CONSERVED CHARGES IN ASYMPTOTICALLY (ANTI)-DE SITTER SPACETIME

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1 CONSERVED CHARGES IN ASYMPTOTICALLY (ANTI)-DE SITTER SPACETIME A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY İBRAHİM GÜLLÜ IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN PHYSICS AUGUST 2005

2 Approval of the Graduate School of Natural and Applied Sciences. Prof. Dr. Canan Özgen Director I certify that this thesis satisfies all the requirements as a thesis for the degree of Master of Science. Prof. Dr. Sinan Bilikmen Head of Department This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science. Assoc. Prof. Dr. Bayram Tekin Supervisor Examining Committee Members Prof. Dr. Ayşe Karasu (METU, PHYS) Assoc. Prof. Dr. Bayram Tekin (METU, PHYS) Assoc. Prof. Dr. Özgür Sarıoğlu (METU, PHYS) Prof. Dr. Atalay Karasu (METU, PHYS) Dr. Hakan Öktem (METU, Applied Math.)

3 I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name Surname : İBRAHİM GÜLLÜ Signature : iii

4 ABSTRACT CONSERVED CHARGES IN ASYMPTOTICALLY (ANTI)-DE SITTER SPACETIME GÜLLÜ, İBRAHİM M.S., Department of Physics Supervisor: Assoc. Prof. Dr. Bayram Tekin August 2005, 77 pages. In this master s thesis, the Killing vectors are introduced and the Killing equation is derived. Also, some information is given about the cosmological constant. Then, the Abbott-Deser (AD) energy is reformulated by linearizing the Einstein equation with cosmological constant. From the linearized Einstein equation, Killing charges are derived by using the properties of Killing vectors. Using this formulation, energy is calculated for some specific cases by using the Schwarzschild-de Sitter metric. Last, the Einstein-Gauss-Bonnet model is studied. The equations of motion are calculated by solving the generic action at quadratic order. Following this, all energy calculations are renewed for this model. Some useful relations and calculations are shown in Appendix (A-B) parts. Keywords: de-sitter spacetime, Killing vector, Conserved charges. iv

5 ÖZ ASİMPTOTİK (ANTİ)-DE SITTER UZAYZAMANINDA KORUNAN YÜKLER GÜLLÜ, İBRAHİM Yüksek Lisans, Fizik Bölümü Tez Yöneticisi: Assoc. Prof. Dr. Bayram Tekin Ağustos 2005, 77 sayfa. Bu master çalışmasında, Killing vektörler tanımlandı ve Killing denklemi çıkarıldı. Ayrıca evrenbilimsel sabit, de-sitter ve Anti-de Sitter uzayları hakkında bilgi verildi. Sonra, Abbott-Deser (AD) enerjisi, evrenbilimsel sabitli Einstein denklemi doğrusallaştırılarak yeniden formüle edildi. Doğrusallaştırılmış Einstein denkleminden, Killing vektörlerin özellikleri kullanılarak Killing yükleri (Deser-Tekin denklemi) çıkarıldı. Schwarzschild-de Sitter metriği kullanılarak özel durumlar için enerji hesaplandı. Son olarak Einstein-Gauss-Bonnet (GB) modeli çalışıldı. İkinci dereceden genel eylem çözülerek hareket denklemleri hesaplandı. Bundan sonra, tüm enerji hesaplamaları bu model için tekrarlandı. Bazı faydalı hesaplamalar ek (A-B) kısımlarında gösterilmiştir. Anahtar Kelimeler: de-sitter uzayzamanı, Killing vektör, Korunan yükler. v

6 to my lovely sister, Emel Egger and admirable brother, İsmail Güllü vi

7 ACKNOWLEDGMENTS I would like to express my sincere feelings to my supervisor, Assoc. Prof. Dr. Bayram Tekin. I am grateful to him for his painstaking care in the course of this project, for his meticulous effort in teaching me very precious, numerous concepts. I could have done nothing without him. I would also like to thank Assoc. Prof. Dr. Özgür Sarıoğlu for his guidance, suggestions and encouragements during the study of this thesis. I am indebted to my family for their moral and financial support, encouragement and love throughout the time that has gone for getting my M.S. degree. I am also grateful to my friends for their friendship and support during my life in Ankara. vii

8 TABLE OF CONTENTS PLAGIARISM iii ABSTRACT iv ÖZ v DEDICATION vi ACKNOWLEDGMENTS vi TABLE OF CONTENTS viii CHAPTER 1 INTRODUCTION KILLING VECTORS MODELS WITH A COSMOLOGICAL CONSTANT DE-SITTER SPACE ANTI DE-SITTER SPACE REFORMULATION OF AD ENERGY CONSERVED CHARGES LINEAR FORM OF THE EINSTEIN EQUATION THE METRIC g µν LINEARIZATION OF THE CHRISTOFFEL SYM- BOL viii

9 4.2.3 LINEAR FORMS OF RIEMANN, RICCI TEN- SORS AND RICCI SCALAR LINEAR FORM OF THE EINSTEIN EQUATION KILLING CHARGES THE ENERGY OF SDS SOLUTIONS THE D = 4 CASE ENERGY FOR D DIMENSIONS THE D = 3 CASE STRING-INSPIRED GRAVITY CONCLUSION REFERENCES APPENDICES A B LINEARIZATION EXPRESSIONS FOR PURE QUADRATIC TERMS KILLING ENERGY EXPRESSION FOR GENERIC QUADRATIC THEORY ix

10 CHAPTER 1 INTRODUCTION Conserved quantities, such as energy-momentum, electric charge, angular momentum, baryon number etc., are important in the description of physical phenomena. In the presence of gravity, definition of certain conserved charges (especially the energy) become rather tricky. Our task in this thesis is to give a review of the techniques of defining conserved charges in asymptotically (Anti)de-Sitter spaces developed by Abbott-Deser (AD) [3] and Deser-Tekin (DT) [1]. We will carry out the computations in great detail. These methods are an extension of the Arnowitt-Deser-Misner (ADM) [4] methods which work for asymptotically flat geometries. We define the global charges primarily in D dimensional quadratic theories. We first present a reformulation of the original definition of conserved charges in cosmological Einstein theory; then we derive the generic form of the energy for quadratic gravity theories in D dimensions and specifically study the ghost-free low energy string-inspired model: Gauss-Bonnet (GB) plus Einstein terms [1]. 1

11 A definition of gauge invariant conserved (global) charges in a diffeomorphisminvariant theory rests on the Gauss law and the presence of asymptotic Killing symmetries. More explicitly, in any diffeomorphism-invariant gravity theory, a vacuum satisfying the classical equations of motion is chosen as the background relative to which excitations and any background gauge-invariant properties (such as energy) are defined. Two important model-independent features of these charges are: First, the vacuum itself has zero charge; secondly, they are expressible as surface integrals [1, 2, 3, 4]. In the first chapter, we will define the Killing vector field and give some useful properties of it. Then we will see how Einstein equations can be linearized to reformulate the AD energy. Using that reformulation, we will calculate the energy in (A)dS spaces. Following that we will look at the Einstein-GB model. We will then state our conclusion and describe some open questions. In the two Appendices, we will give some useful calculations that will help us in this journey. 2

12 CHAPTER 2 KILLING VECTORS Tensor calculus is largely concerned with how quantities change under coordinate transformations. It is of particular interest when a quantity does not change, i.e. remains invariant, under coordinate transformations. For example, coordinate transformations which leave a metric invariant are of importance since they contain information about the symmetries of a Riemannian manifold. In an ordinary Euclidean space, there are two sorts of transformations: descrete ones, like reflections, and continuous ones, like translations and rotations. In most applications, these latter types are the more important ones and they can in principle be obtained systematically by obtaining the so-called Killing vectors of a metric, which we now discuss below. A metric g ab is invariant under the transformation x a x a if g ab(y) = g ab (y) for all coordinates y c. An infinitesimal coordinate transformation is x a x a = x a + ɛx a (x) where X a denotes the vector field and x a denotes a vector in that field. 3

