Ricci Tensor of Diagonal Metric

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1 University of Massachusetts - Amherst ScholarWorks@UMass Amherst Astronomy Department Faculty Publication Series Astronomy 1996 Ricci Tensor of Diaonal Metric K. Z. Win University of Massachusetts - Amherst Win, K. Z., "Ricci Tensor of Diaonal Metric" Astronomy Department Faculty Publication Series. Paper This Article is brouht to you for free and open access by the Astronomy at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Astronomy Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact scholarworks@library.umass.edu.

2 February 1996 Ricci Tensor of Diaonal Metric arxiv:r-qc/ v1 8 Feb 1996 K. Z. Win Department of Physics and Astronomy University of Massachusetts Amherst, MA Abstract Efficient formulae of Ricci tensor for an arbitrary diaonal metric are presented.

3 Introduction Calulation of the Ricci tensor is often a cumbersome task. In this note useful formulae of the Ricci tensor are presented in equations 1 and 2 for the case of the diaonal metric tensor. Application of the formulae in computin the Ricci tensor of n-sphere is also presented. Derivation The sin conventions and notation of Wald [1 will be followed. Latin indices are part of abstract index notation and Greek indices denote basis components. The Riemann curvature tensor R abc d is defined by a b ω c b a ω c = R abc d ω d. The Ricci tensor in a coordinate basis is R λν = ρ Γ ρ λν λγ ρ νρ Γσ λρ Γρ σν + Γρ λν Γσ ρσ where the Christoffel symbols are Γ ρ λν = ρσ λ νσ + ν λσ σ λν /2. For the rest of this note we assume that ρν is diaonal unless otherwise indicated. Also to avoid havin to write down µ ν repeatedly we will always take µ ν and no sum will be assumed on repeated µ and ν. We first note that Γ ρ µν = 0 if µ ρ ν. It means that at least two indices of Γ must be the same for it to be nonvanishin. Also ln = n σ=1 ln σσ where and n are determinant of the metric and dimension of manifold, respectively. Then it is easy to show that 2Γ µ σµ = µµ for all σ and that 2Γ ν µµ = νν ν µµ. Aain µ ν and no sum on µ and ν. These two equations ive 4Γ ν µµ Γµ νν = νν ν µµ µµ µ νν = µ ln νν µ ln νν = 4Γ µ µν Γν µν Usin above facts, we et, with each sum havin upper limit n and lower limit 1, R µν = µ Γ µ µν + νγ ν µν 1/2 µ ν ln Γ µ µρ Γρ µν Γν µρ Γρ νν + 1/2Γ µ µν µ ln + 1/2Γ ν µν ν ln Γ σ µρ Γρ σν =1/2 µ ν ln µµ νν Γ µ µµ Γµ µν Γµ µν Γν µν Γν µµ Γµ νν Γν νµ Γν νν + Γ µ µν µ ln µµ + Γ µ µν µ ln νν + Γ ν µν ν ln µµ + Γ ν µν ν ln νν + 1/2Γ µ µν µ ln + 1/2Γ ν µµ µν ν ln νν µµ νν Γ σ σµ Γσ σν 1

4 =1/2 µ ν ln µµ νν Γ µ µµ Γµ µν Γµ µν Γν µν Γν µν Γµ µν Γν νµ Γν νν Γ σ σµ Γσ σν + Γ µ µµ Γµ µν + Γµ µν Γν µν + Γν µν Γµ µν + Γν νµ Γν νν + 1/2Γ µ µν µ ln + 1/2Γ ν µµ µν ν ln νν µµ νν =1/2 µ ν ln µµ νν Γ σ σµγ σ σν Γµ µν µ ln + 1 µµ νν 2 Γν µν ν ln µµ νν or remember µ ν 4R µν = µ ln νν µ ν ln + µ ν µµ νν µ ln σσ ν ln σσ 1 where µ ν stands for precedin terms with µ and ν interchaned.* A number of useful facts follow from the above formula. First of all R µν = 0 if x µ is an inorable coordinate because µ is in each term of the formula. If there are m inorable coordinates then maximum number of nonzero off-diaonal components of the Ricci tensor is n mn m 1/2. Thus a sufficient condition for the Ricci tensor to be diaonal is that the diaonal metric be a function only of one coordinate. The Robertson-Walker metric with flat spatial sections, ds 2 = dt 2 + at 2 dx 2 + dy 2 + dz 2, satisfies this condition and its Ricci tensor is consequently diaonal. If the diaonal metric is a function only of two coordinates, say x µ and x ν, then all off diaonal components of the Ricci tensor except R µν are zero. For example, for Schwartzschild metric ds 2 = Brdt 2 + Ardr 2 + r 2 dθ 2 + sin 2 θdφ 2 only R θr needs to be explicitly calculated, all other bein zero. It is easily checked usin equation 1 that R θr is also zero. Symmetry aruments can also be iven as to why the Ricci tensor is diaonal in this case. Secondly second derivative terms are identically zero whenever each component of the metric is a product of functions of one sinle coordinate i.e. if, for all µ, µµ = n l=1 fl µ x l where some of f l µ * Alternative forms of equation 1 are may be one. A final trivial corollary 4R µν = [ µ ln νν µ ν ln σσ + µ ν µ ln σσ ν ln σσ and 8R µν = [ µ ln 2 νν σσ 2 µ ν ln σσ + µ ν Pae 178 of Weinber [2. 2

