UTRECHT UNIVERSITYFaculty of Science

Department of Physics and Astronomy

Ivano Lodato

THE NOTHER POTENTIAL:Definition and Applications

MASTER THESIS

Advisor: Prof. B. de Wit

Academic year 2009/2010

Contents

1 Introduction 2

2 The Noether current and Noether potential 42.1 Noether’s theorem . . . . . . . . . . . . . . . . . . . . . 42.2 An important potential . . . . . . . . . . . . . . . . . . . 10

3 Covariant Lagrangians 123.1 Yang-Mills theory example . . . . . . . . . . . . . . . . . 123.2 Diffeomorphism invariant theory example . . . . . . . . . 16

4 Physical applications 234.1 Electric charge and angular momentum . . . . . . . . . . 234.2 The first law of black hole mechanics . . . . . . . . . . . 26

5 Noether potential for Chern-Simons terms 385.1 Ambiguities in the definition of Qµν . . . . . . . . . . . . 385.2 Gauge-theory examples . . . . . . . . . . . . . . . . . . . 405.3 Covariant and mixed Chern-Simons terms . . . . . . . . 435.4 The mixed Chern-Simons term revisited . . . . . . . . . 47

6 Conclusions 58

A Lie-derivative and Killing fields 60A.1 Lie-derivative . . . . . . . . . . . . . . . . . . . . . . . . 60A.2 Killing fields . . . . . . . . . . . . . . . . . . . . . . . . . 63

Bibliography 65

1

Chapter 1

Introduction

In 1973 Hawking, Carter and Bardeen formulated [1] very special rela-tions holding for stationary axisymmetric spacetimes containing a blackhole, which are identical to the four laws of thermodynamics. For thisreason, these relations, now commonly referred to as the Laws of blackhole mechanics, have always been considered to be of crucial impor-tance in order to unveil the secrets of a complete quantistic theoryof gravity. The first law, for instance, is expressed (in natural units~ = GN = c = 1) by the equation

δM =κ

8πδA+ ΩHδJ

that links the variation of mass of a black hole, with its variation in areaand angular momentum due to a quasi static process during which astationary black hole is taken to a new stationary black hole. Althoughat the beginning these laws were seen only as a formal coincidence, thediscovery of the Hawking radiation led to think that black holes behaveindeed like a thermodynamics system.Of course, reliable definitions for the energy and angular momentum ofa black hole already exhisted even before the laws were formulated, andthe Bekenstein-Hawking formula states that the entropy of a black holeis equal, up to a proportionality constant, to the surface area of a blackhole, namely the area of an arbitrary spacelike cross section of the eventhorizon. However, it was believed that a more general expression, whoseBekenstein-Hawking formula is just a first order approximation, musthold, allowing calculation of the entropy of a black hole also in more

2

general context, e.g. string theory. Few year ago, Vafa and Strominger[2] derived such a formula for a class of 5-dimensional supersymmetricextremal black holes in string theory, from a microscopic point of view:the necessity arose to match this result from macroscopic calculations.To this end, a generalized and operative definition of black hole entropywas needed.In 1993, Wald proved two important results: given a Lagrangian theoryin n dimensions, symmetic under a certain group of transformation, theconserved current Jµ associated with this symmetry can, under certainhypotesis, be written as a total derivative term, i.e.

Jµ = ∂νQµν .

Also, the new quantity Qµν , called the Nother potential, is strictly re-lated to the entropy of a black hole, that is a conserved charge of thetheory, when diffeomorphism invariant Lagrangian are considered andfew extra conditions on the fields and the parameter of the transfor-mation are imposed. Obviously this last result turns out to be of greatimportance, since it furnishes a well defined algorithm to calcualate theentropy of a black hole.In the following chapters, we review the most important results relatedto the new concept of Noether potential. Well-known concepts and the-orems are also explained, and a certain number of explicit examples,some of which original, is given and worked out explicitely, making thisreview a self-consistent work.

3

Chapter 2

The Noether current andNoether potential

In this chapter we review some basic concepts that will be useful through-out this work: the Noether’s theorem, that links invariance of an inte-gral under transformations of a Lie group with conserved quantities, ispresented together with the derivation of the general expression for theconserved current. In the second section, we briefly introduce the con-cept of Noether potential, a quantity we will treat extensively, that is ofcrucial importance to calculate physical quantities such as the entropyof a black hole, as we shall see in chapter 4. This chapter serves, then,as theoretical foundation, a well-known starting point of our whole fol-lowing analysis.

2.1 Noether’s theorem

We start by stating the Noether’s theorem in its original formulation,following closely the work [3, 4]: although this approach is quite un-usual, it should give a remarkable insight into the elegance and gener-ality of this important result. A second, but equivalent, formulation ofthe same theorem, expressed in a more modern language, will be givenlately, in order to introduce and specify the notation that will be usedthroughout the work.Consider n independent variables x1, · · · , xn, (x, in compact notation)

4

and m functions of x, u1(x), · · · , um(x) (u in compact notation). Sup-pose that the x’s and u’s are subject to the transformation of a contin-uous group G (Lie group) of dimension r: this means that the groupdepends on r constant parameters ξ1. Obviously, given the invertibilityof the transformations (they belong to a group), n transfomed variablesy and m transformed functions v will be obtained.A function g is invariant under transformations of G if

g

(

x, u,∂u

∂x,∂2u

∂x2, . . .

)

= g

(

y, v,∂v

∂y,∂2v

∂y2, . . .

)

(2.1)

Consequently, an integral I is said to be invariant under the grouptransformation if the relation

I =

∫

f

(

x, u(x),∂u(x)

∂x,∂2u(x)

∂x2. . .

)

dx =

∫

f

(

y, v(y),∂v(y)

∂y,∂2v(y)

∂y2. . .

)

dy

(2.2)holds true for an arbitrary x-interval and the corresponding y-interval.Now, we can calculate the first variation of I, δI, obtained by trans-forming the functions u(x) in the new functions v(x) evaluated at thesame point x, i.e. δu(x) = v(x) − u(x). For an arbitrary, not neces-sarly invariant under this transformation, integral I, according to thecalculus of variations rules, we get:

δI =

∫

δf dx =

∫

[

m∑

j=1

Ej

(

x, u,∂u

∂x, · · ·

)

δuj + Div θ(u, δu, · · · )]

dx

(2.3)The n vector θ is linear in δu and its derivatives and Ej stands forthe j-th Lagrange expression, corresponding to the variation problemδI = 0 when all the variations δu and its derivatives are assumed tovanish at the boundary.Of course, from this integral relation, an important identity can beobtained,i.e.

δf =m

∑

j=1

Ejδuj + Div θ(u, δu, · · · ) , (2.4)

1It is also possible that the group depends on r functions ξ(x), e.g. gaugetransformations, as we will show in the next chapters.

5

that we use in the following.In the particular case of a function f containing only first derivativesof the u’s, the Lagrange expression is given by

Ejδuj = δf − d

dx

(

m∑

j=1

∂f

∂u′jδuj

)

where u′j =duj

dx.

We are now ready to assert the Noether theorem:

Noether’s theorem. The integral I is invariant under transforma-tions of the continuous, r-dimensional group G if and only if r linearlyindependent combinations of the Lagrange expressions become diver-gences.

This means that, when the Lagrange equations Ej = 0 are satisified,we obtain the so called ”laws of conservation”. In the one-dimensionalcase, the divergence becomes a total differential and there exists r firstintegral linearly independent (but they can be non-linearly dependent).We want to give a proof of (the second implication of) this theorem [4].Let G be a Lie group of transformations, and r the dimension of suchgroup. Expanding all yi’s and vj ’s in Taylor series around the zerovalues of the parameters ξ, or ξ(x) = 0, and arranging the 0-th order ofthe expansion to be equal to the original variable xi and function uj(x)we obtain

yi = Ai

(

x, u,∂u

∂x, . . .

)

= xi + ∆xi + · · ·

vj(y) = Bj

(

x, u,∂u

∂x, . . .

)

= uj(x) + ∆uj(x) + . . .

where ∆xi, ∆uj stand for the terms of lowest order (and linear) inξ(x) and its derivatives (it can be proven that this assumption does notimply loss of generality).Now let I be invariant under the infinitesimal transformation

yi = xi + ∆xi

vj(y) = uj(x) + ∆uj (2.5)

6

that belongs to the group G, and it’s such that

0 = ∆I =

∫

f

(

y, v(y),∂v

∂y, . . .

)

dy −∫

f

(

x, u(x),∂u

∂x, . . .

)

dx (2.6)

where the first integral is to be evaluated over the x+ ∆x-interval cor-responding to the x-interval in the second integral. Here by ∆ we meanthe variation of I under infinitesimal transformations of the variablesand the functions that leave I invariant.Notice that this integration may as well be written as an integrationover the x-interval, by means of the following equality, holding for in-finitesimal transformations (2.5):

∫

f

(

y, v(y),∂v

∂y, . . .

)

dy =

∫

f

(

x, v(x),∂v

∂x, . . .

)

dx+

∫

Div (f∆x) dx

Now, consider the following identity, holding for infinitesimal ∆:

∆uj(x) = vj(y)−uj(x) = vj(y)−uj(y)+uj(y)−uj(x) = δξuj(y)+n

∑

j=1

∂uj∂xj

∆xi

where δξuj(y) corresponds to the above defined generic variation δu,when the transformation laws are specified (in this case by (2.5)): thisnotation allows us to write also ∆x = δξx. Using the last two equationsin (2.6) we obtain:

0 =

∫

[δξf + Div(fδξx)] dx . (2.7)

Since this equation is satisfied for integration over any arbitrary in-terval, then the integrand must vanish identically and therefore theinvariance of I can be also written as

δξf + Div(fδξx) = 0 . (2.8)

and using the important identity (2.4) we obtain:

m∑

j=1

Ejδξuj = Div(θ(u, δξu) − fδξx) (2.9)

7

So, for every invariant integral I, if the Lagrange equations are satisfied(Ej = 0 for all j = 1, . . . , m), then we obtain

Div J ′ = Div(θ − fδξx) = 0 (2.10)

However, this is not the most general form of a divergenceless vector(see [4]), that is instead constructed out of

δξL = DivN(ξ, x, u,∂u

∂x, · · · )

so that equation (2.10) becomes

Div J = Div(θ −N) = 0 (2.11)

meaning that J is a conserved quantity, the conserved current, whoseexistence follows from the invariance of I under the group of transfor-mationG. We can state again the Noether theorem in a modern fashion:

Noether’s theorem (second formulation). To every continuoussymmetry of the action, there corresponds a conserved current.

A symmetry of the action is a group of transformation under which theintegral (action):

S =

∫

dnxL

is invariant. Again, the existence of a conserved current is strictly linkedto the fact that the action is invariant under a Lie group of transforma-tion. Notice that the action is a very important quantity that describesphysical systems, so the variables we consider are coordinates of an n di-mensional spacetime (the spatial variables are n−1), and the functionsu are now dynamical fields strictly connected to physical quantities.We indicated by L the Lagrangian density, whereas the Lagrangian ofthe system is given by L =

∫

dn−1xL. Nevertheless we will refer to Las Lagrangian of the system.In the following, we drop the notation used so far: Einstein sum con-vention on the repeated indeces, and, more generally, the formalism ofquantum field theory will be used with letters of the Greek alphabetindicating, unless differently specified, space-time components of ten-sors.

8

Now, if we consider a theory described by a Lagrangian L depending onsome dynamical fields, generically indicated by φ(x), and their deriva-tives, but not explicitely depending on the space-time coordinates x,we can re-write equations (2.6), (2.7), (2.9) as

0 = δξS =

∫

[δξL− ∂µNµ] dnx⇒

δξL − ∂µNµ = 0 (2.12)

Remember that the important identity (2.4) holds, i.e. :

δL = Eδφ+ ∂µθµ(φ, δφ) (2.13)

from which we obtain

∂µ[θµ(φ, δξφ) −Nµ(φ, ξ)] = ∂µJ

µ = −E(φ)δξφ (2.14)

where a summation over all the dynamical fields φ is understood.The quantity

Jµ(φ, ξ) = θµ(φ, δξφ) −Nµ(φ, ξ) (2.15)

is then conserved whenever the equations of motion are satisfied. Thislast equation, together with the expression for δL, eq. (2.13), will beused extensively in the following chapters, where we actually calculatethe conserved currents associated with invariant Lagrangians.We notice that the variation of the Lagrangian under a specific infinites-imal field variation δξφ, ∂µN

µ(φ, ξ), is not necessarly proportional tothe field variations, as we will show in the next chapter. On the otherhand the current is linearly proportional to ξ and possibly its space-time derivatives, and, in the special case in which L is strictly invariantunder the symmetry variation, namely Nµ = 0, then the ξ dependenceresides entirely in the variation δξφ. Furthermore, θµ (and consequentlythe current Jµ) is determined up to possible improvement terms, which,again, do not need to be proportional to δξφ either. We will discussthose ambiguities in chapter 4.Finally, we would like to notice that Noether’s theorem is applicableevery time there exists a differentiable symmetry of the action, not theLagrangian, because one can consider also coordinate variation that

9

gives rise to variation of dnx. Nevertheless we will consider differen-tiable symmetry of the Lagrangian since the variation induced by co-ordinates transformation are taken into account in the term Nµ dueto the specific form the Lagrangian assumes: for gauge theories, forinstance, L is a scalar while for theories of gravity it’s a scalar density2.

2.2 An important potential

In this section introduce the concept of Noether potential, the quantitywhich this entire work is based on3. It is well-known that, on a manifoldM , an exact form is also closed, while the converse is not always true:this means that if a tensor T µ··· can be written as a total derivative, i.e.

