Formulation of Spacetime Thermodynamics

Yuki Yokokura

(Kyoto University)

At Osaka University, ‘11 12/6

Introduction 1 Black Hole Thermodynamics

• 0th law: 𝑇𝐵𝐻 =𝜅𝐵𝐻

2𝜋

• 1st law: δ𝑀𝐵𝐻 = 𝑇𝐵𝐻𝛿𝑆𝐵𝐻 + Ω𝐵𝐻𝛿𝐽 • 2nd law: Generalized Second Law (GSL)

𝛿𝑆 = 𝛿𝑆𝑚𝑎𝑡𝑡𝑒𝑟 + 𝛿𝑆𝐵𝐻 ≥ 0

BH TBH

Hawking Radiation[Hawking 1975]

⇒Thermal equilibrium

< 𝑁𝜔 >=Γ𝜔

exp𝜔𝑇𝐵𝐻

− 1

𝑆𝐵𝐻 =1

4𝐴𝐵𝐻

𝑇𝐵𝐻 =1

8𝜋𝑀𝐵𝐻

Hawking BH temperature BH entropy

Introduction 2 Gibbons-Hawking’s result

・the partition function of a Schwarzschild BH by Euclidean path integral. [Gibbons and Hawking 1977]

・a vacuum solution of the Einstein eq.

⇒absence of conical singularity

⇒a unique equilibrium energy

⇒free energy

x

periodic

x

a

𝑍 = 𝑑[𝑔] exp −𝐼 𝑔 ≅ exp(−𝐼 𝑔0 )

𝑀𝐵𝐻 =𝛽

8𝜋

𝑈 − 𝑇𝑆 = 𝐹 = −𝛽 − 1𝑙𝑜𝑔𝑍 ≅ 𝑀𝐵𝐻 − 𝛽− 11

4𝐴𝐵𝐻

My motivation: Why does a single gravitational configuration create the finite statistical entropy?

⇒A gravitational configuration corresponds to a thermodynamic state?

𝑆 =𝐴𝐵𝐻4= 𝑙𝑜𝑔Ω

Introduction 3 Jacobson’s idea

• Jacobson showed that the Einstein equation can be regarded as the equation of state for spacetime. [Jacobson 1995] For a part of any spacetime,

𝑇𝛿𝑆 = 𝛿𝑄 ⇒ 𝑅𝑎𝑏 −1

2𝑅𝑔𝑎𝑏 + Λ𝑔𝑎𝑏 = 8𝜋𝐺𝑇𝑎𝑏

His assumptions: 1 all energy through the

observer’s horizon = heat 𝛿𝐸 = 𝛿′𝑄

2 the entropy area law

𝛿𝑆 =𝛿𝐴

4

3 the Unruh effect

𝑇𝑈 =𝑎

2𝜋

T

X

external world

system

δE=δ’Q

χ

Our Observer

P

causal horizon

k Jacobson’s Observer

Introduction4 Can a part of any spacetime be really regarded as a

thermodynamic system?

⇒probably, no! But I pointed out unnaturalness of Jacobson’s discussion and tried to reconstruct the discussion in the following ways. What I tried: • Introduce outside observer, rearrange the

discussion and construct the first law for non-equilibrium processes

• Generalize to the f(R) gravity • Introduce ``work term”

• 1 Introduction

• 2 BH entropy from various viewpoints

• 3 Jacobson’s discussion

• 4 problems for Jacobson’s idea

• 5 trying to reconstruct

• 6 summery and discussions

2-1-1 Hawking’s discussion • Dust ⇒gravitational collapse ⇒Schwarzschild BH

Question:

What do we observe in asymptotic flat region?

⇒QFT in time-dependent curved space predicts Hawking radiation!

Event horizon

Singularity

Dust surface

r

t

P

< 𝑁𝜔 >=Γ𝜔

exp𝜔𝑇𝐵𝐻

− 1

𝑇𝐵𝐻 =1

8𝜋𝑀𝐵𝐻

graybody factor

2-1-2 BH entropy from thermodynamic viewpoint

• BH can be equilibrium

with heat bath of temperature TBH.

⇒BH can work as heat bath of TBH in Carnot cycle.

⇒BH has the thermodynamic entropy.

(Hawking’s picture is micro-canonical viewpoint.)

