Conventions, Definitions, Identities, and Other Useful Formulae 1

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Contents

1 Curvature tensors 1

2 Conventions for Diﬀerential Forms2

3 Euler Densities 3

4 Hypersurfaces 3

5 Sign Conventions for the Action4

6 Hamiltonian Formulation 5

7 Conformal Transformations5

8 Small Variations of the Metric6

9 The ADM Decomposition 8

10 Converting to ADM Variables 10

1. Curvature tensors

Consider a d + 1 dimensional manifold M with metric gµν. The covariant derivative on M that is metric- compatible with gµν is ∇µ.

Christoﬀel Symbols 1 Γλ = gλρ (∂ g + ∂ g − ∂ g ) (1) µν 2 µ ρν ν µρ ρ µν Riemann Tensor λ λ λ κ λ κ λ R µσν = ∂σΓ µν − ∂νΓ µσ + Γ µνΓ κσ − Γ µσΓ κν (2)

Ricci Tensor σ λ Rµν = δ λR µσν (3)

Schouten Tensor 1 1 S = R − g R (4) µν d − 1 µν 2 d µν ν ν ∇ Sµν = ∇µS ν (5)

Weyl Tensor λ λ λ λ λ λ C µσν = R µσν + g ν Sµσ − g σ Sµν + gµσ S ν − gµν S σ (6)

1 Commutators of Covariant Derivatives

σ [∇µ, ∇ν] Aλ = RλσµνA (7) λ λ σ [∇µ, ∇ν] A = R σµνA (8)

Bianchi Identity

∇κRλµσν − ∇λRκµσν + ∇µRκλσν = 0 (9) ν ∇ Rλµσν = ∇µRλσ − ∇λRµσ (10) 1 ∇νR = ∇ R (11) µν 2 µ

Bianchi Identity for Weyl

ν ∇ Cλµσν = (d − 2) ∇µSλσ − ∇λSµσ (12)

d − 2 1 d − 1 d + 1 ∇λ∇σC = ∇2R − g ∇2R − ∇ ∇ R − R λR (13) λµσν d − 1 µν 2d µν 2d µ ν d − 1 µ νλ (d + 1) 1 1 + C Rλσ + RR + g RλσR − R2 λµσν d(d − 1) µν d − 1 µν λσ d

2. Conventions for Diﬀerential Forms

p-Form Components 1 A = A dxµ1 ∧ ... ∧ dxµp (14) (p) p! µ1...µp

Exterior Derivative dA(p) = (p + 1) ∂ [ µ1 Aµ2...µp+1 ] (15) µ1...µp+1

1 B := B + permutations (16) [ µ1...µn] n! µ1...µn

Hodge-Star 1 ν1...νp ? A(p) = µ1...µd+1−p Aν1...νp (17) µ1...µd+1−p p!

?? = − 1 p(d+1−p)+1 (18)

Wedge Product

(p + q)! A(p) ∧ B(q) = A [ µ1...µp Bµp+1...µp+q ] (19) µ1...µp+q p! q!

2 3. Euler Densities

Let M be a manifold with dimension d + 1 = 2n an even number. Normalized so that χ(S2n) = 2.

Euler Number Z 2n √ χ(M) = d x g E2n (20) M Z

= e2n (21) M

Euler Density 1 µ µ ν ν µ µ ν ν E2n = µ ...µ ν ...ν R 1 2 1 2 ...R 2n−1 2n 2n−1 2n (22) (8π)n Γ(n + 1) 1 2n 1 2n

1 a a a a e = a ...a R 1 2 ∧ ... ∧ R 2n−1 2n (23) 2n (4π)n Γ(n + 1) 1 2n

Curvature Two-Form 1 Ra = Ra ec ∧ ed (24) b 2 b c d Examples 1 E = Rµνλρ (25) 2 8π µν λρ 1 = R 4π 1 E = Rµναβ Rλργδ (26) 4 128π2 µνλρ αβγδ 1 = RµνλρR − 4 RµνR + R2 32π2 µνλρ µν 1 1 d − 2 d + 1 = CµνλρC − RµνR − R2 32π2 µνλρ 8π2 d − 1 µν 4 d

4. Hypersurfaces

Let Σ ⊂ M be a d dimensional hypersurface whose embedding is described locally by an outward-pointing, unit normal vector nµ. Rather than keeping track of the signs associated with nµ being either spacelike or timelike, µ µν we will just assume that n is spacelike. Indices are lowered and raised using gµν and g , and symmetrization of indices is implied when appropriate.