T µ··· = ∂νWµν···

with W νµ··· antisymmetric in its first two indices, then ∂µTµ··· = 0, but

we are not always assured that, if T µ··· is conserved than a tensor W µν···

exists.Consider now a Lagrangian L invariant under a certain Lie group oftransformations, depending on certain fields φ(x), and implicitely onthe points x of a manifold M . In the previous section we showed howto calculate the associated current Jµ, conserved by virtue the equationsof motion. Suppose now that the theory admits not just an isolatedsolution of the equation of motion, but a variety thereof: if this is thecase, then we are assured that, given ∂µJ

µ = 0, we can always write

Jµ = ∂νQµν (2.16)

The Noether potential Qµν is then a globally defined antisymmetrictensor, locally constructed out of the fields φ and the transformationparameters ξ.Notice that for a given current Jµ, eq. (2.16) determines the Noetherpotential up to the addition of terms of the form ∂µA

µν··· with Aµν···

2see Appendix A.13In the original paper [5], Wald called this quantity the Noether charge. We

prefer to use the name Noether potential to distinguish it from the charges one candetermine by integration of Qµν over spacelike hypersurface, as it will be explainedin chapter 4

10

tensor, antisymmetric in the indeces µ and ν. Of course, one can arguethat if such an ambiguity exists the Noether potential can be always setto be identically zero: however, as we shall see in the following chapter,this is not the case.A lot of details are to be (and will be) added, but, for the moment,we prefer to end this abstract introductory chapter and present, in thenext one, two different pedagogical examples of calculation of Qµν , tobetter understand all the concepts presented in this section.

11

Chapter 3

Covariant Lagrangians

In this chapter we continue our analysis concerning the Noether poten-tial: we first calculate Qµν for a simple example of non-abelian gaugetheory, for which the Lagrangian is invariant under the Lie group oftransformation considered and then give another example, in the con-text of diffeomorphism invariant theories, described by Lagrangiansonly invariant modulo a total derivative term. This chapter is, then,of great importance, since it should clarify the procedure to follow inorder to obtain the Noether potential Qµν starting from a covariantLagrangian.

3.1 Yang-Mills theory example

In this section we present an explicit calculation of Noether potentialobtained from a Yang-Mills Lagrangian, invariant under gauge trans-formations. So far, no physical quantity can be related, in general, toit (in this respect the example we present in the last section assumesa completely different importance, as we will show in chapter 4): nev-ertheless, this pedagogical example will be useful to better understandhow it is possible to write a conserved current Jµ as a total derivativeterm, (2.16), by means of the equation of motion.Consider a Yang-Mills theory described by a gauge-invariant LagrangianL(Fµν ,∇ρFµν , ψ,∇µψ) depending on the field strength Fµν , a matterfield ψ and their first derivatives [6]. All the considered quantities arewritten as Lie-algebra valued quantities in the relevant representation

12

of the matter field ψ.We have, for infinitesimal gauge transformations:

δξAµ = ∇µ ξ(x) ,δξψ = ξ ψ ,

δξFµν = [ξ, Fµν ] ,

δξ∇µψ = ξ∇µψ .(3.1)

where, given δξψ, δξ∇µψ is straightforwardly obtained just as a resultof the formalism we are using. Analogously, without any calculation,we already know that:

δξ(∇ρFµν) = [ξ,∇ρFµν ] (3.2)

It’s worth remembering that the covariant derivatives we used in eqs(3.1) have different definitions, depending on which entity is appliedon. Also, the variation of the gauge field Aµ, written as a covariantderivative of the parameter ξ , is just for matter of notation: in fact,we assume the parameter ξ to be a covariant quantity, and write ∇µ ξ(x),because it is convenient in actual calculations.Using these definitions, we can write the first variations of the othercovariant quantities in a more handy way, namely

δ(∇ρFµν) = ∇ρδFµν + [Fµν , δAρ] , δ(∇µψ) = ∇µδψ − δAµψ (3.3)

Also the following relation holds

δFµν = ∇µδAν −∇νδAµ (3.4)

where again we consider δAµ to be a covariant quantity just for a matterof convenience in the explicit calculations. This relation is also analo-gous to the Palatini equation, that we use in next section, for the fieldstrength Fµν .Now we want to calculate the current Jµ and the Noether potentialQµν for such a theory. We first start from a generic variation of theLagrangian:

δL = Lµν δFµν + Lρ,µν δ(∇ρFµν) + Lψδψ + Lµψδ(∇µψ) (3.5)

where

Lµν =∂L∂Fµν

, Lρ,µν =∂L

∂∇ρFµν, Lψ =

∂L∂ψ

, Lµψ =∂L∂∇µψ

(3.6)

13

Notice that the identity L[µ,ρν] = 0 holds, implied by the Bianchi iden-tity for the field strength.We can analyze each term individually. Using (3.1), (3.3) and (3.4), weobtain:

Lµψδ∇µψ = ∇µ(Lµψδψ) − (∇µLµψ )δψ − LµψδAµψ (3.7)

LµνδFµν = ∇µ(2LµνδAν) − 2∇µ(Lµν)δAν (3.8)

Lρ,µνδ∇ρFµν = 2Lρ,µν∇ρ∇µδAν + Lρ,µν [Fµν , δAρ] (3.9)

The first term on the right-hand side of (3.9) still requires few additionalmanipulations to be put in a standard form: after partial integrationsand re-arrangements of indices we get a contribution to the equation ofmotion for the gauge field Aµ and two terms that we analyze in detail,i.e.

2∇µ[Lµ,ρν∇ρδAν − (∇ρLρ,µν)δAν ]= ∇µ[∇ρ(2Lµ,ρνδAν) − 2 (∇ρLµ,ρν)δAν ] −∇µ[2 (∇ρLρ,µν)δAν ]= ∂µ∂ρ(2Lµ,ρνδAν) −∇µ[4 (∇ρL(µ,ρ)ν)δAν ]

= ∂µ∂ρ(2L(µ,ρ)νδAν) −∇µ[2 (∇ρL(µ,ρ)ν)δAν ] −∇µ[2 (∇ρL(µ,ρ)ν)δAν ]

= ∇µ[∇ρ(2L(µ,ρ)νδAν) − 2 (∇ρL(µ,ρ)ν)δAν ] −∇µ[2 (∇ρL(µ,ρ)ν)δAν ]

= ∇µ[2L(µ,ρ)ν∇ρδAν ] −∇µ[2 (∇ρL(µ,ρ)ν)δAν ] (3.10)

Notice that the change ∇µ → ∂µ in the third line is allowed since theLagrangian we are considering is gauge invariant (δξL = 0) and we aregoing to evaluate these boundary terms for gauge transformation, thenall the terms generated from the variation δξL are invariant as well: thisin turns means that all the terms are scalar under gauge transformation,so that the covariant derivative is equivalent to the normal one.Now, substituting (3.10) in (3.9), and using also eqs. (3.7), (3.8) in(3.5), we have:

δL = Eψ δψ + EµA δAµ + ∂µθ

µ(Aν , ψ, δAν , δψ)

14

where

EµA = 2∇νLµν − 2∇ν∇ρLρ,µν + Lµ,ρσ[Fρσ, ·] − Lµψ·ψ = 0 (3.11)

Eψ = Lψ −∇µLµψ = 0 (3.12)

θµ = Lµψ δψ + 2Lµν δAν − 2∇ρL(µ,ρ)νδAν + 2L(µ,ρ)ν∇ρδAν

Since the Lagrangian is gauge invariant

δξL = 0 ⇒ Nµ = 0

from eq. (2.15) we then obtain:

Jµ = θµ(Aν , ψ, δξAν , δξψ)

= Lµψ ξ ψ + 2Lµν∇νξ − 2∇ρL(µ,ρ)ν∇νξ + 2L(µ,ρ)ν∇ρ∇νξ (3.13)

that is an invariant quantity (remember that the original Lagrangianis written as a trace of quadratic products of field strengths).Now, we are assured that, if the equations of motion are satisfied, wecan write Jµ = ∂νQ

µν . In fact:

∂νQµν = Jµ + Eµ

A ξ

since, as a result of the covariance of ξ we imposed in (3.1), the Noetherpotential Qµν is an invariant quantity, given by:

Qµν = 2Lµν ξ − 2 (∇ρLµ,ρν)ξ + Lρ,µν∇ρξ (3.14)

In order to obtain this final result we needed the Bianchi identity forthe field-strength Fµν and

Lµ,ρνδξFρν = −Lµ,ρν [Fρν , ξ] = 2Lµ,ρν∇ρ∇νξ

from (3.1) and (3.3). As we already anticipated, the Noether poten-tial is a local function of the fields and the transformation parameter.Moreover the fact that ∂µJ

µ = 0 is now trivial, given the antisymmetryof Qµν .We just showed that for Yang-Mills theory, invariant under gauge trans-formations, i.e. Nµ = 0, the conserved current is proportional to the

15

field variations δξφ. This feature has a certain importance since thereare cases of gauge theories. e.g. Lagrangians containing Chern-Simonsterms, for which Nµ is different from zero, and we must force our the-ory to show it, as we will see in chapter 5. On the other hand, forLagrangians describing theories of gravity invariant under diffeomor-phism, the current is never proportional to the field variation and wecould question whether those cases have a physical relevance or not. Wereturn to this point in th next chapter. For the moment, we present anexplicit example on calculation of Qµν for theories of gravity.

3.2 Diffeomorphism invariant theory ex-

ample

In this section we consider a diffeomorphism invariant theory of gravi-tation, in the sense that the Lagrangian is covariant under the actionof a general coordinate transformation. In general, the covariance of aLagrangian under a symmetry transformation δξ can be expressed inits infinitesimal form by the identity:

δξL(φ) =∂L∂φ

δξφ (3.15)

that for diffeomorphism transformation reads

LξL(φ) =∂L∂φ

Lξφ = δξL(φ) (3.16)

The symbol Lξ represents the Lie derivative along the vector field ξ.For this Lagrangians, the current is never proportional to the field vari-ations, since δξL = ∂µN

µ is not vanishing. To be more precise,

δξL = LξL = ∂µNµ = ∂µ(ξ

µL) (3.17)

that is equivalent to say that the Lagrangian is not a scalar, as it is forgauge theories, but a scalar density of weight 1.Let us start the explicit calculations: consider a Lagrangian dependingon the Riemann tensor, the metric, a matter field ψµν (with no partic-ular symmetry) and its first derivative, that, for convenience we write

16

as: √−gL(gµν , Rµνρσ, ψµν ,∇ρψµν) . (3.18)

where, in this notation, L is not a density of weight 1 (since√−g is)

but it’s a scalar.The quantities under examination transform, under diffeomorphism, asfollows:

Lξgµν = ∇µξν + ∇νξµ (3.19)

Lξψµν = ξσ∇σψµν + ∇µξσψσν + ∇νξσψµσ (3.20)

Lξ(∇ρψµν) = ξσ∇σ∇ρψµν + ∇µξσ∇ρψσν + ∇νξσ∇ρψµσ (3.21)

On the other hand, the generic variations of the Christoffel symbolsand the Riemann curvature are given by

δΓ σνρ = 1

2gσλ(∇ρδgνλ + ∇νδgρλ −∇λδgρν) (3.22)

δRµνρσ = R λµν[ρ δgσ]λ + 2∇[µ∇[ρδgσ]ν] (3.23)

An important distinction has been made between symmetry variations(3.19)-(3.21) and generic variations (3.22), (3.23): in fact, since nei-ther Γ nor R are independent fields of the theory, they contribute toθµ(φ,Lξφ) and the equations of motion only by means of the indepen-dent fields and their symmetry variations. Notice that (3.23) is obtainedfrom the Palatini equation

δR βµ ρσ = ∇ρδΓ β

σµ −∇σδΓ βρµ . (3.24)

and the generalization of Riemann tensor definition when we apply[∇µ,∇ν ] on arbitrary tensor, not necessarly a vector field ξρ.We still need the following transformation rules

Lξ

√−g = ∂µ(ξ

µ√−g) = ∇µ(ξµ

√−g) (3.25)

LξL = ξµ∂µL = ξµ∇µL . (3.26)

combining which we obtain

Lξ(√−gL) = ∂µ(ξ

µ√−gL) = ∇µ(ξµ

√−gL) (3.27)

17

On the other hand, under generic variation δ the Lagrangian transformsas

δ(√−gL) = (δ

√−g)L +

√−gδL = 1

2

√−ggµνLδgµν

+√−g

(

LµνρσδRµνρσ + Lµνδgµν + Lµνψ δψµν + Lρ,µνψ δ(∇ρψµν))

(3.28)

which is obtained using the well-known identity δg = g gµνδgµν and thechain rule. We used the notation:

Lµν =∂L∂gµν

, Lµνρσ =∂L

∂Rµνρσ

, Lρ,µνψ =∂L

∂∇ρψµν, Lµνψ =

∂L∂ψµν

.

