AdS Black Holes Thermodynamicsand Hamilton-Jacobi Formalism

Bo Shi

Master Thesis

Supervised by Prof. Erik. P. Verlinde

Instituut voor Theoretische Fysica

August 2010

“Working hard is the only thing you can do”

Abstract

In this thesis, we propose a general metric ansatz, which has two space-timesolutions. And we use the correspondence between the thermodynamics ofSchwarzschild black holes in AdSd+1 space and N = 4 super Yang-Millstheory on the d dimensional boundary to relate the partition function andspace-time path integral formulation, which enables us to derive the thermalquantities. A validity for this ansatz is proved by rederiving the phase tran-sition argued by E. Witten previously. Using Hamilton-Jacobi formalism, wealso study the holographic renormalization group(RG) equation. And aftersubstituting the identified thermal quantities into this equation, we reach athermodynamical interpretation from the RG as the equation of state. Itgives a relation between energy and pressure on a layer of AdS space-time,with the squared energy term as a correction.

Contents

1 Introduction 1

2 Black Hole Thermodynamics 32.1 Classical black holes . . . . . . . . . . . . . . . . . . . . . . . 32.2 Quantum black hole . . . . . . . . . . . . . . . . . . . . . . . 4

3 On the AdS-Schwarzschild Black Hole Phase Transition 63.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Phase transition and entropy . . . . . . . . . . . . . . . . . . . 8

4 Generalized Calculation 104.1 Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Hamiltonian constraint . . . . . . . . . . . . . . . . . . . . . . 114.3 Anti-de Sitter space-time . . . . . . . . . . . . . . . . . . . . . 124.4 AdS-Schwarzschild black hole space-time . . . . . . . . . . . . 144.5 Rederive thermal phase transition . . . . . . . . . . . . . . . . 15

5 Thermodynamics 165.1 General form . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Quantities for both space-times . . . . . . . . . . . . . . . . . 185.3 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Renormalization Group Equation 206.1 A little bit of string theory . . . . . . . . . . . . . . . . . . . . 206.2 Derivative-like expansion . . . . . . . . . . . . . . . . . . . . . 216.3 Reduced Callan-Symanzik equation . . . . . . . . . . . . . . . 23

7 Conclusion 26

i

A Schematic of the Reasoning 27

B Penrose Process 28

C Ricci Scalar for Diagonalized Metric 29

D Anti de Sitter Space 31

E Derivative Expansion in General Form 32

F Callan-Symanzik Equation 33

G Hamilton-Jacobi formalism 34

Acknowledgements 36

ii

Chapter 1

Introduction

Thermodynamical properties of Black holes have been studied tensely sinceBekenstein’s paper appeared. It proposed the conjecture between entropyand black hole event horizon area. Now the subject grows very rich. It isfollowed by a lot of profound ideas, such as black hole radiation and holo-graphic principle. Recently, Erik Verlinde argues that gravity is an emergedforce with statistical origin. It drags the attention back again to the ther-modynamical properties of the black holes. In this sense, it is very usefulto study this subject concerning some specific black hole space-times. Thethermodynamical approach of black holes is reviewed in general, and so doesthe work performed by E. Witten on thermodynamical phase transition char-acterized by different horizons. Inspired by Witten, we introduce a metricansatz which has solutions of space-times such as AdS and AdS-Schwarzschildblack hole. However, the different reasoning here is that we keep the radiusfinite and explore the thermodynamics of black holes as a function of theradial parameter.

In this thesis, the Hamilton-Jacobi(H-J) equation plays an essential role.As indicated by Jan de Boer, Erik Verlinde and Herman Verlinde(dBVV),in five dimensional AdS space-time, the AdS/CFT correspondence with thehelp of Hamilton-Jacobi equation, can yield the so-called renormalizationgroup(RG) equation, which illustrates the relation between the holographicprinciple and renormalization theory. From this approach, we derive thethermodynamical interpretation for this RG equation, which is an originalresult.

1

We work with (d + 1) dimensional gravity, which has the metric func-tion only depending on one of the coordinates, except the angular coordi-nates within the (d− 1) ball. The geometrical structure of this space-time is(R1× S1× Sd−1) (we assume the (d− 1) dimension space-time has sphericalsymmetry). We start from the classical space-time action. It of course con-firms the classical Hamilton-Jacobi formalism on a layer of the space-timeproduced by the ADM decomposition. We reduce this H-J equation into aRG differential equation with respect to the main ansatz. This equation isexpected to have a thermodynamical interpretation.

The contents of the thesis are ordered as follows. Starting from chaptertwo, we review some former works on thermodynamics of black holes. Withthe analogy between the four properties of classical black holes and the nor-mal four laws of thermodynamics, we set up a stage for later discussion. Wehighlight the identity given by S. W. Hawking, which connects the parti-tion function to the space-time path integral in canonical ensemble [7]. Inchapter three, we review the calculation that Witten performed for thermalphase transition, and the entropy-law [4], which is considered as a general-ization to Hawking’s arguments [7]. In chapter four, we propose an ansatz,and then expand the calculation to a more general form. We calculate theLagrangian, Hamiltonian constraint as well as the flow equations. Serving asa verification, we rederive the phase transition that is coherent with Witten’scalculation. The following chapter deals with the thermodynamical features.With the derivation of the thermodynamical quantities for our concerningspace-times, we identify the derivatives of the classical space-time action toenergy and pressure, which enables the thermodynamical interpretation forlater discussion. In chapter six, we carry out the standard procedure givenby dBVV on the holographic renormalization group. We derive a reducedCallan-Symanzik equation as the flow equation of the metric fields, whichlooks like the equation of state in thermodynamical limits. In the end, weconclude that the geometrical (gravitational) properties of the space-timeare closely related to the statistical interpretations of it, which is emphasizedand deepened by Erik Verlinde in his revolutionary paper [11].

