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Einstein-Palatini Action for General Rel- Ativity Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 16 October 2018 doi:10.20944/preprints201810.0367.v1 Peer-reviewed version available at Axioms 2019, 8, 24; doi:10.3390/axioms8010024 GENERAL RELATIVITY WITH A POSITIVE COSMOLOGICAL CONSTANT Λ AS A GAUGE THEORY MARTA DUDEK Institute of Mathematics, University of Szczecin Wielkopolska 15, 70-451 Szczecin, Poland, EU e-mail: [email protected] JANUSZ GARECKI Institute of Mathematics and Cosmology Group, University of Szczecin Wielkopolska 15, 70-451 Szczecin, Poland, EU e-mail: [email protected] In the paper we show that the general relativity action (and Lagrangian) in recent Einstein-Palatini formulation is equivalent in four dimensions to the action (and Lan- grangian) of a gauge field. We begin with a bit of information of the Einstein-Palatini (EP) action, then we present how Einstein fields equations can be derived from it. In the next section, we consider Einstein-Palatini action integral for general relativity with a positive cosmological constant Λ in terms of the corrected curvature Ωcor. We will see that in terms of Ωcor this action takes the form typical for a gauge field. Finally, we give a geometrical interpretation of the corrected curvature Ωcor. Keywords: action integral, fiber bundle, connection in a principal fiber bundle and its curvature, pull-back of forms, Lie groups and their algebras. 1 Introduction: Einstein-Palatini action for general rel- ativity In this section we would like to remind briefly Einstein- Palatini formalism for general relativity (GR). The Einstein field equations can be derived by postulating the Einstein-Hilbert action to be the true action for GR. Albert Einstein firstly used only metric as a independent variable to do variation of this action. The connection in this approach to EH action is the metric and symmetric Levi-Civita connection. Later Einstein and Palatini proposed to take the metric and affine connection as independent variables in the action princi- ple. This method allowed to compute the field equations for a more general metric affine connection rather than the Levi-Civita connection. Here the spacetime admitted torsion when matter Lagrangian explicite depended on connection.Thus, the Einstein-Palatini formalism gave us a powerful tool for theories of gravitation which have more general Riemann-Cartan geometry. In this section, we are going to present Einstein-Palatini action in recent formulation and how the Einstein fields equations can be computed from it. 1 © 2018 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 16 October 2018 doi:10.20944/preprints201810.0367.v1 Peer-reviewed version available at Axioms 2019, 8, 24; doi:10.3390/axioms8010024 The Einstein-Palatini action with cosmological constant Λ in this new formulation [3] is defined as follows 1 Z Λ S = #i ^ #j ^ Ωkl − #i ^ #j ^ #k ^ #l η ; (1) EP 4κ 6 ijkl D where Ω is the curvature of ! (spin connection) and κ = 8πG=c4. All indices take values (0; 1; 2; 3). D means an established 4-dimensional compact domain in spacetime. In the above formula #a denote 1-forms of the Lorentzian coreper in term of which the spacetime i k looks locally Minkowskian: g = ηik# ⊗ # , ηik = diag(1; −1; −1; −1). p ηijkl is completely antysymmetric Levi-Civita pseudotensor: η0123 = jgj, where g := det(gik). In a Lorentzian coreper jgj = 1. Spin connection ! is a general metric connection (or Levi-Civita connection) in Lorentzian coreper. For the geometrical units G = c = 1 the above formula has the extended form 1 Z Λ Z S = η #i ^ #j ^ Ωkl − η #i ^ #j ^ #k ^ #l: (2) EP 32π ijkl 192π ijkl D D Let us denote by S1 and S2 two integrals in the above formula of the Einstein-Palatini action and compute their variations. From the integral 1 Z 1 Z S = η #i ^ #j ^ Ωkl = Ωi ^ η j; (3) 1 32π ijkl 16π j i D D j where ηi is the 2-form defined in Appendix 1, we get 1 Z δS = δ Ωi ^ η j : (4) 1 16π j i D i j i Below we give the method