[PPT] - Quasilocal Smarr Relations in collaboration with Yein Lee(Kyung Hee PowerPoint Presentation

SLIDE 1

Quasilocal Smarr Relations

in collaboration with Yein Lee(Kyung Hee U.), Mattew Richards(U. of McMaster), Sean Stotyn(U. of Calgary)1

Miok Park

Korea Institute for Advanced Study (KIAS), Seoul, S. Korea Strings and Fields 2019, at YITP

August 19, 2019

1arXiv:1809.07259 for AF (ver2 will be updated), arXiv:190X.XXXXX for AAdS

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Content

1

Introduction a conserved charge in curved spacetime? what is Smarr Relation? why we need Quasilocal Formalism?

2

Quasilocal Formalism Quasilocal Quantities : how they are defined

3

Quasilocal Smarr Relation in Asymptotically Flat spacetimes QL Smarr Relation by Euler Theorem Example

4

Quasilocal Smarr Relation in Asymptotically AdS spacetimes QL Smarr Relation for AAdS by Euler Theorem Example

5

Conclusion and Future Works

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 1 / 19

SLIDE 3

Introduction a conserved charge in curved spacetime?

Conserved charge in curved spacetime

There are difficulties to calculate a gravitational conserved energy in curved spacetime. In SR ∇a(Tabξb) = 0, where na is the unit normal to Σ and ta a time-like Killing field.

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 2 / 19

SLIDE 4

Introduction a conserved charge in curved spacetime?

Conserved charge in curved spacetime

There are difficulties to calculate a gravitational conserved energy in curved spacetime. In SR ∇a(Tabξb) = 0, where na is the unit normal to Σ and ta a time-like Killing field. This guarantees that total energy is conserved.

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 2 / 19

SLIDE 5

Introduction a conserved charge in curved spacetime?

Conserved charge in curved spacetime

There are difficulties to calculate a gravitational conserved energy in curved spacetime. In SR ∇a(Tabξb) = 0, → E =

Σ

Tabnatb where na is the unit normal to Σ and ta a time-like Killing field. This guarantees that total energy is conserved. In GR Rµν − 1 2Rgµν = 8πTµν the energy properties of matter are represented by Tµν, but a gravitational field energy is not included in Tµν.

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 2 / 19

SLIDE 6

Introduction a conserved charge in curved spacetime?

Conserved charge in curved spacetime

There are difficulties to calculate a gravitational conserved energy in curved spacetime. In SR ∇a(Tabξb) = 0, → E =

Σ

Tabnatb where na is the unit normal to Σ and ta a time-like Killing field. This guarantees that total energy is conserved. In GR Rµν − 1 2Rgµν = 8πTµν the energy properties of matter are represented by Tµν, but a gravitational field energy is not included in Tµν. But some methods have been developed such as ADM method, Komar method, AD(T) method, and etc.

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 2 / 19

SLIDE 7

Introduction what is Smarr Relation?

Smarr relation by Larry Smarr in 1973 2

ds2

RN =

1 − 2Mr − Q2

R2

du2 + 2dudr + 2a sin2 θ

R2 (2Mr − Q2)dudφ − 2a sin2 θdrdφ − R2dθ2 + sin2 θ R2

∆a2 sin2 θ − (a2 + r 2)2
dφ2,

R2 ≡ r 2 + a2 cos2 θ, ∆ ≡ r 2 + a2 − 2Mr + Q2, a ≡ J M The black hole’s area is written as S = 4π

2M2 + 2(M4 − J2 − M2Q2)

1 2 − Q2

,

2Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 3 / 19

SLIDE 8

Introduction what is Smarr Relation?