2

Chapter 2

Black Hole Thermodynamics

In this chapter, we give an brief introduction for both classical and quan-tum black hole thermodynamics. The subject is mostly contributed by J. D.Bekenstein and S. W. Hawking, which forms the basis of modern statisticalinterpretation of the black holes.

2.1 Classical black holes

It is found that a black hole experiences a remarkable tendency to increaseits horizon surface area when it is undergoing any transformation, such asPenrose process [10]. Later an even more strong proof is given by Hawkingthat the black-hole surface area cannot decrease in any process. As an exam-ple, one can consider that when two black holes are merging, the area of theresulting black hole cannot be smaller than the sum of initial areas [8], whichmakes a huge hint in the direction of the second law in thermodynamics, ormore precisely the entropy. However, this analogy was vigorously pursued inearly 1970, for several flaws were standing there embarrassing physicists.

1. The temperature of a black hole vanishes.

2. Entropy is dimensionless, whereas horizon area is a length squared.

3. The horizon area of every black hole is separately non-decreasing, whereasonly the total entropy is non-decreasing in thermodynamics.

3

It is found by S. W. Hawking in 1975 that when one includes the quantumeffect to the theory, all these flaws can be eliminated. However, if we onlyconsider the classical case now, we can formulate four laws of Black holemechanics bearing analogy with thermodynamics:

1. surface gravity is always constant over the horizon of a stationary blackhole;

2. for a rotating charged black hole, there is an identity,

dM = κdA/8πG+ ΩdJ + ΦdQ (2.1)

where J and Q are the angular momentum and charge of the black hole,Ω and Φ are the angular velocity and electric potential of the horizon

3. the horizon area can never decrease assuming Cosmic Censorship anda positive energy condition (which is also known as Hawking’s areatheorem);

4. the surface gravity of the horizon cannot be reduced to zero in a finitenumber of steps [17].

2.2 Quantum black hole

This name seems awkward in the first look, since we study black hole in theframe of Einstein’s general relativity, which is a classical theory. However,when one tries to eliminate the flaws given in previous section, it is necessaryto consider the term ”quantum gravity”, though such a term is not wellunderstood so far.

Hawking temperature

As in the classical theory, the surface gravity is known as temperature, andwith cooperation of the Hawking effect, one can construct the true Hawkingtemperature, which is given by,

TH =hκ

2π. (2.2)

4

Moreover, if one measures the BH entropy in units of the squared Plancklength, which is given by,

L2p =

hG

c3(2.3)

one derives that,

SBH =A

4hG(2.4)

which indicates that the entropy of a black hole is given by one fourth thearea of event horizon in squared Planck lengths.

Quantum gravitational statistical mechanics

Another thing we use afterwards, is the Gibbons-Hawking approach whichproposes a formulation of quantum gravitational statistical mechanics thatallows us to compute the black hole entropy from the classical Einstein action.This method imitates the standard methods of introducing thermodynam-ical ensembles in this quantum gravitational stage. The central identity isrepresented by,

Z = Tr exp(−βH) =

∫DgDφe−I[g,φ] (2.5)

where g, and φ are metric and matter fields respectively, and I is the classicalEinstein action in Euclidean signature.

5

Chapter 3

On the AdS-SchwarzschildBlack Hole Phase Transition

This phase transition was first introduced by S. W. Hawking, then extendedby E. Witten who argued that this transition happened in n dimension. Inthis chapter, we carry out his work with detailed calculation, as a startingpoint for our discuss on this very subject. And this chapter also serves as abackground for further logic and calculation.

3.1 Geometry

As we know, the AdS/CFT correspondence was used to identify boundarymanifold as asymptotic infinity of the bulk. The boundary is S1 × Sn−1, sothe bulk will either be a quotient of AdS space by a subgroup of SO(1, n+1),which is isomorphic to Z (with Euclidean signature), or AdS-Schwarzschildblack hole [4]. For the first case, the metric can be written as,

ds2 =dr2

( r2

b2+ 1)

+ (r2

b2+ 1)dt2 + r2dΩ2. (3.1)

The Einstein equation with a negative cosmology constant reads,

Rij −1

2Rgij + Λgij = 0. (3.2)

6

Here we normalize (3.1), in such a way that the dimension doesn’t appearexplicitly in metric [4], so that the Einstein equations reads,

Rij = −nb−2gij. (3.3)

On the other hand, the relation between Ricci scalar and the cosmologicalconstant is simply derived by (3.2), which will be useful for later simplifyingthe space-time action,

1

2R =

Λ(n+ 1)

n− 1. (3.4)

Then we insert the the normalization condition (3.3), and the cosmologicalconstant yields,

Λ = −n(n− 1)

2b2. (3.5)

For a Schwarzschild black hole in AdS space, the metric in Euclidean signa-ture is

ds2 = (r2

b2+ 1− ωnM

rn−2)dt2 +

dr2

( r2

b2+ 1− ωnM

rn−2 )+ r2dΩ2, (3.6)

where ωn = 16piGN(n−1)V ol(Sn−1)

, and the horizon is just the solution of the equation

(r2

b2+ 1− ωnM

rn−2) = 0. (3.7)

Hawking argued that the singularity at the horizon is nothing but a singular-ity at the origin of polar coordinates, which can be eliminated by coordinatetransformation. We can achieve that by regarding the imaginary time as anangular coordinate with a special period for (3.6). The metric is smooth andcomplete, if and only if the period of t is

β0 =4πb2r+

nr2+ + (n− 2)b2

. (3.8)

t can be expressed by sine function which has a period of 2π. One canconstruct thermal states on this space-time by also identify this period withthe inverse of the actual temperature. It shows that the d + 1 dimensionalSchwarzschild black hole has a natural temperature associated with it. Andthere exists a minimum temperature (or maximum β0).