13 Differentiating x a gives x a x b = xa x b + ɛ Xa x b = δ a b + ɛ b X a, and the metric transforms Then x a x a will be an isometry if g ab (x) = x c x a x d x b g cd(x ). g ab (x) = x c x d x a x g cd(x ) ( b ) ( ) = x a (xc + ɛx c ) x b (xd + ɛx d ) g cd (x e + ɛx e ) We expand g cd (x e + ɛx e ) using the Taylor s theorem, g ab (x) = (δ c a + ɛ a X c )(δ d b + ɛ b X d ){g cd (x) + ɛx e e g cd (x) +...} g ab (x) = g cd (δ c a + ɛ a X c )(δ d b + ɛ b X d ) + ɛx e e g cd (x)(δ c a + ɛ a X c )(δ d b + ɛ b X d ), = g ab (x) + ɛ{g ad b X d + g bd a X d + X e e g ab } + O(ɛ 2 ), g ad b X d + g bd a X d + X e e g ab = 0. Where we use Einstein sum convention. Now define: L x g ab = X c c g ab + g ca b X c + g cb a X c. In the first term we make c e transformation, in the second and third term c d that gives us L x g ab = X e e g ab + g ad b X d + g bd a X d = 0. 4

14 This is called Lie derivative and since in the expression for a Lie derivative of a tensor, all occurrences of the partial derivatives may be replaced by covariant derivatives. Therefore we can make the a a substitution, that is X e e g ab + g ad b X d + g bd a X d = 0. Since we work in a metric compatible system b g ab = 0, we have g ad b X d + g bd a X d = 0. Finally we have b X a + a X b = 0, (2.1) where X a is a vector field that leaves the metric invariant and such a vector field is called a Killing vector [5, 6]. The importance of Killing vectors comes from the symmetry considerations. Translational or rotational symmetries will give us conserved quantities. With the first, we can get energy and momentum; and with the latter, angular momentum. 5

15 CHAPTER 3 MODELS WITH A COSMOLOGICAL CONSTANT A characteristic feature of general relativity is that the source for the gravitational field is the entire energy-momentum tensor. In non-gravitational physics, only changes in energy from one state to another are measurable; the normalization of the energy is arbitrary. For example, the motion of a particle with potential energy V (x) is precisely the same as that with a potential energy V (x) + V 0, for any constant V 0. In gravitation, however, the actual value of the energy matters, not just the differences between states. This behavior opens up the possibility of vacuum energy: an energy density characteristic of empty space. One feature that we might want the vacuum to exhibit is that it must not pick out a preferred direction; it will still be possible to have a nonzero energy density if the associated energy-momentum tensor is Lorentz invariant in locally inertial coordinates. Lorentz invariance implies that the corresponding energy-momentum tensor should be proportional to the metric, T (vac) ˆµˆν = ρ vac ηˆµˆν, (3.1) (where ˆµˆν denote locally inertial coordinates and ρ vac is the vacuum energy 6

16 density), since ηˆµˆν is the only Lorentz invariant (0,2) tensor. This generalizes straightforwardly from inertial coordinates to arbitrary coordinates as T (vac) µν = ρ vac g µν. (3.2) Comparing with the perfect-fluid energy-momentum tensor T µν = (ρ + p)u µ U ν + pg µν, we find that the vacuum looks like a perfect fluid with an isotropic pressure opposite in sign to the energy density, p vac = ρ vac. (3.3) The energy density should be constant throughout spacetime, since a gradient would not be Lorentz invariant. If we decompose the energy-momentum tensor into a matter piece T (M) µν and a vacuum piece T (vac) µν = ρ vac g µν, the Einstein s equation is R µν 1 2 Rg µν = 8πG(T (M) µν ρ vac g µν ), (3.4) where R µν and R are Ricci tensor and scalar, G is Newton s constant. Soon after inventing general relativity, Einstein tried to find a static cosmological model, since that was what astronomical observations of the time seemed to imply. The result was the Einstein static universe. In order for this static cosmology 1 to solve the field equation with an ordinary matter source, it was necessary to add a new term called the cosmological constant, Λ, which enters as R µν 1 2 Rg µν + Λg µν = 8πGT µν. (3.5) 1 We now know that even with a cosmological constant, static universe is not possible. 7

17 The cosmological constant is precisely equivalent to introducing a vacuum energy density ρ vac = Λ 8πG. (3.6) The terms cosmological constant and vacuum energy are essentially interchangeable [7]. Maximally symmetric solutions of the cosmological Einstein equation with T µν, are de Sitter and Anti-de Sitter spaces which we now briefly describe. 3.1 DE-SITTER SPACE De Sitter space corresponds to a four-dimensional surface in a flat five-dimensional space with metric (, +, +, +, +) described by z z z z z 2 4 = 3 Λ, Λ > 0. (3.7) The symmetries of this space are then the ten rotations and boosts of this fivedimensional embedding space. Rotations among the z 1 z 4 clearly result in spacelike Killing vectors. However, boosts which mix z 1 z 4 with z 0 can lead to Killing vectors which are timelike. The Killing vector which corresponds to a mixing of z 4 and z 0 is ξ a = ( z 4, 0, 0, 0, z 0 ). (3.8) Now ξ 2 = z z 2 0, (3.9) 8

18 so ξ 2 < 0, (3.10) if and only if z 4 > z 0. (3.11) This is a distinctive feature of de Sitter space which implies the existence of a cosmological horizon. To be more specific, consider the de Sitter space metric in the form dτ 2 = dt 2 + f 2 (t)[dx 2 + dy 2 + dz 2 ], (3.12) where 1 f(t) = exp Λt. (3.13) 3 The Killing vector for this symmetry is Now ξ µ = ( 1, 1 Λx). (3.14) 3 ξ 2 = Λf 2 x 2, (3.15) which is timelike in the region 1 3 Λf 2 x 2 < 1. (3.16) This ξ µ generates a Killing energy. However the restriction (3.16) limits the region of applicability of this quantity. In order for E( ξ) to act like an energy, the surface 9

19 of integration must lie inside the event horizon defined by 1 3 Λf 2 x 2 = 1 [3, 8]. (3.17) 3.2 ANTI DE-SITTER SPACE Anti-de Sitter space is the covering space for the four-dimensional surface, in a flat five-dimensional space with metric (-,+,+,+,-), described by z0 2 + z1 2 + z2 2 + z3 2 z4 2 = 3, Λ < 0. (3.18) Λ Once again the symmetries of this space are just the rotations and boosts in the five-dimensional embedding space. Here, however, there is a global timelike Killing vector corresponding to the rotation mixing z 0 and z 4, ξ a = ( z 4, 0, 0, 0, z 0 ), (3.19) and ξ 2 = z 2 4 z 2 0 < 0. (3.20) Thus, ξ a is timelike everywhere [note that the condition (3.17) excludes the point z 4 = z 0 = 0], and there is no cosmological horizon [3]. 10

20 CHAPTER 4 REFORMULATION OF AD ENERGY 4.1 CONSERVED CHARGES We first look at how conserved charges arise in a generic gravity theory coupled to a covariantly conserved bounded matter source τ µν. Consider the following equations of motion that come from a Lagrangian Φ µν (g, R, R, R 2,...) = κτ µν, (4.1) where Φ µν is the Einstein tensor of a local, invariant, but otherwise arbitrary, gravity action and κ is an effective coupling constant. We work in generic D dimensions. Now we will decompose our metric into the sum of two parts: g µν = ḡ µν + h µν, (4.2) where ḡ µν solves (4.1) for τ µν = 0 and a deviation part h µν that vanishes sufficiently rapidly at infinity and it is not necessarily small everywhere. Separating the field equations (4.1) into a part linear in h µν and collecting all other non-linear terms and the matter source τ µν in T µν that constitute the total 11

21 source, one obtains O(ḡ) µναβ h αβ = κt µν, (4.3) Φ µν (ḡ, R, R, R 2,...) = 0, by assumption; the operator O(ḡ) depends only on the background metric ḡ µν. The linearization of the (source free) Einstein equation R µν 1 2 g µνr + Λg µν = 0, (4.4) follows as R L µν 1 2 h µν R 1 2ḡµνR L + Λh µν + O(h 2 ) +... = 0, the background terms will be zero because ḡ µν itself satisfies the field equation (4.4). Here, [ µ, ν ]V λ = R µνλ σ V σ and R µν R µλν λ and the constant curvature vacuum ḡ µν has Riemann, Ricci and scalar curvatures that are R µλνβ = (D 1)(D 2) (ḡ µνḡ λβ ḡ µβ ḡ λν ) ; (4.5) R µν = R µλνβ ḡ βλ, = = = (D 1)(D 2) (ḡ µνḡ λβ ḡ µβ ḡ λν )ḡ βλ, (D 1)(D 2) (ḡ µνd δµḡ λ λν ), (D 1)(D 2)ḡµν(D 1) = ; (4.6) (D 2)ḡµν R = R µν ḡ µν, = (D 2)ḡµνḡ µν = (D 2) D. (4.7) 12

