Spacetime Thermodynamics from Geometric Point of View

Yu Tian （田雨）Department of Physics,

Beijing Institute of Technology

OUTLINE

• Brief Introduction to Thermodynamics of Black Holes, de Sitter and Other Spacetimes

• Spacetime Thermodynamics from Higher Dimensional Global Embedding Minkowski Spacetimes (GEMS)

• Possible Thermodynamics of Causal Diamonds, Its Relation to GEMS, and Implications

Black Hole Thermodynamics

Kerr-Newman black hole:

The first law of black holes (in natural units):

, , 2 4

dU TdS dJ dQ

AU M T S

2 22 2 2 2 2 2 2 2 2 2 2

2

2 2 2 2 2 2 2

2( )sin ( sin )

2 , cos , /

Mr Qds dt dr d r a d a d dt

r Mr a Q r a a J M

Black holes are really like black bodies in thermodynamics?S.W. Hawking’s answer: yes![Commun. Math. Phys. 43 (1975) 199]

Hawking Radiation

Schwarzschild black hole:

Spectrum for static observer at infinity outside the black hole

/

1, ( for Schwarzschild black hole)

1 2 4TN T

e M

12 2 2 2 2 2 22 2

1 1 ( sin )M M

ds dt dr r d dr r

(Heuristic) physical picture:Particle-antiparticle pair creation (quantum tunneling)

Can this picture be physically realized?M.K. Parikh & F. Wilczek’s answer: yes![Phys. Rev. Lett. 85 (2000) 5042]

Unruh Effect

Rindler transformation for Minkowski spacetime (in L and R wedges)

2 2 2 2 2 2 2ds a d d dy dz

x

t

const.

const.

horizon

at 0

W.G. Unruh, Phys. Rev. D 14 (1976) 870.

For observer staying at

/

1,

1 2T

aN T

e

1/ a

00/PT T g

“+” for fermions

, 1/ .P P Pa For observer staying at

Tolman relation:

Comparison Between the Two Effects

• Schwarzschild BH

• Maximal Kruskal extension

• Horizon intrinsic to the spacetime

• Information lost due to the horizon

• Thermal spectrum detected by static observer

• Rindler Spacetime

• Maximal Minkowski extension

• Horizon associated with a particular observer

• Information lost due to the horizon

• Thermal spectrum detected by static observer

4-dim de Sitter spacetime in static coordinates:12 2

2 2 2 2 2 2 22 2

1 1 ( sin )r r

ds dt dr r d dR R

For inertial observer,

G.W. Gibbons & S.W. Hawking, Phys. Rev. D 15 (1977) 2738.

1 .2T R

de Sitter Thermodynamics

33R

What is GEMS?

4-dim de Sitter spacetime in FRW coordinates:1k

diag(1, 1, 1, 1, 1)

, 0, , 4AB

A B

For anti-de Sitter spacetime:diag(1, 1, 1, 1,1)

, 0, , 4AB

A B

For Schwarzschild black hole:

2

diag(1, 1, 1, 1, 1, 1)

A BAB

AB

ds dz dz

C. Fronsdal, Phys. Rev. 116 (1959) 778.

Is this embedding global?

S. Deser and O. Levin, Phys. Rev. D 59 (1999) 064004.

Kruskal transformation for 2r M

4

4

1sinh2 4

1cosh2 4

rM

rM

r tv e

M M

r tu e

M M

Mapping Hawking into Unruh

H. Narnhofer et al’s key observation [Int. J. Mod. Phys. B 10 (1996) 1507]

Inertial Observer in de Sitter = Rindler Observer in GEMS

static metri

c

horizon of inertial observer in dS = Rindler horizon dS in GEMS

( 0)r

• The 4-dimensional Side

Hawking Temperature

• The 5-dimensional Side

Unruh Temperature

The entropies in two sides are also matched!

H

1 1

2T

R R

5 U

4

1 1

2( 0)

a TR R

a

Generalization to constantly accelerated observer in (A)dS?

S. Deser and O. Levin, Class. Quant. Grav. 14 (1997) L163.

pseudo-circular motion in (A)dS

const. in static (A)dSr

2 25 4 2 for (A)dSPa a R T

Generalization to various black holes?