7

3.2 Phase transition and entropy

With the normalized cosmological constant (3.5), the classic Einstein actionreads:

I =n

8πb2GN

∫dn+1x

√g. (3.9)

The action is reduced to the space-time volume times some constant. Forboth space-times we mentioned above, there are two regularized volumes,

V1(R) =

∫ β

0

dt

∫ R

0

dr

∫ n−1

S

dΩrn−1, (3.10)

V2(R) =

∫ β0

0

dt

∫ R

r+

dr

∫ n−1

S

dΩrn−1. (3.11)

V2(R) is easy to calculate, and the integration for V1(R), gives

V1(R) = V ol(Sn−1)β0

∫ R

0

dr

√Rn +Rn−2b2

(Rn −Rn+) + (Rn−2 −Rn−2

+ )b2rn−1. (3.12)

One can then find the difference between these two Euclidean actions iden-tified with the same physical period in imaginary time, where we simply

substitute that β = β0

√Rn+Rn−2b2

(Rn−Rn+)+(Rn−2−Rn−2+ )b2

. This formula follows that

when one adjusts β such that the geometry of the hyper-surface r = R is thesame in the two case; We compute (3.12), it gives:

V1(R) = V ol(Sn−1)β0

√R2

b2+ 1− ωnM

Rn−2

R2

b2+ 1

Rn

n. (3.13)

and then take the R to infinity,

∆I =n

8πb2GN

limR→∞

(V2(R)− V1(R)) =V ol(Sn−1)(b2rn−1

+ − rn+1+ )

4Gn(nr2+ + (n− 2)b2)

(3.14)

where

(V2(R)− V1(R))/β0 = (

√R2

b2+ 1− ωnM

Rn−2

R2

b2+ 1

− 1)Rn

n−rn+n. (3.15)

8

When R→∞, (3.15) becomes ωnMb2

2n− rn+

n, which yields the answer in (3.14).

This is positive for small r+, showing that the phase transition found in [7]occurs for all n. Then one can compute the energy, as well as entropy,

E =∂I

∂β0

=(n− 1)V ol(Sn−1)(rn+b

−2 + rn−2+ )

16πGN

= M. (3.16)

and the entropy of the black hole reads,

S = β0E − I =1

4GN

rn−1+ V ol(Sn−1). (3.17)

The entropy can also be written as

S =A

4GN

, (3.18)

with A the volume of the horizon , which is the surface at r = r+ [4]. Thereason of these identifications will be discussed in chapter 4 in detail. And inlater discussion we will let R be finite, so the equation will eventually havea radial dependence.

9

Chapter 4

Generalized Calculation

In this chapter, a generalized calculation is performed with respect to the pre-vious chapter, and a verification is worked out for the same subject, whichstands for our set-up, as in following chapters we use this set-up to disscusthe holographic RG equation.

4.1 Ansatz

We propose a diagonalized metric as follows,

ds2d+1 = dρ2 + β2(ρ)dt2 +R2(ρ)dΩ2

d−1, (4.1)

where dΩ2d−1 is (d − 1) dimensional sphere, and t has a periodicity of 2π.

What we need here is the classical Einstein action of the space-time. Thesphere metric reads:

dΩ2d−1 = dθ2

0 + sin2(θ0)dθ21 + . . .+ sin2(θ0) . . . sin2(θd−3)dθ2

d−2. (4.2)

So we start by computing the Ricci scalar. Normally we need to compute thederivative of the metric, then work out Christoffel symbol, followed by con-tracting the Ricci tensor. However for our ansatz, the metric is diagonalized,so it is simpler to get,

R = −2(d− 1)Rβ

Rβ+ (d− 1)(d− 2)

R2

R2+ 2

β

β+ 2(d− 1)

R

R+

R

R2, (4.3)

10

where tilde is introduced to distinguish with radial coordinate. Next, wemultiple the square root of metric determinant, which will be important forour later calculation, it can be simplified as,

√gR = Rd−32(d− 1)RRβ + (d− 1)(d− 2)R2β + βR+ T (4.4)

where T is the total derivative term, which we get by partial integrating. Weisolated it out for further convenience, and reads,

T = 2dβRd−1 + 2(d− 1)d(Rd−2Rβ). (4.5)

It does not contribute to the Lagrangian, however, we will need it for fullspace-time action. It gives the boundary term of the action. For moreinformation about the calculation, please consult Appendix A.

4.2 Hamiltonian constraint

It is always useful to work out the Hamilton formulation, and gain insightfrom the classical theory, like it already did for turning classical mechanicsinto quantum mechanics. The bulk action of this metric is given by

Ibulk =1

16πGN

∫dρ

∫dt

∫dθd−2(

√g(R− 2Λ)− T ). (4.6)

Carry out the integration of space-time volume, so we have the Lagrangianform as,

Ibulk =

∫dρL (4.7)

So the Lagrangian for this bulk action(with respect to ρ) becomes a functionof R, β, and their derivatives, which will be eliminated later. The Lagrangianreads,

L =V ol(S1 × Sd−1)

16πGN

(d−1)Rd−32RRβ+(d−2)βR2+β[dR2

b2+(d−2)]. (4.8)

We can also derive the canonical momentums with respect to β(ρ) and R(ρ),which are useful for constructing the Hamiltonian. They are obviously cal-

11

culated by,

πβ =∂L∂β

=V ol(S1 × Sd−1)

16πGN

2(d− 1)Rd−2R (4.9)

πR =∂L∂R

=V ol(S1 × Sd−1)

16πGN

2(d− 1)Rd−3[Rβ + (d− 2)βR]. (4.10)

Now we can formulate the Hamiltonian using,

H =∑

πq q − L, (4.11)

where q represents R and β. So the Hamiltonian reads,

H =V ol(S1 × Sd−1)

16πGN

(d− 1)Rd−32RRβ + (d− 2)βR2 − β[(d− 2) + dR2

b2].

