As the name suggests, these notes contain a summary of important conventions, definitions, identities, and various formulas that I often refer to. They may prove useful for researchers working in General Relativity, Supergravity, String Theory, Cosmology, and related areas

McNees, Robert A, IV

Loyola eCommons

Loyola University ChicagoLoyola eCommonsPhysics: Faculty Publications and Other Works Faculty Publications2010Conventions, Definitions, Identities, and OtherUseful FormulaeRobert A. McNees IVLoyola University Chicago, rmcnees@luc.eduThis Notes is brought to you for free and open access by the Faculty Publications at Loyola eCommons. It has been accepted for inclusion in Physics:Faculty Publications and Other Works by an authorized administrator of Loyola eCommons. For more information, please contactecommons@luc.edu.This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.Copyright © 2010 Robert A McNees IVRecommended CitationWhen citing these notes please include a link to the website: http://jacobi.luc.edu/notes.html.Conventions, Definitions, Identities,and Other Useful FormulaeVContents1 Curvature tensors 12 Conventions for Differential Forms 23 Euler Densities 34 Hypersurfaces 35 Sign Conventions for the Action 46 Hamiltonian Formulation 57 Conformal Transformations 58 Small Variations of the Metric 69 The ADM Decomposition 810 Converting to ADM Variables 101. Curvature tensorsConsider a d + 1 dimensional manifold M with metric gµν . The covariant derivative on M that is metric-compatible with gµν is ∇µ.Christoffel SymbolsΓλµν =12gλρ (∂µgρν + ∂νgµρ − ∂ρgµν) (1)Riemann TensorRλµσν = ∂σΓλµν − ∂νΓλµσ + ΓκµνΓλκσ − ΓκµσΓλκν (2)Ricci TensorRµν = δσλRλµσν (3)Schouten TensorSµν =1d− 1(Rµν − 12 dgµνR)(4)∇νSµν = ∇µSνν (5)Weyl TensorCλµσν = Rλµσν + gλν Sµσ − gλσ Sµν + gµσ Sλν − gµν Sλσ (6)1Commutators of Covariant Derivatives[∇µ,∇ν ]Aλ = RλσµνAσ (7)[∇µ,∇ν ]Aλ = RλσµνAσ (8)Bianchi Identity∇κRλµσν −∇λRκµσν +∇µRκλσν = 0 (9)∇νRλµσν = ∇µRλσ −∇λRµσ (10)∇νRµν = 12∇µR (11)Bianchi Identity for Weyl∇νCλµσν = (d− 2)(∇µSλσ −∇λSµσ)(12)∇λ∇σCλµσν = d− 2d− 1[∇2Rµν − 12dgµν∇2R− d− 12d∇µ∇νR−(d+ 1d− 1)R λµ Rνλ (13)+ CλµσνRλσ +(d+ 1)d(d− 1) RRµν +1d− 1 gµν(RλσRλσ − 1dR2)]2. Conventions for Differential Formsp-Form ComponentsA(p) =1p!Aµ1...µp dxµ1 ∧ . . . ∧ dxµp (14)Exterior Derivative (dA(p))µ1...µp+1= (p+ 1) ∂ [µ1Aµ2...µp+1 ] (15)B [µ1...µn] :=1n!(Bµ1...µn + permutations)(16)Hodge-Star (?A(p))µ1...µd+1−p=1p!µ1...µd+1−pν1...νp Aν1...νp (17)? ? =(− 1 )p(d+1−p)+1 (18)Wedge Product (A(p) ∧B(q))µ1...µp+q=(p+ q)!p! q!A [µ1...µp Bµp+1...µp+q ] (19)23. Euler DensitiesLet M be a manifold with dimension d+ 1 = 2n an even number. Normalized so that χ(S2n) = 2.Euler Numberχ(M) =∫Md 2nx√g E2n (20)=∫Me2n (21)Euler DensityE2n = 1(8pi)n Γ(n+ 1)µ1 ...µ2n ν1 ...ν2n Rµ1µ2ν1ν2 . . . Rµ2n−1µ2nν2n−1ν2n (22)e2n =1(4pi)n Γ(n+ 1)a1 ...a2nRa1a2 ∧ . . . ∧Ra2n−1a2n (23)Curvature Two-FormRab =12Rab c d ec ∧ ed (24)ExamplesE2 = 18piµνλρRµνλρ (25)=14piRE4 = 1128pi2µνλρ αβγδ Rµναβ Rλργδ (26)=132pi2(RµνλρRµνλρ − 4RµνRµν +R2)=132pi2CµνλρCµνλρ − 18pi2(d− 2d− 1) (RµνRµν − d+ 14 dR2)4. HypersurfacesLet Σ ⊂M be a d dimensional hypersurface whose embedding is described locally by an outward-pointing, unitnormal vector nµ. Rather than keeping track of the signs associated with nµ being either spacelike or timelike,we will just