Smarr relation by Larry Smarr in 1973 2

ds2

RN =

1 − 2Mr − Q2

R2

du2 + 2dudr + 2a sin2 θ

R2 (2Mr − Q2)dudφ − 2a sin2 θdrdφ − R2dθ2 + sin2 θ R2

∆a2 sin2 θ − (a2 + r 2)2
dφ2,

R2 ≡ r 2 + a2 cos2 θ, ∆ ≡ r 2 + a2 − 2Mr + Q2, a ≡ J M The black hole’s area is written as S = 4π

2M2 + 2(M4 − J2 − M2Q2)

1 2 − Q2

,

M = S 16π + 4πJ2 S + Q2 2 + πQ4 S 1

2

= M[S, J, Q]

2Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 3 / 19

SLIDE 9

Introduction what is Smarr Relation?

Smarr relation by Larry Smarr in 1973 2

ds2

RN =

1 − 2Mr − Q2

R2

du2 + 2dudr + 2a sin2 θ

R2 (2Mr − Q2)dudφ − 2a sin2 θdrdφ − R2dθ2 + sin2 θ R2

∆a2 sin2 θ − (a2 + r 2)2
dφ2,

R2 ≡ r 2 + a2 cos2 θ, ∆ ≡ r 2 + a2 − 2Mr + Q2, a ≡ J M The black hole’s area is written as S = 4π

2M2 + 2(M4 − J2 − M2Q2)

1 2 − Q2

,

M = S 16π + 4πJ2 S + Q2 2 + πQ4 S 1

2

= M[S, J, Q] dM = TdS + ΩdJ + ΦdQ

2Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 3 / 19

SLIDE 10

Introduction what is Smarr Relation?

Smarr relation by Larry Smarr in 1973 2

ds2

RN =

1 − 2Mr − Q2

R2

du2 + 2dudr + 2a sin2 θ

R2 (2Mr − Q2)dudφ − 2a sin2 θdrdφ − R2dθ2 + sin2 θ R2

∆a2 sin2 θ − (a2 + r 2)2
dφ2,

R2 ≡ r 2 + a2 cos2 θ, ∆ ≡ r 2 + a2 − 2Mr + Q2, a ≡ J M The black hole’s area is written as S = 4π

2M2 + 2(M4 − J2 − M2Q2)

1 2 − Q2

,

M = S 16π + 4πJ2 S + Q2 2 + πQ4 S 1

2

= M[S, J, Q] dM = TdS + ΩdJ + ΦdQ = ⇒ M = 2TS + 2ΩJ + ΦQ

2Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 3 / 19

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Introduction what is Smarr Relation?

Smarr relation by Larry Smarr in 1973 3

Euler theorem states that if a function f(x, y, z) obeys the scaling relation f(αpx, αqy, αkz) = αrf(x, y, z), (1) then it satisfies rf(x, y, z) = p ∂f ∂x

x + q

∂f ∂y

y + k

∂f ∂z

z.

(2) M in terms of S, L and Q satisfies this relation having a following scaling M ∝ [L], S ∝ [L]2, J ∝ [L]2, Q ∝ [L], (3) then the Euler’s theorem yields M = ∂M ∂S

S +

∂M ∂J

J +

∂M ∂Q

Q.

(4)

3Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 4 / 19

SLIDE 12

Introduction what is Smarr Relation?

Smarr relation by Larry Smarr in 1973 3

Euler theorem states that if a function f(x, y, z) obeys the scaling relation f(αpx, αqy, αkz) = αrf(x, y, z), (1) then it satisfies rf(x, y, z) = p ∂f ∂x

x + q

∂f ∂y

y + k

∂f ∂z

z.

(2) M in terms of S, L and Q satisfies this relation having a following scaling M ∝ [L], S ∝ [L]2, J ∝ [L]2, Q ∝ [L], (3) then the Euler’s theorem yields M = 2TS + 2ΩJ + ΦQ. (4)

3Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 4 / 19

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Introduction what is Smarr Relation?

Smarr relation by Bardeen, Carter, and Hawking

In 1973, this work resulted in “The Four laws of black hole mechanics4” by Bardeen, Carter, and Hawking

4J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 5 / 19

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Introduction what is Smarr Relation?