22 Using (4.7), we get: Rµν L 1 L 1 2ḡµνR 2 h 2DΛ µν (D 2) + Λh µν + O(h 2 ) +... = 0. Collecting the terms we get R L µν 1 2ḡµνR L 1 2 h µνr L h µν (D 2) + O(h2 ) +... = 0. We define all terms of second and higher order in h µν and the matter source τ µν to be the gravitational energy-momentum tensor and write G L µν as or G L µν R L µν 1 2ḡµνR L (D 2) h µν κt µν. (4.8) The left hand side of (4.8) obeys the background Bianchi identity µ (R µν L 1 2ḡµν R L (D 2) hµν ) = 0 (4.9) µ T µν = 0. (4.10) Therefore, we have a background conserved energy-momentum tensor. However, the derivative in (4.9) is a background covariant, not an ordinary derivative; furthermore, only integrals over divergences of contravariant vector densities have invariant content, so (4.8) and (4.10) cannot be used directly to construct conserved quantities. To overcome this problem we contract T µν with a Killing vector ξ µ that is: µ (T µν ξν ) = ( µ T µν ) ξ ν T µν ( µ ξν + ν ξµ ) = 0, 13

23 since T µν = T νµ and with the help of the Killing equation, we have the above equation. Now the quantity ( ξ ν T µν ) is a vector density whose covariant divergence becomes an ordinary one, and gives the desired conservation law, µ ( ξ ν T µν ) = µ ( ξ ν T µν ) = 0, or µ ( ḡ T µν ξν ) µ ( ḡ T µν ξν ) = 0. (4.11) Therefore the conserved Killing charges are expressed as Q µ ( ξ) = M d D 1 x ḡ T µν ξν = Σ ds i F µi. (4.12) Here M is a spatial (D 1) hypersurface and Σ is its (D 2) dimensional boundary and i ranges over (1, 2,..., d 2) and F µi is an antisymmetric tensor whose explicit form will be written below [1, 4]. 4.2 LINEAR FORM OF THE EINSTEIN EQUATION In order to write the spatial volume integrals as surface integrals, we need to carry out the linearization of the relevant tensors. In this part that is what we shall do. 14

24 4.2.1 THE METRIC g µν We will take the signature to be (, +, +, +,...). We know that any metric must satisfy g µν g να = δ µ α where δg µν = h µν, g µν = ḡ µν + δg µν, (ḡ µν + δg µν )g να = δ µ α g µν = ḡ µν δg µν LINEARIZATION OF THE CHRISTOFFEL SYMBOL We linearize Γ µ αβ = 1 2 gµν ( α g βν + β g αν ν g αβ ) with the use of g µν ḡ µν δg µν and g βν ḡ βν + δg βν. That is, δγ µ αβ = 1 2 δgµν ( α ḡ βν + β ḡ αν ν ḡ αβ ) + 1 2ḡµν (δ α g βν + δ β g αν δ ν g αβ ), δγ µ αβ = 1 2ḡµν ( α δg βν + β δg αν ν δg αβ ). (4.13) LINEAR FORMS OF RIEMANN, RICCI TENSORS AND RICCI SCALAR The Riemann tensor is R µ αβν = β Γ µ αν ν Γ µ αβ + Γσ ανγ µ σβ Γσ αβγ µ σν, (4.14) which can be linearized as δr µ αβν = β δγ µ αν ν δγ µ αβ + (δγσ αν)γ µ σβ + Γσ ανδγ µ σβ (δγσ αβ)γ µ σν Γ σ αβδγ µ σν. 15

25 To find the partial derivatives of the Christoffel symbol we can use covariant derivative, β (δγ µ αν) = β δγ µ αν + Γ µ βσ δγσ αν Γ σ βαδγ µ σν Γ σ βνδγ µ σα, β δγ µ αν = β (δγ µ αν) Γ µ βσ δγσ αν + Γ σ βαδγ µ σν + Γ σ βνδγ µ σα, and ν (δγ µ αβ ) = νδγ µ αβ + Γµ νσδγ σ αβ Γ σ ναδγ µ σβ Γσ νβδγ µ σα, ν δγ µ αβ = ν (δγ µ αβ ) Γµ νσδγ σ αβ + Γ σ ναδγ µ σβ + Γσ νβδγ µ σα. If we insert these into our linearized Riemann tensor, we get δr µ αβν = β (δγ µ αν) Γ µ βσ δγσ αν + Γ σ βαδγ µ σν + Γ σ βνδγ µ σα ν (δγ µ αβ ) +Γ µ νσδγ σ αβ Γ σ ναδγ µ σβ Γσ νβδγ µ σα + (δγ σ αν)γ µ σβ + Γσ ανδγ µ σβ (δγ σ αβ)γ µ σν Γ σ αβδγ µ σν yielding; δr µ αβν = β (δγ µ αν) ν (δγ µ αβ ). (4.15) µ β contraction gives the linear Ricci tensor δr µ αµν = µ (δγ µ αν) ν (δγ µ αµ) = δr αν, or by α µ δr µν = α (δγ α µν) µ (δγ α αν). 16

26 Now let us write this in terms of δg αβ δr µν = [ 1 2ḡασ α ( µ δg νσ + ν δg µσ ] σ δg µν ) [ 1 2ḡασ µ ( α δg σν + ν δg ασ ] σ δg αν ). In the last term if we make the σ α exchange, it will cancel the fourth term δr µν = 1 2 { σ µ δg νσ + σ ν δg µσ σ σ δg µν µ ν ḡ ασ δg ασ } and we are left with δr µν = 1 2 { σ µ δg νσ + σ ν δg µσ δg µν µ ν δg}, (4.16) where δg = h = ḡ ασ h ασ and σ σ. The linear part of the Ricci scalar reads δr = δ(g µν R µν ) = (δr µν )ḡ µν R µν δg µν, = 1 2 ( h µν µ ν h + σ ν h σµ + σ µ h σν )ḡ µν 2 (D 2) Λḡ µνh µν, = 1 2 ( h h + σ µ h σµ + σ 2 ν h σν ) (D 2) Λh, or by taking ν µ δr = 1 2 ( 2 h + 2 σ 2 µ h σµ ) (D 2) Λh, = h + σ 2 µ h σµ Λh. (4.17) (D 2) 17

27 4.2.4 LINEAR FORM OF THE EINSTEIN EQUATION Now, we are ready to write (4.8) in terms of the deviation part of the metric. That is, G µν L = R µν L 1 2ḡµν 2 R L (D 2) Λhµν = 1 2 ( h µν µ ν h + σ ν h σµ + σ µ h σν ) 1 2ḡµν ( h + σ α h σα 2 (D 2) Λh) 2 (D 2) Λhµν. (4.18) We can also show that this equation is background covariantly constant: µ G µν L = 1 2 ( µ h µν ν h + µ σ ν h σµ + µ σ µ h σν ) 1 2 ( ν h + ν σ α h σα ) + 1 (D 2) Λ ν 2 h (D 2) Λ µ h µν. In the fourth term change the places of the covariant derivatives to get the first term with an opposite sign: [ µ, σ ] µ h σν = µ σ µ h σν σ µ µ h σν, µ σ µ h σν = [ µ, σ ] µ h σν + σ h σν. In the last term, we make the σ µ transformation µ σ µ h σν = [ µ, σ ] µ h σν + µ h µν. 18

