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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.