assume that nµ is spacelike. Indices are lowered and raised using gµν and gµν , and symmetrizationof indices is implied when appropriate.First Fundamental Form / Induced Metric on Σhµν = gµν − nµnν (27)Projection onto Σ⊥Tµ ... ν ... = hµλ . . . hσν . . . T λ ... σ ... (28)Second Fundamental Form / Extrinsic Curvature of ΣKµν = ⊥(∇µnν) = h λµ h σν ∇λnσ =12£nhµν (29)3Trace of Extrinsic CurvatureK = ∇µnµ (30)‘Acceleration’ Vectoraµ = nν∇νnµ (31)Surface-Forming Normal Vectorsnµ =1√gνλ ∂να∂λα∂µα ⇒ ⊥∇[µnν ] = 0 (32)Covariant Derivative on Σ compatible with hµνDµTα ... β ... = ⊥∇µTα ... β ... ∀ T = ⊥T (33)Intrinsic Curvature of (Σ, h)[Dµ,Dν ]Aλ = RλσµνAσ ∀ Aλ = ⊥Aλ (34)Gauss-Codazzi⊥Rλµσν = Rλµσν −KλσKµν +KµσKνλ (35)⊥(Rλµσν nλ)= DνKµσ −DσKµν (36)⊥(Rλµσν nλnσ)= − LnKµν +K λµ Kλν +Dµaν − aµaν (37)Projections of the Ricci tensor⊥ (Rµν) = Rµν +Dµaν − aµaν − LnKµν −KKµν + 2K λµ Kν λ (38)⊥ (Rµν nµ) = DµKµν −DνK (39)Rµνnµnν = − LnK −Kµν Kµν +Dµaµ − aµaµ (40)Decomposition of the Ricci scalarR = R−K2 −Kµν Kµν − 2LnK + 2Dµaµ − 2 aµaµ (41)Lie Derivatives along nµ£nKµν = nλ∇λKµν +Kλν∇µnλ +Kµλ∇νnλ (42)⊥ (£nFµ ... ν ...) = £nFµ ... ν ... ∀ ⊥F = F (43)5. Sign Conventions for the ActionThese conventions follow Weinberg, keeping in mind that he defines the Riemann tensor with a minus signrelative to our definition. They are appropriate when using signature (−,+, . . . ,+). The d + 1-dimensionalNewton’s constant is 2κ2 = 16piGd+1. The sign on the boundary term follows from our definition of the extrinsiccurvature.4Gravitational ActionIG =12κ2∫Md d+1x√g(R− 2 Λ)+1κ2∫∂Md dx√γ K (44)=12κ2∫Md d+1x√g(R+K2 −Kµν Kµν − 2 Λ)(45)Gauge Field Coupled to ParticlesIM = − 14∫Md d+1x√g FµνFµν (46)−∑nmn∫dp(−gµν(xn(p)) dxµn(p)dpdxνn(p)dp)1/2(47)+∑nen∫dpdxµn(p)dpAµ(xn(p)) (48)Gravity Minimally Coupled to a Gauge FieldI =∫Md d+1x√g[12κ2(R− 2 Λ)− 14FµνFµν]+1κ2∫∂Md dx√γ K (49)6. Hamiltonian FormulationThe canonical variables are the metric hµν on Σ and its conjugate momenta piµν . The momenta are definedwith respect to evolution in the spacelike direction nµ, so this is not the usual notion of the Hamiltonian as thegenerator of time translations.Bulk Lagrangian DensityLM =12κ2(K2 −Kµν Kµν +R− 2 Λ)(50)Momentum Conjugate to hµνpiµν =∂LM∂ (£nhµν)=12κ2(hµν K −Kµν) (51)Momentum ConstraintHµ = 1κ2⊥ (nνGµν) = 2Dνpiµν = 0 (52)Hamiltonian ConstraintH = − 1κ2nµnνGµν = 2κ2(piµν piµν − 1d− 1 pi2)+12κ2(R− 2 Λ) = 0 (53)7. Conformal TransformationsThe dimension of spacetime is d+ 1. Indices are raised and lowered using the metric gµν and its inverse gµν .Metricĝµν = e2σ gµν (54)5ChristoffelΓ̂λµν = Γλµν + Θλµν (55)Θλµν = δλµ∇νσ + δλν ∇µσ − gµν ∇λσ (56)Riemann TensorR̂λµρν = Rλµρν + δλν ∇µ∇ρσ − δλρ∇µ∇νσ + gµρ∇ν∇λσ − gµν ∇ρ∇λσ (57)+ δλρ∇µσ∇νσ − δλν ∇µσ∇ρσ + gµν ∇ρσ∇λσ − gµρ∇νσ∇λσ (58)+(gµρ δλν − gµν δλρ)∇ασ∇ασ (59)Ricci TensorR̂µν = Rµν − gµν ∇2σ − (d− 1)∇µ∇νσ + (d− 1)∇µσ∇νσ (60)− (d− 1) gµν∇λσ∇λσ (61)Ricci ScalarR̂ = e−2σ(R− 2 d∇2σ − d (d− 1)∇µσ∇µσ)(62)Schouten TensorŜµν = Sµν −∇µ∇νσ +∇µσ∇νσ − 12gµν∇λσ∇λσ (63)Weyl TensorĈλµρν = Cλµρν (64)Normal Vectorn̂µ = e−σ nµ n̂µ = eσ nµ (65)Extrinsic CurvatureK̂µν = eσ(Kµν + hµν nλ∇λσ)(66)K̂ = e−σ(K + dnλ∇λσ)(67)8. Small Variations of the MetricConsider a small perturbation to the metric of the form gµν → gµν + δgµν . All indices are raised and loweredusing the unperturbed metric gµν and its inverse. All quantities are expressed in terms of the perturbation tothe metric with lower indices, and never in terms of the perturbation to the inverse metric. As in the previoussections, ∇µ is the covariant derivative on M compatible with gµν and Dµ is the covariant derivative on ahypersurface Σ compatible with hµν .Inverse Metricgµν → gµν − gµα gνβ δgαβ + gµα gνβ gλρ δgαλ δgβρ + . . . (68)6Square Root of Determinant√g → √g(1 +12gµνδgµν + . . .)