28 Moreover [ µ, σ ] µ h σν = R µσ µ λ λ h σν + R µσ σ λ µ h λν + R µσ ν λ µ h σλ = R σλ λ h σν R µλ µ h λν = = + (D 1)(D 2) (δν µḡ σλ ḡ µλ δσ) ν µ h σλ (D 2) (ḡ σλ λ h σν ḡ µ µλ h λν ) + (D 1)(D 2) ( ν h λ h νλ ) (D 2) ( σ h σν λ h λν ) + (D 1)(D 2) ( ν h λ h νλ ). If we make the λ σ substitution the term in the first parenthesis will vanish and we will get µ σ µ h σν = (D 1)(D 2) ( ν h λ h νλ ) + µ h µν. In the fifth and sixth terms of µ G µν, we use the same procedure. Making suitable index substitutions we get ν h = (D 2) ν h + ν h, ν σ µ h σµ = (D 1)(D 2) ν h (D 1)(D 2) µ h µν (2D 1) + µ σ ν h σµ. When these results are inserted into (4.18), one will have µ G µν L = Λ (D 2)(D 1) ( ν h λ h νλ ) Λ (D 2) ν h + (2D 1)Λ (D 2)(D 1) µ h µν 19

29 Λ (D 2)(D 1) ν Λ h + (D 2) ν h (D 2) µ h µν = 0. In the first term the covariant derivative of h will cancel with the fourth term and we change the indices of the remaining part of the first term: λ µ. Thus we conclude that the energy-momentum tensor is background covariantly constant, that is µ G µν L = 0, (or µ T µν = 0). (4.19) 4.3 KILLING CHARGES Let us recall that there are two facets of a proper energy definition: First, identification of the Gauss law, whose existence is guaranteed by gauge invariance; second, choice of the proper vacuum, possessing sufficient Killing symmetries with respect to which global, background gauge-invariant, generators can be defined; these will always appear as surface integrals in the asymptotic vacuum [2]. In converting the volume integrals to surface integrals, let us follow a route, which will be convenient in the higher curvature cases. We take the energymomentum tensor in a Killing vector field and collect terms in the covariant derivative to get surface terms: 2 ξ ν G µν L = 2 ξ ν R µν L ξ ν ḡ µν R L 4Λ (D 2) ξ ν h µν = ξ ν { ρ ρ h µν µ ν h + σ ν h σµ + σ µ h σν } ξ µ { ρ ρ h + σ ν h σν 4Λ h} (D 2) (D 2) ξ ν h µν 20

30 where we used (4.16) and (4.17). We rename the indices for convenience; second term: ν ρ third term: σ ν, ν ρ fourth term: σ ρ sixth term: σ ρ. Thus our equation becomes 2 ξ ν G µν L = ξ ν ρ ρ h µν ξ ρ µ ρ h + ξ ρ ν ρ h νµ + ξ ν ρ µ h ρν + ξ µ ρ ρ h ξ µ ρ ν h ρν + (D 2) ξ µ h 4Λ (D 2) ξ ν h µν. To collect all terms, we use the commutator relation of a vector that gives us the Riemann tensor. In the first, fourth, fifth and sixth terms the Killing vectors are taken inside the covariant derivative with extra terms that will come from the derivative of the Killing vectors. In the second and third terms, places of derivatives must change, after that the Killing vectors can be taken inside the derivative with two additional terms, the second comes from exchange of derivatives. After these calculations we are left with 2 ξ ν G µν L = ρ ( ξ ν ρ h µν ) + ( ρ ξν )( ρ h µν ) ρ ( ξ ρ µ h) + ( ρ ξρ )( µ h) + ρ ( ξ ρ ν h µν ) ( ρ ξρ )( ν h µν ) + ρ ( ξ ν µ h ρν ) ( ρ ξν )( µ h ρν ) + ρ ( ξ µ ρ h) ( ρ ξµ )( ρ h) ρ ( ξ µ ν h ρν ) + ( ρ ξµ )( ν h ρν ) + (D 2) hνµ ξν + (D 2) h ξ µ 4Λ (D 2) hνµ ξν + (D 2)(D 1) (hνµ ξν h ξ µ ) 21

31 The fourth, sixth and the eighth terms vanish because of the Killing equation 2 ξ ν G µν L = ρ { ξ ν ρ h µν ξ ρ µ h + ξ ρ ν h µν + ξ ν µ h ρν + ξ µ ρ h ξ µ ν h ρν } + (D 2) h ξ µ (D 2) hνµ ξν + (D 2)(D 1) (hνµ ξν h ξ µ ) +( ρ ξν )( ρ h µν ) ( ρ ξµ )( ρ h) + ( ρ ξµ )( ν h ρν ). We will look at the last three terms closely: ( ρ ξν )( ρ h µν ) = ρ (h µν ρ ξν ) h µν ( ρ ρ ξν ). Operating on (2.1) with µ, one gets µ µ ξν + µ ν ξµ = 0. The second term can be written in the commutator form that is ξν + [ µ, ν ] ξ µ = 0, or simply ξν = (D ξ λ. 2)ḡνλ Using this relation, we have ( ρ ξν )( ρ h µν ) = ρ (h µν ρ ξν ) + (D 2) hµν ḡ νλ ξλ, and ( ρ ξµ )( ρ h) = ρ (h µ ξρ ) + (D 2) h ξ µ. 22

32 Using the property of a Killing vector, shown in the appendix (see (9.3)) ( ρ ξµ )( ν h ρν ) = ρ (h ρν µ ξν ) Finally collecting these results, we have (D 2)(D 1) ( ξ ν h µν ξ µ h). 2 ξ ν G µν L = ρ { ξ ν ρ h µν ξ ρ µ h + ξ ρ ν h µν + ξ ν µ h ρν + ξ µ ρ h ξ µ ν h ρν + h µν ρ ξν + h µ ξρ h ρν µ ξν }. (4.20) Since the charge densities are surface terms, the Killing charges become Q µ ( ξ) = 1 4Ω (D 2) G D Σ ds i { ξ ν i h µν ξ i µ h + ξ i ν h µν + ξ ν µ h iν + ξ µ i h ξ µ ν h iν + h µν i ξν + h µ ξi h iν µ ξν }. (4.21) Here ds i detḡ dω i where i ranges over (1, 2,..., D 2); the charge is normalized by dividing with the (D-dimensional) Newton s constant G D and the solid angle dω D 2. Before calculating the conserved charges Q 0, we check our expression to see whether it is gauge invariant or not, and we will look if it goes to the ADM charges in the limit of an asymptotically flat background ( j j ) in which case our timelike Killing vector is ξ µ = (1, 0) [1]. First we will look at the gauge-invariance. Under an infinitesimal diffeomorphism, generated by a vector δ ζ, the deviation part of the metric transforms as δ ζ h µν = µ ζ ν + ν ζ µ [1]. (4.22) 23

33 First we will look at the linear Ricci scalar: R L = (g µν R µν ) L, Now using (4.16), we have δ ζ R L = δ ζ (ḡ µν R L µν R µν h µν ) = ḡ µν δ ζ R L µν (D 2)ḡµν δ ζ h µν. δ ζ R L = ḡ µν 1 2 δ ζ( h µν µ ν h + σ ν h σµ + σ µ h σν ) 2)ḡµν ( (D µ ζ ν + ν ζ µ ) = δ ζ h + σ µ δ ζ h σµ 4Λ (D 2) µ ζ µ. With the help of (4.22), we can write δ ζ R L = ḡ µν ( µ ζ ν + ν ζ µ ) + σ µ ( σ ζ µ + µ ζ σ ) 4Λ (D 2) µ ζ µ = 2 µ ζ µ + σ µ σ ζ µ + σ ζ σ 4Λ (D 2) µ ζ µ. We look examine the second and third terms carefully: Therefore, we have [ µ, σ ]ζ µ = µ σ ζ µ σ µ ζ µ, µ σ ζ µ = σ µ σ ζ µ = (D 2) ζ σ + σ µ ζ µ. (D 2) σ ζ σ + µ ζ µ. 24

34 In the third term the same calculations can be done to get σ ζ σ = (D 2) β ζ β + σ ζ σ We have the background gauge invariance of the linear Ricci scalar δ ζ R L = 0. Therefore δ ζ G L µν = δ ζ R L µν (D 2) δ ζh µν. Using (4.16) and (4.22), we have δ ζ Gµν L = 1 2 ( µ ζ ν ν ζ µ µ ν β ζ β µ ν α ζ α + σ ν σ ζ µ + σ ν µ ζ σ + σ µ σ ζ ν + σ µ ν ζ σ ) (D 2) ( µ ζ ν + ν ζ µ ). Just sa before, let us look at the terms that are in the second line: The fifth term is: σ ν σ ζ µ = (D 2)(D 1) (ḡ νµ σ ζ σ ḡ σµ σ ζ ν ) + σ σ ν ζ µ. The sixth term is: σ ν µ ζ σ = The third term is: σ µ σ ζ ν = (D 2)(D 1) ( ν ζ µ µ ζ ν ) + σ µ ν ζ σ. (D 2)(D 1) (ḡ µν σ ζ σ ḡ σν σ ζ µ ) + σ σ µ ζ ν. 25

