CALCULATING THE CURVATURE TENSOR FOR THE 3D GODEL-LIKE SPACETIMES

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1 CALCULATING TE CURVATURE TENSOR FOR TE 3 GOEL-LIKE SPACETIMES Abstract. We compute the Riemann curvature tensor for these 3 spacetimes. As the Weyl tensor vanishes the Ricci tensor is the only relevant tensor for these spaces, for this reason we contract and produce the non-vanishing Ricci tensor components and the Ricci Scalar. Using the Lorentz group we may transform the Ricci tensor to its simplest form; this is necessary to begin the Cartan-Karlhede equivalence algorithm. The coframe and its dual frame, and the connection coefficients To begin we write down the coframe of one-forms, by defining F ± = ± : m µ = θ 1 µ = dr, n µ = θ µ = 1 (dt + F dφ), l µ = θ 3 µ = 1 (dt + F +dφ). Although we have not calculated the dual coframe, we may define it due to the properties with a dual frame basis, Notice that this takes the metric ẽ ν αθ α = ẽ ν αe α µdx µ = δ ν µdx µ = dx µ g µ ν = l ( µ n ν) + m µ m ν = θ 3 ( µ θ ν) + θ1 µθ 1 ν Thus g(e ) = θ 3 and g(e 3 ) = θ, meaning that if one has a or 3 index on the top of a tensor when we apply the metric to lower the index it becomes a 3 or respectively and the term is multiplied by a negative, i.e., A A 3, A 3 A where A is some arbitrary one-form relative to the coframe basis. The connection coefficients relative to this particular coframe are simply: Γ 11 = Γ 113 = Γ 33 = Γ 3 = 0 Γ 1 = F +, Γ 331 = F Γ 31 =, Γ 13 =, Γ 31 = With these connection coefficients we may continue to calculate the curvature twoform by combining these with θ α s to make the connection one-forms ω α β : 1

2 CALCULATING TE CURVATURE TENSOR FOR TE 3 GOEL-LIKE SPACETIMES ω 1 = ω 1 = F + θ + θ3, ω 1 3 = ω 13 = θ + F θ3, ω 3 3 = ω 3 = θ1. Notice that ω is not needed as ω = ω 3 = ω 3 = ω 3 3. The Riemann and Ricci tensor components We may calculate the components of the Riemann tensor, using the second Cartan structure equation dω α β + ω α γω γ β = Rα β where R α β = Rα βγδ θγ θ δ is defined as the Curvature two-form. Taking the connection 1-forms above we apply the exterior derivative, dω 1 = + dω 1 3 = + dω 3 3 = 0. ( F ) + F F + 4 F ] + 4 θ 1 θ ( ) F +F θ 1 θ 3 ( ) F +F θ 1 θ ( F ) + F 4 + F 4 + F ] 4 θ 1 θ 3 Alternatively all non-vanishing wedge products of the connection 1-forms yields: ω 1 ω = ω 1 ω 3 = F + 4 θ1 θ 4 θ1 θ 3 ω 1 3 ω 3 3 = ω 13 ω 3 = 4 θ1 θ + F 4 θ1 θ 3 ) ] ω 3 1 ω 1 3 = ω 1 ω 13 = F +F 4 ( θ θ 3. Adding these sums together in the appropriate manner, the Riemann tensor components are read off of the coefficients of the basis θ α θ β