First Fundamental Form / Induced Metric on Σ

hµν = gµν − nµnν (27)

Projection onto Σ µ ... µ σ λ ... ⊥ T ν ... = h λ . . . h ν ...T σ ... (28) Second Fundamental Form / Extrinsic Curvature of Σ 1 K = ⊥(∇ n ) = h λh σ∇ n = £ h (29) µν µ ν µ ν λ σ 2 n µν

3 Trace of Extrinsic Curvature µ K = ∇µn (30)

‘Acceleration’ Vector µ ν µ a = n ∇νn (31)

Surface-Forming Normal Vectors 1 nµ = ∂µα ⇒ ⊥∇[ µnν ] = 0 (32) p νλ g ∂να∂λα

Covariant Derivative on Σ compatible with hµν

α ... α ... DµT β ... = ⊥ ∇µT β ... ∀ T = ⊥ T (33)

Intrinsic Curvature of (Σ, h) λ λ σ λ λ [Dµ, Dν] A = R σµνA ∀ A = ⊥ A (34)

Gauss-Codazzi

⊥Rλµσν = Rλµσν − KλσKµν + KµσKνλ (35) λ ⊥ Rλµσν n = DνKµσ − DσKµν (36)

λ σ λ ⊥ Rλµσν n n = − Ln Kµν + Kµ Kλν + Dµaν − aµaν (37)

Projections of the Ricci tensor

λ ⊥ (Rµν) = Rµν + Dµaν − aµaν − Ln Kµν − KKµν + 2Kµ Kν λ (38) µ µ ⊥ (Rµν n ) = D Kµν − DνK (39) µ ν µν µ µ Rµνn n = − Ln K − K Kµν + Dµa − aµa (40)

Decomposition of the Ricci scalar

2 µν µ µ R = R − K − K Kµν − 2 Ln K + 2 Dµa − 2 aµa (41)

Lie Derivatives along nµ

λ λ λ £nKµν = n ∇λKµν + Kλν∇µn + Kµλ∇νn (42) µ ... µ ... ⊥ (£n F ν ...) = £n F ν ... ∀ ⊥F = F (43)

5. Sign Conventions for the Action

These conventions follow Weinberg, keeping in mind that he defines the Riemann tensor with a minus sign relative to our definition. They are appropriate when using signature (−, +,..., +). The d + 1-dimensional 2 Newton’s constant is 2κ = 16πGd+1. The sign on the boundary term follows from our definition of the extrinsic curvature.

4 Gravitational Action Z Z 1 d+1 √ 1 d √ IG = 2 d x g R − 2 Λ + 2 d x γ K (44) 2 κ M κ ∂M Z 1 d+1 √ 2 µν = 2 d x g R + K − K Kµν − 2 Λ (45) 2 κ M

Gauge Field Coupled to Particles Z 1 d+1 √ µν IM = − d x g F Fµν (46) 4 M Z µ ν 1/2 X dxn(p) dx (p) − m dp −g (x (p)) n (47) n µν n dp dp n Z µ X dxn(p) + e dp A (x (p)) (48) n dp µ n n

Gravity Minimally Coupled to a Gauge Field Z Z d+1 √ 1 1 µν 1 d √ I = d x g 2 (R − 2 Λ) − F Fµν + 2 d x γ K (49) M 2 κ 4 κ ∂M

6. Hamiltonian Formulation

µν The canonical variables are the metric hµν on Σ and its conjugate momenta π . The momenta are deﬁned with respect to evolution in the spacelike direction nµ, so this is not the usual notion of the Hamiltonian as the generator of time translations.