(3.29)At this point, again, we just need to analyze each term individually.We have

LµνρσδRµνρσ = Lµνρσ(R λµνρ δgσλ + 2∇µ∇ρδgσν) = Lρσλ(µR

ν)ρσλ δgµν

+2∇µ(Lµνρσ∇ρδgσν) − 2(∇µLµνρσ)∇ρδgσν= (Lρσλ(µR

ν)ρσλ + 2∇(ρ∇σ)Lσνρµ)δgµν

+2∇µ(Lµνρσ∇ρδgσν −∇ρLρνµσδgσν) (3.30)

Lρ,µνψ δ∇ρψµν = Lρ,µνψ δ(∂ρψµν − Γ λρµ ψλν − Γ λ

ρν ψµλ)

= Lρ,µνψ (∇ρδψµν − δΓ λρµ ψλν − δΓ λ

ρν ψµλ)

= ∇µ(Lµ,ρσψ δψρσ) − (∇ρLρ,µνψ )δψµν

−12Lρ,µνψ gλβ

(

∇ρδgµβ + ∇µδgρβ −∇βδgρµ)

ψλν

−12Lρ,µνψ gλβ

(

∇ρδgνβ + ∇νδgρβ −∇βδgρν)

ψµλ (3.31)

obtained by just exploiting the symmetries of Lµνρσ and δgµν . Noticealso that each term in the last two lines of (3.31) can be in turn writtenas

Lρ,µνψ gλβ∇ρδgµβ ψλν = ∇ρ(Lρ,µνψ ψβν δgµβ) −∇ρ(Lρ,µσψ ψνσ)δgµν

= ∇µ(Lµ,ρσψ ψβσ δgρβ) −∇ρ(Lρ,(µ|σ|ψ ψν)σ)δgµν(3.32)

18

Using this last equality in (3.31), and substituting, together with (3.30),in (3.28) we obtain

δ(√−gL) =

√−g

[

Eg δgµν + Eψ δψµν + ∇µθµ(gµν , δgµν , ψµν , δψµν)]

(3.33)with

Eg = 12gµνL + Lµν + Lρσλ(µR

ν)ρσλ + 2∇(ρ∇σ)Lσνρµ

+12∇λ

[

Lλ,(µ|ρ|ψ ψν)ρ + Lλ,ρ(µψ ψ ν)ρ + L(µ,|λρ|

ψ ψν)ρ

+L(µ,|ρλ|ψ ψ ν)

ρ − L(µ,ν)ρψ ψλρ −L(µ,|ρ|ν)

ψ ψ λρ ] = 0 (3.34)

Eψ = ∇ρLρ,µνψ − Lµνψ = 0

(3.35)

and θµ is given by

θµ(φ, δφ) = 2(Lµνρσ∇ρδgσν −∇ρLρνµσδgσν) + Lµ,ρσψ δψρσ

−12

[

Lµ,ρσψ ψνσ δgρν + Lρ,µνψ ψσν δgρσ − Lρ,σνψ ψµν δgρσ

+Lµ,ρνψ ψ σµ δgνσ + Lρ,νµψ ψ σ

µ δgρσ −Lρ,σνψ ψ µσ δgρν

]

(3.36)

Notice that both θµ and Nµ (and then Jµ) are vector densities of weight1, since the Lagrangian itself transforms as a scalar density.Now, evaluating θµ for δφ=Lξφ, if the equations of motion are satisfied,we obtain the identity

√−g∇µθµ(φ,Lξφ) −∇µ(

√−gξµL) = 0 = ∇µ(

√−gθµ(φ,Lξφ) −

√−gξµL)

=⇒ ∇µ(√−gJµ) =

√−g∇µJµ = 0 (3.37)

from which we see that both Jµ and√−gJµ are covariantly conserved,

while only the latter is conserved, i.e. ∂µ(√−gJµ) = 0. Notice, also,

that this result is completely general and holds for every gravitationalaction of the form S =

∫

dnx√−gL. Let’s just analyze Jµ, that ex-

19

plicitely reads:

Jµ = 2Lµνρσ[

∇ν∇ρξσ +Rλνρσξλ]

− 4∇ρLµνρσ∇(νξσ)

+Lµ,ρσψ [ξλ∇λψρσ + ∇ρξλψλσ + ∇σξλψρλ]

−12(∇λξρ + ∇ρξλ)

[

Lµ,ρσψ ψλσ + Lρ,µσψ ψλσ −Lρ,λσψ ψµσ

+Lµ,σρψ ψ λσ + Lρ,σµψ ψ λ

σ − Lρ,σλψ ψ µσ

]

− ξµL (3.38)

where, to obtain the first two terms in the current from the first twoterms in (3.36) we used the following manipulations:

Lµνρσ∇ρLξgσν = Lµνρσ(∇ρ∇σξν + ∇ρ∇νξσ)= Lµνρσ(∇σ∇ρξν +Rλνρσξ

λ + ∇ρ∇νξσ)= Lµνρσ(∇ν∇ρξσ +Rλνρσξ

λ) (3.39)

where we used the symmetry property Lµ[νρσ] = 0, and

∇ρLρνµσLξgσν = 2∇ρLµσρν∇(σξν) = 2∇ρLµνρσ∇(σξν) (3.40)

that holds since Lαβγδ = Lγδαβ .Now, of course, it seems impossible to write Jµ as a exact total deriva-tive. However, we are missing the relation imposed by the generalcovariance of the (scalar) Lagrangian L: in fact, the condition (3.16)has to be satisfied for both

√−gL, exploiting which we obtained thecurrent (3.38), and L itself. We already know LξL from (3.26), so wewant to calculate δξL, i.e.

δξL = LµνρσδξRµνρσ + Lµνδξgµν + Lµνψ δξψµν + Lρ,µνψ δξ∇ρψµν= Lµνρσ

(

ξα∇αRµνρσ + 4Rµνρα∇σξα)

+ 2Lµν∇µξν

+Lµνψ[

ξσ∇σψµν + ∇µξσψσν + ∇νξσψµσ]

+Lρ,µνψ

[

∇ρ(

ξσ∇σψµν + ∇µξσψσν + ∇νξσψµσ)

−12gλβ

(

∇ρδξgµβ + ∇µδξgρβ −∇βδξgρµ)

ψλν

−12gλβ

(

∇ρδξgνβ + ∇νδξgρβ −∇βδξgρν)

ψµλ

]

(3.41)

20

where we used the explicit expression for the Lie derivative of the Rie-mann tensor and (3.20)-(3.22). After a good exercise of bookkeeping,we can re-write this equation in the form

δξL = ξσLµν∇σgµν + ξσLµνψ ∇σψµν + ξσLρ,µνψ ∇σ∇ρψµν + ξσLµνρα∇σRµνρα

+2Lµν∇µξν + 4LµνρσR λµνρ ∇σξλ + Lµνψ

[

∇µξσψσν + ∇νξσψµσ]

+Lρ,µνψ

[

∇µξσ∇ρψσν + ∇νξσ∇ρψµσ + ∇ρξσ∇ρψµν]

(3.42)

where the first line is simply

ξσLµν∇σgµν+ξσLµνψ ∇σψµν+ξσLρ,µνψ ∇σ∇ρψµν+ξσLµνρα∇σRµνρα = ξσ∇σL = LξL

Now, in view of (3.16) we get, from (3.42)

2Lµν∇µξν + 4LµνρσR λµνρ ∇σξλ + Lµνψ

(

∇µξσψσν + ∇νξσψµσ)

+Lρ,µνψ

[

∇µξσ∇ρψσν + ∇νξσ∇ρψµσ + ∇ρξσ∇ρψµν]

= 0 (3.43)

Observe that we can write (3.43) in the form Xµν∇µξν = 0 and sincethis identity holds for arbitrary ξν, then it follows Xµν = 0. But thisimplies also Xµνξν = 0, i.e.

2Lµνξν − 4LµνρσR λρσν ξλ + Lµνψ ξσψσν + Lνµψ ξσψνσ

+Lµ,νρψ ξσ∇σψνρ + Lρ,µνψ ξσ∇ρψσν + Lρ,νµψ ξσ∇ρψνσ = 0 . (3.44)

Now, we can use this additional constraint on the current, together withthe equations of motion: adding 2 ξνEg = 0 and subtracting (3.44) toJµ, using also the relation Eψ = 0, we obtain

Jµ = ∇νQµν (3.45)

and Qµν reads

Qµν = 2Lµνρσ∇ρξσ − 4∇ρLµνρσξσ−

[

L[µ,|σρ|ψ ψν]ρ + Lσ,[µ|ρ|ψ ψν]ρ − L[µ,ν]ρ

ψ ψσρ

+L[µ,|ρσ|√−gQµν

ψ ψ ν]ρ + Lσ,ρ[µψ ψ ν]

ρ − L[µ,|ρ|ν]ψ ψ σ

ρ

]

ξσ(3.46)

21

where, to obtain the first line, we used the identity 2Lρσλ[µRν]

ρσλ ξν +2∇ν∇ρLνρµσξσ = 0, and the the symmetry of Rµν .Now remember that the conserved current is given by

√−gJµ, so theNoether potential equals

√−gQµν , since

√−gJµ =

√−g∇νQµν = ∇ν(

√−gQµν) = ∂ν(

√−gQµν) .

We showed an explicit example of calcualtion of Qµν for a diffeomor-phism invariant theory: in the next chapter, we’ll see how this is relatedto physical quantities, such as the entropy of a black hole.

22

Chapter 4

Physical applications

So far we introduced the concept of Noether potential, and explicitelycalculated this quantity in few examples: Yang-Mills theory and a dif-feomorphism invariant theory. We also anticipated that the calculationof Qµν is not just a pure algebraic exercise: in this chapter we showthe link between the Noether potential and physical relevant quantitiescharacterizing a black hole. For this reason, this result constitutes thecentral part of the entire work, considering the possibility of extensionsand applications to more general theories it offers, not to mention theincredibily simple algorithm it proves to exist to calculate such impor-tant physical quantities, as the entropy of a black hole.We mainly use results and concepts given in [7, 8, 9, 10, 11, 12, 13].

4.1 Electric charge and angular momen-

tum

The Noether potential is a very intriguing and useful quantity to defineand explicitely calculate conserved quantities. For instance, it’s not dif-ficult to show that under certain hypotesis the electric charge enclosedin a surface or the angular momentum of a space-time can be straight-forwardly obtained. Let us specifically analyze the first case. Considera Yang-Mills Lagrangian, strictly invariant under gauge transformation(e.g. the one considered in section 3.1): as we already stressed, in thiscase the conserved current Jµ is proportional to the field variationsδξφ, since θµ is. Now, it is immediate to verify that ∂νQ

µν can be writ-

23

ten as the sum of the current, proportional to the field variations, andthe generalized Yang-Mills-Maxwell equation of motion. This meansthat, if the equations of motions for the gauge field is satisfied and thesymmetric background condition is imposed, δξφ = 0, then Qµν is a con-served quantity. Furthermore, integral of Qµν over a n− 2-dimensionalspacelike surface gives the enclosed electric charge:

q =

∫

Ξ

dΣµνQµν(φ, ξ) (4.1)

where dΣµν is the surface element of Ξ. Of course this definition ofcharge corresponds with the usualFor now, we give an example to clarifythe previous argument. The current (3.13) is proprortional to the fieldsvariations δξAµ and δξψ. We also showed that

∂νQµν = Jµ + Eµ

A ξ ,

so, if the symmetric background condition are imposed, Jµ = 0 and theNoether potential is conserved on the generalized Yang-Mills-Maxwellequation of motion Eµ

A. Furthermore the electric charge defined in (4.1)coincides with the “usual” charge defined by means of the equation ofmotion. To show this, consider non-linear electrodynamics describedby a Lagrangian L(Fµν , ∂ρFµν): the equation of motion for the gaugefield reads

EµA = 2 ∂νLµν − 2 ∂ν∂ρLρ,µν = 0 (4.2)

that is the analogous of (3.11) for the abelian theory. Also, since wewant to obtain a conserved electric charge, the presence of any chargedfield ψ has been suppressed. Now, adding a source JµS

1 on the righthand side of (4.2) and integrating over a spacelike hypersurface V withboundary Ξ, we obtain:

∫

V

dΩµEµA =

∫

V

dΩµJµS = q .

Using Gauss’ theorem this last equality can be written as

q =

∫

Ξ

dΣµν(2Lµν − 2 ∂ρLρ,µν) (4.3)

1This source-current does NOT coincide with the Noether current, although, inspecific cases these two quantities can be somehow related

24

This is the conserved electric charge enclosed in Ξ, obtained by follow-ing the “usual” procedure. Now, consider the Noether potential for thistheory that reads exactly like (3.14), but with the substitution ∇µ → ∂µ:imposing the symmetric field configuration dξAµ = ∂µξ = 0 and sub-stituting in (4.1) we obtain for the conserved charge q the same result,as we wanted to show. Notice that for more complicated theories (e.gYang-Mills theory) the “usual” procedure fails to give the conservedelectric charge, while the definition (4.1) holds true independently fromthe theory under examination.Analogously, we can define another conserved charge in a very simpleway: it can be proven [9] that whenever ξµ is a killing vector field gen-erating a periodic isometry of the spacetime that lies in the integrationsurface Ξ, then the integral

J =

∫

Ξ

dΣµνQµν (4.4)

is conserved. The proof is not very hard and goes as follows: consider an-dimensional covariant theory of gravity, and the conserved current ofthe form (2.15). Suppose also to choose the n−1 dimensional spacelikeCauchy surface C to be foliated by compact n−2-dimensional surfacesΞC over some interval I = [a, b] of R. The equality

∫

C

dΩµJµ =

∫

I

∫

ΞC

dΩµJµ =

∫

Ξ1

dΣµνQµν(ξµ) −

∫

Ξ2

dΣµνQµν(ξµ)

(4.5)holds, from Gauss’ theorem, where the two surfaces Ξ1,2 are the bound-ary of C, and dΩµ is the hypersurface element of ΞC . Suppose now thatthe Killing vector ξµ generates a periodic symmetry along a direction αof ΞC : this means that ξα lies on this surface, since both x′α and xα lieon ΞC . Now dΩµ is proportional to the vector nµ, normal to ΞC , as wewill show, and from (2.15) and (3.17) we know that, when evaluated ina symmetric background, the conserved current under diffeomorphismis just given by −ξµL. Therefore, the integral (4.5) is proportional tothe product ξµnµ, that vanishes because the two vectors are orthogonaland not null. As a consequence, the integral of the Noether potential(4.4) does not depend on the surface Ξ chosen, and then it is a con-served quantity, the angular momentum of the space-time. To put itsimply, if we are considering a Schwarzschild spacetime in coordinates

25

(t, r, θ, φ), its angular momentum does not vary with the radial distancer from the origin of the spacetime.Of course, the definition (4.4) coincides with the Komar integral for theangular momentum, as it is straightforward to check: consider the ex-pression of the Noether potential for a diffeomorphism invariant theory,eq. (3.46)

Qµν = 2Lµνρσ∇ρξσ − 4∇ρLµνρσξσ (4.6)

where the dependence of the Lagrangian from the matter field has beensuppressed. The expression for Lµνρσ can be explicitely evaluated forgeneral relativity and, as we will show at the end of this chapter, equals

Lµνρσ = − 1

16πGN

gµ[ρgσ]ν

from which we obtain the expression for the angular momentum ingeneral relativity

JGR = − 1

16πGN

∫

Ξ

dΣµν∇[µξν] (4.7)

that is the Komar integral ([14]) for the angular momentum in generalrelativity, as we wanted to show.