(4.12)

We propose the Hamiltonian constraint H = 0, in case our metric is static.Substituting the canonical momentums, we can eliminate the flow velocities

inside the constraint. For convenience we setA = V ol(S1×Sd−1)16πGN

and B =

2A(d− 1)Rd−2. Now the momentums are represented by flow velocities,

πβ = BR (4.13)

πR = B[β + (d− 2)βR

R]. (4.14)

And the Hamiltonian constraint is now in terms of conjugate momentumsand two metric functions R, β, it reads,

πβ[2πR − (d− 2)βπβR

] =B2β

R[(d− 2) + d

R2

b2]. (4.15)

This Hamiltonian constraint is a very interesting equation, which have mo-mentum square on the left hand side, and pure geometrical parameters onthe other hand side. This formulation will be our central discussion later.

4.3 Anti-de Sitter space-time

It can be proved that there are two good fitted solutions for our ansatz, andthey are Anti-de Sitter space and AdS-Schwarzschild black hole space-time.

12

For the former case,

ds2d+1,AdS =

dr2

h0(r)+ h0(r)β2

0dt2 + r2dΩ2

d−1 (4.16)

= dρ2 + β2(ρ)dt2 +R2(ρ)dΩ2d−1; t ∈ [0, 2π] (4.17)

h0(r) = 1 +r2

b2. (4.18)

From above equations, comparing the similar terms, we get the identificationsas,

dρ

dr=

√h0 (4.19)

R(ρ) = r (4.20)

β(ρ) = β0

√h0. (4.21)

These equations connect the ansatz to the actual space-time. Then it isnatural to also calculate the flow velocities as,

R =dR

dρ=dR

dr

dr

dρ=

√h0 (4.22)

β =β0

2h′0 =

βR

b2√h0

(4.23)

where x′ is the derivative with respect to r. Above functions are all we needto construct the classical action for our target space-time. Now we can derivethe bulk action as,

IAdSbulk =−2d

16πGNb2

∫dd+1x

√g = −2Aβ

Rd

b2√h0

. (4.24)

And the boundary term (4.4) is given by,

Ibdry = 2A[(d− 1)Rd−2Rβ +Rd−1β]. (4.25)

So the whole on-shell action reads,

IAdS = 2A(d− 1)Rd−2β√h0. (4.26)

Actually, we can check the validity of our ansatz and the identifications wemade by substituting the results to the equations (4.13),(4.14),and (4.15).One can prove that they are the allowed solutions. So our ansatz is consistentwith AdS space-time, and the Hamiltonian constraint. Then it is reasonalbeto further use the ansatz on AdS-Schwarzschild black hole space-time.

13

4.4 AdS-Schwarzschild black hole space-time

In this case, we make the following identifications as last section,

ds2d+1,AdS =

dr2

h(r)+ h1(r)β2

1dt2 + r2dΩ2

d−1 (4.27)

= dρ2 + β2(ρ)dt2 +R2(ρ)dΩ2d−1; t ∈ [0, 2π] (4.28)

h1(r) = 1 +r2

b2− (1 +

r2+

b2)(rd−2

+

rd−2) = 1 +

r2

b2− ωM

rd−2(4.29)

where r+ is the horizon of the black hole, given by

h1(r) = 1 +r2

b2− ωM

rd−2= 0 (4.30)

where we have defined the mass of the black hole as ωM = rd−2+ (1 +

r2+

b2),

which will be clear in further calculation that it is the real mass of the blackhole. And ω = 16πGN

(d−1)V ol(Sd−1). Further by differentiate R and β with respect

to ρ, we get the flow velocities,

R =√h(R) (4.31)

β =β0

2h′1 =

h′1(R)

h′1(r+)(4.32)

We can calculate the action for AdS-Schwarzschild black hole space-time, as

IA−Sbulk = −A 4

b2

Rd − r2+

h′1(r+)(4.33)

IA−Sbdry |r=R = 2A[(d− 1)Rd−2√h(R)β +Rd−1 h

′1(R)

h′1(r+)] (4.34)

IA−S = 2A(d− 1)Rd−2√h1(R)β + 2rd−1

+ . (4.35)

Here we identify the boundary with AdS action sharing the same reason inprevious section. R is large but finite here, and also a very important point isthat on both space-times, we have β identified, which is achieved by setting,

β0/β1 =√h1/

√h0. (4.36)

This is also a reason that we propose this special ansatz.

14

4.5 Rederive thermal phase transition

The same as chapter 2, now we use our formalism to derive the phase changecharacterized by different event horizons. Given two metric function R(r)and β(r), firstly we identify both action at r = R and then let R→∞. Thesubtraction of action is,

IAdS(R, β)− IA−S(R, β) = 2A(d− 1)Rd−2β[√h0(R)−

√h1(R)]− rd−1

+ .(4.37)

This is a rather complicated expression for generic R and β. In next step,we take R to infinity where the boundary sits, and it yields,

limR→∞

(IAdS(R, β)− IA−S(R, β)) =2rd−1

+ (b2 − r2+)

dr2+ + (d− 2)b2

. (4.38)

One can easily see that:

1. when the horizon r+ < b, positive;

2. when the horizon r+ > b, negative.

It is consistent with the solutions that Witten provided. In the following cal-culations, we keep the radial variable finite, so the action stays as a functionof R. We want to study the radial evolution of the action and to relate it tothe renormalization flow.

15

Chapter 5

Thermodynamics

In this chapter, the thermodynamical quantities are derived explicitly. Mo-tivated by Hawking and Witten, we arrive at various quantities, which willbe used in next chapter to extract thermal interpretation from the dual CFTdifferential equation.