3 CALCULATING TE CURVATURE TENSOR FOR TE 3 GOEL-LIKE SPACETIMES 3 R 1 = 1 F + θ 1 θ θ 1 θ 3 R 1 3 = 1 4 θ 1 θ + 1 F + θ 1 θ 3 R 3 3 ( = 1 ) ] θ θ 3. 4 Thus the non-vanishing and algebraically independent components of the Riemann tensor, relative to the coframe, will be R 1 1, R 1 13, R and R To build the Ricci tensor we contract along the first and third index of the Riemann tensor Ric αβ = R γ αγβ Computing (αβ) = {(1, 1), (1, ), (1, 3), (, ), (, 3), (3, 3)} exhausts all algebraically independent components of the tensor, from this list the non-zero components are: Ric 11 = R 113 = 1 Ric = R 11 = 1 F + Ric 3 = R 113 R 33 = 1 ] Ric 33 = R 1313 = 1 F + where we have lowered the upper-index in the Riemann tensor to exploit the antisymmetry of the first pair of the indices. There is one more component that is relevant to the construction of the Riemann tensor, and it comes from contracting the Ricci tensor, R = Ric α α - the Ricci scalar: R = Ric 11 Ric 3 = 1 4

4 4 CALCULATING TE CURVATURE TENSOR FOR TE 3 GOEL-LIKE SPACETIMES To bring this in line with the quantities defined in 1, ] we note Φ 00 = 1 Ric abl a l b = Ric. Φ = 1 Ric abn a n b = Ric 33. Φ 10 = 1 Ric ab m a l b = Ric 1. Φ 1 = 1 Ric ab m a n b = Ric 13. Φ 11 = 1 6 (Ric abm a m b + Ric ab n a l b ) = 1 6 (Ric 11 + Ric 3 ). Λ = R For the Godel-like spacetimes we find that Φ 10 = Φ 1 = 0 and the non-zero components are: Φ 00 = 1 F + 4 Φ = 1 F 4 + Φ 11 = 1 1 Λ = 1 4 To normalize the components of the Ricci tensor we must use the Lorentz freedom to bring the components in line with the table in 1] on page 7. To summarize the frame freedoms we have a boost: θ 1 = θ 1, θ = Aθ, θ3 = 1 A θ 3, and null rotations about n a and l a : θ 1 + Bθ, θ = θ, θ3 = θ 3 + Bθ 1 + B θ, θ 1 + Cθ 3, θ = θ + Cθ 1 + C θ3, θ3 = θ 3. Questions to Ask at the First Step of the C.K. algorithm Prior to the canonical form, answer these questions: (1) Give me an argument why our null rotations about n a and l a have already been exhausted. That is, why does B = C = 0 for null rotations produce part of the canonical form. () When is Φ 00 = Φ prior to making a boost? Give a differential equation in terms of and. (3) When does Φ 00 or Φ vanish? If Φ = 0 can we change our coframe so that Φ 00 Φ in the new coframe?

5 CALCULATING TE CURVATURE TENSOR FOR TE 3 GOEL-LIKE SPACETIMES 5 (4) Assuming this is the case, given a parameter for the boost that will set Φ = 1. (5) When Φ 00 Φ what do we choose for the Lorentz boost parameter A so that in the new coframe Φ 00 = Φ? (6) When do both Φ 00 and Φ both vanish so that boosts have no effect? After we have used as much of the frame freedom as possible we will have two cases Φ 00 = Φ or Φ 00 = 0 and Φ = 1 and two more sets of questions to answer: Case 1: Φ 00 = Φ. (1) When is Φ 00 = Φ = 3Φ 11 with Φ 00 0? What is the Segre type in this case? () When is Φ 00 = Φ 0 and Φ 11 Φ 00? What is the Segre type in this case? (3) When is Φ 00 = Φ = 0 with Φ 11 0? What is the Segre type in this case? (4) When is Φ 00 = Φ = Φ 11 = 0? What is the Segre type in this case? Case : Φ 00 = 0, Φ = 1. (1) When is Φ 11 0? What is the Segre type in this case? () When is Φ 11 = 0? What is the Segre type in this case? References 1] F.C. Sousa, J.B. Fonseca, and C. Romero, Equivalence of three dimensional spacetimes, Class Quant Grav (008). ] G.S. all, T. Morgan, and Z. Perjes Three-imensional Space-Times, Gen. Rel. Grav (1987).