35 The fourth term is: σ µ ν ζ σ = (D 2) µ ζ ν + (D 2)(D 1) ( µ ζ ν ḡ σ µν ζ σ ) + (D 2) ν ζ µ + µ ν σ ζ σ. Collecting all these, terms we end up with δ ζ G L µν = 0, which means that G L µν is gauge-invariant. Therefore, we have δ ζ R L µν = δ (D 2) ζh µν. Hence δ ζ Q µ = 0; that is, the Killing charge is indeed background gaugeinvariant. Now we will examine (4.21) in the limit of an asymptotically flat background, which should yield the ADM charge. Let us just look at the mass to begin with. With ξ µ = (1, 0), we have ξ i = 0, ξ 0 = 1 and ξ 0 = 1 in flat space with the metric η µν = diag( 1, 1, 1, 1). We have h = h 00 + h ii. Being in Cartesian coordinates, we can take the covariant derivatives to partial derivatives ( j j ). Hence, M = 1 4Ω (D 2) G D Σ ds i { ξ 0 0 h i0 ξ 0 i h 00 ξ 0 i h ξ 0 0 h i0 + j h ij }. Expanding the last term we have M = = 1 4Ω (D 2) G D 1 4Ω (D 2) G D Σ Σ ds i { i h 00 i h jj + i h 00 + j h ij } ds i { j h ij i h jj } which is the usual ADM mass [6]. 26

36 CHAPTER 5 THE ENERGY OF SDS SOLUTIONS Having established the energy formula for asymptotically (A)dS spaces, we can now evaluate the energy of Schwarzschild-de Sitter (SdS) solutions. In static coordinates, the line element of D-dimensional SdS reads ( ds 2 = 1 ( r ) 0 r )D 3 r2 dt 2 l 2 ( + 1 ( r ) 1 0 r )D 3 r2 dr 2 + r 2 dω 2 l 2 D 2, (5.1) where l 2 (D 2)(D 1)/ > 0. The background (r 0 = 0) Killing vector is ξ µ = ( 1, 0), which is timelike everywhere for AdS (l 2 < 0), but remains timelike for ds (l 2 > 0) only inside the cosmological horizon: ḡ µν ξµ ξν = (1 r2 l 2 ) [1]. 5.1 THE D = 4 CASE Let us concentrate on D = 4 first and calculate the surface integral (4.21) not at r, but at some finite distance r from the origin; this will not be gauge-invariant, since energy is to be measured only at infinity. Nevertheless, for ds space (which has a horizon that keeps us from going smoothly to infinity), let 27

37 us first keep r finite as an intermediate step. The integral becomes E(r) = r 0 2G (1 r2 l 2 ) (1 r 0 r r2 l 2 ). (5.2) For AdS, take r and we get the usual mass r 0. For ds, we can only consider 2G small r 0 limit, which do not change the location of the background horizon, that also yields r 0 2G [1]. 5.2 ENERGY FOR D DIMENSIONS In this case h 0. From the line element (5.1) g 00, g rr, ḡ 00, ḡ rr and the corresponding h terms can be calculated. These are g 00 = ( 1 ( r ) 0 r )D 3 r2 l 2, g rr = and the background metric components will be ḡ 00 = ( 1 r2 l 2 ), ḡ rr = since r 0 = 0 for the background. From (4.2) ( 1 ( r ) 1 0 r )D 3 r2, (5.3) l 2 ( 1 r2 l 2 ) 1, (5.4) h 00 = ( r 0 r )D 3, ḡ 00 ḡ 00 h 00 = h 00 = ( r0 r ) D 3 (1 r2 l 2 ) 2. (5.5) h rr = ( r 0 r ) D 3 (1 ( r 0 r ) D 3 r2 )(1 r2 ), hrr = ( r0 r )D 3 (1 r2 ) l 2 (1 ( r l 2 l 2 0 r ) D 3 r2 ). (5.6) l 2 Using (4.21), we have Q 0 = 4π 16πG D r lim r D 2 { ξ 0 0 h r0 ξ r 0 h 00 + h 00 r ξ0 h rr 0 ξr + ν h rν } 28

38 where the constant factors come from the normalization constant and the integration element ds i. Q 0 = 4π 16πG D r lim r D 2 { ξ 0 0 h r0 ξ r 0 h 00 + h 00 r ξ0 h rr 0 ξr + 0 h r0 + i h ri } and Q 0 = 4π 16πG D r lim r D 2 { ξ r 0 h 00 + h 00 r ξ0 h rr 0 ξr + i h ri }, playing with indices Q 0 = 4π 16πG D lim r D 2 {ḡ 00 ḡ rr r h 00 + h 00 ḡ 00 0 ξr + h rr ḡ rr 0 ξr + i h ri }, r and the terms in the middle cancel with each other since h = 0. Therefore we are left with Q 0 = = = = = 4π 16πG D 4π 16πG D 4π 16πG D 4π 16πG D 4π 16πG D lim r D 2 {ḡ 00 ḡ rr r h 00 + i h ri } r r lim r D 2 {ḡ 00 ḡ 00 ḡ 00 ḡ rr r h 00 + i h ri } r lim r D 2 { r h 00 + i h ri } lim r D 2 { r h Γ 0 r r0h 00 + r h rr + Γ r ijh ij + Γ i irh rr } lim r r D 2 { r h 00 + h 00 ḡ 00 r ḡ 00 + r h rr hrr ḡ rr r ḡ rr hrr ḡ ij r ḡ ij }. The last term can be expanded 29

39 Q 0 = 4π lim r D 2 { r h 00 + h 00 ḡ 00 r ḡ 00 + r h rr πG r D 2 hrr ḡ rr r ḡ rr hrr ḡ rr r ḡ rr hrr ḡ ij r ḡ ij }, where ḡ ij r ḡ ij = 1 r 2 when i = j r, therefore we have Q 0 = 4π 16πG D lim r D 2 { r h 00 +h 00 ḡ 00 r ḡ 00 + r h rr +h rr ḡ rr r ḡ rr + 1 r 2 hrr ḡ ij r ḡ ij }. Lets look at our expression term by term in the limit of r : r h 00 = (D 3) rd 3 0 r D 2 r D 2 r h 00 = (D 3)r D 3 0. The second term r D 2 h 00 ḡ 00 r ḡ 00 = r D 2 ( r 0 1 2r r )D 3 (1 r2 ) l 2 l 2 the third term = rd ( 1 1 r 2 r h rr = (D 3)( r 0 r )D 4 ( r 0 r2 )(1 r2 l ) 1 2 (1 ( r 0 r ) D 3 r2 +( r 0 r )D 3 ( 2r l ) 1 2 (1 ( r 0 r ) D 3 r2 ) l 2 We will operate on this with r D 2, l 2 ) +( r 0 r )D 3 (1 r2 l ){(D 3)( r0 r ) D 4 ( r0 ) + 2r r 2. 2 (1 ( r 0 r ) D 3 r2 ) l l = 2 2rD 3 l 2 } l 2 ) 0 ; r D 2 r h rr = (D 3)r D r D 3 0 2r D 3 0, = (D 3)r D

40 The fourth term is h rr ḡ rr r ḡ rr = ( r 0 r ) D 3 ( 2r l 2 ) (1 ( r 0 r ) D 3 r2 l 2 ), and 1 2 hrr ḡ ij r ḡ ij = r D 2 h rr ḡ rr r ḡ rr = 2r D 3 0 (D 2) r ( r 0 r ) D 3 (1 r2 l 2 ) (1 ( r 0 r ) D 3 r2 l 2 ), r D hrr ḡ ij r ḡ ij = (D 2)r D 3 0. Hence, adding all the above we get the energy in D-dimensions E = (D 2) r0 D 3. (5.7) 4G D Here r 0 can be arbitrarily large in the AdS case but must be small in ds [1]. 5.3 THE D = 3 CASE Let us note that analogous computations can also be carried out in D = 3; the proper solution to consider is ds 2 = (1 r 0 r2 l 2 )dt2 + (1 r 0 r2 l 2 ) 1 dr 2 + r 2 dφ 2, (5.8) for which the energy is E = r 0 /2G again, but now r 0 is a dimensionless constant and the Newton constant G has dimensions of 1/mass [1, 9]. 31