5.1 General form

As we discussed in the “Introduction” part, in order to count the number ofthe states on the space-time layers and also identify various thermodynamicalquantities. We need to import a statistical ensemble into the gravity theory.So to speak, the partition function of the dual conformal field theory in theboundary have contribution from the path integral of the space-time actionwith respect to the metric fluctuations, which is studied in [12]. Here weignore the matter fields and bulk scalar fields, since we want to focus onthe dominant part of this contribution. After this clarification, we can writedown the relation between free energy and the classical space-time action (inEuclidean signature), and it reads,

I = − logZ =F

T, (5.1)

where T is the temperature. As we know already, temperature is identifiedwith the inverse period of the imaginary time, which is 2πβ in our notation,

16

so we have,I = 2πβ(U − TS) = 2πβU − S. (5.2)

Using the standard thermodynamical laws, we can get further formulas bydifferentiating the action, and it yields,

∆I = 2π(U∆β − βP∆V ). (5.3)

Now we connect the classical space-time action and the dual thermodynam-ical quantities. The metric has spherical symmetry SO(d − 1), so it is easyto write the volume as,

V = V ol(Sd−1)Rd−1. (5.4)

One can also get the infinitesimal change of space-time volume as a functionof radius,

∆V = (d− 1)V ol(Sd−1)Rd−2∆R = C∆R. (5.5)

Combining (5.3)and(5.5), we now can construct the various thermodynam-ical quantities from the on-shell action. We assume that the saddle pointapproximation is valid, so the quantities are as follows:

1. Energy

U =1

2π

∂I

∂β; (5.6)

2. Entropy

S = U/T − I = β∂I

∂β− I; (5.7)

3. Pressure

P = − 1

2πβ

1

C

∂I

∂R. (5.8)

17

5.2 Quantities for both space-times

Using the formulas derived in previous section, we can easily compute thethermodynamical quantities for both AdS space-time and AdS-Schwarzschildblack hole space-time. For AdS space-time we have,

UAdS = −V ol(Sd−1)

8πGN

(d− 1)Rd−2√h0(R) (5.9)

PAdS =1

8πGN

1

R√h0(R)

[(d− 2) + (d− 1)R2

b2] (5.10)

SAdS = 0. (5.11)

These quantities will serve as the zero point, because that the only meaningfulthermodynamical quantities come from the relative difference. So to speak,we get the bounded energy, entropy and pressure after subtract AdS solution,which is relatively measured. And for AdS-Schwarzschild black hole space-time, we have,

UA−S = −V ol(Sd−1)

8πGN

(d− 1)Rd−2√h(R) (5.12)

PA−S =1

8πGN

1

R√h(R)

[(d− 2) + (d− 1)R2

b2− d− 2

2

ωM

Rd−2] (5.13)

SA−S = 2Ard−1+ =

V ol(Sd−1)rd−1+

4GN

=A′

4GN

, (5.14)

where A′ is the event horizon area of AdS-Schwarzschild black hole. Oncemore, we can see the quarter law of entropy.

5.3 Equation of state

We start from the space-time Lagrangian, and then calculate the canonicalconjugate momentums, together with Hamilton constraint, for this metricansatz. In Thermodynamical interpretation, we expect the equation of statepopping up, which gives the relation between various thermodynamical quan-tities. Because it gets rid of the limit of both the general relativity and quan-tum mechanics, and can describe the physical world in a sense of statisticalway. As a matter of fact, the pressure we get from previous section can be

18

interpreted as an entropic force. If we consider a system within equilibrium,the formula

T∆S = P∆V (5.15)

holds. We can see that the pressure is presented by entropic changes, whichmeans the pressure is exactly an entropic force. In (d+ 1) dimension, whichwe work on, ∆V will depend on radius and dimension, and as in our ansatz,the space-time has spherical symmetry, which gives that (5.5). So we canchange the above equation to be,

T∆S = P (d− 1)V ol(Sd−1)Rd−2∆R = PC∆R = Φ∆R (5.16)

where we put the entropic force together with the space which already exists,and leave the one dimension radius coordinate to be there ready for emerging.If we use Kelvin Temperature, the left hand side is positive defined, so theleft hand side, both variables should be with same signature. In other words,the entropic force pressure is in the same direction as the emerged radialdirection. In this way, the gravity is supposed working to slow down themechanics for space-time emerging process, which is also appearring as anentropic force. If we use this proposal in cosmology, we can argue that theexpanding universe, is the same as emerging, gravity is just like the elasticforce which hold the stuff together, which is the resistance to the expandingand of course have an entropic origin. And in my opinion, there is no gravityat the big bang starting point, though it is just a theory. So to speak,the gravity plays a role to shrink the space-time only when the big bangsingularity starting to create universe.

19

Chapter 6

Renormalization GroupEquation

In this chapter we want to derive the RG equation for our discussion. Thevalidity of this approach is argued by the dBVV paper which uses Hamilton-Jacobi formalism to derive standard Callan-Symanzik equation. As a matterof fact, we have a simpler ansatz, that has no matter fields. So the RG equa-tion shows in a briefer way. And then we will interpret it thermodynamically.

6.1 A little bit of string theory

Firstly, I would like to introduce some definitions that will show up in laterdiscussions. Strictly speaking string theory is the most prominent quantumgravity theory now. And there is not just one type of string theory, but five.They are type I, type IIA, type IIB, SO(32), E8 ∗E8. It is proposed later byWitten that these 5 types of Superstring theories are connected by a web ofdualities, which emerge them into one whole theory called M-theory.

Superstring theory contains many p-branes, which are objects with pspatial dimensions, in addition to normal fundamental strings which is alsocalled 1-branes. There are a class of p-branes that we call it D-branes, intypeI and typeII superstring theroy. Putting it simply, D-branes are theobjects on which fundamental strings can end. An interesting cosmologicalscenario given on D-branes is that we experience four dimension include

20

time, is because we are confined to live on D3-branes! And these branes areembedded in higher-dimensional space-time, such as 10 dimension.