4.2 The first law of black hole mechanics

In the following we show how, by using the first law of black hole me-chanics, we can define the entropy of a black hole as the integral overa spacelike hypersurface of the Noether potential: this procedure hasthe advantage to be easily generalizable to black holes in any numberof spacetime dimensions.Consider a general theory of gravity with LagrangianL satisfying (3.16),describing a spacetime containing a stationary black hole, in 4 space-time dimensions: although we are restricting the proof to this particularcase, generalizations are straightforward and almost always possible.Suppose, furthermore, that the theory admits a continuous variety ofsolutions φ connected infinitesimally by any variation δ, and let, fromnow on, φ be any of these solutions, and δφ any infinitesimal variation

26

around it. We want to calculate the variation of the current under suchvariation δ on the solution, i.e.:

δJµ = δθµ(φ,Lξφ) − ξµδL = δθµ(φ,Lξφ) − ξµ∂νθν(φ, δφ)

= δθµ(φ,Lξφ) − Lξθµ(φ, δφ) + ∂ν [ξ

νθµ(φ, δφ) − ξµθν(φ, δφ)] (4.8)

where, in the second line, the equation of motion and the Lie derivativeof the vector density θµ have been used.Define the quantity

ωµ(φ, δ1φ, δ2φ) = δ1θµ(φ, δ2φ) − δ2θ

µ(φ, δ1φ) (4.9)

from which be obtain, using (4.8), the expression

ωµ(φ, δφ,Lξφ) = δJµ − ∂ν(ξνθµ − ξµθν) (4.10)

Now, the crucial observation [11] is that, if a timelike Killing vector fieldξµ exists, then the Hamiltonian H generates the evolution along theintegral curves2 of ξµ and we indicate its variation due to the presenceof a continuous family of solutions by δH . As it turns out, in thecovariant phase-space approach, δH is realted to ω(φ, δφ,Lξφ) by therelation [15]:

δH =

∫

C

dΩµ ωµ(φ, δφ,Lξφ) (4.11)

with C arbitrary Cauchy surface extending from asymptotic infinity tothe event horizon of a stationary black hole with volume element

dΩµ = nµ

√

gind3 d3x (4.12)

where nµ represents a vector normal to the hypersurface C and gind3 is

the determinant of the metric induced on C.Despite the appearance, δH does not depend upon the choice of C: infact, it can be proven (see [11]) that if δφ and Lξφ satisfy the linearizedequations of motion, the dependence of δH on C disappears, and theHamiltonian itself be written as a surface term, as we will see in amoment.Now, using (4.10) and (4.11), we get the important equality

δH = δ(

∫

C

dΩµ Jµ)

+

∫

C

δΩµ ∂ν(ξµθν − ξνθµ) . (4.13)

2The integral curves of a vector field V µ are the curves xµ(t) satisfying dxµ

dt= V µ

27

We already know that δJµ = δ∂νQµν , but there’s no reason in principle

why the variation δ and the derivative operator should commute: if,however, the linearized equation of motion are satisfied, then the com-mutation is possible. To show this, consider the solution φ and supposeφ+ δφ, with δφ infinitesimal, is another solution of the equation of mo-tion. This would induce a variation in the current Jµ → Jµ + δJµ andagain (2.14) has to be satisfied, namely

∂µJµ = −Ei(φ)Lξφ

i → ∂µ(Jµ + δJµ) = −Ei(φ+ δφ)Lξφ

i

where the summation over all the dynamical fields is made explicit fornotation convenience.Expanding around the solution φ we get

Ei(φ+ δφ) = Ei(φ) +∂Ei∂φj

∣

∣

∣

φδφj

from which it follows that

∂µδJµ = −∂Ei

∂φj

∣

∣

∣

φδφjLξφ

i = −Elinij (φ)δφjLξφ

i

so that if the linearized equation of motion Elinij (φ) are satisfied, then

also the current δJµ is conserved (on the equation of motion) and wecan write

δJµ = ∂νδQµν (4.14)

Substituting this result in (4.13), we get

δH =

∫

∂C

dΣµν

(

δQµν + ξµθν − ξνθµ)

. (4.15)

where dΣµν can be rewritten also as

dΣµν = ǫµν

√

gind2 d2x (4.16)

with gind2 determinant of the induced metric on the 2-dimensional sur-

face ∂C. Also, the quantity ǫµν is called binormal and it is defined, forthe case at hand, as follows. Consider a 2-dimensional space normal toa (2-dimensional) surface ∂C: in this space there exists a linearly inde-pendent antisymmetric tensor of the second rank, ǫµν indeed, and it’s

28

normalized according to ǫµνǫµν = −2. As one can imagine, the explicit

evaluation of such a tensor depends on the details of the surface ∂C,so we won’t dwell on it, for the moment.Notice that the last two term in (4.15) are proportional to the vectorfield ξµ and not to its curl ∇[µξν]: this will simplify the final expressionwe aim to obtain, as we shall see shortly.Suppose now that, given θµ it is possible to find a quantity Λµ suchthat

∫

∂C

dΣµν

(

ξµθν − ξνθµ)

= δ

∫

∂C

dΣµν

(

ξµΛν − ξνΛµ)

(4.17)

In this case, it is immediate to see that the Hamiltonian can be writtenas a surface integral, and then it loses its dependence on the choice ofC, i.e.

H =

∫

∂C

dΣµν

(

Qµν + ξµΛν − ξνΛµ)

(4.18)

Suppose, furthermore, that ξµ is a Killing vector that characterizes theinvariance of the full background, namely Lξφ = 0 for all the indepen-dent fields φ, solutions of the equation of motion. In this case, theparameter ξµ characterizes a residual symmetry of the particular solu-tion, and then it may change continuously as well as the solution itself:given the arbitrariness in the choice of the solution φ, the invariance ofthe full background holds for each and every solution. This means thatcalling δ the combined variation of the solution φ and its correspondentKilling vector ξ, the condition

δ(

Lξφ)

= 0 (4.19)

must be satisfied. Of course, in view of this further assumption one canquestion whether taking into account also the Killing vector variation,if a more general expression than (4.8) can be proved. Let us considerthe combined variation δ of the current Jµ: obviously it reduces to δif it acts on a Lagrangian, that depends only on the fields, or on theKilling vector ξµ:

δJµ(φ, ξ) = δθµ(φ,Lξφ) − δξµL(φ) − ξµδL(φ)

= δθµ(φ,Lξφ) + Jµ(φ, δξ)− θµ(φ,Lδξφ) − ξµ∂νθν(φ, δφ)

29

where we used (2.15). Observe however that

δθµ(φ,Lξφ) − θµ(φ,Lξφ) = θµ(φ+ δφ,Lξ+δξ(φ+ δφ)) − θµ(φ,Lξφ) − θµ(φ,Lξφ)

= δθµ(φ,Lξφ) + θµ(φ+ δφ,Lδξ(φ+ δφ)) − θµ(φ,Lδξφ)

= δθµ(φ,Lξφ) + O(δ2)

and analogously

δJµ − Jµ(φ, δφ) = Jµ(φ+ δφ, ξ + δξ) − Jµ(φ, ξ)− Jµ(φ, δφ)

= δJµ(φ, ξ) − Jµ(φ+ δφ) − Jµ(φ, δφ)

= δJµ(φ, ξ) + O(δ2) .

This leads then to the expression

δJµ(φ, ξ) = δθµ(φ,Lξφ) − ξµ∂νθν(φ, δφ) + O(δ2) ,

identical to (4.8), up to terms of higher order in δ, and then it is themost general expression we can obtain.If the symmetric background condition is imposed, ωµ(φ, δφ,Lξφ) and,consequently, δH vanish, so

∫

∂C

dΣµν

(

δQµν + ξµθν − ξνθµ)

= 0 . (4.20)

Take the boundary of C to be the union of two 2-dimensional spaces,i.e.

∂C = Ξ∞ ∪ Ξhor (4.21)

where Ξ∞ is the boundary of the Cauchy surface at infinity, that isa two-sphere for asymptotically flat spacetimes, and Ξhor is an arbi-trary spacelike cross section of the event horizon ∆. This is an infiniteredsfhift hypersurface swept by Ξhor in time, representing the boundaryof all events which can be observed by an external inertial observer. Itis worth mentioning that under certain assumptions the event horizonis also a Killing horizon, a null hypersurface where the Killing vectorfield becomes lightlike. In the following, we will deal with static orstationary black hole for which event horizon and Killing horizon coin-cide3.

3To be more precise, the identification between the two different horizons isalways possible for static black hole solutions [16]. However if one considers sta-

30

Now, we can split the expression (4.20) in two different integrals: let’sanalyze for the moment the integral over the 2-sphere Ξ∞.It is well known that if the spacetime is stationary (i.e. it is possibleto find a coordinate system in which all the elements gαβ of the met-ric tensor are time-independent), then there exists a timelike vectorfield, tµ, called Killing vector field, associated with the invariance ofthe spacetime.For our following considerations, it is important that this Killing vectorfield represents also time translation in the asymptotically flat region4.In this case, we can always specify conditions at infinity for the fieldssuch that H in (4.18) approaches a finite limit at infinity and Λµ existsand satisfies (4.17). Consequently, we define the canonical energy E ofan asymptotically flat spacetime to be

E ≡ λ

∫

Ξ∞

dΣµν

(

Qµν(tµ) + tµΛν − tνΛµ)

(4.22)

Obviously, this can be considered as a conserved charge, due to thespacetime invariance under asymptotic time translation.The constant λ appearing in (4.22) is due to the scaling invariance ofthe metric: in fact, when evaluating a solution gµν of Einstein’s equa-tion satisfying the asymptotically flat condition, the solution gλµν suchthat gλµν → λ ηµν at spatial infinity, with ηµν Minkowsky flat metric,is the same physical solution. Obviously, this ambiguity reflects itselfin a multiplicative factor in the current, the Noether potential and,consequently, in all the conserved charges we can define, such as theenergy E . However, the physical quantities should not depend on suchambiguity, meaning that we can absorb the constant λ on the otherphysical constant of the theory, such as the Newton constant GN , thatis, then, “non constant”. To make this argument clearer let us consider

tionary axisymmetric solutions with a discrete reflection symmetry that changesthe sign of the time coordinate t and the angular coordinate ϕ (parametrizing therotation around the axis of symmetry) contemporarly, e.g. the Kerr metric, thenthe identification is still correct [17, 18].

4If t is the proper time of an observer at infinity, and xµ(t) are his coordinates,then asymptotically tµ → ∂xµ

∂t, so the vector field tµ becomes orthogonal to the sur-

faces t = const, and then it is the generator of a time translation in the asymptoticalregion.

31

the Einstein-Hilbert action in units c = 1:

S = − 1

16πGN

∫ √−gR d4x

with R Ricci scalar. As a consequence of the scaling invariance of themetric,

√−g scales as λ2 in 4 space-time dimensions, and the Ricciscalar as λ−1: then the action is also scaling invariant with constant λ.Nevertheless we can define a “scaled” Newton constant Gλ

N = GN/λabsorbing the ambiguity. If we insist to impose the natural units con-dition GN = 1 , then the value of λ is unequivocally fixed to be 1

2: this

can be easily proven noticing that λ = 12

is the value that leads to theBekenstein-Hawking area formula for vacuum general relativity, as wewill show at the end of this chapter, and, contemporarly, to the correctvalue for the canonical energy associated with an asymptotically flatspacetime, the ADM mass ([19], see [11] for the proof). From now on,we impose GN = c = 1 and, consequently, fix then the value of thescaling constant λ.We are now tempted to analogously define the canonical angular mo-mentum J , of an axially symmetric (asymptotically flat) spacetime, i.e.a spacetime invariant under rotation around an invariant axis. We callthe Killing vector associated to this symmetry ϕµ. Again, we need theKilling vector ϕµ to be an asymptotic rotation. If this is the case, wecan choose the surface Ξ∞ to be everywhere tangent to ϕµ, and sincedΣµν is orthogonal to the surface at infinity, the the contribution fromthe terms linear in ξµ = ϕµ in eq. (4.20) vanishes, so that:

J ≡ −1

2

∫

Ξ∞

dΣµν Qµν(ϕµ) (4.23)

this quantity is the conserved charge one obtains when invariance underasymptotic rotation is satisfied, and corresponds to the charge linked toperiodic isometry of the space-time we defined in the previous section,evaluated at spatial infinity.Again, this definition turns out the give the correct result for the angu-lar momentum in vacuum general relativity, Komar formula [20]. Letus point out that the difference in sign between the definitions (4.23)and (4.22) is due to the Lorentz signature of the spacetime under con-sideration, and it is present also in the definitions of energy and angular

32

momentum for a particle in special relativity.At this point, it is sufficient to consider a stationary axially symmetricblack hole solution (e.g. Kerr solution), with Killing vector ξµ given by

ξµ = tµ + Ωahor ϕ

µa (4.24)

where the summation over the index a starts when the spacetime di-mension is equal or larger than five. Notice that for many known so-lutions, as Kerr’s, the Killing vectors tµ and ϕµ represent asymptotictime translation or rotation, and their combination ξµ is timelike atinfinity, so all the hypotesis so far imposed are satisfied.To finally get the First law of black hole mechanics (in natural units~ = c = G = 1), i.e.