BH TBH

BH

V’

W

-Q

𝑆𝐵𝐻 𝑀𝐵𝐻 = 𝑑𝑀1

𝑇𝐵𝐻 𝑀=1

4𝐴𝐵𝐻 = 𝑙𝑜𝑔Ω(𝑀𝐵𝐻)

𝛿𝑆𝐵𝐻 =𝛿𝑄

𝑇𝐵𝐻

2-2-1 Gibbons-Hawking’s discussion

• Image a BH in heat bath of T.

⇒Canonical viewpoint

⇒𝐹(𝑇;𝑀) = −𝑇𝑙𝑜𝑔𝑍(𝑇;𝑀)

⇒equilibrium condition 𝑑𝐹

𝑑𝑀= 0

⇒𝑀 =1

8𝜋𝑇

⇒𝐹 𝑇 =1

16𝜋𝑇

⇒𝑆 𝑇 = −𝜕𝐹

𝜕𝑇=

1

16𝜋𝑀2 =1

4𝐴𝐵𝐻(𝑇)

BH T

2-2-2 Gibbons-Hawking’s derivation HOW TO? ・Euclidean QFT ・WKB approximation ・Pure gravity

𝐹 𝑇;𝑀 = −𝑇𝑙𝑜𝑔𝑍(𝑇;𝑀) ≅ 𝑇𝐼[𝑔0(𝑇;𝑀)]

𝐼 𝑔0 𝑇;𝑀 = −1

16𝜋 𝑑4𝑥 𝑔𝑅𝑉

−1

8𝜋 𝑑3𝑦 𝑕𝜕𝑉

𝐾 − 𝐾0 −1

16𝜋2α𝐴

τ

ｒ

Conical singularity

a=2M

τ-periodic β(1-a/r)1/2 Deficit angle α

developed figure

ｒ

deficit angle : 𝛼 = 2𝜋 −𝛽

2𝑎

Einstein-Hilbert Gibbons-Hawking Gauss-Bonnet

𝑍 = 𝑑[𝑔] exp −𝐼 𝑔 ≅ exp(−𝐼 𝑔0 )

2-2-3 BH entropy from statistical viewpoint

• In equilibrium

⇒A gravitational configuration corresponds to a thermodynamic state?

𝑆𝐵𝐻 𝑇 =𝐴𝐵𝐻4= 𝑆 𝑀𝐵𝐻 = 𝑘𝐵𝑙𝑜𝑔Ω 𝑀𝐵𝐻 > 0

A single gravitational configuration g0 creates! Finite statistical entropy!

2-2-4 Gravity has a duality?(1)

12

<Other interaction> effective theory of low energy effective theory of QCD finite temperature QCD ≠ (T=0) (equation of state) (Chiral lagrangian) ⇒A single classical configuration of finite temperature chiral lagrangian dose not create finite entropy.

<Gravity> effective theory of low energy effective theory finite temperature string theory = of string theory (T=0) (General relativity) (General relativity) ⇒A single classical configuration of finite temperature general relativity creates finite entropy.

13

viewpoint B: thermodynamic effective theory Gravity = entropic force Einstein eq.=eq. of state

viewpoint A: low energy effective theory Gravity = fundamental interaction Einstein eq.=eq. of motion

⇒Gravity has two different properties simultaneously. ⇒A duality?

2-2-4 Gravity has a duality?(2)

• 1 Introduction

• 2 BH entropy from various viewpoints

• 3 Jacobson’s discussion

• 4 problems for Jacobson’s idea

• 5 trying to reconstruct

• 6 summery and discussions

3-1 Jacobson’s idea

• Jacobson showed that the Einstein equation can be regarded as the equation of state for spacetime. [Jacobson 1995] For a part of any spacetime,

𝑇𝛿𝑆 = 𝛿𝑄 ⇒ 𝑅𝑎𝑏 −1

2𝑅𝑔𝑎𝑏 + Λ𝑔𝑎𝑏 = 8𝜋𝐺𝑇𝑎𝑏

His assumptions: 1 all energy through the

observer’s horizon = heat 𝛿𝐸 = 𝛿′𝑄

2 the entropy area law

𝛿𝑆 =𝛿𝐴

4

3 the Unruh effect

𝑇𝑈 =𝑎

2𝜋

T

X

external world

system

δE=δ’Q

χ

External Observer

P

causal horizon

k Jacobson’s Observer

3-2 System, External world, and Heat • In general, heat is transfer of energy which cannot be

identified and controlled by an external observer.

• ⇒ In spacetime thermodynamics, heat can be defined as energy flow through any causal horizon.

• ⇒The system is defined as the region inside the horizon, and the external world as the outside.