Come back to AdS/CFT. The most successful Ads/CFT correspondenceis described by the duality between type IIB string theory compactified onAdS5 × S5, and four-dimensional N = 4 super Yang-Mills theory [6].Thisduality arises by considering D3 branes in type IIB string theory. Taking alow-energy limit of the string theory with the D3-branes yields N = 4 superYang-Mills theory, whereas the same low-energy limit applied to the dualsupergravity description of the D3 branes leaves us with AdS5×S5. And theso-called Poincare patch of AdS5 × S5 is described by the metric

ds2 = dr2 + e2r/bηµνdxµdxν . (6.1)

where b = (g2YMN)1/4 is the radius of AdS5, We know that four-dimensional

N = 4 super Yang-Mills theory is a conformal invariant field theory. If wedenote a set of bulk fields by φi , and by Oi the corresponding operators ofthe boundary theory. The usual statement that relates boundary correlationfunctions to bulk quantities is

〈e∫φiOi〉CFT = e−S[φi]sugra . (6.2)

φi and gµν solve the supergravity equations of motion and behave for large ras

φi(r →∞) ∼ er(∆−4)φi, gµν(r →∞) ∼ e2rgµν . (6.3)

(6.2) is not yet a complete prescription for the correlation functions, becausethe supergravity action will in general diverge. To deal with this problem, weproceed very much as in field theory. Namely we first introduce a finite cutoff,subtract divergent terms and then take the cutoff to infinity. As cutoff, wetruncate AdS at r = r0. This has the interpretation of a UV cutoff in theboundary theory. The AdS/CFT correspondence now becomes [5],

〈e∫φiOi〉CFT,metricgµν = e− limr0→∞S[φi,ds2=dr2+e2r/Lηµνdxµdxν ]sugra,r≤r0−singular terms.

(6.4)So, asymptotically, the duality enables us to explore the holographic renor-malization.

6.2 Derivative-like expansion

We expand the local on-shell action, such that there is a direct correspon-dence between the classical evolution equations of (d + 1) dimensional su-

21

pergravity and the RG equations of the dual d dimensional large N gaugetheory. It is shown in [1] that the supergravity Hamiltonian constraint canbe reduced to a flow equation for the classical supergravity action I, whichfurther more, in the asymptotic limit, can be cast in the form of a standardCallan-Symanzik equation. The system in dBVV paper is a 5 dimensionalgravity theory, minimally coupled to a set of scalars φi, with potential V (φ),and kinetic term, which depend on the fields.

The corresponding canonical momenta will be denoted by πµν and πi. Themomenta can, as we treat in previous chapter, be related to derivatives of theclassical action i with respect to the boundary conditions. In that case, theychoose the boundary at a slice at fixed values of r = R, and I[φ, g] will bethe classical supergravity action, evaluating on a solution of the Hamiltonianconstraint, with boundary condition that has extra boundary terms in orderto have a good variational meaning, and the boundary terms have to be takeninto account in the evaluation of the classical action I. For our case, we havea metric without coupling to the scalar fields, so there is no potential. UsingH-J formulation method given by [1], We can simply start by splitting upthe on-shell action as

I = Iloc + Γ. (6.5)

And according to H-J formulation

π =1√g

∂I

∂φ. (6.6)

So it is naturally to haveπ = πloc + πΓ. (6.7)

In this way, we can also split the Hamiltonian constraint into a derivativeexpansion form,

H = H(0)loc +HΓ. (6.8)

If we think an arbitrary action which has AdS background, and equal topure AdS action add some term, which we call Γ, then we can split up thearbitrary action as

I = IAdS + Γ, (6.9)

and the corresponding conjugate momentums read,

πβ =δI

δβ(6.10)

πR =δI

δR. (6.11)

22

So we have,

πβ = π0β + π′β (6.12)

πR = π0R + π′R, (6.13)

where π′ = δΓδx

, x = β,R. After subtracting the Hamiltonian constraint, weget

π0βπ′R + π′β[π0

R − (d− 2)π0β

β

R] = −π′β[π′R −

(d− 2)

2

βπ′βR

]. (6.14)

And then we can correspondingly derive the Hamilton-Jacobi equation bychanging the canonical momentums to the derivatives of the classical action,.

6.3 Reduced Callan-Symanzik equation

Hamilton-Jacobi equation

Substitute the pure AdS solution in chapter 4, and rearrange to simplify theequation. We split up the background and the non-local part now. It reads,

B√h0π

′R +

BRβ√h0b2

π′β = −π′β[π′R −(d− 2)

2

βπ′βR

]. (6.15)

Using the fact that the momentum is nothing but the derivative of non-localaction with respect to the field(metric field in our case), the constraint canbe cast as the so called H-J equation. And the H-J equation takes the formas,

B√h0∂Γ

∂R+

BRβ√h0b2

∂Γ

∂β= −∂Γ

∂β[∂Γ

∂R− (d− 2)

2

β

R

∂Γ

∂β] (6.16)

We noticed that, the second term on the LHS can be related to the first termby differentiating,

d√h0

dR=

R√h0b2

. (6.17)

So let us explore that if ∂Γ∂β

and ∂Γ∂R

are linear independent. If it is, then theformalism can be simply identified. Comparing the left and right hand sides,

23

we have,

∂Γ

∂β= −B

√h0 (6.18)

d− 2

2

∂Γ

∂β= B

d√h0

dRβ. (6.19)

After solving these two equations, we find that it cannot be prevented to letβ as a function of R. So to speak, β and R are not independent variables.We assume that β depend on R now.