δM =κS2πδS + Ωa

horδJa (4.25)

we just need to identify the mass M with the energy E in (4.22) andgeneralize the expression (4.23) for the case of a rotational Killing vectorfields ϕµa . Substituting in (4.20) we obtain

∫

Ξhor

dΣµν

(

δQµν + ξµθν − ξνθµ)

= δM − ΩahorδJa (4.26)

that leads to the identification

κS2πδS =

1

2

∫

Ξhor

dΣµν

(

δQµν + ξµθν − ξνθµ)

(4.27)

explicity revealing the link between the Noether potential and the en-tropy of a black hole, as we desired to show.Notice that, to obtain (4.26) we used the obvious property Qµν(ξµ) =Qµν(τµ) + Ωa

horQµν(ϕµa), given the linearity of the Noether potential in

the Killing vector field and its derivatives.In equation (4.25) we used the symbol κS to indicate the surface grav-ity of a black hole, that is defined 5 by the equation (ξµ is the Killingvector field)

∇µ(ξνξν) = −2κSξµ . (4.28)

5In case of static asymptotically flat black hole solution, the surface gravity hasbeen given a more physical definition: it is the force that an observer at infinity hasto apply on a test body on the event horizon to maintain it at rest. This force isredshifted, then finite, despite the infinite value it assumes in the local frame of thetest body.

33

Comparing (4.25) with the First law of thermodynamics, it is imme-diate to identify the temperature of the thermodynamics system withthe surface gravity of the black hole, up to a proportionality constant(2π)−1. It is possible to prove, to make the analogy even stronger,that for stationary black hole, the surface gravity κS is constant on thewhole event horizon ∆ (see [20], for instance): this result is known asthe zero-th law of black hole mechanics, and can be interpreted as thethermal equilibrium between the Killing horizon of the black hole, thatis also radiating like a black body, and the environment. Of course, thisinterpretation has been proven correct by the discovery of the existenceof such a radiation, today commonly referred as Hawking radiation.Different consideration are to be made if dealing with extremal blackholes, since their surface gravity vanishes: we return to this point atthe end of this chapter.Going back to equation (4.27) we want to summarize the conditionsunder which we obtained this powerful result: φ and δφ are solutionsof Ei(φ) = Elin

ij (φ) = 0 and ξµ is a Killing vector field of a stationaryspacetime such that Lξφ = 0 for all independent fields φ. However,adding an extra feature to our spacetime, we are able to simplify theequations even more.Consider the case, in fact, when the spacetime exibits the presence ofa 2-dimensional spacelike surface Ξ0, called bifurcation surface, charac-terized by the fact that the Killing vector vanishes there, ξµ = 0. It canbe proven that, if the black hole has constant and non-vanishing surfacegravity, then Ξ0 is contained inside the event horizon ∆ or its maximalanalitical extension [21, 22]. If we choose the arbitrary Cauchy surfacesuch that Ξhor ≡ Ξ0, then we can impose, on the left hand side of (4.27)the condition ξµ = 0 already mentioned and ∇[µξν] = κSǫµν . This lastequation comes from the decomposition of the tensor ∇[µξν], i.e.

∇[µξν] = κSǫµν + τ[µξν] (4.29)

into the sum of a tensor orthogonal, ǫµν , and a tensor tangential, τ[µξν],to the surface Ξ0 (here τµ is a vector field tangential to Ξ0). The coef-ficient κS is determined by contracting (4.29) with ξν , using (4.28) andthe fact that, on Ξ0, the Killing vector is lightlike, ξνξν = 0, togetherwith the explicit form of the binormal. Obviously, on the bifurcation

34

surface, (4.29) reduces to

∇[µξν] = κSǫµν

so exploiting all these properties in (4.27) we obtain the result

δ

∫

Ξ0

dΣµν Qµν

∣

∣

∣

ξµ=0, ∇[µ ξν]=κSǫµν

=κSπδS (4.30)

From this last equality, it is straightforward to obtain

S =π

κS

∫

Ξ0

√

gind2 d2x Qµν ǫµν

∣

∣

∣

ξµ=0, ∇[µ ξν]=κSǫµν

(4.31)

where the definition of the surface element dΣµν as been explicitelyused. As we shall prove at the end of this section, this particular ex-pression reproduces the Bekenstein-Hawking area formula for station-ary black hole in general relativity, i.e.

S =A

4GN

. (4.32)

with A is the area of the 2-dimensional surface Ξhor, and S does notdepend on our choice of the horizon cross section.Consider, now, the explicit expression for the Noether potential (3.46)obtained in section 3.2, after imposing the conditions on the Killingvector field obtained by replacing Ξhor with the bifurcation surface Ξ0.Firstly, we see that the only term that contributes to the Noether poten-tial, evaluated at the bifurcation surface, is 2Lµνβγ∇[βξγ] = 2κS Lµνβγǫβγ ,then, when the Lagrangians depends on the Riemann tensor but notits derivatives, the entropy is given by

S = 2π

∫

Ξ0

√

gind2 d2x Lµνρσ ǫµνǫρσ (4.33)

This result is easily generalizable in the case L depends also on deriva-tives of the Riemann tensor [11]: if we call Eµνρσ

R the equation of motionthat one would obtain from the curvature Rµνρσ if it were viewed as anindependent field of the theory, we get the general result for the en-tropy of stationary, axially symmetric black hole in an asymptoticallyflat spacetime, containing a bifurcation surface Ξ0 [11, 6, 12], i.e.

S = 2π

∫

Ξ0

√

gind2 d2x Eµνρσ

R ǫµνǫρσ (4.34)

35

So, in order to evaluate the entropy of a black hole satisfying all theconditions just mentioned, we just have to calculate the equation ofmotion for the Riemann tensor, as if it were an independent field of outtheory, and integrate the result on the bifurcation surface, since theNoether potential, for diffeomorphism invariant theory, can be writtenas [11]:

Qµν = Ξµναξα + 2EµνβγR ∇[βξγ] . (4.35)

But we can still be more general: in fact, it can be proven [12] that notonly the integral (4.34) is independent from the choice of the integrationsurface, but the integrand itself doesn’t change. To see this, rememberthat the Killing vector field ξµ is tangent to the horizon and then itgenerates a flow on ∆, corresponding to a transformation of the horizonin itself. In this case, if a tensor, such as Ξµναξα in (4.35), vanishesat one point, it will vanish at all points sharing the same generatorof the flow. Also, the bifurcation surface has the property that allthe generators of the horizon terminate there, so, if all the fields areregular at the bifurcation surface, then all the tensor proportional to ξαvanishes, so it is again possible to simplify (4.29), taking into accountjust the term proportional to the binormal ǫµν . If this is the case, then,we can generalize the expression for the entropy as:

S = 2π

∫

Ξhor

√

gind2 d2x Eµνρσ

R ǫµνǫρσ (4.36)

This is also the equation we must use to calculate the entropy of ex-tremal black holes, since they do not posses a bifurcation surface. Also,this integral is well behaved and, in general, non-vanishing in the ex-tremality limit κS → 0.Unfortunately, such a simple and unique recipe does not exist (yet) formore general cases, for instance Lagrangian containing Chern-Simonsterms, as we will see in the next section. Also, we want to point out thefact that all possible ambiguities in the definition of Qµν do not modifythe physical quantity, the entropy, that in our interpretation is, togetherwith the energy E and the angular momentum J , a conserved chargeof our system. Again, this feature is not always present if one considersnot-covariant Lagrangian: in that case, in fact, the improvement termscan be physical quantities, as we shall see in the next chapter.Finally, as promised, we want to show that expression (4.31) is con-sistent with the area-law (4.32) for stationary black hole in general

36

relativity. To this end, take the relation satisfied by the Lagrangiann8πL = 1

2R, with R Ricci scalar, and consider the variation of both

sides under variation of the Riemann tensor and also its derivatives.We obtain

8πGNL = −1

2R → 8πGNE

µνρσR δRµνρσ = −1

2gµ[ρgσ]νδRµνρσ

→ 8πGNEµνρσR = −1

2gµ[ρgσ]ν

Also, we know that Qµν = 2EµνρσR ǫρσ (we changed the normalization

of ∇[µξν], but accordingly, we got rid of the factos κ−1S in (4.31)), from

which we obtain

Qµν = − 1

8πGN

gµ[ρgσ]νǫρσ = − 1

8πGN

ǫµν .

Substituting this result in (4.31) we get

S = − 1

8GN

∫

Ξ0

√h d2x ǫµνǫ

µν

=1

4GN

∫

Ξ0

√h d2x =

A

4GN

that is what we wanted to show. Of course, once we fix the scaling con-stant λ by calibrating it using the area law, the charges defined abovemaintain the same normalization also for more complicated theories.

37

Chapter 5

Noether potential forChern-Simons terms

So far, we analyzed Lagrangians that were covariant and invariant undersome continuous transformation (e.g. gauge or diffeomorphism symme-tries). In this chapter we extend our range of analysis to not-covariantLagrangian, in particular Chern-Simons terms, in the context of gaugeand diffeomorphism invariant theories, and mixed theories invariant un-der both transformations. We give a certain number of examples, andfor some of them the prescription to be used for obtaining the correctcharges.

5.1 Ambiguities in the definition of Qµν

So far we focused on calculating Qµν , given a Lagrangian invariantunder a certain Lie group of transformation: the Noether potentialturns out to be strictly related to meaningful physical quantities suchas the electric charge or the entropy of a black hole. However thereexists sources of ambiguities in the form of θµ (and then Jµ), that leadto ambiguities in the Noether potential. For instance, if we add a totalderivative to the Lagrangian,

L → L + ∂µIµ

38

we don’t vary the dynamics of the theory, since the equations of motionare unaffected, but we must take into account the corresponding shift

θµ → θµ + δIµ

Furthermore, we can add to θµ a total derivative

θµ → θµ + ∂νBµν(φ, δφ)

where Bµν is an antisymmetric covariant tensor, linear in the field vari-ations1 δφ that is armless for the conservation of the current.Now suppose that we can extract a total derivative term from δξIµ(this is always the case for diffeomorphism symmetry), i.e.

δξIµ = ∂νCµν(ξ, I) +Dµ(ξ, I) (5.1)

where Cµν(ξ, A) is an antisymmetric tensor linear to the parameter ξ.If this is the case, the current Jµ is determined up to the followingambiguities

Jµ → Jµ + ∂νBµν(φ, δξφ) + ∂νC

µν(ξ, I) .

Then, for a given Jµ, eq. (2.16) determines the Noether potential upto the following ambiguities:

Qµν → Qµν +Bµν(φ, δξφ) + Cµν(ξ, I) + ∂ρGµνρ (5.2)

where Gµνρ is an antisymmetric tensor in (at least) the indices µ and ν.Of course, all of those ambiguities (due often to the use of different al-gorithm for computing Qµν) give rise to different results, that approachthe same Noether potential when evaluated in a symmetric background,δξφ = 0, are completely armless and, as expected, do not influence thephysical charges. However, things get more complicated when one con-siders non covariant Lagrangian: while under certain hypotesis [23] it’sstill possible to obtain the first law of the black hole mechanics, we areno longer assured that improvement terms are physically unimportant:in other words, in order to obtain sensible physical results from the

1This feature turns out to be a necessary condition for covariant improvementterms, since θµ is covariant and linear in the variations δφ, that are, on the otherhand, covariant terms themselves

39

Noether potential one can be forced to choose non covariant improve-ment terms to add to the current, that do not vanish in the symmetricbackground. So, in the following sections, we consider Chern-Simonsterms, to be added to a covariant Lagrangian, and their contributionto the Noether potential: we will then try to find out what criterionhas to be used in order to get the correct Noether charge.

5.2 Gauge-theory examples

Consider, as a first example of non covariant Lagrangian, an abelianChern-Simons term in 5 space-time dimensions

LCS = εµνρστAµFνρFστ (5.3)

The quantity εµνρστ is the total antisymmeric Levi-Civita symbol, thatis gauge invariant. The transformation laws for the fields are

εµνρστδFµν = 2εµνρστ∂νδAρ δξAµ = ∇µξ δξFµν = 0

This Lagrangian is gauge invariant, up to a total derivative, and itsvariation under general and gauge transformations is given by:

δLCS = εµνρστ (δAµFνρFστ + 2AµδFνρFστ )

= εµνρστ [−3FνρFστδAµ + ∂µ(4AρδAνFστ )] ,

δξLCS = εµνρστ∂µ(ξFνρFστ ) ⇒Jµ = −εµνρστ (ξFνρFστ − 4Aρ∂νξFστ ) (5.4)

where to obtain δξL we made use of the Bianchi identity ∂[µFνρ] = 0.The equation of motion for δAµ is

3εµνρστFνρFστ = 0 . (5.5)

Of course we are tempted to use this equality2 in (5.4), to obtain, upto term proportional to the equation of motion, the value

Qµν = 4 εµνρστξAρFστ .

2Notice that the multiplicative factor it’s not relevant if we consider just theLagrangian (5.3)

40

However, as we will see in a moment, the correct result has a differentmultiplicative factor, 6 instead of 4, so using the equation of motioncarelessy could lead to mistake. At this point, then, becomes necessaryto add an improvement term of the form

2c εµνρστ∂ν(ξAρFστ ) (5.6)

to the current (5.4). The final expression for Jµ is

Jµ = −εµνρστ (ξFνρFστ − 4Aρ∂νξFστ ) + 2c εµνρστ∂ν(ξAρFστ )

= (c− 1)εµνρστξFνρFστ + (4 + 2c)εµνρστ (∂νξ)AρFστ

= −3εµνρστ ξFνρFστ + (4 + 2c)εµνρστ∂ν(ξAρFστ ) . (5.7)

This time, the multiplicative factor in the first term of the last equal-ity is equal to the one appearing in the equation of motion, but theambiguity is still present. Nevertheless, if we consider a “full” theoryincluding also an invariant Maxwell term, i.e.

L =1

4FµνF

µν + εµνρστAµFνρFστ

the relative multiplicative factors of all terms become relevant. The“full” current reads:

Jµ = −εµνρστξFνρFστ + 4 εµνρστ (∂νξ)AρFστ + (∂νξ)Fµν (5.8)

and the equation of motion equals

3 εµνρστFνρFστ + ∂νFµν = 0 (5.9)

Adding again the relevant improvement term (5.6), we obtain

Jµ = −3 εµνρστξFνρFστ + (4 + 2c)εµνρστ∂ν(ξAρFστ ) + ∂νξFµν

= −(

3 εµνρστξFνρFστ + ξ∂νFµν

)

+ ∂ν

[

(4 + 2c)εµνρστξAρFστ + ξF µν]

= ∂ν

[

εµνρστ (4 + 2c)ξAρFστ + ξF µν]

This time, we could use directly the equation of motion (5.9) for Aµ,to obtain the last equality.