Smarr relation by Bardeen, Carter, and Hawking

In 1973, this work resulted in “The Four laws of black hole mechanics4” by Bardeen, Carter, and Hawking M = − 1 8π

∂V

dSµνDµkν = − 1 8πG

V

dSµRµ

νξν

= −2

Σ

dSµT µ

νξν −

1 8πG

H

dSµνDµξν

4J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 5 / 19

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Introduction what is Smarr Relation?

Smarr relation by Bardeen, Carter, and Hawking

In 1973, this work resulted in “The Four laws of black hole mechanics4” by Bardeen, Carter, and Hawking M = − 1 8π

∂V

dSµνDµkν = − 1 8πG

V

dSµRµ

νξν

= −2

Σ

dSµT µ

νξν −

1 8πG

H

dSµνDµξν = ΦHQ + κA 8π

4J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 5 / 19

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Introduction what is Smarr Relation?

Smarr relation by Bardeen, Carter, and Hawking

In 1973, this work resulted in “The Four laws of black hole mechanics4” by Bardeen, Carter, and Hawking M = − 1 8π

∂V

dSµνDµkν = − 1 8πG

V

dSµRµ

νξν

= −2

Σ

dSµT µ

νξν −

1 8πG

H

dSµνDµξν = ΦHQ + κA 8π (Same as Smarr Relation!!)

4J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 5 / 19

SLIDE 17

Introduction what is Smarr Relation?

Smarr relation by Bardeen, Carter, and Hawking

In 1973, this work resulted in “The Four laws of black hole mechanics4” by Bardeen, Carter, and Hawking M = − 1 8π

∂V

dSµνDµkν = − 1 8πG

V

dSµRµ

νξν

= −2

Σ

dSµT µ

νξν −

1 8πG

H

dSµνDµξν = ΦHQ + κA 8π (Same as Smarr Relation!!) Differential form is interpreted as the first law of black hole thermodynamics

4J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 5 / 19

SLIDE 18

Introduction what is Smarr Relation?

Smarr relation by Bardeen, Carter, and Hawking

In 1973, this work resulted in “The Four laws of black hole mechanics4” by Bardeen, Carter, and Hawking M = − 1 8π

∂V

dSµνDµkν = − 1 8πG

V

dSµRµ

νξν

= −2

Σ

dSµT µ

νξν −

1 8πG

H

dSµνDµξν = ΦHQ + κA 8π (Same as Smarr Relation!!) Differential form is interpreted as the first law of black hole thermodynamics δM = κ 8π δA + ΦHδQ

4J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 5 / 19

SLIDE 19

Introduction what is Smarr Relation?

Black hole thermodynamics

To completely characterize the black hole thermodynamic properties, we need to check

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 6 / 19

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Introduction what is Smarr Relation?

Black hole thermodynamics

To completely characterize the black hole thermodynamic properties, we need to check the first law of black hole thermodynamics

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 6 / 19

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Introduction what is Smarr Relation?

Black hole thermodynamics

To completely characterize the black hole thermodynamic properties, we need to check the first law of black hole thermodynamics and also Smarr relation

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 6 / 19

SLIDE 22

Introduction why we need Quasilocal Formalism?

Quasilocal Formalism

Why do we need a quasilocal frame?

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 7 / 19

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Introduction why we need Quasilocal Formalism?

Quasilocal Formalism

Why do we need a quasilocal frame?

1

Approaching infinity is not always an appropriate theoretical idealization, and is never satisfied in reality

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 7 / 19

SLIDE 24

Introduction why we need Quasilocal Formalism?

Quasilocal Formalism

Why do we need a quasilocal frame?

1

Approaching infinity is not always an appropriate theoretical idealization, and is never satisfied in reality

2

black hole thermodynamics in a quasilocal frame can be applied to gravitational and matter fields within a bounded, finite spatial region

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 7 / 19

SLIDE 25

Introduction why we need Quasilocal Formalism?