Bulk Lagrangian Density

1 2 µν LM = K − K Kµν + R − 2 Λ (50) 2 κ2

Momentum Conjugate to hµν µν ∂LM 1 µν µν π = = 2 h K − K (51) ∂ (£nhµν) 2 κ Momentum Constraint 1 H = ⊥ (nνG ) = 2 Dνπ = 0 (52) µ κ2 µν µν

Hamiltonian Constraint 1 1 1 H = − nµnνG = 2 κ2 πµν π − π2 + R − 2 Λ = 0 (53) κ2 µν µν d − 1 2 κ2

7. Conformal Transformations

µν The dimension of spacetime is d + 1. Indices are raised and lowered using the metric gµν and its inverse g .

Metric

2 σ gbµν = e gµν (54)

5 Christoﬀel

λ λ λ Γb µν = Γ µν + Θ µν (55) λ λ λ λ Θ µν = δ µ ∇νσ + δ ν ∇µσ − gµν ∇ σ (56)

Riemann Tensor

λ λ λ λ λ λ Rb µρν = R µρν + δ ν ∇µ∇ρσ − δ ρ ∇µ∇νσ + gµρ ∇ν∇ σ − gµν ∇ρ∇ σ (57) λ λ λ λ + δ ρ ∇µσ ∇νσ − δ ν ∇µσ ∇ρσ + gµν ∇ρσ ∇ σ − gµρ ∇νσ ∇ σ (58) λ λ α + gµρ δ ν − gµν δ ρ ∇ σ ∇ασ (59)

Ricci Tensor

2 Rbµν = Rµν − gµν ∇ σ − (d − 1) ∇µ∇νσ + (d − 1) ∇µσ ∇νσ (60) λ − (d − 1) gµν∇ σ ∇λσ (61)

Ricci Scalar

−2 σ 2 µ Rb = e R − 2 d ∇ σ − d (d − 1) ∇ σ ∇µσ (62)

Schouten Tensor

1 λ Sbµν = Sµν − ∇µ∇νσ + ∇µσ ∇νσ − gµν∇ σ ∇λσ (63) 2

Weyl Tensor

λ λ Cb µρν = C µρν (64)

Normal Vector

µ −σ µ σ nb = e n nbµ = e nµ (65)

Extrinsic Curvature

σ λ Kbµν = e Kµν + hµν n ∇λσ (66)

−σ λ Kb = e K + d n ∇λσ (67)

8. Small Variations of the Metric

Consider a small perturbation to the metric of the form gµν → gµν + δgµν. All indices are raised and lowered using the unperturbed metric gµν and its inverse. All quantities are expressed in terms of the perturbation to the metric with lower indices, and never in terms of the perturbation to the inverse metric. As in the previous sections, ∇µ is the covariant derivative on M compatible with gµν and Dµ is the covariant derivative on a hypersurface Σ compatible with hµν.

Inverse Metric µν µν µα νβ µα νβ λρ g → g − g g δgαβ + g g g δgαλ δgβρ + ... (68)

6 Square Root of Determinant √ √ 1 g → g 1 + gµνδg + ... (69) 2 µν

Variational Operator

2 δ(gµν) = δgµν δ (gµν) = δ(δgµν) = 0 (70) µν µα νβ 2 µν µλ νρ µα νβ λρ δ(g ) = −g g δgαβ δ (g ) = δ −g g δgλρ = 2 g g g δgαλ δgβρ (71) 1 1 F(g + δg) = F(g) + δF(g) + δ 2F(g) + ... + δ nF(g) + ... (72) 2 n!