41

From this equation, the contribution to the Noether potential from theLagrangian (5.3) is then given by

QµνCS = εµνρστ (4 + 2c)ξAρFστ + ξF µν . (5.10)

The constant c can be finally evaluated in the following way. We con-sider the conserved charge for abelian gauge theories, with ξ satisfyingthe symmetric background condition, in this case ∂µξ = 0, so ξ con-stant. But this means that Jµ(δφ = 0) = 0, since after the additionof the improvement term and the use of the equation of motion, thecurrent is proportional to δξφ. Then our conserved charge is given by:

∫

∂C

dΣµν

[

εµνρστ (4 + 2c)ξAρFστ + ξF µν]

= 0 (5.11)

with ξ constant.Notice, however, that we can re-write (5.9) as

∂ν

(

6 εµνρστξAρFστ + ξF µν)

= 0 (5.12)

just multiplying by a constant ξ (satisfying δξAµ = 0). Integratingover a n−1 dimensional Cauchy surface C we get the conserved chargerelated to the residual gauge symmetry, namely

∫

∂C

dΣµν

(

6 εµνρστξAρFστ + ξF µν)

= 0 (5.13)

Confronting (5.11) and (5.13) we immediately get c = 1, that is thedesidered result.At this point we notice that the procedure we just followed is identi-cal to the one used in section 4.1 to obtain the electric charge for aYang-Mills theory: we are forced to add a non covariant improvementterm to the current and then we used the equation of motion to fixthe last degree of freedom of our theory. We already anticipated thepossible presence of such a feature: when dealing with non covariantLagrangian the (non covariant) improvement terms can be physicallyrelevant, they really do change the final physical result (in this case,the difference resided in a multiplicative factor).In retrospect, this result is not surprising: we already know, in fact,that a sufficient condition for the theory to admit a conserved charge

42

related to the residual abelian gauge symmetry of the fields is Jµ ∝ δξφ.In the case at hand, this condition is enough to obtain the correct result,since, already in the second line of (5.7), the current is proportional tothe field variation δξAµ = ∂µξ only if c = 1. This is indeed the generalrecipe we extrapolate from this example:when dealing with Lagrangian invariant (up to a boundary term) underabelian gauge symmetry, the current has to be proportional to the fieldvariation δξφ.In this way, the current identically vanishes for symmetric solutionsδξφ = 0, and the Noether potential is conserved on the equation ofmotion.Lastly, we point out that the relevant improvement term (5.6) is notproportional to the field variation, i.e. δAµ, but instead to the parame-ter ξ: this feature is due, again, to the non covariance of the Lagrangianunder examination, leading also to the physical importance of the im-provement term considered.

5.3 Covariant and mixed Chern-Simons terms

The first example we analyze in this section is the Chern-Simons La-grangian in (5.3), under general coordinates transformation. This La-grangian can be added to the Lagrangian (3.18) in the previous chapter,and it won’t modify the equations of motion we already found, and willgive a genuine new contribution to the Noether potential. Notice alsothat, while under gauge transformation LCS was non covariant, now ittransforms like a density of weight 1, as it should, so it is a covariantLagrangian.Under diffeomorphism the fields transform as follows

LξAµ = ξλ∇λAµ + ∇µξλAλ = ξλFλµ + ∇µ(ξλAλ) (5.14)

LξFµν = ∇µξλFλν + ∇νξλFµλ + ξµ∇λFµν = −2∇[µ(ξλFν]λ) (5.15)

where the second equality of (5.15) is obtained using the Bianchi iden-tity for the field strength. Also, the second term in the variation of thegauge field it is equivalent to a gauge transformation with parameterξ = ξλAλ, that we just analyzed in the previous section. Neverthelesswe will consider the whole transformation law, since this time Nµ is

43

now different.Consider now the variation of this Lagrangian under general and dif-feomorphism transformation. We get:

δξLCS = ∂µ(ξµLCS) , (5.16)

δLCS = εµνρστ (δAµFνρFστ + 2AµδFνρFστ )

= εµνρστ [−3FνρFστδAµ + ∂µ(4AρδAνFστ )] ⇒Jµ = 4 εµνρστ

[

ξλFλν + ∂ν(ξλAλ)

]

AρFστ −ξµενρστλAνFρσFτλ(5.17)

Now we will use the equality, holding in n dimensions

A[µ1µ2···Bµi··· · · ·C ···µnµn+1] = 0 . (5.18)

For the case at hand, this equality reads:

ε[µνρστξλ] = 16εµνρστξλ−ενρστλξµ+ερστλµξν−εστλµνξρ+ετλµνρξσ−ελµνρσξτ = 0

(5.19)Using this identity, after few rearrangements of indeces, we get

4 εµνρστξλAρFστFλν − ενρστλξµAνFρσFτλ = −εµνρστ (ξλAλ)FνρFστ

from which, straightforwardly

Jµ = −εµνρστ (ξ · A)FνρFστ + 4 ∂ν(ξ · A)AρFστ

= −3 εµνρστ (ξ · A)FνρFστ + 4 εµνρστ∂ν

[

(ξ ·A)AρFστ

]

(5.20)

where the notation ξ ·A = ξλAλ is understood.At this point, using the equation of motion in (5.5), we get the desideredresult, i.e. the contribution to the Noether potential coming from theLagrangian LCS:

QµνCS = 4 εµνρστ (ξ · A)AρFστ . (5.21)

The fact that the Lagrangian (5.3) is a covariant quantity under diffeo-morphism saved us from including any improvement terms. In the pre-vious example, because of the non covariance of the same Lagrangian,under gauge symmetry, we had to add a (non covariant) improvement

44

term, in order to extrapolate a total derivative from the current, af-ter using the equation of motion, and, hence, obtain the result for thecontribution to the Noether potential. No new recipe for adding im-provement terms is then unveiled, as we could expect from the verybeginning.To end this section we analyze a mixed Chern-Simons term: the La-grangian this time contains contemporarly both the Riemann tensorand the gauge field, and is of the form

LCS = εµνρστAµRαβ

νρ Rσταβ . (5.22)

Notice, also, that this Lagrangian is non covariant under gauge trans-formation because of the explicit presence of the gauge field Aµ. Wewant to evaluate the conserved current obtained under diffeomorphism.The transformation laws are eqs. (5.14), (3.23) and (3.19). From thevariations of the Lagrangian we get:

δξLCS = ∂µ(ξµLCS) ,

δLCS = εµνρστ[

δAµRνργǫRσταβ gγαgǫβ + AµδRνργǫR

γǫστ + AµR

αβνρ δRσταβ

+AµRνργǫRσταβ δ(gγαgǫβ)

]

= εµνρστ[

δAµRαβ

νρ Rσταβ + 2Aµ(Rλ

νρα δgβλ + 2∇ν∇αδgβρ)R αβστ

−2R αβνρ R γ

στα δgγβ

]

(5.23)

where to obtain the last equality we used the identity

δgαβ = −gαρδgρσgσβ (5.24)

that is straightforwardly obtained from the identity gαβgβγ = δγα.

Also, if we change contemporarly the indeces (ν, ρ) in (στ), the overallsign remains unchanged, so the terms proportional to the variation ofthe Riemann tensor are identical. To obtain the current we manipulatethe terms containing derivatives of the metric variation in the followingway:

εµνρστAµRαβ

στ ∇ν∇αδgβρ = ∇ν(εµνρστAµR αβστ ∇αδgβρ) − εµνρστ∇ν(AµR αβ

στ )∇αδgβρ= ∇µ(−εµνρστAνR αβ

στ ∇αδgβρ − 12εανρστFναR

µβστ δgβρ)

+12εµνρστ∇α(FµνR αβ

στ )δgβρ (5.25)

45

Again, we used the Bianchi identity ∇[ρRαβ

στ ] = 0. From this expressionwe immediately get the current

Jµ = −4 εµνρστAνRαβ

στ ∇αLξgβρ + 2 εανρστFανRµβ

στ Lξgβρ − ξµLCS(5.26)

and the equation of motion for the gauge field Aµ and the metric gµν ,i.e.

εµνρστR αβνρ Rσταβ = 0 (EoM for Aµ) (5.27)

4 ερστλ(µ∇α(FρσR ν)ατλ ) = 0 (EoM for gµν) (5.28)

Using (5.19), we obtain the following useful identity

ξµLCS = εµνρστ (ξ · A)R αβνρ Rσταβ − 4 εµνρστAρξ

γRνγαβRαβ

στ

where we made use of the definition of Rµνρσ and the antisymmetry ofthe Riemann tensor in the last two indices α and β.Also, the other two terms inthe current can be rewritten as:

εανρστFανRµβ

στ Lξgβρ = −2 εανστ(ρFανRβ)µ

στ ∇βξρεµνρστAνR

αβστ ∇αLξgβρ = −εµνρστAρR αβ

στ

[

(∇ν∇αξβ − Rανβγξγ)

−12Rαβνγξ

γ]

Now, using these last three equations in (5.26), we get:

Jµ = 4 εµνρστAρRαβ

στ

[

∇ν∇αξβ − Rανβγξγ − 1

2Rαβνγξ

γ]

−4 εανστ(ρFανRβ)µ

στ ∇βξρ − εµνρστ (ξ · A)R αβνρ Rσταβ

+4 εµνρστAρξγRνγαβR

αβστ (5.29)

This expression can be easily manipulated: for example, all the termscontaining two Riemann tensor, one of which is also contracted with ξγ

sum up to zero, since R[ανβ]γξγ = 0, Rαβνγ = Rνγαβ and R αβ

στ R γναβ =

−R αβστ R γ

νβα .Furthermore we can add and subract the terms

4 εµνρστ∇ν(

AρRαβ

στ

)

∇αξβ , 4 εανστ(ρ∇β(

FανRβ)µ

στ

)

ξρ , 2 εανστµ∇β(

FανRρβ

στ

)

ξρ

46

with ∇[αFβγ] = 0. Now, using the following identities

εανστβ∇β(

FανRρµ

στ

)

ξρ = 0 ,

4 εανστ(µ∇β(

FανRρ)β

στ

)

ξρ = −2 εανστρ∇β(

FανRβµ

στ

)

ξρ + 2 εανστµ∇β(

FανRρβ

στ

)

ξρ

4 εανστ [µ∇β(

FανRβ]ρ

στ ξρ

)

= −2 εανστβ∇β(

FανRρµ

στ ξρ

)

+ 2 εµνρστFνρRαβ

στ ∇αξβ

−2 εανστµ∇β(

FανRρβ

στ

)

ξρ

the current is given by:

Jµ = −4 εανστ(µ∇β(

FανRρ)β

στ

)

ξρ − εµνρστ (ξ · A)R αβνρ Rσταβ

+∇ν(

4 εµνρστAρRαβ

στ ∇αξβ − 2 ερστλαFρσRνµ

τλ ξα

−4 ερστλ[µFρσRν]α

τλ ξα

)

It is sufficient to recognize the first line to be the sum of the equationsof motion (5.27), (5.28), to obtain the Noether potential

Qµν = 4 εµνρστAρRαβ

στ ∇αξβ − 2 ερστλαFρσRνµ

τλ ξα

−4 ερστλ[µFρσRν]α

τλ ξα (5.30)

As we expected, since the Chern Simons Lagrangian (5.22) is diffeomor-phism covariant and it’s a scalar density of weight 1, as it is supposedto be, we didn’t have to add any improvement term, in order to obtainthe Noether potential.

5.4 The mixed Chern-Simons term revis-

ited

So far, we analyzed Chern-Simons Lagrangians that were either covari-ant or non covariant under gauge or general coordinate transformation.We also explicitely showed that it is possible for the same Lagrangian,e.g. (5.3), to be at the same time covariant under a group of transfor-mation but non covariant under a different one: in the latter case, we

47

could need to add not-covariant improvement terms in order to get theNoether potential, in the former case the Noether potantial is simplyobtained by following the standard procedure explained in the prevoiuschapters.We could ask then, the following question: are the Lagrangians con-sidered covariant under local Lorentz transformation? The answer isobviously yes, they are not only covariant but also invariant undertransformations of the local Lorentz group. However, we can add toLCS a boundary term such that the theory becomes explicitely de-pendent on Lorentz indices, covariant under gauge and local Lorentztransformation, but not under diffeomorphism anymore, without, ofcourse, varying the dynamics of the system. We will see shortly if andwhat kind of modifications the addition of the boundary term leads to,but, first of all, we want to briefly introduce the quantities we shall useextensively in this section3, i.e. vielbeine and spin connection fields.Consider a field φ(x) , whose nature we don’t need to specify, and itstransformation under the local Lorentz group:

φ(x) → φ(x) + 12ǫab(x)Mabφ(x) (5.31)

where Mab are the generators of the Lorentz group M and their repre-sentation depends, of course, on the representation of the Lorentz groupof the field φ. We already know that these generators are antisymmetricin the two indices a and b, and satisfied the usual commutator algebra

[Mab,Mcd] = −δacMbd + δadMbc − δbdMac + δbcMad (5.32)

Now, in order to express the covariant derivative of the field under localLorentz transformation, we need the gauge field associated with such agroup: we call this new gauge field spin connection field ω ab

µ , and wehave

Dµφ = ∂µφ− 12ω abµ Mabφ (5.33)

From this definition of the spin connection field together with (5.32)one easily obtains the transformation rule of ω ab

µ under local Lorentztransformations:

δǫωabµ = Dµǫ

ab = ∂µǫab − ω a

µ cǫcb − ω b

µ cǫac (5.34)

3For further references see [24, 25]

48

where, obviously, only the Lorentz indices are affected by Lorentz trans-formations. Notice that the spin connection field contains both Lorentzand world indices, both running from 1 to the dimension n of the spacetime we are considering. The presence of the world index µ allows usto define in a very natural way the transformation rule under diffeo-morphism, i.e.