Quasilocal Formalism

Why do we need a quasilocal frame?

1

Approaching infinity is not always an appropriate theoretical idealization, and is never satisfied in reality

2

black hole thermodynamics in a quasilocal frame can be applied to gravitational and matter fields within a bounded, finite spatial region

3

Intrinsically, thermodynamics for AdS black hole should be studied at a quasilocal frame, becuase THawking(| − ξµξµ|1/2 = 1) (5) | − ξµξµ|1/2 = 1 r → ∞ for AAdS (6) because N(r) ∼ r 2, r → ∞ (7) .

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 7 / 19

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Introduction why we need Quasilocal Formalism?

Black Hole Thermodynamics

For electrically charged and AF dM = THawkingdS + ΦdQ, M = 2TS + ΦQ For electrically charged and AAdS dM = THawkingdS + ΦdQ + VdP, M = 2TS + ΦQ − 2PV

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 8 / 19

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Introduction why we need Quasilocal Formalism?

Black Hole Thermodynamics

For electrically charged and AF dM = THawkingdS + ΦdQ, M = 2TS + ΦQ For electrically charged and AAdS dM = THawkingdS + ΦdQ + VdP, M = 2TS + ΦQ − 2PV How they will change in a quasilocal frame???? M?, THawking?, S?, Φ?, Q? (8)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 8 / 19

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Quasilocal Formalism Quasilocal Quantities : how they are defined

Tolman Temperature

Hawking Temperature THawking = 1 2π κ (9) Tolman Temperature TTolman = 1 N(R)THawking (10)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 9 / 19

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Quasilocal Formalism Quasilocal Quantities : how they are defined

Brown-York Quasilocal Formalism

The renormalized gravity action Srenormalized = SEH + SGH + SMM-counterterm = 1 16πG

M

d4x

−gR +

ǫ 8πG

∂M

d3x √ −h

K − ˆ

K

The energy-momentum boundary stress tensor

τ ab = 2 √ −h δSrenormalized δhab = − 2 √ −h

πab − ˆ

πab

The quasilocal quantities

ε = uaubτ ab, ji = −σiaubτ ab, sij = σi

aσj bτ ab

where σij is a two-dimensional induced metric.

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 10 / 19

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Quasilocal Formalism Quasilocal Quantities : how they are defined

Entropy in QLF from Euclidean Formulation

Define quasilocal free energy FR ≡ TRIE,(rh,R) (11) for example, IE = −

M

√g

1

16πG R − 1 4F 2

−
∂M

√ hF µνnµAν − 1 8πG

∂M

√ h(K − ˆ K)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 11 / 19

SLIDE 31

Quasilocal Formalism Quasilocal Quantities : how they are defined

Entropy in QLF from Euclidean Formulation

Define quasilocal free energy FR ≡ TRIE,(rh,R) = E − ΦRQ − TRS (11) for example, IE = −

M

√g

1

16πG R − 1 4F 2

−
∂M

√ hF µνnµAν − 1 8πG

∂M

√ h(K − ˆ K) quasilocal entropy S = −FR − (E − ΦRQ) TR = A 4G (12)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 11 / 19

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Quasilocal Formalism Quasilocal Quantities : how they are defined

Entropy in QLF from Euclidean Formulation

Define quasilocal free energy FR ≡ TRIE,(rh,R) = E − ΦRQ − TRS (11) for example, IE = −

M

√g

1

16πG R − 1 4F 2

−
∂M

√ hF µνnµAν − 1 8πG

∂M

√ h(K − ˆ K) quasilocal entropy S = −FR − (E − ΦRQ) TR = A 4G (12) Electric charge and potential ΦR(r) = Aµ(r)uµ(r)

rh

R

, QM =

F.

(13)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 11 / 19

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Quasilocal Smarr Relation in Asymptotically Flat spacetimes QL Smarr Relation by Euler Theorem

Construction Smarr Relation at a quasilocal frame

How can we find Smarr Relation at a quasilocal frame???