Christoﬀel (All Orders) 1 δ Γλ = gλρ ∇ δg + ∇ δg − ∇ δg (73) µν 2 µ ρν ν µρ ρ µν 2 λ λα ρβ δ Γ µν = −g g δgαβ ∇µ δgρν + ∇ν δgµρ − ∇ρ δgµν (74) n δ n Γλ = δ n−1 gλρ ∇ δg + ∇ δg − ∇ δg (75) µν 2 µ ρν ν µρ ρ µν

Riemann Tensor

λ λ λ δ R µσν = ∇σδ Γ µν − ∇νδ Γ µσ (76)

Ricci Tensor

λ λ δ Rµν = ∇λδ Γ µν − ∇νδ Γ µλ (77) 1 = ∇λ∇ δg + ∇λ∇ δg − gλρ ∇ ∇ δg − ∇2 δg (78) 2 µ λν ν µλ µ ν λρ µν

Ricci Scalar

µν µ ν λρ δ R = −R δgµν + ∇ ∇ δgµν − g ∇µ δgλρ (79)

Surface Forming Normal Vector 1 1 δ n = n nνnλδg = δg nν + c (80) µ 2 µ νλ 2 µν µ

1 1 1 c = n nνnλδg − δg nν = − h λ δg nν (81) µ 2 µ νλ 2 µν 2 µ λν

Extrinsic Curvatures 1 δ K = nαnβδg K + δg nρ n Kλ + n K λ (82) µν 2 αβ µν λρ µ ν ν µ 1 − h λ h ρnα ∇ δg + ∇ δg − ∇ δg 2 µ ν λ αρ ρ λα α λρ

1 1 δ K = − Kµν δg − nµ ∇νδg − gνλ ∇ δg + D cµ (83) 2 µν 2 µν µ νλ µ

7 9. The ADM Decomposition

The conventions and notation in this section (and the next) are diﬀerent than what was used in the preceding sections. We consider a d-dimensional spacetime with metric hab.

We start by identifying a scalar ﬁeld t whose isosurfaces Σt are normal to the timelike unit vector given by

ua = −α ∂at , (84) where the lapse function α is 1 α := . (85) p ab −h ∂at ∂bt

An observer whose worldline is tangent to ua experiences an acceleration given by the vector

c d ab = u · ∇cub , (86) which is orthogonal to ua. The (spatial) metric on the d − 1 dimensional surface Σt is given by

σab = hab + uaub . (87)

The intrinsic Ricci tensor built from this metric is denoted by Rab, and its Ricci scalar is R. The covariant derivative on Σt is deﬁned in terms of the d dimensional covariant derivative as

c ed c DaVb := σa σb ∇cVe for any Vb = σb Vc . (88)

The extrinsic curvature of Σt embedded in the ambient d dimensional spacetime (the constant r surfaces from the previous section) is 1 θ := −σ cσ dd∇ u = −d∇ u − u a = − £ σ . (89) ab a b c d a b a b 2 u ab

This definition has an additional minus sign, compared to the extrinsic curvature Kµν for the constant r surfaces of the previous section. This is merely for compatibility with the standard conventions in the literature. Now we consider a ‘time flow’ vector field ta, which satisfies the condition

a t ∂at = 1 . (90)

a The vector t can be decomposed into parts normal and along Σt as

ta = α ua + βa , (91)

a a b where α is the lapse function (85) and β := σ bt is the shift vector. An important result in the derivations that follow relates the Lie derivative of a scalar or spatial tensor (one that is orthogonal to ua in all of its indices) along the time ﬂow vector ﬁeld, to Lie derivatives along ua and βa. Let S be a scalar. Then

£tS = £α uS + £βS = α£uS + £βS. (92)

Rearranging this expression then gives 1 £ S = £ S − £ S . (93) u α t β Similarly, for a spatial tensor with all lower indices we have

£tWa... = α£uWa... + £βWa... . (94)

8 This is not the case when the tensor has any of its indices raised. In a moment, these identities will allow us to express certain Lie derivatives along ua in terms of regular time derivatives and Lie derivatives along the shift vector βa. Next, we construct the coordinate system that we will use for the decomposition of the equations of motion. The adapted coordinates (t, xi) are deﬁned by

a a ∂tx := t . (95)

i The x are d dimensional coordinates along the surface Σt. If we deﬁne ∂xa P a := , (96) i ∂xi a a then it follows from the deﬁnition of the coordinates that Pi ∂at = 0 and we can use Pi to project tensors onto Σt. For example, in the adapted coordinates the spatial metric, extrinsic curvature, and acceleration and shift vectors are