5 of equation 1 is that the Ricci tensor is diaonal in 2-dimensions. This fact also follows trivially from the fact that in 2-dimensions, the Ricci tensor is the metric tensor not necessarily diaonal up to a factor of a scalar function. When n = 3, it is easy to find an example of a diaonal metric which results in a non-diaonal Ricci tensor. For example, ds 2 = dx 2 + xdy 2 + ydz 2 ives 4R xy = 1/xy. We now calculate the diaonal components of Ricci tensor. We have R µµ = σ Γ σ µµ 2 µ ln Γ σ µρ Γρ σµ + Γρ µµ ρ ln = 2 µ ln µµ + σ Γ σ µµ 2 µ ln Γ µ µρ Γρ µµ or Γ σ µρ Γρ µσ + Γµ µµ µ ln + Γ σ µµ =1/2 µ 2 ln µµ + σ Γ σ µµ Γ µ 2 µµ Γ µ µσγ σ µµ Γ σ σµγ σ σµ + Γ µ µµ µ ln µµ + Γ µ µµ µ ln Γ σ µµγ µ σµ Γ σ µµ + µµ [ σ Γ σ µµ 2Γσ µµ Γµ µσ Γ σ 2 σµ =1/4 µ ln µµ 2 µ µ ln + µµ + Γ σ µµ =1/4 µ ln µµ 2 µ µ ln + [ 1/2 σ σσ σ µµ µµ + σσ σ µµ µµ = 1 4 µ ln µµ 2 µ µ ln 1 µµ 4 + σσ σ µµ µµ 2 µ ln 2 σσ σσ /2 σ µµ [ 2 σ σσ σ µµ + µ ln σσ 2 4R µµ = µ ln µµ 2 µ µ ln [ µ ln σσ 2 + µµ 2 µµ σσ σ µµ where 2 σ acts everythin on its riht. The efficiency of this formula is obvious for Schwartzschild metric iven earlier. To compute R tt we note that all terms with t are zero. 3 2

6 Thus 4R tt = σ t 2 tt σσ σ tt. Because tt = Br only the σ = r A A + B B + B 2A term survives and we et, after very little computation, R tt = B Ar B 4A where prime indicates differentiation with respect to r. Other diaonal components are just as easy to compute. Application to n-sphere To illustrate the use of above formulae we compute Ricci tensor of n-sphere which in spherical coordinates has diaonal metric 11 = 1 and µ+1,µ+1 = µµ sin 2 x µ for 1 µ n 1. First note that ln µµ = µ 1 ν=1 2 ln sinx ν and thus ρ µµ = 0 if n ρ µ 1. These facts will be used repeatedly in what follows. We first compute the off diaonal components of Ricci tensor. Accordin to the eneral arument iven earlier second derivative terms will be zero. Let µ > ν. Then equation 1 becomes 4R µν = σ>µ [ ν ln µµ µ ln σσ µ ln σσ ν ln σσ =4 σ>µcot x ν cotx µ cot x µ cotx ν = 0 We next compute the diaonal components. We have, from equation 2, 4R µµ = [ 2 µ 2 ln σσ + µ ln σσ 2 2 σσ σ µµ σ>µ σ<µ µµ = 2 2 µ cotx µ 4 cot 2 x µ µ 1 ln µ 1 µ 1,µ 1 µ 1 µµ σ>µ µµ 4 sinx µ 1 σσ σ µ 1,µ 1 sin 2 x µ 1 = 4 sin 2 4 cos2 x µ x σ>µ µ sin 2 x µ [ µ 1 ln σσ 2 µ 1 ln µµ + 2 µ 1 µ 1,µ 1 µ 1 µµ σ>µ 1 sin 2 x µ 1 σ ln σσ σ µ 1,µ 1 Cf. pae 178 of Weinber and note that his R abc d is minus of Wald s. Also see problem 6.2 of [1. 4