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 12 / 19

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Quasilocal Smarr Relation in Asymptotically Flat spacetimes QL Smarr Relation by Euler Theorem

QL Smarr Relation for AF by Euler Theorem

Euler theorem says E ∝ [L]D−3, S ∝ [L]D−2, A ∝ [L]D−2, Q ∝ [L]D−3,

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 13 / 19

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Quasilocal Smarr Relation in Asymptotically Flat spacetimes QL Smarr Relation by Euler Theorem

QL Smarr Relation for AF by Euler Theorem

Euler theorem says E ∝ [L]D−3, S ∝ [L]D−2, A ∝ [L]D−2, Q ∝ [L]D−3, Smarr Relation in a quasilocal frame (D − 3)E = (D − 2) ∂M ∂S

S − (D − 2)

∂M ∂A

A + (D − 3)

∂M ∂Q

Q.

(14)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 13 / 19

SLIDE 36

Quasilocal Smarr Relation in Asymptotically Flat spacetimes QL Smarr Relation by Euler Theorem

QL Smarr Relation for AF by Euler Theorem

Euler theorem says E ∝ [L]D−3, S ∝ [L]D−2, A ∝ [L]D−2, Q ∝ [L]D−3, Smarr Relation in a quasilocal frame (D − 3)E = (D − 2)TS − (D − 2)PA + (D − 3)ΦQ. (14)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 13 / 19

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Quasilocal Smarr Relation in Asymptotically Flat spacetimes QL Smarr Relation by Euler Theorem

QL Smarr Relation for AF by Euler Theorem

Euler theorem says E ∝ [L]D−3, S ∝ [L]D−2, A ∝ [L]D−2, Q ∝ [L]D−3, Smarr Relation in a quasilocal frame (D − 3)E = (D − 2)TS − (D − 2)PA + (D − 3)ΦQ. (14) The first law of thermodynamics for charged black holes is dE = TdS − PdA + ΦQ

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 13 / 19

SLIDE 38

Quasilocal Smarr Relation in Asymptotically Flat spacetimes Example

Example : Charged BH in n-dimensions

ds2 = −N(r)2dt2 + h(r)2dr 2 + r 2dΩn−2, N(r) = 1 h(r) =

1 −

µ r n−3 + q2 r 2(n−3)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 14 / 19

SLIDE 39

Quasilocal Smarr Relation in Asymptotically Flat spacetimes Example

Example : Charged BH in n-dimensions

ds2 = −N(r)2dt2 + h(r)2dr 2 + r 2dΩn−2, N(r) = 1 h(r) =

1 −

µ r n−3 + q2 r 2(n−3) Tolman temperature : TTolman =

1 N(R)THawking

Entropy : S =

ωn−2rn−2

h

4G

quasilocal area : A = ωn−2Rn−2 electric charge : Q = ±

(n−2)(n−3)

8π

qωn−2 electric potential : Φ(R) =

1 N(R)

1

8π n−2 n−3

q

rn−3

h

−

q Rn−3

Miok Park (KIAS)

Quasilocal Smarr Relations August 19, 2019 14 / 19

SLIDE 40

Quasilocal Smarr Relation in Asymptotically Flat spacetimes Example

Example : Charged BH in n-dimensions

ds2 = −N(r)2dt2 + h(r)2dr 2 + r 2dΩn−2, N(r) = 1 h(r) =

1 −

µ r n−3 + q2 r 2(n−3) Tolman temperature : TTolman =

1 N(R)THawking

Entropy : S =

ωn−2rn−2

h

4G

quasilocal area : A = ωn−2Rn−2 electric charge : Q = ±

(n−2)(n−3)

8π

qωn−2 electric potential : Φ(R) =

1 N(R)

1

8π n−2 n−3

q

rn−3

h

−

q Rn−3

From the Brown-York formalism with the background subtraction method,

Quasilocal energy : E = (n−2)