Lξωabµ = ∂µξ

νω abν + ξν∂νω

abµ (5.35)

with ξµ(x) is the parameter of the general coordinate transformation.This time, only the world index is affected by this transformation, asit was possible to foresee. The transformation rule for Dµφ under dif-feomorphism it is basically the same as (5.35). Now, we would like tore-write the covariant derivative Dµ in such a way that it transformsas a vector under local Lorentz transformations and as a scalar underdiffeomorphism: in order to do so we need to introduce a new quantity,whose transformation is exactly the opposite of the term containingderivatives of ξ. As it turns out, this is not only possible but alsostraightforward: we introduce the vielbein field e µ

a to define the newcovariant derivative as follows:

Dαφ = e µa Dµφ (5.36)

This new define covariant derivative satisfies all of our requests: sinceit has one Lorentz index, it transform as a vector under Lorentz trans-formation, and, on the contrary, transforms as a scalar under diffeo-morphism, since it does not contain any world index, i.e.

δǫDαφ = 12ǫcd

(

δacδdb − δadδc

b + δabMcd

)

Dβφ (5.37)

where in the first two terms we could specify the representation for thegenerators to be the vector representation because Dβφ already con-tains a vector index, while, since the representation of the field φ isnot specified, we left the general form Mcd of the generators in the lastterm.The vielbein field, that can be seen as the gauge field associated to dif-feomorphism, on the other hand, transforms as a vector under Lorentztransformation and as a contravariant vector under general coordinatetransformation. Furthermore, it is possible to prove that it is not sin-gular and the inverse e β

m is such that

eaµeµ

b = δab , eµ

bebν = δµ

ν

49

Also, a simple relation exists that links the flat Minkowsky metric ηabwith the metric tensor gµν of the whole spacetime, i.e.

gµν ≡ e aµ e

bν ηab .

Next, we can define a covariant derivative ∇µ not only for Lorentz fieldsφ, but also for world vector and tensor, using the vielbein to convertthe Lorentz index in world index and viceversa. The results read:

∇µAν = DµAν − ΓµνρAρ , ∇µAν = DµA

ν + ΓµρνAρ (5.38)

where Γ’s coincide, if the torsion tensor is identically zero (from nowon, we assume this condition is always satisfied) with the Christoffelsymbol, and in this formalism are given by:

Γµνρ = ea

ρDµeνa = −eνaDµe

ρa

The new defined covariant derivative exactly coincides with the onewe used throughout our entire work, but, so far, no Lorentz indicesand, consequently, no spin connection field appeared in our calculation.However, the definition of the covariant derivative ∇ in its more generalform contains both spin connection fields and the Christoffel symbols,and reduces to Dµ if there are no world indices involved. It can besimpy proven that:

∇µeνa = 0 , ∇µgνρ = 0 ,

namely both the vielbein and the metric are covariantly constant, if thetheory under examination is torsion free.Notice that the second equality is the usual definition of covariantderivative of the metric tensor, from which one can obtain the expres-sion of Γ as a function of the metric and its derivatives, while thefirst equality represents a relation between the spin connection and thevielbein field, that can be made explicit by exploiting the null torsioncondition. The result is

ωµab = 12eνb

(

∂µeaν−∂νeaµ)

−12eνa

(

∂µebν−∂νebµ)

−12eσbe

ρaecµ

(

∂ρecσ−∂σecρ)

(5.39)The last step of our brief review is to define the curvature tensor,

by means of the commutator of covariant derivatives: if we use Dα we

50

obtain the curvature associated with Lorentz transformation, given bythe equation

Rµνab(ω) = ∂µων

ab − ∂νωµab − ωµ

ac ωνcb + ων

ac ωµcb . (5.40)

This expression looks analogous to the expression of the Riemann tensorobtained by using the commutator of ∇’s,

Rµνρσ = ∂µΓνρ

σ − ∂νΓµρσ − Γµρ

λΓνλσ + Γνρ

λΓµλσ (5.41)

and in fact they are related by the expression

Rµνρσ = Rµν

abeρaebσ .

Naturally, Bianchi identities holds for Rµνab too, and read

∇[µRνρ]ab = 0 , R[µν

abeρ]b = 0 .

These are mostly all the tools we will use in the following. Other prop-erties will be given along with the calculations.As we already said in section 1.3, the addition of a total derivativeterm to L, as weel as the addition of covariant improvement term tothe current, does not change the dynamics of the system and the phys-ical quantities are unaffected. However, we just found that, if the La-grangian is non covariant, we may need to add specific non covariantimprovement terms in order to obtain the correct result for Noetherpotential, and, more importantly, those improvement terms turn out tobe physically relevant. What we want to know is if the other source ofambiguity in the current, the addition of a non covariant total deriva-tive term to the Lagrangian leads to any difference with respect to theresult (5.30).Consider now the Lagrangian (5.22) covariant under diffeomorphismtransformation and suppose to add a non covariant term4

LCS = εµνρστAµRνρabRστab →

LCS − ∂ν

(

2 εµνρστAµRστab ωρab + 4

3εµνρστAµωρ

ab ωσac ωτcb

)

4Notice that we re-write (5.22) using two Lorentz indices instead of world indices,as before. However this does not constitues a source of difference since both Lorentzand world indices run from 1 to the dimension n of the spacetime, in this case 5,and are contracted.

51

The resulting Lagrangian, that we will call again LCS for simplicity ofnotation, can be written in a more handy form as:

LCS = 2 εµνρστFµν ωρab

(

∂σωτab − 23ωσac ωτ

cb

)

, (5.42)

where the following identities were used:

εµνρστRστab = εµνρστ(

∇σ ωτab + ∂σωτab

)

(5.43)

εµνρστ ωρab ωνac∂σωτcb = 1

3εµνρστ∂σ

(

ωτcb ωρab ωνa

c)

(5.44)

εµνρστ ωρab ωσ

cd(ωνac ωτdb) = εµνρστ ωρab ωσ

cd(−ωτbd ωνca) = 0(5.45)

The proofs of these identities are straightforward: take the last, for in-stance. If we simultaneously use the changes ν ↔ τ , a↔ b and c↔ d,the overall sign changes, and given the antisymmetry of the spin con-nection field in the Lorentz indices, the result follows. Notice also that,since all the world indices are antisymmetrized by the Levi-Civita sym-bol, no expression containing an odd number of Christoffel symbols,with all lower indices saturated by εµν··· can appear explicitely, giventhe symmetry of Γ in its lower (world) indices.The Lagrangian (5.42) is now non covariant under diffeomorphism, be-cause of explicit presence of the spin connection fields. Also, this timewe must consider the variation δǫ;ξ = δǫ + Lξ of LCS under both dif-feomorphism and local Lorentz transformation. To do so, we need thefollowing transformation laws:

δǫ;ξeµa = ǫaceµc + ∇µ(ξλeλa) + ξλωλ

aceµc (5.46)

δǫ;ξ ωµab = ∇µǫab − Rµν

ab(ω)ξν + ∇µ(ξλωλab) (5.47)

where to obtain the first equation we used (5.33) together with thevector representation of the Lorentz generators (Mab)c

d = δbaδcd−δcaδbd

while the second one comes from eqs. (5.34), (5.35) and (5.40). Also,from varying the null torsion condition we can obtain the followingidentity:

δωµab = ∇[ρ

[

eµceaρebσ δeσ]

c − ebρ δeµ]a + eaρ δeµ]

b]

. (5.48)

Obviously, this identity is such that if we use (5.46) in it we get (5.47).First of all, we calculate as usual the variation of the Lagrangian (5.42)

52

under combined Lorentz and diffeomorphism transformation. We have

LξLCS = ∂µ(ξµLCS) (5.49)

δǫLCS = 2 εµνρστFµν δǫ

[

ωρab

(

∂σωτab − 23ωσac ωτ

cb

)

]

= −2 εµνρστFµν∂σǫab∂ρωτab = −2 εµνρστ∂µ

(

Fνρ∂σǫab ωτab

)

.(5.50)

where, to obtain the second equality in δǫLCS one notices that all theother terms, not proportional to the derivative of ǫab vanish identically.We want to evaluate also θµ, but before doing this, we give few usefulgeneral identities that will be used later on, i.e.

εµνρστ(

Rστab − ωσ

ab∇τ)

Aab = εµνρστAab∂σωτab + ∂σ

(

εµνρστ ωτabAab

)

(5.51)

ωσab(∂τ −∇τ )Aab = 2ωσ

ab ωτbcAac (5.52)

that hold whenever Aab = −Aba. Now, under a generic variation, theLagrangian transforms as follows

δLCS = 2 εµνρστ[

δFµν ωρabBστab + Fµνδ(ωρ

abBστab)]

(5.53)

where we use the short hand notation Bστab = ∂σωτab − 23ωσac ωτ

cb.

As usual, all the varied quantities in the previous equation must beexpressed in terms of variation of the independent fields of our theory,gµν and Aµ: this is our method to calculate both θµ and the equations ofmotion for those fields. Obviously the addition of a total derivative term(even a non covariant one) in a Lagrangian can’t modify the equationof motion, eqs. (5.27) and (5.28) we calculated in the previous section.On the other hand, in this case, it’s simpler to consider the vielbein tobe the independent field of the theory, instead of gµν , since we alreadyknow the relation between δωµ

ab and δeµa (5.48).

Now, consider the first term in (5.53):

2 εµνρστδFµν ωρabBστab = 4 εµνρστ∂µ(δAνωρ

abBστab) − (. . . )δAν (5.54)

Notice that the dots indicate the equation of motion for Aµ, (5.28).The other term in the generic variation of LCS can be manipulated as

53

follows:

2 εµνρστFµνδ(ωρabBστab) = 4 εµνρστFµν∂σωτabδωρ

ab + 2 εµνρστFµν∂σ(ωρab δωτab)

−43εµνρστFµνδ(ωρ

ab ωσac ωτcb)

= 2 εµνρστFµν(2 ∂σωτab − 2ωσac ωτcb)δωρ

ab

+2 εµνρστ∂σ(Fµν ωρab δωτab)

= 2 εµνρστFνρRστabδωµab − 2 εµνρστ∂µ(Fνρ ωσ

ab δωτab)

(5.55)

where we have used (5.44) to obtain the second equality , and thedefinition (5.40) in the third equality.The term proportional to the spin curvature still isn’t of the desiredform, since the spin connection field is not an independent field of thetheory, but we can re-write it using the relation (5.48), i.e.

2 εµνρστFνρRστabδωµab = 2 εµνρστFνρRστ

ab∇α(eµceaαebβ δeβ]c − eb

α δeµ]a + eaα δeµ]b)

= ∇[α[2 εµνρστFνρRστ

ab(eµceaαeb

β δeβ]c − eb

α δeµ]a + eaα δeµ]b)]

−(. . . )δeµa (5.56)

since the vielbein are covariantly constant. We are not interested inthe equation of motion for eµ

a, since the dynamics of the theory isalready known, so we just need to evaluate the contribution to θµ,by specifying the variation of the vielbein to be given by eq. (5.46).For simplicity we analyze the contribution from Lorentz and generalcoordinate transformation separately. First, substituting δǫeµ

a in thesecond line of (5.56) we get

∇[α[2 εµνρστFνρRστ

ab(eµcea

αebβǫcdeβ]

d − ebαǫaceµ]

c + eaα ǫbceµ]

c)]

= ∇[α[2 εµνρστFνρRστ

ab(eaαeb

βǫµβ] − ebαǫaµ] + ea

α ǫbµ])]

= ∇µ(2 εµνρστFνρRστabǫab) = ∂µ(2 ε

µνρστFνρRστabǫab) (5.57)

where to obtain the last line we have used the antisymmetry of Rστab

and ǫab in their Lorentz indices, and the last equality holds becausethe flat indices are dummy and the Christoffel symbols are antisym-metrized, then they do not give any contribution.Also, using δξeµ

a, we evaluate the contribution to the current of the

54

term (5.56) due to diffeomorphism transformation:

∇[α

[

2 εµνρστFνρRστab

(

eµcea

αebβ(∇β]ξ

c − ξλωλβ]c) − eb

α(∇µ]ξa − ξλωλµ]a)

+eaα(∇µ]ξb − ξλωλµ]b)

)]

= ∇a[

2 εµνρστFνρRστab(∇bξµ − ξλωλbµ)

]

+ ∇a[

2 εµνρστFνρRστab(∇µξb − ξλωλµb)

]

−∇µ[

2 εµνρστFνρRστab(∇aξb − ξλωλab)

]

= ∇a[

4 εµνρστFνρRστab∇(µξb)

]

−∇µ[

2 εµνρστFνρRστab(∇aξb − ξλωλab)

]

= ∇µ[

4 ενρστλFνρRστµα∇(λξα) + 2 εµνρστFνρRστ

ab(ξλωλab −∇aξb)]

= ∂µ

[

4 ενρστλFνρRστµα∇(λξα) + 2 εµνρστFνρRστ

ab(ξλωλab −∇aξb)]

(5.58)

The antisymmetry of the spin connection field in its last two indiceshas been used to obtain the third equality.We are now ready to write the full conserved current:

Jµ = 4 εµνρστδǫ;ξAνωρabBστab − 2 εµνρστFνρ ωσ

ab δǫ;ξ ωτab

+2 εµνρστFνρRστabǫab + 4 ενρστλFνρRστ

µα∇(λξα)

+2 εµνρστFνρRστab(ξλωλab −∇aξb) − ξµLCS

+2 εµνρστFνρ∂σǫab ωτab (5.59)

Of course this expression can be re-written in a more handy form, using(5.47) and (5.14)5, and using the following identity