S. Deser and O. Levin, Class. Quant. Grav. 15 (1998) L85; Phys. Rev. D 59 (1999) 064004.

For Schwarzschild black hole:

• The surface gravity is

• A static observer P at r detects a local temperature

• The corresponding observer in the GEMS is effectively a Rindler observer with proper acceleration

1

4M

00

1

28 1

P

TT

g MM

r

6

1 12

24 1

Pa TM

Mr

For RN-AdS black hole:

• The thermal spectrum is

• The chemical potential is

• But where are the chemical potential in the GEMS?

+

qQ

r

( ) /

1

1TN

e

Generalization to Stationary Motions

What lesson can we learn from the Rindler transformation?

The inertial observer in Minkowski spacetime follows an integral curve of Killing vector field (ignoring Pi without loss of generality)

The Rindler observer follows a (timelike) integral curve of Killing vector field

xK x tt x

Ht

Recall that besides H (Pi) and Ki, there are also other independent Killing vector fields in Minkowski spacetime:

Any linear combination of the above fields are Killing vector field, which gives a set of integral curves leading to all stationary motions and the corresponding Rindler-like transformation.

ji ijk k

J xx

J.R. Letaw, Phys. Rev. D 23 (1981) 1709; J.R. Letaw and J.D. Pfautsch, Phys. Rev. D 24 (1981) 1491.

Quantization and vacuum structures in all stationary coordinate systems of Minkowski spacetime:

6 classes of stationary coordinate systems have 2 types of vacua:

1. without horizon: Minkowski vacuum;

2. with horizon: Fulling vacuum.

J.I. Korsbakken and J.M. Leinaas, Phys. Rev. D 70 (2004) 084016.

Geometric aspects of all the stationary coordinate systems & presence of the chemical potential:

emergence of ergosphere = deviation of positive norm & positive frequency

How to apply it to the black hole case?

For spherically symmetric black holes, the independent Killing vector fields are H and Ji, whose linear combinations give integral curves leading to all stationary motions. Without loss of generality,

zH H J

circular motions around the BH with uniform angular velocity

• General stationary motions:

• Transform to rest frame of the detector:

• The metric becomes stationary and axisymmetric but not asymptotically flat:

Generalization for Schwarzschild BH

H.Z. Chen, Y. Tian, Y.-H. Gao & X.-C. Song, JHEP 0410 (2004) 011.

• We can obtain the thermal spectrum (for example, using the Damour-Ruffini method) detected by an observer at rest in this spacetime, with chemical potential:

m: magnetic quantum number

m

• To get the local quantities, we divide by the red shift factor and finally get:

with

• On the GEMS side, the detector is in an effectively 4-dimensional Rindler motion superposed with a circular motion in the transverse directions, which is a stationary motion in the 6-dimensional Minkowski spacetime;

• The thermal spectrum detected by this GEMS detector can be obtained by the method of Korsbakken and Leinaas, whose temperature and chemical potential exactly match the results obtained above.

The GEMS Side of Schwarzschild BH

Further generalization to stationary motions in RN black holes, matching the whole spectrum including the chemical potential:

H.-Z. Chen & Y. Tian, Phys. Rev. D 71 (2005) 104008.

Thermal Time Hypothesis

• Basic idea: The foundation of (the flow of) physical time is thermodynamics (or statistics), but not dynamics, i.e. the flow of physical time dependents on the quantum statistical states of the system under consideration.

• Key points:1. Modular flow () gives the flow of physical time;2. If there is a flow of geometric (proper) time s proportional to the modular flow, then an inverse temperature can be defined such that 2s .

• Applications (for example, in cosmology):C. Rovelli, Class. Quantum Grav 10 1567 (1993).

Thermal time hypothesis (C. Rovelli, to understand the concept of time in quantum gravity):

A Spacetime Region O Vacuum State

Local Observable Algebra A Hilbert Space HAct on

Single-Parameter () Automorphism (Modular Flow)

Belong to

Act

on

Indu

ce A

ctio

n on

Com

pare

KMS Condition

Unruh Effect Revisited

• To thoroughly understand Unruh effect is the key to understand general spacetime thermodynamics;

• There is still no experimental test of Unruh effect;

• Unruh effect revisited from the viewpoint of thermal time hypothesis (and its extension to the causal diamond case): P. Martinetti & C. Rovelli, Class. Quant. Grav. 20 (2003) 4919 [gr-qc/0212074].