41 CHAPTER 6 STRING-INSPIRED GRAVITY In flat backgrounds, the ghost freedom of low energy string theory requires the quadratic corrections to Einstein s gravity to be of the Gauss-Bonnet (GB) form, an argument that should carry over to the AdS backgrounds. Below we construct and compute the energy of various asymptotically (A)dS spaces that solve the generic Einstein plus quadratic gravity theories, particularly the Einstein-GB model [1, 10]. At quadratic order, the generic action is I = d D x g{ 1 κ R + αr2 + βr 2 µν + γ(r 2 µνρσ 4R 2 µν + R 2 )} (6.1) In D = 4, the GB part (γ terms) is a surface integral and plays no role in the equations of motion. In D > 4, on the contrary, GB is the only viable term, since non-zero α, β produce ghosts [11]. Here κ = 2Ω D 2 G D, where G D is the D-dimensional Newton s constant [1]. After lengthy calculations, that are shown in the appendix B, we reach the equations of motion 32

42 1 κ (R µν 1 2 g µνr) + 2αR(R µν 1 4 g µνr) + (2α + β)(g µν µ ν )R + 2γ[RR µν 2R µσνρ R σρ + R µσρτ R ν σρτ 2R µσ R ν σ 1 4 g µν(r 2 τλσρ 4R 2 σρ + R 2 )] + β (R µν 1 2 g µνr) + 2β(R µσνρ 1 4 g µνr σρ )R σρ = τ µν. (6.2) In the absence of matter, flat space is a solution of these equations; but more important is that (A)dS is also a solution [1]. The cosmological constant can be found using Eq. (6.2): 1 κ ( R µν 1 R) + 2α R( R 1 µν R) 2ḡµν 4ḡµν + (2α + β)(ḡ µν µ ν ) R + 2γ[ R R µν 2 R µσνρ Rσρ + R µσρτ Rν σρτ 2 R µσ Rν σ 1 4ḡµν( R 2 τλσρ 4 R 2 σρ + R 2 )] + β ( R µν 1 2ḡµν R) + 2β( R µσνρ 1 4ḡµν R σρ ) R σρ = 0. (6.3) The terms that have covariant derivatives will be zero by using (4.5), (4.6) and (4.7). The other terms can be calculated one by one: 1 κ ( R µν 1 R) = 1 2ḡµν κ ( 2 (D 2) Λḡ µν 1 2 and the second one is 2DΛ (D 2) ) = 1 κ Λḡ µν, 2α R( R µν 1 ( R) 4DΛ 2 = 2α 4ḡµν (D 2) 1 4D 2 Λ 2 ) 2 4 (D 2) 2 ḡ µν 33

43 The next one is a bit longer than the others: = 2Dα Λ2 (D 4) (D 2) 2 ḡµν. I = [ R R µν 2 R µσνρ Rσρ + R µσρτ Rν σρτ 2 R µσ Rν σ 1 4ḡµν( R 2 τλσρ 4 R 2 σρ + R 2 )] = 4DΛ 2 8Λ 2 (D 2) 2 ḡµν (D 2) 2 (D 1) (ḡ µνḡ σρ ḡ µρ ḡ σν )ḡ σρ 4Λ 2 + (D 2) 2 (D 1) (ḡ µρḡ 2 στ ḡ µτ ḡ σρ )(ḡνḡ ρ στ ḡνḡ τ σρ ) 8Λ2 (D 2) ḡµσḡ σ 2 ν 1 4DΛ 2 (D 3). 4 (D 2)(D 1)ḡµν If we look carefully to the second and the fourth terms, we can see that they are of the same type and can be added together I = 4DΛ 2 (D 2) 2 ḡµν 16Λ2 (D 2) 2 ḡµν 8Λ 2 + DΛ2 (D 3), (D 2) 2 (D 1)ḡµν (D 2)(D 1)ḡµν and we get I = Λ 2 ḡ µν {4D(D 1) 16(D 1) + 8 D(D 3)(D 2)} (D 2) 2 (D 1) = Λ 2 ḡ µν (D 2) 2 (D 1) (D3 9D D 24), and finally we have Similarly (D 4)(D 3) I = (D 2)(D 1) Λ2 ḡ µν. 2β( R µσνρ 1 R σρ ) R σρ 4Λ 2 = 2β 4ḡµν (D 2) 2 (D 1) (ḡ µνḡ σρ ḡ µρ ḡ σν )ḡ σρ 34

44 1 4DΛ 2 4 (D 2) 2 ḡµν = 2β Λ2 (D 4) (D 2) 2 ḡµν. Adding all these terms and equating the sum to zero, one can find 1 κ Λḡ µν 2Dα Λ2 (D 4) (D 4)(D 3) (D 2) 2 ḡµν 2γ (D 2)(D 1) Λ2 ḡ µν 2β Λ2 (D 4) (D 2) 2 ḡµν = 0, and 1 κ (D 4) (D 4)(D 3) = (Dα + β) + γ (D 2) 2 (D 2)(D 1), (6.4) where Λ 0 [1, 12]. Several comments are in order here: In the string-inspired Einstein-GB model (α = β = 0 and γ > 0), only AdS background (Λ < 0) is allowed (the Einstein constant κ is positive in our conventions). String theory is known to prefer AdS to ds [1, 13] and we can see why this is so in the uncompactified theory. Another interesting limit is the traceless theory (Dα = β), which, in the absence of a γ term, does not allow constant curvature spaces unless the Einstein term is also dropped. For D = 4, the γ term drops out, and the pure quadratic theory allows (A)dS solutions with arbitrary Λ. For D > 4, (6.4) leaves a two-parameter set (say α, β) of allowed solutions [1]. Now we will linearize the total energy-momentum tensor T µν to first order in h µν and define the total energy-momentum tensor T µν as we did before. With the help of equations in appendix A, the total energy-momentum tensor can be calculated (shown in appendix B) T µν (h) = T µν (ḡ) + G L µν { 1 κ + 4ΛDα (D 2) + 4Λβ } 4Λγ(D 3)(D 4) + (D 1) (D 1)(D 2) 35

45 ( +(2α + β) ḡ µν µ ν + ) (D ( 2)ḡµν +β G µν L ) L (D 1)ḡµνR 2 h µν { 1 κ R L (D 4) γ(d 4)(D 3) + (Dα + β) + (D 2) 2 (D 2)(D 1) Using (6.4), one has T µν (ḡ) = 0 and the last term also vanishes, yielding }. (6.5) { T µν = Gµν L 1 κ + 4ΛD } (D 2) (2α + β 2 (D 1) ) ( +(2α + β) ḡ µν µ ν + ) R L (D ( 2)ḡµν +β G µν L ) L. (6.6) (D 1)ḡµνR This is a background conserved tensor ( µ T µν = 0) and it is checked explicitly in appendix B. An important aspect of (6.6) is the sign change of the 1 κ term relative to Einstein theory, due to the GB contributions. Hence in the Einstein- GB limit, we have T µν = G µν L /κ, with the overall sign exactly opposite to that of the cosmological Einstein theory. However, this does not mean that E is negative there [1, 14]. There remains now to obtain a Killing energy expression from (6.6), namely, to write ξ ν T µν as a surface integral. The first term is the usual AD piece (4.21), which we derived in chapter 4. The term in the middle (which has the coefficient 2α + β), is easy to handle. First we take the indices up and than operate on this equation with a Killing vector, say ξ ν, ξ µ R L ξ ν µ ν R L + (D 2) ξ µ R L. (6.7) 36