• ⇒A conventional observer is

defined as an observer in the

external world, who measures

the thermodynamic quantities.

T

X

external world

system

δE=δQ

Observer

causal horizon

3-3 Jacobson’s setup • Take a local inertial frame near any point P

⇒uniformly accelerating observer

⇒Rindler horizon for him

⇒spacetime thermodynamic system

・By using an accelerating observer𝜒in the system:

estimate the energy flow 𝛿𝐸 with affine parameter 𝜆:

(Near horizon limit 𝑥 → 0, 𝜒 → 𝑘)

・Unruh temperature

T

X

external world

system

δE=δQ

χ

External Observer

P

causal horizon

dΣ

k Jacobson’s Observer

𝛿𝑄 = 𝛿𝐸 = 𝑇𝑎𝑏𝜒𝑎𝑑Σ𝑏 ≈ −𝑥 − 1 𝑇𝑎𝑏𝑘

𝑎𝑘𝑏𝜆𝑑𝜆 𝑑𝐴

𝑇 = 𝑥 cosh 𝑡 , 𝑋 = −𝑥 sinh 𝑡 ,

𝑇𝑈 =𝑥 − 1

2𝜋

𝛿𝑄

𝑇≈ −2𝜋 𝑇𝑎𝑏𝑘

𝑎𝑘𝑏𝜆𝑑𝜆 𝑑𝐴

3-4 Jacobson’s derivation1

• Entropy change=area change

• The affine-parameterized Raychaudhuri eq.

4

AS

G ⇒

4

AS

G

HA d dA

F

ak

P

Expansion 𝜃 =1

Δ𝐴

𝑑Δ𝐴

𝑑𝜆

a b

abH

A R k k d dA

𝑑𝜃

𝑑𝜆= −

1

2𝜃2− 𝜎2− 𝑅𝑎𝑏𝑘

𝑎𝑘𝑏

Assumption: Local equilibrium 𝜃 = 𝜎 = 0

𝜃 = −𝜆𝑅𝑎𝑏𝑘𝑎𝑘𝑏

• This holds for any point.

⇒

• This holds for any null vector.

⇒

• Energy conservation

• Bianchi id.

⇒Einstein eq.

𝛿𝑆 =𝛿𝑄

𝑇𝑈⟹−

1

4𝐺 𝑑𝜆𝑑𝐴𝜆𝑅𝑎𝑏𝑘

𝑎𝑘𝑏 = −2𝜋 𝑑𝜆 𝑑𝐴𝜆𝑇𝑎𝑏𝑘𝑎𝑘𝑏

𝑅𝑎𝑏𝑘𝑎𝑘𝑏 = 8𝜋𝐺𝑇𝑎𝑏𝑘

𝑎𝑘𝑏

𝑅𝑎𝑏 + 𝑓𝑔𝑎𝑏 = 8𝜋𝐺𝑇𝑎𝑏

𝑅𝑎𝑏 −1

2𝑅𝑔𝑎𝑏 + Λ𝑔𝑎𝑏 = 8𝜋𝐺𝑇𝑎𝑏

𝛻𝑏 𝑅𝑎𝑏 −

1

2𝑔𝑎𝑏𝑅 = 0

𝛻𝑏𝑇𝑎𝑏 = 0

3-4 Jacobson’s derivation2

• 1 Introduction

• 2 BH entropy from various viewpoints

• 3 Jacobson’s discussion

• 4 problems for Jacobson’s idea

• 5 trying to reconstruct

• 6 summery and discussions

4-1 unnatural observer

T

X

external world

system

δE=δ’Q

χ

External Observer

P

causal horizon

dΣ

k Jacobson’s Observer

Jacobson’s observer = observer in the system

Observer in thermodynamics = observer out of the system

Jacobson’s formulation cannot be applied to BH thermodynamics.

If closed system, 𝜃 ≠ 0. ⇒non-stationary!

4-2 What is the entropy?

• The Carnot cycle cannot be constructed.

⇒Thermodynamic entropy cannot be introduced!

• If Information entropy or entanglement entropy

⇒Entropy can diverge!

system

𝜃 =2

𝑟inflatspacetime

Ex1. spherical light Ex2. “light box”

𝜃 = 0 on faces

𝜃 ≠ 0 on the corners

Jacobson’s system = open system with𝜃 = 0

4-3 Is the Unruh effect true? • If an observer is accelerating uniformly forever,

he can feel the Unruh temperature.