8π Sn−2Rn−3

1 −
1 −

µ rn−2 + q2 r2(n−3)

surface pressure : p = (n−3)

8πr

1
1−

µ rn−2 + q2 r2(n−3)

1 −

µ 2rn−3

− 1
r=R

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 14 / 19

SLIDE 41

Quasilocal Smarr Relation in Asymptotically Flat spacetimes Example

Example : Charged BH in n-dimensions

The first law of thermodynamics is satisfied dE = TdS + ΦRdQ + PdA

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 15 / 19

SLIDE 42

Quasilocal Smarr Relation in Asymptotically Flat spacetimes Example

Example : Charged BH in n-dimensions

The first law of thermodynamics is satisfied dE = TdS + ΦRdQ + PdA The Smarr relation is satisfied (n − 3)E = (n − 2)TS + (n − 3)ΦRQ + (n − 2)PA

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 15 / 19

SLIDE 43

Quasilocal Smarr Relation in Asymptotically AdS spacetimes QL Smarr Relation for AAdS by Euler Theorem

QL Smarr Relation for AAdS by Euler Theorem

The cosmological constant is identified as the pressure P = − Λ 8πG = (n − 1)(n − 2) 16πl2 (15) Euler theorem says E ∝ [L]D−3, S ∝ [L]D−2, A ∝ [L]D−2, Q ∝ [L]D−3, P ∝ [L]−2,

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 16 / 19

SLIDE 44

Quasilocal Smarr Relation in Asymptotically AdS spacetimes QL Smarr Relation for AAdS by Euler Theorem

QL Smarr Relation for AAdS by Euler Theorem

The cosmological constant is identified as the pressure P = − Λ 8πG = (n − 1)(n − 2) 16πl2 (15) Euler theorem says E ∝ [L]D−3, S ∝ [L]D−2, A ∝ [L]D−2, Q ∝ [L]D−3, P ∝ [L]−2, Smarr Relation in a quasilocal frame (D − 3)E = (D − 2) ∂E ∂S

S − (D − 2)

∂E ∂A

A + (D − 3)

∂E ∂Q

Q − 2

∂E ∂P

P

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 16 / 19

SLIDE 45

Quasilocal Smarr Relation in Asymptotically AdS spacetimes QL Smarr Relation for AAdS by Euler Theorem

QL Smarr Relation for AAdS by Euler Theorem

The cosmological constant is identified as the pressure P = − Λ 8πG = (n − 1)(n − 2) 16πl2 (15) Euler theorem says E ∝ [L]D−3, S ∝ [L]D−2, A ∝ [L]D−2, Q ∝ [L]D−3, P ∝ [L]−2, Smarr Relation in a quasilocal frame (D − 3)E = (D − 2)TS − (D − 2)PA + (D − 3)ΦQ − 2PV

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 16 / 19

SLIDE 46

Quasilocal Smarr Relation in Asymptotically AdS spacetimes QL Smarr Relation for AAdS by Euler Theorem

QL Smarr Relation for AAdS by Euler Theorem

The cosmological constant is identified as the pressure P = − Λ 8πG = (n − 1)(n − 2) 16πl2 (15) Euler theorem says E ∝ [L]D−3, S ∝ [L]D−2, A ∝ [L]D−2, Q ∝ [L]D−3, P ∝ [L]−2, Smarr Relation in a quasilocal frame (D − 3)E = (D − 2)TS − (D − 2)PA + (D − 3)ΦQ − 2PV The first law of thermodynamics for charged black holes is dE = TdS − PdA + ΦQ

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 16 / 19

SLIDE 47

Quasilocal Smarr Relation in Asymptotically AdS spacetimes Example

Example : Charged BH in n-dimensions

ds2 = −N(r)2dt2 + h(r)2dr 2 + r 2dΩn−2, N(r) = 1 h(r) =

r 2

l2 + k − µ r n−3 + q2 r 2(n−3) , N0(r) =

r 2

l2 + k (16)