a b σij = Pi Pj σab (97) a b θij = Pi Pj θab (98) b aj = Pj ab (99) a a βi = Pi βa = Pi ta . (100) The line element in the adapted coordinates takes a familiar form: ∂xa ∂xa ∂xb ∂xb h dxadxb = h dt + dxi dt + dxj (101) ab ab ∂t ∂xi ∂t ∂xj a a i b b j = hab t dt + Pi dx t dt + Pj dx (102) a 2 a i a b i j = t tadt + 2tadtPi dx + habPi Pj dx dx (103) 2 i 2 i i j = − α + β βi dt + 2βidtdx + σijdx dx (104) a b 2 2 i i j j ⇒ habdx dx = − α dt + σij dx + β dt dx + β dt . (105)

Thus, in the adapted coordinate system we can express the components of the (d dimensional) metric hab and its inverse hab as 2 i j ! −α + β βi σijβ h = (106) ab j σijβ σij 1 1 i ! − 2 2 β hab = α α (107) 1 i ij 1 i j α2 β σ − α2 β β 2 det(hab) = −α det(σij) (108) Obtaining the components of the inverse is a short algebraic calculation. Note that the spatial indices ‘i, j, . . .’ ij in the adapted coordinates are lowered and raised using the spatial metric σij and its inverse σ . In adapted coordinates there are several results concerning the projections of Lie derivatives of scalars and tensors which will be important in what follows. The ﬁrst, which is trivial, is that the Lie derivative of a scalar S along the time-ﬂow vector ta is just the regular time-derivative ∂xa ∂S £ S = ta∂ S = = ∂ S. (109) t a ∂t ∂xa t a a Next, we consider the projector Pi applied to the Lie derivative along t of a general vector Wa, which gives

a Pi £tWa = ∂tWa ∀ Wa . (110)

9 The important point is that this applies not just to spatial vectors but to any vector Wa, as a consequence of the result a Pi £tua = 0 . (111) Finally, we can show that the Lie derivative along ta of any contravariant spatial vector satisﬁes i a i i i a P a£tV = ∂tV ∀ V = P aV . (112) This follows from a lengthier calculation than what is required for the ﬁrst two results. Given these results, we can express various geometric quantities and their projections normal to and along Σt in terms of quantities intrinsic to Σt and simple time derivatives. First, the extrinsic curvature is 1 θ = − P aP b£ σ (113) ij 2 i j u ab 1 1 = − P aP b £ σ − £ σ (114) 2 i j α t ab β ab 1 ⇒ θ = − ∂ σ − D β + D β . (115) ij 2α t ab a b b a a Since θab is a spatial tensor, projections of its Lie derivative along u can be expressed in a similar manner 1 P aP b£ θ = ∂ θ − £ θ . (116) i j u ab α t ab β ab Now we present the Gauss-Codazzi and related equations in adapted coordinates: 1 1 P aP bdR = R + θθ − 2θ kθ − ∂ θ − £ θ − D D α (117) i j ab ij ij i jk α t ij β ij α i j ad b j Pi Rabu = Diθ − D θij (118) 1 1 dR uaub = ∂ θ − βi∂ θ − θijθ + D Diα (119) ab α t i ij α i 2 2 dR = R + θ2 + θijθ − ∂ θ − βi∂ θ − D Diα . (120) ij α t i α i

10. Converting to ADM Variables

The metric is often presented in the form a b 2 i i j habdx dx = httdt + 2htidtdx + hijdx dx . (121)

We would like to relate these components to the ADM variables: the lapse function α, the shift vector βi, and the spatial metric σij. This is a fairly straightforward exercise in linear algebra. Comparing with (105), we ﬁrst note that

σij = hij . (122) ij ij The inverse spatial metric, σ , is literally the inverse of hij, which is not the same thing as h ij −1 −1 ij σ = (σij) = (hij) 6= h . (123) For the shift vector we have j ik ik l i hti = σijβ → σ htk = σ σklβ = β (124) i ij ⇒ β = σ htj . (125) Finally, for the lapse we obtain 2 ij α = σ htihtj − htt . (126)