• However, such an observer does not exist!

• Cf. A uniformly rotating

observer dose not feel

the Unruh effect. [Davies, Dray, and Manogue 1996]

T

X

accelerating observer

P

4-4 Why is there no work term?

Thermodynamic 1st law: δ𝑀𝐵𝐻 = 𝑇𝐵𝐻𝛿𝑆𝐵𝐻 + Ω𝐵𝐻𝛿𝐽

Thermodynamic 1st law: 𝛿𝑈 = 𝛿′𝑄 + 𝛿′𝑊

Jacobson’s assumption: 𝛿𝐸 = 𝛿′𝑄+?

• 1 Introduction

• 2 BH entropy from various viewpoints

• 3 Jacobson’s discussion

• 4 problems for Jacobson’s idea

• 5 try to reconstruct

• 6 summery and discussions

5-1 an observer out of the local system near the BH event horizon

• Stationary BH(𝜃 = 0) = equilibrium thermodynamic system

⇒a surface patch = a local thermodynamic system

⇒an external observer near the patch = an natural observer

<question>

What form dose the thermodynamic 1st law of the local system take in the following process?

ＢＨ１ ＢＨ２

Equilibrium state1 S1 Equilibrium state2 S2

Non-equilibrium process

Hawking radiation Gravitational wave

matter flow

stretched horizon

ＢＨ

𝜃 = 0

5-2 introduce external observer • The external observer:

• Affine parameter𝜆⇒proper time𝜏 = 𝑥𝑡

T

X

external world

system δE=δQ

u Our Observer

P

causal horizon

n

k

𝑇 = 𝑥 cosh 𝑡 , 𝑋 = 𝑥 sinh 𝑡

𝛿𝐸 = 𝑑𝜏 𝑑𝐴𝑇𝑎𝑏𝑢𝑎𝑛𝑏

𝑆(𝜏)

𝜏2

𝜏1

≈ 𝑥 − 1 𝑑𝑡𝑡2

𝑡1

𝑑𝐴𝑇𝑎𝑏𝑘𝑎𝑘𝑏

𝑆(𝑡)

𝑇𝑈 =𝑥 − 1

2𝜋

Expansion 𝜃 =1

Δ𝐴

𝑑Δ𝐴

𝑑𝜏

𝑑𝜃

𝑑𝜏= 𝑥 − 1𝜃 −

1

2𝜃 2− 𝜎 2− 𝑅𝑎𝑏𝑘

𝑎𝑘 𝑏

𝑢 =𝜕

𝜕𝜏=1

𝑥

𝜕

𝜕𝑡

Local equilibrium 𝜃 = 𝜎 = 0at 𝜏 = 𝜏1, 𝜏2

stretched horizon

5-3 1st law for Einstein’s gravity

δＥ＝δ’Ｑ (matter only)

External local temperature

・event horizon⇒time-asymmetry ・σ～dynamical gravitational effect ⇒dissipation

The negative coefficient only for

dynamical processes

𝑇𝑒𝑥𝛿𝑆 = 𝛿′𝐷 + 𝛿′𝑄 = 𝛿𝑈

``local ADM energy “= gravitational energy + matter energy

1st law 2nd law

5-4 1st law for f(R) gravity < f(R) gravity >

・Action ・BH entropy

Additional new term

𝑇𝑒𝑥𝛿𝑆 = 𝛿′𝐷 + 𝛿′𝑄 = 𝛿𝑈

The coefficients depend on spacetime points.

• 1 Introduction

• 2 BH entropy from various viewpoints

• 3 Jacobson’s discussion

• 4 problems for Jacobson’s idea

• 5 try to reconstruct

• 6 summery and discussions

6 Summery and Discussions • BH can be equilibrium with heat bath by Hawking radiation. ⇒BH has thermodynamic entropy. • Gibbons-Hawking’s result indicates that a gravitational

configuration corresponds to a thermodynamic state. ⇒Gravity has a duality? • Jacobson showed that the Einstein equation can be regarded as

the equation of state for spacetime. ⇒ However, probably, a part of any spacetime cannot be regarded as a thermodynamic system. What I tried: • Introduce outside observer, rearrange the discussion and

construct the first law for non-equilibrium processes • Generalize to the f(R) gravity • Introduce ``work term”

Outlook

• Micro counting finite temperature BH entropy

• Understanding Gibbons-Hawking’s result

• Information Paradox

⇒Back reaction from Hawking radiation is important.

Thank You!