46 In the first term the covariant derivative must be taken outside to get surface terms: ξ µ α α R L = α ( ξ µ α R L ) ( α ξµ )( α R L ) = α ( ξ µ α R L ) α (R L α ξµ ) + R L ( ξµ ) = α { ξ µ α R L R L α ξµ } (D 2) R L ξ µ. In the second term of (6.7), we can easily change the places of covariant and contravariant derivatives because of the Ricci scalar R L, that is ξ ν µ ν R L = ξ ν ν µ R L, and making the ν α substitution, we have ξ α α µ R L = α ( ξ α µ R L ) ( α ξα )( µ R L ) = α ( ξ α µ R L ), where α ξα is zero because of the Killing equation. Inserting these results into (6.7), the surface terms can be taken out ξ µ R L ξ ν µ ν R L + (D 2) ξ µ R L = α { ξ µ α R L R L α ξµ } (D 2) R L ξ µ + (D 2) R L ξ µ α ( ξ α µ R L ) = α { ξ µ α R L ξ α µ R L + R L µ ξα }. The last term in (6.6) can be written as a surface term plus extra terms: ξ ν G µν L = ξ ν α α Ḡ µν L = α { ξ ν α G µν L } ( α ξν )( α G µν L ), 37

47 where we have put the Killing vector inside the covariant derivative. In the second term we can freely move the α indices and afterwards, we can also take terms inside the covariant derivative with an extra term. Hence, we get ξ ν G µν L = α { ξ ν α G µν L G µν L α ξν } + G µν L ξν. Now we can add and subtract the terms α { ξ ν µ G αν L } and α {G αν L µ ξν } ξ ν G µν L = α { ξ α ν G µν L ξ µ ν GL αν G µν L α ξν + GL αν µ ξν } +G µν L ξν + α { ξ µ ν GL αν } α {GL αν µ ξν }. If we expand the last two terms we can see that: (i) When the covariant derivative hits on the Killing vector ξ ν, it will be zero in the first one with the use of (2.1), because α and ν are symmetric in G αν L. (ii) With the help of Bianchi identity (4.19), the term ( α G αν L )( µ ξν ) is zero. Hence we are left with ξ ν G µν L = α { ξ α ν G µν L ξ µ ν GL αν G µν L α ξν + GL αν µ ξν } +G µν L ξν + ξ ν α µ GL αν GL αν α µ ξν. (6.8) Using (B.1), (B.2) and (B.3), we can write ξ ν G µν L as a surface term. Collecting everything, the final form of the conserved charges for the generic quadratic theory reads Q µ ( ξ) = { 1 κ + 8Λ (D 2) (αd + β)} d D 1 x ḡ ξ 2 ν G µν L +(2α + β) ds i ḡ{ ξµ i R L ξ i µ R L + R L µ ξi } +β ds i ḡ{ ξν i G µν L ξ µ ν GL iν G µν L i ξν + GL iν µ ξν }. (6.9) 38

48 Now let us compute the energy of an asymptotically SdS geometry that might be a solution to our generic model. Should such a solution exist, we only require its asymptotic behavior to be h 00 + ( ) r0 D 3 ( ), h rr r0 D O(r 2 r r 0). (6.10) It is easy to see that for asymptotically SdS spaces the second and the third lines of (6.9) do not contribute, since for any Einstein space, to linear order R L µν = D 2 h µν, (6.11) which in turn yields R L = ḡ µν R L µν [/(D 2)]h = 0 and thus G L µν = 0 in the asymptotic region. Therefore the total energy of the full (α, β, γ) system, for geometries that are asymptotically SdS, is given only by the first term in (6.9), E D = { 1 + 8Λκ } (D 2) (αd + β) r (D 2) 2 0 D 3, D > 4, (6.12) 4G D where γ is implicitly assumed not to vanish. For D = 4, equivalently from (6.5), it reads (for models with an explicit Λ) E 4 = {1 + κ(4α + β)} r 0 2G 4 [1]. (6.13) In D = 3, the GB density vanishes identically and the energy expression has the same form of the D = 4 model, with the difference that r 0 comes from the metric (5.8) [1]. 39

49 From (6.12), the asymptotically SdS solution seemingly has negative energy, in the Einstein-GB model: E = (D 2) r0 D 3. (6.14) 4G D While this is, of course, correct in terms of the usual SdS signs, their exact form is [14] ds 2 = g 00 dt 2 + g rr dr 2 + r 2 dω D 2, (6.15) g 00 = g 1 rr = 1 + r 2 4κγ(D 3)(D 4) 1 ± [1 + 8γ(D 3)(D 4) rd 3 0 r D 1 ] 1 2. (6.16) Note that there is a branching here, with qualitatively different asymptotics: Schwarzschild and Schwarzschild-AdS, ( ) r0 D 3, g 00 = 1 r ( ) r0 D 3 r 2 = r κγ(d 3)(D 4). (6.17) [Here we restored γ, using κγ(d 3)(D 4) = l 2.] The first solution has the usual positive (for positive r 0 of course) ADM energy E = +(D 2)r D 3 0 /4G D, since the GB term does not contribute when expanded around flat space. On the other hand, the second solution which is asymptotically SdS, has the wrong sign for the mass term. However, to actually compute the energy here, one needs our energy expression (6.9), and not simply the AD formula which is valid only 40

50 for cosmological Einstein theory. Now from (6.17), we have h 00 ( ) r0 D 3 ( ), h rr r0 D 3 + O(r 2 r r 0), (6.18) whose sign is opposite to that of the usual SdS. This sign just compensates the flipped sign in the energy definition, so the energy (6.12) reads E = (D 2)r D 3 0 /4G D and the AdS branch, just like the flat branch, has positive energy, after the GB effects are taken into account also in the energy definition. Thus, for every Einstein-GB external solution, energy is positive and AdS vacuum is stable [1, 14]. 41

51 CHAPTER 7 CONCLUSION We have defined the energy of generic Einstein plus cosmological term plus quadratic gravity theories as well as pure quadratic models in all D, for both asymptotically flat and (A)dS spaces. The higher derivative terms do not change the form of the energy expression in flat backgrounds. On the other hand, for asymptotically (A)dS backgrounds (which are generically solutions to these equations, even in the absence of an explicit cosmological constant), the energy expressions (6.9) essentially reduce to the AD formula [up to higher order corrections that vanish for spacetimes that asymptotically approach (A)dS at least as fast as SdS spaces] [1]. Among quadratic theories, we have studied the string-inspired Einstein-GB model in more detail. This one, in the absence of an explicit cosmological constant, has both flat and AdS vacua. The AdS vacuum has specific cosmological constant and some of these are negative. These constants are determined by the Newton s constant and the GB coefficient. The constants that are negative being fixed from the string expansion to be positive. The explicit spherically 42

52 symmetric black hole solutions in this theory consist of two branches: asymptotically Schwarzschild spaces with a positive mass parameter or asymptotically Schwarzschild-AdS spaces with a negative one. The asymptotically Schwarzschild branch has the usual positive ADM energy. Using the compensation of two minus signs in the solution and in the correct energy definition, we noted that the AdS branch has likewise positive energy and that the AdS vacuum was a stable zero energy state [1]. In this thesis, out of conserved charges our explicit examples were related to the energy. In fact, with this formalism, we can easily study the angular momenta of black holes in both asymptotically AdS and flat spacetimes. Once a background Killing vector is given, our formalism provides us with a conserved charge. For the case of angular momentum, we just take (in four dimensions) the Killing vector to be ξ µ = (0, 0, 0, 1). For the details, we refer the reader to [15]. 43

53 REFERENCES [1] S. Deser, B. Tekin, Phys. Rev. D67, (2003). [2] S. Deser, B. Tekin, Phys. Rev. Lett. 89, (2002). [3] L.F. Abbott and S. Deser, Nucl. Phys. B195, 76 (1982). [4] R. Arnowitt, S. Deser and C. Misner, Phys. Rev. 116, 1322 (1959); 117, 1595 (1960); in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962). [5] R. d Inverno, Introducing Einstein s Relativity (Clarendon Press, Oxford, 1995). [6] S. Weinberg, Gravitation and Cosmology: Principles and Applications of The General Theory of Relativity (Wiley, New York, 1972). [7] S. Carroll, Spacetime and Geometry (Addison Wesley, 2004). [8] E. Witten, Quantum Gravity in de-sitter space, hep-th/ ; A. Strominger, J. High Energy Phys. 10, 034 (2001). [9] S. Deser and R. Jackiw, Ann. Phys. (N.Y.)153, 405 (1984). [10] B. Zwiebach, Phys. Lett. 156B, 315 (1985). [11] K.S. Stelle, Phys. Rev. D16, 953 (1977). [12] M. Cvetic, S. Nojiri and S.D. Odintsov, Nucl. Phys. B628, 295 (2002). [13] J.M. Maldacena and C. Nunez, Int. J. Mod. Phys. A16, 822 (2001). [14] D.G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656 (1985). [15] S. Deser, İ. Kanık and B. Tekin, Conserved Charges of Higher D Kerr-AdS Spacetimes, Class. Quan. Grav. 22, 3383 (2005). 44