(4 εµνρστξλFλν − 2 ελνρστξµFλν)ωρabBστab = −2 εµνρτλFλνξ

σ(3ω[ρabBστ ]ab)

= −2 εµνρτλFνρξσ(3ω[σ

ab∂τωλ]ab − 2ω[σab ωτac ωλ]

cb) (5.60)

obtained from (5.19), where the antisymmetrization of the three indices

5Remember that the gauge field does not transform under Lorentz transforma-tion

55

(σ, τ, λ) is considered. Finally, the current reads

Jµ = 4 εµνρστ∂ν(ξλAλ)ωρ

ab(∂σωτab − 23ωσac ωτ

cb)

−2 εµνρτλFνρ ξσ(3ω[σ

ab∂τωλ]ab − 2ω[σab ωτac ωλ]

cb)

+2 εµνρστFνρ ωσabRabτ

cξc

+4 ενρστλFνρRστµα∇(λξα) − 2 εµνρστFνρRστ

ab∇aξb+2 εµνρστFνρ ∂σǫ

ab ωτab

+2 εµνρστFνρRστab(ǫab + ξλωλab)

−2 εµνρστFνρ ωσab∇τ (ǫab + ξλωλab) (5.61)

Now, from this current we want to extract the Noether potential, byusing the equation of motion for the metric and the gauge field: let’sanalyze again each term individually. We have:

4 εµνρστ∂ν(ξλAλ)ωρ

abBστab = ∂ν(4 εµνρστξλAλωρ

abBστab)

−4 εµνρστξλAλ∂ν [ωρabBστab] (5.62)

and the second term is equivalent to the equation of motion for themetric,

−4 εµνρστξλAλ[∂νωρab∂σωτab − 2

3∂ν(ωρ

ab ωσac ωτcb)]

= −4 εµνρστξλAλ(∂νωρab∂σωτab − 2∂νωρ

ab ωσac ωτcb)

= −4 εµνρστξλAλRνρabRστab

that holds because of the identities (5.44), (5.45).Consider now the fourth line in the expression of the current:

4 ενρστλFνρRστµα∇(λξα) − 2 εµνρστFνρRστ

ab∇aξb= ∇λ(2 ενρστλFνρRστ

µαξα) − 2 ενρστλ∇λ(FνρRστµα)ξα

∇α(2 ενρστλFνρRστµαξλ) − 2 ενρστλ∇α(FνρRστ

µα)ξλ

∇α(−2 εµνρστFνρRσταβξβ) + 2 εµνρστλ∇α(FνρRστ

αβ)ξβ

= ∇ν [2 ερστκνFρσRτκµαξα − 2 ερστκµFρσRτκ

ναξα

+2 ερστκλFρσRστµνξλ] + 4 ερστκ(µ∇α(FρσRτκ

λ)α)ξλ

= ∂ν(−4 ερστκ[µFρσRτκν]αξα + 2 ερστκλFρσRστ

µνξλ)

−4 ενρστ(µ∇α(FνρRστλ)α)ξλ (5.63)

56

where we used the Bianchi identity for both the Field strength andthe spin curvature. Notice that the last line is exactly the equation ofmotion for the metric.Next, we manipulate the last three lines of equation (5.61): we use theidentities (5.51) and (5.52), obtaining

2 εµνρστFνρ(Rστab − ωσ

ab∇τ )(ǫab + ξλωλab) + 2 εµνρστFνρ ∂σǫab ωτab

= ∂σ[2 εµνρστFνρ ωτ

ab(ǫab + ξλωλab)] + 2 εµνρστFνρ∂σ(ωτabǫab)

+2 εµνρστFνρξλωλab ∂σωτ

ab

= ∂ν [4 εµνρστFρσ ωτ

ab(ǫab + 12ξλωλab)] + 2 εµνρστFνρ ξ

λωλab ∂σωτab .

(5.64)

We add the last term to all the remaining terms of the current, i.e.

−2 εµνρτλFνρ ξσ(3ω[σ

ab∂τωλ]ab − 2ω[σab ωτac ωλ]

cb)

+2 εµνρστFνρ ξλωλab ∂σωτ

ab + 2 εµνρστFνρ ωσabRabτ

cξc (5.65)

Now, by using the explicit expression for the spin curvature (5.40), thelast expression is the sum of terms containing products of spin connec-tion and terms of the form ω∂ω: as it turns out, both the contributionsvanish identically.Finally we write the desired result: the Noether potential for the La-grangian (5.42) reads:

Qµν = 4 εµνρστξλAλωρab(∂σωτab − 2

3ωσac ωτ

cβ)

−4 ερστκ[µFρσRτκν]αξα + 2 ερστκλFρσRστ

µνξλ

+4 εµνρστFρσ ωτab(ǫab + 1

2ξλωλab) (5.66)

As it is immediate to realize, since the Lagangian (5.42) doesn’t containany explicit (Riemann or spin) curvature, the term proportional to thecurl of ξµ is absent, while the others covariant terms, proportional toF ∧R don’t undergo any variation. On the other hand, the addition ofexplicit spin connection fields introduces a new parameter, ǫab, whosecontribution to the entropy must still be analyzed.

57

Chapter 6

Conclusions

In this work of thesis, we presented the concept of Noether potentialand the most important results related to it.In the first part of the work, the Noether theorem and a theorem byWald, defining Qµν , are treated, together with a simple abelian gaugetheory example. After that, in chapter 3, we show how this quantityturns out to be closely related to the entropy of a black hole, when dif-feomorphism invariant theory of gravity are considered, under specificassumptions for the fields that we extensively treated in the same chap-ter. This constitutes the central part of the whole work: an explicitequation for the entropy of a black hole is obtained, exploiting the firstlaw of black hole mechanics, and the validity and the applicability ofthis formula are discussed. We also discussed at length a simple ex-ample of an implicit Lagrangian theory of gravity, showing what is theprocedure to follow to get the Noether potential.In the last part, various original examples are treated, in the attempt togeneralize the results obtained previously, when non covariant improve-ment terms in the current are considered or total derivative terms areadded to a covariant Lagrangian, in order to get the ”correct” result forthe conserved charge. The last example we gave is, in this respect, stillincomplete: the non covariant terms we added to the Lagrangian (5.22)introduced a new parameter ǫab related to the invariance of (5.42) un-der not only general coordinate, but also Lorentz transformations. Westill need to fully understand what kind of condition must be imposedin order to get the conserved charges, when a new invariance of theLagrangian is introduced.

58

Also, we presented an explicit definition of the conserved charge re-lated to abelian gauge theories and we worked out the calculationsexplicitely, while an analogous definition for non abelian gauge theoriesis still missing.

59

Appendix A

Lie-derivative and Killingfields

We want to give in this appendix some useful definitions, propertiesand identities of Lie derivatives and Killing vector fields, that havebeen used in the previous chapters: of course the main intent for thispart of the work is not to produce a complete review about these argu-ments, but to collect, and in some case, prove important relations usedso far. In this way, we should be able to produce an (almost) auto-consistent review, containing a unique notation throughout the entirework. In this appendix we follow mainly the notation used in [14, 19].

A.1 Lie-derivative

Consider a diffeomorphism φ : M → M from the manifold M to itself:such a map, with its inverse, allows us to compare arbitrary tensors atdifferent points on the manifolds. Consequently, a new derivative op-erator on tensor fields can be defined, expressing the rate of change ofthe tensors itself due to the diffeomorphism. However, the continuityof such derivative operator requires a continuous family of diffeomo-prphism, parametrized by the real variable t, φt: a point p of M , underthe action of φt, describes a curve in the manifold, and this is true forevery point p. If ξµ(x) is the vector field tangent to all those curves,the Lie derivative Lξ of generic tensor is its rate of change along the

60

integral curves of ξµ: calling ∆tTµ1···µn

ν1···νmis the difference between

the pullback of the tensor evaluated at φ(p) and its value at p, we have

LξTµ1···µn

ν1···νm= lim

t→0

(∆tTµ1···µn

ν1···νm

t

)

.

This new derivative operator is equivalent to the directional derivativewhen applied on functions

Lξf = ξµ∂µf ,

while it is simply proven that

LξVµ = [ξ, V ]µ .

From these last two result we can evaluate the expression for the Liederivative of a one-form, since

Lξ(ωµVµ) = ξν∂ν(ωµV

µ)

andLξ(ωµV

µ) = (Lξω)µVµ + ωµ(LξV )µ .

We finally obtainLξωµ = ξν∂νωµ + (∂µξ

ν)ων (A.1)

The generalization of these results, the Lie derivative of an arbitrarytensor field T α1···αk

β1···βlis given by

LξTα1···αk

β1···βl= ξγ∂γT

α1···αk

β1···βl−

k∑

i=1

T α1···γ···αk

β1···βl∂γξ

αi

+l

∑

j=1

T α1···αk

β1···γ···βl∂βiξγ (A.2)

Notice, that the same expression can be written using the substitution∂µ → ∇µ, without any change in form, for a torsion free theory. Now,then, it is immediate to obtain

Lξgαβ = ∇αξβ + ∇βξα (A.3)

61

given the constraint ∇γgαβ = 0.The Lie-derivative can be applied also on quantities that are not tensorsbut transform similarly: those are called tensor densities of weight W .Under diffeomorphism, they transform like ordinary tensors, exceptthat in addition the W -th power of the Jacobian J appears as a factor:

T ′µ1···µnν1···νm

= JW∂x′µ1

∂xα1· · · xβ1

∂x′ν1· · · T α1···

β1··· .

Important examples are the Levi-Civita alternating symbol ǫµνρσ thatis a tensor density of weight 1 and the metric determinant g, that isa scalar density of weight 2: we are interested in evaluating the Lie-derivative of the square root of the metric (that is a tensor density ofweight 1), as a useful example. If the metric elements undergo a genericvariation gµν → gµν + δgµν then

det(gµν + δgµν) = det[gµρ(1 + gρσδgσν)] = det(gµν) · det(1 + gρσδgσν)

= det(gµν) + det(gµν) · Tr(δgσνgρσ) = det gµν + det gµνδgσν)g

νσ

This last relation can be written as

δg = ggµνδgµν (A.4)

From the condition ∇αgβγ = 0 and (A.4) with δ correspondent to thederivative operator, we get the useful result

∂αg = ggβγ∂αgβγ = ggβγ(

Γαβρgργ + Γαγ

ρgβρ

)

= gδρβΓαβ

ρ + gδργΓαγ

ρ = 2 gΓαγγ

⇒ ∇αg = ∂αg − 2 gΓαγγ = 0 (A.5)

Now, the Lie derivative reads

Lξg = ggµνLξgµν = 2g∂µξµ + ξµ∂µg

and since

δ√g =

δg

2√g

=1

2

√ggµνδgµν

we obtain the desired result

Lξ

√g =

1

2

√ggµνLξgµν = ∂µ(ξ

µ√−g) = ∇µ(ξµ

√−g) . (A.6)

62

Another useful result is the general expression for the Lie-derivative ofa vector density Iµ, i.e.

LξIµ = Lξ(√−gIµ) = (Lξ

√−g)Iµ +

√−g(LξIµ)

= ∂ν(ξν√−g)Iµ +

√−g(ξν∂ν Iµ − Iν∂νξµ)

= ∂ν(ξνIµ) − Iν∂νξµ

= ∂ν(ξνIµ − ξµIν) + ξµ∂νIν , (A.7)

where Iµ is a vector.

A.2 Killing fields

In the previous section we studied the possibility to define the variationof a generic tensor field under diffeomorphism, a map from a manifoldM to itself: in this section we are interested in defining the conceptof symmetry of a certain manifold by exploiting the concept of Liederivative. It is obvious, for instance, that if the metric elements gµνis independent from a coordinate xµi for certain i, then the translationalong this coordinates is a symmetry of the metric (isometry), since∂µigµν = 0. Of course, this is a very simple case of symmetry that we

would like to generalize: to this end we consider the geodesics equationas a function of the four momentum pµ, i.e.

pν∇νpµ = 0

that can be re-written as

mdpµdτ

=1

2(∂µgνρp

νpρ) ,

where τ is the parameter along the geodesics. This last equation clearlystates that everytime the metric is independent from a certain coordi-nate xµi , then the momentum component pµi

is conserved, that is a re-statement of what we just said about translation along xµi

. However,we can write the conserved momentum component as pµi

= ξνpν = ξνpν

with ξµ = (∂µi)µ = δµµi

in the simplest case, and we say that ξµ is thegenerator of the isometry. But if pµi

is constant along the geodesics,then its directional derivative vanishes, pν∇ν(ξµpµ) = 0, i.e.

pν∇ν(ξµpµ) = pµpν∇(µξν) = 0

63

simply by using the geodesics equation. We then obtained the desiredresult: if ξµ is such that ∇(µξν) = 0 then ξµpν is conserved along thegeodesics. But this requirement correspond to the vanishing of the Liederivative of the metric, eq. (A.3), also known as Killing’s equation:the vector field ξµ is called, then, a Killing vector. Obviously thiscondition can be generalized for generic tensor Tµ1···µn

and in this caseTµ1···µn

pµ1 · · · pµn is a conserved quantity.This definition allows us to related Killing vectors to Riemann tensor:as it is simply proved, if ξµ satisfies the Killing equation (A.3) then wecan derive the following equation

∇ν∇ρξµ = Rρµνδξδ (A.8)

In fact, using (A.3) in the definition of the Riemann tensor

∇µ∇νξρ −∇ν∇µξρ = −Rµνρσξσ (A.9)

we get∇µ∇νξρ + ∇ν∇ρξµ = −Rµνρσξ

σ

If we re-write the same equation, taking the cyclic permutations of thethree indeces (µνρ), and then we sum two of them and subtract theequation correspondent to the permutation (ρµν), we get

2∇ν∇ρξµ = −(Rµνρδ +Rνρµδ − Rρµνδ)ξδ = 2Rρµνδξ

δ

that is the result (A.8). In this last equation we used the symmetryproperty of the Riemann tensor R[ρµν]δ = 0.

64

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