + rh2

4πrhℓ2

Ωc

i = (ℓ2 − rc

2 )aiℓ2 rc

2 + ai2( )

Ωhi = (ℓ

2 − rh2 )ai

ℓ2 rh2 + ai

2( )Vc =

rcAcD −1

+ 8π(D −1)(D − 2)

aii∑ Ji . Vh =

rhAhD −1

+ 8π(D −1)(D − 2)

aii∑ Ji .

2m = 1

ℓ2rc(ℓ2 − rc

2 ) (rc2 + ai

2 )i∏ =

1ℓ2rh

(ℓ2 − rh2 ) (rh

2 + ai2 )

i∏

Cosmological Horizon Black Hole Horizon

Odd  Dim’l Kerr-‐dS Black  HolesM = mωD−2

4π Ξ jj∏

1Ξii

∑ −12

⎛⎝⎜

⎞⎠⎟

Ji =maiωD−2

4πΞi Ξ jj∏

Sc =ωD−24rc

rc2 + ai

2

Ξii∏ =

Ac4

Sh =ωD−24rh

rh2 + ai

2

Ξii∏ =

Ah4

Tc = −

rc2πℓ2

(ℓ2 − rc2 )

rc2 + ai

2i∑ +

12πrc

Th =rh2πℓ2

(ℓ2 − rh2 )

rh2 + ai

2i∑ −

12πrh

Ωc

i = (ℓ2 − rc

2 )aiℓ2 rc

2 + ai2( )

Ωhi = (ℓ

2 − rh2 )ai

ℓ2 rh2 + ai

2( )Vc =

rcAcD −1

+ 8π(D −1)(D − 2)

aii∑ Ji . Vh =

rhAhD −1

+ 8π(D −1)(D − 2)

aii∑ Ji .

2m = 1

ℓ2rc(ℓ2 − rc

2 ) (rc2 + ai

2 )i∏ =

1ℓ2rh

(ℓ2 − rh2 ) (rh

2 + ai2 )

i∏

(Reverse)  Isoperimetric  Inequality?

R = (D −1)VωD−2

⎛⎝⎜

⎞⎠⎟

1 D−1ωD−2A

⎛⎝⎜

⎞⎠⎟1 D−2

R ≤1VA

Kerr-(A)dS Black Hole

Vh =

rhAhD −1

+ 8π(D −1)(D − 2)

aii∑ Ji =

rhAhD −1

1+ ℓ2 ± rh

2

(D − 2)ℓ2rh2

ai2

Ξii∑

⎡

⎣⎢

⎤

⎦⎥

Ah =

ωD−2rh1−ε

rc2 + ai

2

Ξii∏

RD−1 = rh 1+z

D − 2⎡⎣⎢

⎤⎦⎥1rh1−

(rh2 + ai

2 )Ξii

∏⎡

⎣⎢

⎤

⎦⎥

12−D

= 1+ zD − 2

⎡⎣⎢

⎤⎦⎥

(rh2 + ai

2 )rh2Ξii

∏⎡

⎣⎢

⎤

⎦⎥

12−D

≥ 1+ zD − 2

⎡⎣⎢

⎤⎦⎥

2D −1

1Ξii

∑ +ai2

rh2Ξii

∑⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

D−14−2D

= 1+ zD − 2

⎡⎣⎢

⎤⎦⎥1+ 2z

D −1⎡⎣⎢

⎤⎦⎥

D−14−2D

≡ F(z)

F(0) = 1 dF(z)dz

> 0 F(z) ≥1 R ≥1

Cvetic/Gibbons/Kubiznak/Pope  PRD84  (2011)  024037  

Cosmic  Volume

Can  we  understand  cosmic  volume  without  black  hole  volume?Yes!    With  cosmic  solitons!

Clarkson/MannPRL  96  (2006)  051104

• Geometry  depends  on  relative  size  of  the  soliton and  the• No  black  hole  horizon!    • Can  now  have  a  cosmological  horizon  surrounding  soliton• Obtained  a  number  of  results  depending  on  mass/energy  of  the  soliton and  its  size  relative  to  the  cosmic  horizon

Mbarek/MannPLB  765  (2017)  352

Soliton:  a  bubble  in  spacetime!