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 17 / 19

SLIDE 48

Quasilocal Smarr Relation in Asymptotically AdS spacetimes Example

Example : Charged BH in n-dimensions

ds2 = −N(r)2dt2 + h(r)2dr 2 + r 2dΩn−2, N(r) = 1 h(r) =

r 2

l2 + k − µ r n−3 + q2 r 2(n−3) , N0(r) =

r 2

l2 + k (16) Tolman temperature : TTolman =

1 N(R)THawking

Entropy : S =

ωn−2rn−2

h

4G

quasilocal area : A = ωn−2Rn−2 electric charge : Q = ±

(n−2)(n−3)

8π

qωn−2 The electric potential : ΦR =

1 N(R)

32π(n−2)

(n−3)

q

Rn−3 − q rn−3

h

pressure : P = (n−1)(n−2)

16πl2

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 17 / 19

SLIDE 49

Quasilocal Smarr Relation in Asymptotically AdS spacetimes Example

Example : Charged BH in n-dimensions

From the Brown-York formalism with the background subtraction method, Quasilocal energy : E = 2(n−2)

κ

ωn−2Rn−3

N(R) − N0(R)
surface pressure :

p =

−l2(n−3)(kr n

h(R3r n−2 h

r 3

h Rn)+q2R3r 6 h)−R2r n+2 h

((n−3)Rr n−2

h

(n−2)rhRn) κl2r n−3

h

Rn−1N(R)

The thermodynamic volume : V = ∂E

∂P = 16πωn−2 (n−1)κ

(Rn−1−r n−1

h

) N(R)

− Rn−1

N0(R)

Miok Park (KIAS)

Quasilocal Smarr Relations August 19, 2019 18 / 19

SLIDE 50

Quasilocal Smarr Relation in Asymptotically AdS spacetimes Example

Example : Charged BH in n-dimensions

From the Brown-York formalism with the background subtraction method, Quasilocal energy : E = 2(n−2)

κ

ωn−2Rn−3

N(R) − N0(R)
surface pressure :

p =

−l2(n−3)(kr n

h(R3r n−2 h

r 3

h Rn)+q2R3r 6 h)−R2r n+2 h

((n−3)Rr n−2

h

(n−2)rhRn) κl2r n−3

h

Rn−1N(R)

The thermodynamic volume : V = ∂E

∂P = 16πωn−2 (n−1)κ

(Rn−1−r n−1

h

) N(R)

− Rn−1

N0(R)

The first law of thermodynamics is satisfied

dE = TdS + ΦRdQ + PdA + VdP

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 18 / 19

SLIDE 51

Quasilocal Smarr Relation in Asymptotically AdS spacetimes Example

Example : Charged BH in n-dimensions

From the Brown-York formalism with the background subtraction method, Quasilocal energy : E = 2(n−2)

κ

ωn−2Rn−3

N(R) − N0(R)
surface pressure :

p =

−l2(n−3)(kr n

h(R3r n−2 h

r 3

h Rn)+q2R3r 6 h)−R2r n+2 h

((n−3)Rr n−2

h

(n−2)rhRn) κl2r n−3

h

Rn−1N(R)

The thermodynamic volume : V = ∂E

∂P = 16πωn−2 (n−1)κ

(Rn−1−r n−1

h

) N(R)

− Rn−1

N0(R)

The first law of thermodynamics is satisfied

The Smarr relation for AF (n − 3)E = (n − 2)TS + (n − 3)ΦRQ + (n − 2)PA The Smarr relation for AAdS (n − 3)E = (n − 2)TS + (n − 3)ΦRQ + (n − 2)PA − 2PV Future works Quasilocal Smarr relation for hairy black holes or dyonic black holes in an asymptotically flat spacetime? How about employing other quasilocal formalism? Can we modify Komar formula so as to derives the Smarr relation in a quasilocal frame? Interpretation of thermodynamic volume for AdS Black holes

Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 19 / 19