Take O to be one of the Rindler wedges (say, the R wedge X > |T|).

The world line of a Rindler observer is a pseudo-circle:2 2 2T X a

The well-known modular flow associated with the R wedge is along the world line of Rindler observers, which can be written as

1

1

sinh

cosh

T a

X a

1

1

sinh

cosh

T a as

X a as

The same world line parameterized in the proper time s is

So we get 2a, which agrees with the familiar result of Unruh temperature.

Causal Diamond of a Non-Eternal Observer

Information lost due to the finite lifetime of the observer

Associated thermodynamics?

Martinetti & Rovelli’s generalization of the thermal time hypothesis: local temperature

causal diamond

Diamond’s Temperature

How to obtain the modular flow associated with the causal diamond |X|+|T| < L?

Through conformal transformationsP.D. Hislop & R. Longo, Comm. Math. Phys. 84 (1982) 71.

L

L

O

Higher dimensional diamond:0x x L

( ,0,0,0), ( ,0,0,0)i fx L x L

The conformally transformed modular flow is still along pseudo-circles, which is world lines of uniformly accelerated observers (now with finite lifetime). The world line of an observer from xi to xf with constant proper acceleration a is

Modular flow associated with the above causal diamond can be written as

2 2 2 2 2 1( ) sinhT X a L a T a as

2 2

2

2 2

sinh

cosh 1

cosh 1

LT

a L

aLX

a L

Differentiating the above two expressions for T and using the definition of local temperature finally gives

2 20a L s

L

Unruh-like effect?

characteristic temperature of the causal diamond

Generalization to the (A)dS Case

Y. Tian, JHEP 0506 (2005) 045.

Consistency check: A uniformly accelerated observer in dS spacetime in the viewpoint of thermal time hypothesis.

An observer with r const. in static dS spacetime is of a const. acceleration

2 2

ra

R R r

The causal “diamond” of this observer is the region embraced by the corresponding static horizon, which looks unlike a diamond.

How to obtain the modular flow associated with this region?

Through conformal mapping

The most intuitive conformal mapping from dS spacetime to Minkowski spacetime is the (pseudo-)stereographic projection from the embedding point of view:

44

Projection center: (0,0,0,0, ) 2

Projection hyperplane:

P R Rx

RR

A two-dimensional sketch map of the conformally flat coordinates on the dS spacetime. All the points on the plane except those on the hyperbola, which is actually the conformal boundary of the dS spacetime, are points on the dS spacetime. The diamond embraced by the dashed lines is the region covered by the static coordinates. The solid line segment is the world line of the inertial observer, while the solid segment of a hyperbola is the world line of the observer staying at r R/2.

The world line of the observer with const. r is

22 2 24

4R

T X X Rr

Direct application of the known result in Minkowski spacetime with L 2R gives the corresponding modular flow:

2 2

2 2

2 2

2 sinh

cosh /

2 /

cosh /

RT

R R r

Rr R rX

R R r

The above modular flow leads to a temperature

This result agrees with that from the conventional approach (horizon + Tolman relation) or the GEMS approach.

Further generalization: A uniformly accelerated observer with finite lifetime in dS spacetime in the viewpoint of thermal time hypothesis.

2 2

2 sinh2

cosh

RRL MR

R R r

An observer with r const. & < t < in static dS spacetime is associated with a reduced causal diamond with

The corresponding modular flow is

Finally we obtain a simple form of local temperature in terms of the static time t:

Comparison with GEMS

Simple observation:A uniformly accelerated observer with finite lifetime in the dS spacetime corresponds to a uniformly accelerated observer with finite lifetime in the GEMS.

2 25 1/a R r

The local temperature of a uniformly accelerated observer with finite (proper) lifetime < s < in the 5-dim Minkowski spacetime:

2 21 /s t r R

Compatible: Conformal Mapping & GEMS (both using the thermal time hypothesis)

Open Questions

• Does the GEMS approach make physical sense?

• Does the thermal time hypothesis make physical sense?

• Is there satisfactory definition of entropy for causal diamonds?

• Can the thermodynamics of causal diamonds be really established?

Thank You!