of computing δ Ω j ^ ηi in Lorentzian coreper # with i spin connection ! k. i j i j i j ρδ Ω j ^ ηi = ρ δΩ j ^ ηi + Ω j ^ δηi ; (5) 1 1 where ρ denotes 2κ = 16π . Let us take the standard expressions i i i k Ω j = d! j + ! k ^ ! j (6) and j jp ηi = g ηip 1 = gjp#k ^ η 2 ipk 1 = gjp#k ^ #l ^ η 2 ipkl 1 = gjlη #m ^ #n: (7) 2 ilmn 2 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 16 October 2018 doi:10.20944/preprints201810.0367.v1 Peer-reviewed version available at Axioms 2019, 8, 24; doi:10.3390/axioms8010024 ηip, ηipk, ηipkl are forms firstly introduced by A. Trautman [11] (see Appendix 1). Then, we have i j ρδ Ω j ^ ηi = ρ = ρ δd!i + δ!i ^ !k + !i ^ δ!k ^ η j + Ωi ^ δ gjlη #m ^ #n : (8) j k j k j i 2 j ilmn il il Now we use the fact that δ and d comute and that g = η = const and ηilmn = const i in the coreper # , so the subintegral expression in δS1 has the following form i j ρδ Ω j ^ ηi h 1 i = ρ dδ!i ^ η j + δ!i ^ !k ^ η j + !i ^ δ!k ^ η j + Ωil ^ η ^ δ#m ^ #n : j i k j i k j i 2 ilmn (9) Because δ#m ^ #n = δ#m ^ #n + #m ^ δ#n (10) and i j i j i j d δ! j ^ ηi = d δ! j ^ ηi + δ! j ^ dηi ; (11) so i j ρδ Ω j ^ ηi = h i j i j i k j i k ji = ρ d δ! j ^ ηi + δ! j ^ dηi + δ! k ^ ! j ^ ηi + ! k ^ δ! j ^ ηi h1 i + ρ Ωilη ^ δ#m ^ #n + #m ^ δ#n ; (12) 2 ilmn 1 c4 where ρ = 2κ = 16πG . Next, we use two laws given below: pq !p ^ !q = (−1) !q ^ !p; (13) p d !p ^ !q = d!p ^ !q + (−1) !p ^ d!q (14) i j and we change some indices. Then the variation of ρδ Ω j ^ ηi has the form i j ρδ Ω j ^ ηi = h 1 i = ρ dδ!i ^ η j + δ!i ^ 2dη j + !j ^ η k − !k ^ η j + δ#i ^ Ωklη ^ #m j i 2 j i k i i k klim 1 = δ#i ^ ρΩklη ^ #m + δ!i ^ 2ρDη j + dρδ!i ^ η j (15) klim 2 j i j i where j j j k k j Dηi = dηi + ! k ^ ηi − ! i ^ ηk : (16) 3 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 16 October 2018 doi:10.20944/preprints201810.0367.v1 Peer-reviewed version available at Axioms 2019, 8, 24; doi:10.3390/axioms8010024 Finally, we get the following result i j i kl m i j i j ρδ Ω j ^ ηi = δ# ^ ρΩ ηklim ^ # + ρδ! j ^ Dηi + d ρδ! j ^ ηi : i kl i j = δ# ^ ρΩ ηkli + ρδ! j ^ Dηi + an exact form: (17) The second integral from the Einstein-Palatini action −Λ Z S = η #i ^ #j ^ #k ^ #l (18) 2 192π ijkl D has the following variation −Λ Z δS = 24δ#i ^ η 2 192π i D −Λ Z = δ#i ^ η : (19) 8π i D It was computed in a similar way as the variation for the first integral S1. −Λ Z δS = δ η #i ^ #j ^ #k ^ #l 2 192π ijkl D −Λ Z = η δ #i ^ #j ^ #k ^ #l (20) 192π ijkl D where δ #i ^ #j ^ #k ^ #l = = δ#i ^ #j ^ #k ^ #l + vi ^ δ#j ^ #k ^ #l + #i ^ #j ^ δ#k ^ #l + #i ^ #j ^ #k ^ δ#l = δ#i ^ #j ^ #k ^ #l − δ#j ^ #i ^ #k ^ #l + δ#k ^ #i ^ #j ^ #l − δ#l ^ #i ^ #j ^ #k (21) and i j k l i j k l δ # ^ # ^ # ^ # ηijkl = 4δ# ^ ηijkl# ^ # ^ # : (22) Thus, the variation δS2 finally equals −Λ Z δS = 4δ#i ^ η #j ^ #k ^ #l (23) 2 192π ijkl D j k l The expression ηijkl# ^ # ^ # is equal 6ηi (see Appendix), so using this substitution we obtain as follows −Λ Z δS = 24δ#i ^ η 2 192π i D −Λ Z = δ#i ^ η (24) 8π i D 4 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 16 October 2018 doi:10.20944/preprints201810.0367.v1 Peer-reviewed version available at Axioms 2019, 8, 24; doi:10.3390/axioms8010024 Joining both variations δS1, δS2 with the third variation δSm of the matter action R A A i Sm = Lmat(φ ; Dφ ;# ), which we write in the form [11] D Z A A i δSm = δ Lmat(φ ; Dφ ;# ) D Z 1 = δ#i ^ t + δ!i ^ sj + δφA ^ L + an exact form ; (25) i 2 j i A D we obtain for total action, gravity and matter δS = δS1 + δS2 + δSm Z h 1 1 i = δ#i ^ Ωkl ^ η − Λη + 8πt + an exact form 8π 2 kli i i D Z h1 1 i + δ!i ^ Dηj + sj + δφA ^ L + an exact form : (26) 2 j 8π i i A D i 1 i j A A The form of the variation δLm = δ# ^ ti + 2 δ! j ^ si + δφ ^ L + an exact form j A defines the 3-forms ti, si , L of the energy-momentum, classical spin and the left hand side of the equations of motion for matter respectively. In the above formulas φA means tensor-valued matter form and DφA its absolute exterior derivative.
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