RG equation

Start from (6.16), we can manage to get RG equation which describes therenormalization flow between the different layers of space-time. In general,one can propose a rescaling variable, denoting as “a”, which is a function ofR and β. However, after working on this general variable, we finally noticethat the best way to rescaling is using R as the variable. Actually, this pointis also indicated by the previous section, that we found that β depends onR. So we can then in this direction (radially), derive the Beta function forinverse temperature variable β, which is given by,

R∂β

∂R=

R2β

R2 + b2. (6.20)

We define,

∆(R) =R2

R2 + b2, (6.21)

which gives the rescaling information. One should also notice, in this case,we have,

∂Γ

∂R=∂Γ

∂β

∂β

∂R(6.22)

where, ∂β∂R

= RβR2+b2

. Then we head back to H-J equation and do the substi-tutions. The RG equation we get reads,

(R∂

∂R+ ∆(R)β

∂

∂β)Γ = γ

∂Γ

∂β

∂Γ

∂β. (6.23)

24

where,

γ = β(d− 4)R2 + (d− 2)b2

2B√h0(R2 + b2)

. (6.24)

For more information about this equation, please consult the general form ofCallan-Symanzik equation in appandix F.

Thermodynamical interpretation

Identify the various thermodynamical quantities. For this RG equation, wehave,

T =1

2πβ(6.25)

U =1

2πββ∂Γ

∂β(6.26)

P = − 1

2πβC

∂Γ

∂R(6.27)

where C = ∆V∆R

= (d−1)V ol(Sd−1)Rd−2. Substitute back to the RG equation,we get,

−(d− 1)PV + ∆(R)U = γU2. (6.28)

Now we arrive to a pure thermal equation which is involved by pressure,space-time volume, and energy. We can actually interpreted this as theequation of state for our system. And it is quite fit as an equation we areused to

PV ∼ U. (6.29)

However here we have dimension factor as well as a quadratic term in energy.After we take R→∞, the equation of state yields,

U =1

d− 1PV. (6.30)

It is actually a pleasant equation for us, which confirms the cosmologicalderivation. If this can be explained by future work, it should indicate deeplyfor the origin of the gravity in a basis of statistical mechanics.

25

Chapter 7

Conclusion

We reviewed the thermodynamical properties of the AdS black hole . Afterinputting a diagonalized Euclidean metric ansatz, we pursued all the ther-mal quantities, which illustrated again the connection between gravity andstatistical reasoning. We also extended Witten’s calculation, and got the ex-pected phase transition as well as the right behavior of the classical Einsteinaction under space-time with or without a black hole in it. Hamilton-Jacobiformalism was applied to study both thermal translation and the holographicproperties under AdS/CFT framework. It was proved again, that Hamilton-Jacobi equation for this general AdSd+1 metric can reduce to RG equationfor the dual space-time on d dimensional boundary. The connection betweengravity and statistical feature was emphasized, and explored. Finally, Wegave a thermodynamical interpretation for the RG equation as an equationof state. As a matter of fact, there are still details needed to fill in the the-ory. However, the exciting part is that this deep connection is getting moreand more serious notices now, which is led by E. Verlinde’s emerged gravityproposal.

26

Appendix A

Schematic of the Reasoning

We start from classical action, and we have to ways to go. We can eithergo the direction by formulate Hamilton constraint by Lagrangian derivedand further Hamilton-Jacobi equation. Or we can use correspondence toidentify the thermal quantities by differentiate the non-local action. Bothways converge to the Hamilton-Jacobi formalism. Because the dependenceof Hamilton-Jacobi equation on differentiation of classical action agree withthe thermal quantities only by a quotient, which enable us to interpret theRG equation for CFT who is living on the boundary of the AdS space, as anequation of states. Though deeper explanation is needed in future.

27

Appendix B

Penrose Process

Penrose process states that energy can be extracted from a rotating blackhole. This energy is located out side of the event horizon in ergosphere.Putting it simply, the matter which falls in the ergosphere splitting into twopart. One for into the black hole and the other just escape to infinity, whichcan get more energy than the initial state. In such a way, the angular momen-tum of the black hole is decreasing, finally rotating black hole become staticas Schwarzschild black hole. Or otherwise, it can also add more energy(mass)to the black hole to make its event horizon increasing.

28

Appendix C

Ricci Scalar for DiagonalizedMetric

The on-diagonal Ricci tensor reads [9]:

4Rµµ = (∂µln|gµµ|−2∂µ)∂µln|g

gµµ|−

∑σ 6=µ

[(∂µln|gσσ|)2+(∂σln|g|g2µµ

+2∂σ)gσσ∂σgµµ].

(C.1)From above equation, we can see that the first term on the RHS is vanisheddue to the metric. and second term gives:

4Rµµ = −∑σ 6=µ

[(∂µln|gσσ|)2 + (∂σln|g|g2µµ

+ 2∂σ)gσσ∂σgµµ]. (C.2)

29

Rρρ = − ββ

+ (d− 1)R

R (C.3)

Rtt = −(d− 1)ββR

R+ ββ (C.4)

Rθ0θ0 = −RRββ

+ (d− 2)R2 +RR+ R∗θ0θ0 (C.5)

Rθ1θ1 = −Rθ0θ0 × sin2(θ0) + R∗θ1θ1 (C.6)

... (C.7)

Rθiθi = −Rθ0θ0 × sin2(θ0) . . . sin2(θi−1) + R∗θiθi (C.8)

... (C.9)

Rθd−2θd−2= −Rθ0θ0 × sin2(θ0) . . . sin2(θd−3) + R∗θd−2θd−2

(C.10)

where the derivative is with respect to ρ. And the second term of aboveequation on the RHS give:

d−2∑i=0

gθiθiR∗θiθi =R

R2(C.11)

where R is the n-1 dimensional unit sphere’s ricci scalar. And it is easy tocarry out to be (d− 1)(d− 2). After contracting with the metric, we get thericci scalar,

R = −2(d− 1)Rβ

Rβ+ (d− 1)(d− 2)

R2

R2+ 2

β

β+ 2(d− 1)