7 [ 2 =4n µ cotxµ 1 sin 2 x µ 1 σ>µ 1 4 cotxµ µ 1 µ 1,µ 1 µ 1,µ 1 sin 2x µ 1 σσ σ µ 1,µ 1 =4n µ [2 cotx µ 1 n + 1 µ 4 cotx µ 1 sin 2x µ cos2x µ 1 sin 2 x µ 1 σ ln σσ σ µ 1,µ 1 =4n µ 4 [ cos 2 x µ 1 n µ cos 2 x µ 1 + cos 2 x µ 1 sin 2 x µ 1 sin 2 x µ 1 σ ln σσ σ µ 1,µ 1 =4 [ n µ cos 2 x µ 1 n µ sin 2 x µ 1 σ ln =sin 2 x µ 1 [ 4n µ + 1 σσ σ µ 1,µ 1 σσ σ µ 1,µ 1 Thus [ 4R µ+1,µ+1 = sin 2 x µ 4n µ 2 σ<µ µµ σσ σ µµ We now rewrite the expression inside the square brackets. Break up the sum into σ = µ 1 and σ < µ 1 and use µµ = µ 1,µ 1 sin 2 x µ 1. Then [ =4n µ µ 1 ln 2 µµ + 2 µ 1 µ 1,µ 1 µ 1 µµ 4 sinx µ 1 σσ σ µµ =4n µ µ 1 ln 2 µ 1 ln µµ + 2 µ 1 µ 1,µ 1 µ 1,µ 1 sin 2x µ 1 σσ σ µ 1,µ 1 sin 2 x µ 1 [ =4n µ σ>µ 1 sin 2 x µ 1 µ 1 ln σσ 2 2 cotx µ µ 1 sin 2x µ 1 σσ σ µ 1,µ 1 5

8 =4n µ [2 cotx µ 1 n µ cotx µ 1 sin 2x µ 1 4 cos 2x µ 1 sin 2 x µ 1 σσ σ µ 1,µ 1 =4n µ 4 [cos 2 x µ 1 n µ cos 2 x µ 1 + cos 2 x µ 1 sin 2 x µ 1 sin 2 x µ 1 σσ σ µ 1,µ 1 =4 [n µ cos 2 x µ 1 n + 1 µ + sin 2 x µ 1 + cos 2 x µ 1 sin 2 x µ 1 σσ σ µ 1,µ 1 =4n + 1 µ sin 2 x µ 1 sin 2 x µ 1 =4R µµ σσ σ µ 1,µ 1 Next 4R 11 = [ σ> ln σσ + µ ln σσ 2 1 = 4 sin 2 x 1 cot 2 x 1 σ>1 = 4n 1 All precedin calculations amount to provin that R ρµ = n 1 ρµ. Let {va ρ } be the dual basis of the tanent space where ρ labels a particular basis vector. Then R ab = R ρσ v ρ a vσ b = n 1 ρσv ρ a vσ b = n 1 ab This fact about n-sphere is also provable usin the machinery of symmetric spaces. See chapter 13 of [2. Equations 1 and 2 were derived when attempt was made to establish any connection between diaonality of R νσ and that of σν. Finally we remark that Dinle [3 has listed values of T σ ν for the case of 4-dimensional diaonal metric. Acknowledements I would like to thank Professor Jennie Traschen for much encouraement and discussion and for useful comments on the manuscript. 6

9 References 1. Wald, R. M. 1984, General Relativity Chicao: University of Chicao Press. 2. Weinber, S. 1972, Gravitation and Cosmoloy: Principles and Applications of The General Theory of Relativity New York: Wiley. 3. Dinle, H. 1933, Values of Tµ ν and the Christoffel Symbols for a Line Element of Considerable Generality, Proc. Nat. Acad. Sci., 19,