(69)Variational Operatorδ(gµν) = δgµν δ2(gµν) = δ(δgµν) = 0 (70)δ(gµν) = −gµα gνβ δgαβ δ 2(gµν) = δ(−gµλ gνρ δgλρ)= 2 gµα gνβ gλρ δgαλ δgβρ (71)F(g + δg) = F(g) + δF(g) + 12δ 2F(g) + . . .+ 1n!δ nF(g) + . . . (72)Christoffel (All Orders)δ Γλµν =12gλρ(∇µ δgρν +∇ν δgµρ −∇ρ δgµν)(73)δ 2 Γλµν = −gλα gρβ δgαβ(∇µ δgρν +∇ν δgµρ −∇ρ δgµν)(74)δ n Γλµν =n2δ n−1(gλρ) (∇µ δgρν +∇ν δgµρ −∇ρ δgµν)(75)Riemann Tensorδ Rλµσν = ∇σδ Γλµν −∇νδ Γλµσ (76)Ricci Tensorδ Rµν = ∇λδ Γλµν −∇νδ Γλµλ (77)=12(∇λ∇µ δgλν +∇λ∇ν δgµλ − gλρ∇µ∇ν δgλρ −∇2 δgµν)(78)Ricci Scalarδ R = −Rµν δgµν +∇µ(∇ν δgµν − gλρ∇µ δgλρ)(79)Surface Forming Normal Vectorδ nµ =12nµ nνnλδgνλ =12δgµν nν + cµ (80)cµ =12nµ nνnλδgνλ − 12δgµν nν = −12h λµ δgλνnν (81)Extrinsic Curvaturesδ Kµν =12nαnβδgαβKµν + δgλρ nρ(nµKλν + nνKλµ)(82)− 12hµλ hνρnα(∇λδgαρ +∇ρδgλα −∇αδgλρ)δ K = −12Kµν δgµν − 12nµ(∇νδgµν − gνλ∇µδgνλ)+Dµcµ (83)79. The ADM DecompositionThe conventions and notation in this section (and the next) are different than what was used in the precedingsections. 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 byua = −α∂at , (84)where the lapse function α isα :=1√−hab ∂at ∂bt. (85)An observer whose worldline is tangent to ua experiences an acceleration given by the vectorab = uc · d∇cub , (86)which is orthogonal to ua. The (spatial) metric on the d− 1 dimensional surface Σt is given byσab = hab + uaub . (87)The intrinsic Ricci tensor built from this metric is denoted by Rab, and its Ricci scalar is R. The covariantderivative on Σt is defined in terms of the d dimensional covariant derivative asDaVb := σacσbe(d∇cVe)for any Vb = σbcVc . (88)The extrinsic curvature of Σt embedded in the ambient d dimensional spacetime (the constant r surfaces fromthe previous section) isθab := −σacσbd(d∇cud)= −d∇aub − uaab = −12£uσab . (89)This definition has an additional minus sign, compared to the extrinsic curvature Kµν for the constant r surfacesof 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 conditionta ∂at = 1 . (90)The vector ta can be decomposed into parts normal and along Σt asta = αua + βa , (91)where α is the lapse function (85) and βa := σabtb is the shift vector. An important result in the derivationsthat follow relates the Lie derivative of a scalar or spatial tensor (one that is orthogonal to ua in all of itsindices) along the time flow vector field, to Lie derivatives along ua and βa. Let S be a scalar. Then£tS = £αuS + £βS = α£uS + £βS . (92)Rearranging this expression then gives£uS =1α(£tS −£βS). (93)Similarly, for a spatial tensor with all lower indices we have£tWa... = α£uWa... + £βWa... . (94)8This is not the case when the tensor has any of its indices raised. In a moment, these identities will allow us toexpress certain Lie derivatives along ua in terms of regular time derivatives and Lie derivatives along the shiftvector 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∂txa := ta . (95)The xi are d dimensional coordinates along the surface Σt. If we definePia :=∂xa∂xi, (96)then it follows from the definition of the coordinates that Pia∂at = 0 and we can use Pia to project tensors ontoΣt. For example, in the adapted coordinates the spatial metric, extrinsic curvature, and acceleration and shiftvectors areσij = PiaPjbσab (97)θij = PiaPjbθab (98)aj = Pjbab (99)βi = Piaβa = Piata . (100)The line element in the adapted coordinates takes a familiar form:habdxadxb = hab(∂xa∂tdt+∂xa∂xidxi)(∂xb∂tdt+∂xb∂xjdxj)(101)= hab(tadt+ Piadxi)(tbdt+ Pjbdxj)(102)= tatadt2 + 2tadtPiadxi + habPiaPjbdxidxj (103)=(− α2 + βiβi)dt2 + 2βidtdxi + σijdxidxj (104)⇒ habdxadxb = − α2dt2 + σij(dxi + βidt)(dxj + βjdt). (105)Thus, in the adapted coordinate system we can express the components of the (d dimensional) metric hab andits inverse hab ashab =(−α2 + βiβi σijβjσijβj σij)(106)hab =(− 1α21α2βi1α2βi σij − 1α2βiβj)(107)det(hab) = −α2 det(σij) (108)Obtaining the components of the inverse is a short algebraic calculation. Note that the spatial indices ‘i, j, . . .’in the adapted coordinates are lowered and raised using the spatial metric σij and its inverse σij .In adapted coordinates there are several results concerning the projections of Lie derivatives of scalars andtensors which will be important in what follows. The first, which is trivial, is that the Lie derivative of a scalarS along the time-flow vector ta is just the regular time-derivative£tS = ta∂aS =∂xa∂t∂S∂xa= ∂tS . (109)Next, we consider the projector Pia applied to the Lie derivative along ta of a general vector Wa, which givesPia£tWa = ∂tWa ∀ Wa . (110)9The important point is that this applies not just to spatial vectors but to any vector Wa, as a consequence ofthe resultPia£tua = 0 . (111)Finally, we can show that the Lie derivative along ta of any contravariant spatial vector satisfiesP ia£tVa = ∂tVi ∀ V i = P iaV a . (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 alongΣt in terms of quantities intrinsic to Σt and simple time derivatives. First, the extrinsic curvature isθij = − 12PiaPjb£uσab (113)= − 12PiaPjb(1α(£tσab −£βσab))(114)⇒ θij = − 12α(∂tσab −(Daβb +Dbβa)). (115)Since θab is a spatial tensor, projections of its Lie derivative along ua can be expressed in a similar mannerPiaPjb£uθab =1α(∂tθab −£βθab). (116)Now we present the Gauss-Codazzi and related equations in adapted coordinates:PiaPjb(dRab)= Rij + θθij − 2θikθjk − 1α(∂tθij −£βθij)− 1αDiDjα (117)Pia(dRabub)= Diθ −Djθij (118)dRabuaub =1α(∂tθ − βi∂iθ)− θijθij + 1αDiDiα (119)dR = R+ θ2 + θijθij − 2α(∂tθ − βi∂iθ)− 2αDiDiα . (120)10. Converting to ADM VariablesThe metric is often presented in the formhabdxadxb = httdt2 + 2htidtdxi + hijdxidxj . (121)We would like to relate these components to the ADM variables: the lapse function α, the shift vector βi, andthe spatial metric σij . This is a fairly straightforward exercise in linear algebra. Comparing with (105), we firstnote thatσij = hij . (122)The inverse spatial metric, σij , is literally the inverse of hij , which is not the same thing as hijσij = (σij)−1 = (hij)−1 6= hij . (123)For the shift vector we havehti = σijβj → σikhtk = σikσklβl = βi (124)⇒ βi = σijhtj . (125)Finally, for the lapse we obtainα2 = σijhtihtj − htt . (126)10

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