54 APPENDIX A LINEARIZATION EXPRESSIONS FOR PURE QUADRATIC TERMS In this section, we will carry out the linearization of certain tensors that appear in the equation of motion (6.2). δ(r µρνσ R ρσ ) = (δr µρνσ ) R ρσ + R µρνσ δ(r ρσ ) = (δr µρνσ )ḡ αρ ḡ θσ Rαθ + R µρνσ δ(g αρ g θσ R αθ ) = (δr µρνσ )ḡ αρ ḡ θσ 2 (D 2) Λḡ αθ + R µρνσ { (δg αρ )ḡ θσ Rαθ ḡ αρ (δg θσ ) R αθ + ḡ αρ ḡ θσ δr αθ } Using (4.6) and (4.7), we get { (δ(r µρνσ )R ρσ ) = δr µρνσ ḡ αρ δα σ 2 (D 2) Λ + R µρνσ (δg αρ )ḡ θσ 2 (D 2) Λḡ αθ } ḡ αρ (δg θσ 2 ) (D 2) Λḡ αθ + ḡ αρ ḡ θσ δr αθ = (δr µρνα )ḡ αρ 2 (D 2) Λ } + R µρνσ { (δg σρ 2 ) (D 2) Λ 2 (δgρσ ) (D 2) Λ + ḡαρ ḡ θσ δr αθ. 45

55 Inserting (4.5) in R µρνσ δ(r µρνσ R ρσ ) = 2 4Λ 2 (D 2) ΛRL µν (D 1)(D 2) (h 2 µν ḡ µν h) + (D 1)(D 2) (ḡ µνḡ ρσ ḡ µσ ḡ ρν ) { } δg σρ 4Λ (D 2) + ḡαρ ḡ θσ δr αθ, expanding the parentheses in the second line we get δ(r µρνσ R ρσ ) = 2 (D 2) ΛRL µν 4Λ 2 (D 1)(D 2) 2 h µν + 8Λ 2 (D 1)(D 2) ḡµνh 8Λ (D 1)(D 2) h 2 µν + (D 1)(D 2) RL αθ(ḡ µν ḡ θσ δσ α δµδ θ ν) ρ. 4Λ 2 (D 1)(D 2) 2 ḡµνh By inserting the definition of h = ḡ αθ h αθ δ(r µρνσ R ρσ ) = where 2 (D 2) Λḡ αθ = R αθ, 2 4Λ 2 (D 2) ΛRL µν + (D 1)(D 2) h 2 µν 4Λ 2 (D 1)(D 2) ḡµνh αθ ḡ 2 αθ + (D 1)(D 2) RL αθḡ µν ḡ αθ (D 1)(D 2) RL µν = (RL µν(d 1) Rµν) L 4Λ 2 + (D 1)(D 2) (D 1)(D 2) h 2 µν 2 (D 2) Λḡ αθ αθ (D 1)(D 2)ḡµνh + (D 1)(D 2) RL αθḡ µν ḡ αθ, δ(r µρνσ R ρσ ) = (D 2) (D 1)(D 2) RL µν 4Λ 2 + (D 1)(D 2) h 2 µν + L (D 1)(D 2)ḡµν(R αθḡ θα R αθ h αθ ) 46

56 and the last term in the parentheses is equal to R L. We end up with δ(r µρνσ R ρσ ) = (D 1) RL µν + Now we will linearize R µρσα R ν ρσα : 4Λ 2 (D 1)(D 2) 2 h µν + (D 1)(D 2)ḡµνR L. (A.1) δ(r µρσα R ν ρσα ) = (δr µρσα ) R ν ρσα + R µρσα δr ν ρσα. We follow the same path here δ(r µρσα R ν ρσα ) = (δr µρσα )ḡ ρβ ḡ σθ ḡ αη Rν βθη + R µρσα δ(g ρβ g σθ g αη R ν βθη ) = (D 1)(D 2) (δr µρσα)ḡ ρβ ḡ σθ ḡ αη (ḡ νθ ḡ βη ḡ νη ḡ βθ ) R µρσα (δg ρβ )ḡ σθ ḡ αη Rν βθη R µρσα ḡ ρβ (δg σθ )ḡ αη Rν βθη R µρσα ḡ ρβ ḡ σθ (δg αη ) R ν βθη + R µρσα ḡ ρβ ḡ σθ ḡ αη δr ν βθη. Using (4.5), we have δ(r µρσα R ν ρσα ) = (D 1)(D 2) {(δr µρσα)(δ σ ν δ ρ η ḡ αη δ α ν δ ρ θ ḡσθ ) +(ḡ µσ ḡ ρα ḡ µα ḡ ρσ )ḡ ρβ (δg σθ )ḡ αη Rν βθη } 4Λ 2 (D 1) 2 (D 2) 2 {(δg ρβ )ḡ σθ ḡ αη (ḡ µσ ḡ ρα ḡ µα ḡ ρσ )(ḡ νθ ḡ βη ḡ νη ḡ βθ ) +ḡ ρβ (δg σθ )ḡ αη (ḡ µσ ḡ ρα ḡ µα ḡ ρσ )(ḡ νθ ḡ βη ḡ νη ḡ βθ ) +ḡ ρβ ḡ σθ (δg αη )(ḡ µσ ḡ ρα ḡ µα ḡ ρσ )(ḡ νθ ḡ βη ḡ νη ḡ βθ )}, 47

57 expanding the parentheses we get δ(r µρσα R ν ρσα ) = (D 1)(D 2) {(δr µηνα)ḡ αη (δr µθσν )ḡ θσ +δα β δµ θ ḡ αη δr νβθη δµ η δσ β ḡ σθ δr νβθη } 4Λ 2 (D 1) 2 (D 2) 2 {δgρβ (δµδ θ ρ η δµδ η ρ)(ḡ θ νθ ḡ βη ḡ νη ḡ βθ ) +δg σθ (ḡ ρβ ḡ µσ δρ η δσδ β µ)(ḡ η νθ ḡ βη ḡ νη ḡ βθ ) +δg αη (δµδ θ α β ḡ σθ ḡ µα δσ)(ḡ β νθ ḡ βη ḡ νη ḡ βθ )}. We rewrite the indices according to the Kronecker delta factors δ(r µρσα R ν ρσα ) = (D 1)(D 2) {(δr µηνα)ḡ αη + (δr µθνσ )ḡ θσ +(δr ναµη )ḡ αη ḡ σθ δr νσθµ } 4Λ 2 (D 1) 2 (D 2) 2 {δgρβ (δµ θ δρ η ḡ νθ ḡ βη δµ θ δρ η ḡ νη ḡ βθ δµ η δρ θ ḡ νη ḡ βη + δµ η δρ θ ḡ νη ḡ βθ ) +δg σθ (δρ η δη ρ ḡ µσ ḡ νθ δρ η δ ρ θ ḡµσḡ νη δσ β δµ η ḡ νθ ḡ βη + δσ β δµ η ḡ νη ḡ βθ ) +δg αη (δµ θ δα β ḡ νθ ḡ βη δµ θ δα β ḡ νη ḡ βθ δσ β δν σ ḡ µα ḡ βη + δσ β δβ σ ḡ µα ḡ νη )} = 8Λ (D 1)(D 2) {RL µν (D 1)(D 2) (h µν ḡ µν h} 4Λ (D 1)(D 2) (4hḡ µν 6h µν + 2Dh µν ) collecting terms, we end up with δ(r µρσα R ν ρσα ) = 8Λ 8Λ 2 (D 1)(D 2) RL µν (D 1)(D 2) h 2 µν. (A.2) The calculations for the Riemann tensor squared are similar but longer. We start 48

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