R

R+

R

R2(C.12)

we can do some rearrangement, and simplify the result to,

√gR = Rd−32(d− 1)RRβ + (d− 1)(d− 2)R2β + βR+ T (C.13)

where T is the total derivative term get by partial integral,it reads,

T = 2dβRd−1 + 2(d− 1)d(Rd−2Rβ) (C.14)

30

Appendix D

Anti de Sitter Space

Anti de sitter space often is often abbreviated as AdSn, in general relativity,which refer to the maximally symmetric , vacuum solution of the Einstein’sfield equation with a negative cosmological constant Λ,

Rµν −1

2gµνR + gµνΛ = 0. (D.1)

Where we say maximally symmetric, we mean that the time and space inall directions are mathematically equivalent. This ideal space-time becamefamous when J. Maldacena proposed AdS/CFT correspondence in 1997. TheAdS space can be visualized as the Lorentzian analogue of a sphere in a spaceof one dimension higher. Formally, the AdS space in a particular coordinateswhich covers the entire space has the form like,

ds2 = −(k2r2 + 1)dt2 +1

k2r2 + 1dr2 + r2dΩ2 (D.2)

where if we take imaginary time, the metric will be Euclidean, which is theform we use in our thesis.

31

Appendix E

Derivative Expansion inGeneral Form

We shall make the following premises regarding the counter terms S[2k] [5].

1. The counterterms are covariant(and gauge invariant, if vector fields areinvolved) local expressions in terms of the metric gij and the field φ atthe cut-off boundary as well as their derivatives.

2. The term S[2k] contains exactly k inverse matrices.

3. The counter terms should completely contain the power divergences ofS.

4. The counter terms must be universal, i.e., they must contain the powerdivergences of S for any asymptotically AdS solution of the bulk equa-tions of motion.

The number k of power divergent terms is dictated by premise 3: By powercounting using the leading asymptotic behavior of the fields for r →∞, onecan determine which covariant boundary integrals are generically divergent.In equation (6.5), Γ may contain a logarithmic divergence, but shall be re-garded as finite for the power counting. Finite local terms can be arbitrarilyshifted between Γ and the counter terms. This reflects the usual ambiguityof choosing a renormalization scheme.

32

Appendix F

Callan-Symanzik Equation

Callan-Symanzik Equation is also known as Renormalization Group equa-tion, which was discovered independently by Curtis Callan[1] and Kurt Symanzikin 1970. Later it is used to understand asymptotic freedom. It has the stan-dard form,

(a∂

∂a− βI∂I)〈OI1(x1) · · · OIn(xn)〉 =

n∑i=1

γJiIi 〈OI1(x1) · · · OJi(xi) · · · OIn(xn)〉

(F.1)

where

γJI = ∇IβJ (F.2)

and we also have the identity,

〈OI1(x1) · · · OIn(xn)〉 =1√g

δ

δφI1(x1)· · · 1√g

δ

δφIn(xn)Γ[φ, g]. (F.3)

As mentioned in former paper, the the action S in H-J theory represents thegenerating function of correlation function of local operators OI in the dualCFT. And more important, the n-point functions of operators OI at differentpoints is all contained in the non-local part of the action, we call it Γ.

33

Appendix G

Hamilton-Jacobi formalism

In classical mechanics, we have Hamilton formulation, which is given by twoconjugate differential equations:

−π =∂H

∂q(G.1)

q =∂H

∂π(G.2)

To improve the Hamiltonian for simplification(to get constant generalizedsolutions above), We define a transformed Hamiltonian

K = H(q,∂S

∂q, t) +

∂F

∂t. (G.3)

where K is the transformed Hamiltonian which related to old Hamiltonianby a derivative of the so called generating function, we can simply use theclassical action as generating function. To ensure constant variables, wefurther take K = 0. If Hamiltonian is not an explicit function of t, then theHamilton-Jacobi equation for S becomes

H(q,∂S

∂q) = 0. (G.4)

where

H =∑i

qi∂L

∂qi− L =

∑i

qiπi − L (G.5)

34

and an easy way to connect this to familiar equation is the Schrodingerequation,

(h

i

∂

∂t−H(q,

h

i

∂

∂q)χ = 0. (G.6)

where χ = expihS, and expand (G.6) to lowest order in h, we recover the

Hamilton-Jacobi equation. And it is obvious from WKB approximation,that the lowest order approximation of quantum mechanical path integralconfirm with the classical action.

35

Acknowledgements

Firstly, I would like to thank my supervisor Prof. E. Verlinde for insightfulsuggestion and guidance, during the writing of this thesis. And I also wantto thank Dr. B. van Rees and K. Holsheimer for helpful discussions. As aforeign student, I am also very grateful to Prof. B. Nienhuis, who helps mea lot as a coordinator during this two years. Moreover, I want to thank VeraWu, and especially my mom, who always encourages me through all kinds ofdifficulties, since I was a kid.

It is an amazing two-year stay in Amsterdam, where are full of nice peopleand hardcore physicists, and also where I get a lot of international friends!What I learned here is not only the knowledge in researching, but also todiscover myself and the way to be a better man.

Thank you all, without your help, I could not go further!

36

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[3] Miranda. Cheng, notes on thermodynamic of black holes.

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[5] J. de Boer, Fortsch.Phys. 49,339-358,(2001),hep-th/0101026v1.

[6] J. Maldacena,Adv.Theor. Math.Phys. 2,231,(1998),hep-th/9711200.

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[8] S. W. Hawking,Phys.Rev.Letters.26, 1344,(1971).

[9] K. Z. Win, hep-th/9602015

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[16] T. Jacobson, Introductory Lectures on Black Hole Thermodynam-ics,(1996), Utrecht University.

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38

Black hole

It’s not just “black” holebut also throw

We can never see itThough it is even more real

Enlighten us pleaseGod says no

—–B.Shi

39

Hope you enjoy it!

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