Spherically symmetric solution in Møller gravity Edward Rakhmetov1 Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University PoS(QFTHEP 2013)092 1(2), Leninskie gory, GSP-1, Moscow 119991, Russian Federation E-mail: [email protected] Sergey Keyzerov Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University 1(2), Leninskie gory, GSP-1, Moscow 119991, Russian Federation E-mail: [email protected] In this paper, following C. Møller [1,2], we obtain spherically symmetric solution in the tetrad Møller gravity. We use this solution and the so-called PPN-formalism [3,5] to compare our theory with experiment and to improve restrictions on the theory constants. The XXI International Workshop High Energy Physics and Quantum Field Theory June 23 – June 30, 2013 Saint Petersburg Area, Russia 1 Speaker Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it Spherically symmetric solution… Edward Rakhmetov 1. Introduction Spherically symmetric solutions are very important to understand the slow-motion and weak-field limit of the gravity theory under study and to confront this theory with Solar System experiments [4]. For example, the three classical tests of General Relativity are based on the spherically symmetric Schwarzschild solution and the weak-field limit. In this paper we consider a spherically symmetric solution, which can be obtained in the interesting generalization of General Relativity suggested by C. Møller [1]. It is convenient to use in such analyses so-called PPN-formalism, specially built to study the slow-motion and PoS(QFTHEP 2013)092 weak-field limit of the extended theories of gravity [3]. Save for this little introduction and a short conclusion, this paper is organized as follows. In § 2, we give a brief review of the Møller gravity. In § 3, we obtain the spherically symmetric solution. In § 4, we gain restrictions on the theory parameters, using PPN-formalism and data of Solar System experiments. 2. Møller gravity review It was C. Møller, who first put forward this theory in 1978 [1]. The Møller gravity theory is a metric theory, in which the metric tensor 3 ≡ δ αα α α ≡ α α gµν∑ ()()()()() g µ g ν g µ g ν , δ( αβ ) ≡diag { − 1,1,1,1} (2.1) α=0 is constructed from an orthonormal tetrad: α βµ ≡ δ αβ g()()()µ g (2.2) Here the indices in the brackets are frame (vielbein) indices, which run from 0 to 3, and the summation over the repeated indices is implied. The stress tensor for this vector fields (denoted α by f ()µν ) is, as usually: α ≡ ∂ α f()()µν[] µ g ν (2.3) Where the antisymmetrization over the indices in the square brackets is implied. The coordinate indices can be turned into frame (vielbein) indices as it is presented below: α α ≡ Cµν()() g λ C µνλ (2.4) We can obtain the Riche tensor from the stress tensor: 1 1 1 R()()()()()()()αβ= ∂α f µµβ+ ∂β f µµα+ ∂µ f βαµ 2 2 2 1 1 1 + ∂µ()()()()()()f αβµ− f αβµ f ννµ− f βαµ f ννµ+ (2.5) 2 2 2 1 1 1 −f()()()()()() µνα f νµβ− f µνα f µνβ+ f αµν f βµν 2 2 4 ∂ α ≡ αµ ∂ Where ()()g µ (2.6) Contacting indices in different ways, we can build from the stress tensor three different scalars: 2 Spherically symmetric solution… Edward Rakhmetov αβγ βαγ α βγ ≡ ≡ ≡ (2.7) L1 fαβγ f L2 fαβγ f L3 fαγ fβ Then, in general case, the simplest action quadratic in the partial derivatives is: =( + + + ) S∫ k0 k 1 L 1 k 2 L 2 k 3 L 3 gdx (2.8) X where k0 , k1 , k2 , k3 are arbitrary dimensional constants. Using (2.5) we can obtain more usual expression for Einstein-Møller action (up to a complete divergence): 1 S={ Λ + R + k′′ L + k L} gdx PoS(QFTHEP 2013)092 κ ∫ 1 1 2 2 2 X (2.9) We could not write the action with the terms L1 and L2, if we used just the metric tensor without the vielbein formalism. Thus, we have a generalization of metric gravity. ′ ′ As we can see, Møller gravity theory coincides with General Relativity, if k1 , k2 is equal to 0. Denoting the variation of the action with respect to g()µ α as α 1 δS X ()µ ≡ , (2.10) g δg() µ α we can write the symmetric and the antisymmetric parts of the equations of motion δ µα ≡1 S = X ( ) 0 separately. The symmetric part gives: g δg() µ α ()()[]αβ=− αβαβ −1 ( +Λ++∇) ( ′′′′) αβµ −−( ) µναβ + X4 kRgR3 ÷ 2 2 kkf 1 2µ 2 kkff 1 2 [ µν ] 2 (2.11) +( ′′′′′ +) µναβ() − α βµν + αβ µνλ + µνλ = 2kkff1 2µν 2 kff 2 µν 2 gkff( 1µνλ kff 2 νµλ ) 0 The antisymmetric part of the equations of motion is [][][]αβ=( ′′′′′ −) ∇ αβµ − ∇ µαβ −( −) µναβ = X2 2 k1 k 2µ f 4 k 2 µ f 2 k 1 3 k 2 f f µν 0 (2.12) ′ ′ If k1 , k2 are equal to zero, the antisymmetric part vanishes. The symmetric part of the equations of motion gives us General Relativity. As we can see, in the Møller gravity theory we have more restrictions on frame vectors, than in General Relativity, because of the additional antisymmetric part of the equations of motion. And now the action of the theory is not invariant under the rotations of the frame, unlike in General Relativity. 3.Spherically symmetric solution Let us write the required spherically symmetric metric using isotropic coordinates [4]: 22γ()()r 2 2 α r 2 2 2 2 2 ds= e dt − e{ dr + r( d θ +sin θ d ϕ )} (3.1) Then the metric tensor is: γ e2 0 g = ÷ αβ 2α ÷ (3.2) 0 −e g pq 3 Spherically symmetric solution… Edward Rakhmetov It is convenient to write an ansatz for the frame: = γ 0 = −γ = = α q= − −α q g(0)0 e , g(0) e , g( a )0 0 , g()() aq e g a q , g()() a e g a (3.3) Where cosθ −sin θ cos ϕ −sin θ sin ϕ ÷ ÷ ÷ = − θ = − θ ϕ = − θ ϕ g(1)q r sin ÷, g(2)q r cos cos ÷, g(3)q r cos sin ÷ (3.4) ÷ ÷ ÷ 0 r sinθ sin ϕ −r sin θ cos ϕ is an orthonormal frame in 3-dimensional flat space. Using (3.1)-(3.4) and the equations of motion, after some calculations we obtain three PoS(QFTHEP 2013)092 equations for two functions parameterizing the metric and the frame: α+α+2− 1 α=ξα−γ−γ−αγ− 2 2 − 1 γ 2,,,,,,,,,rrr 4r r( 2 rrrr 2 2 rr 4 r r ) (3.5) α2 +− 1 α + γ α + − 1 γ = −ξ α 2 + γ 2 + − 1 α ,,,,,,,,r2r r 2 r r 2 r r( 2 r r 4 r r ) (3.6) α+−1 α+ − 1 γ+γ+γ=ξ−α+γ−αγ− 2 2 − 1 α ,,,,,,,,,,rrr r r rrrr( 2 rrr 2 rr 2 r r ) (3.7) ξ ≡′′ + Where 2k1 k 2 (3.8) Obviously, this system is overdetermined. Nevertheless, it has a solution in the form of α =( −) +( − ) Aln 1 r0 / r B ln 1 r 1 / r (3.9) γ =( −) +( − ) Cln 1 r0 / r D ln 1 r 1 / r (3.10) Where 1− ξ +( 1 − 2 ξ) ( 1 + ξ) 1− ξ −( 1 − 2 ξ) ( 1 + ξ) A = , B = , 1− 3 ξ 1− 3 ξ 2ξ +( 1 − 2 ξ) ( 1 + ξ) ( 1− 2 ξ) ( 1 + ξ) − 2 ξ C = − , D = (3.11) 1− 3 ξ 1− 3 ξ 1 ( 1− 2 ξ) ( 1+ ξ) 1 ( 1− 2 ξ) ( 1+ ξ) r=ξ − ÷ r r=ξ + ÷ r 0 + ξ ÷ , 1 + ξ ÷ (3.12) 1 2 2 1 2 2 And where r is the Schwarzschild radius. Thus the spherically symmetric metric looks like: 2=( −) 2CDAB( −) 2 2 −( −) 2( −) 2 2 + 2 + 2 ds1 r0 / r 1 r 1 / r dt 1 r 0 / r 1 r 1 / r( dx dy dz ) (3.13) As we can see, the spherically symmetric solution coincides with Schwarzschild metric not only ′ ′ in the trivial case, when our theory coincides with General Relativity and k1 , k2 are equal to ξ ≡′′ + = zero separately, but in the case when 2k1 k 2 0 as well. Up to the second order in inverse radius we obtain for the space-time metric r1+ ξ r2 1 r 1 + ξ r 2 ds2=1 − + ÷ dt 2 − 1 + + ÷( dx 2 + dy 2 + dz 2 ) (3.14) r 2(1+ 2 ξ )r2 1 + 2 ξ r 4(1 + 2 ξ ) r 2 It is convenient to use this expression in the so-called PPN-formalism to compare our theory with experiment. 4 Spherically symmetric solution… Edward Rakhmetov 4.Spherically symmetric solution and the PPN-formalism The main idea of the parameterized post-Newtonian formalism is that the comparison of metric theories of gravity with experiment becomes simple when we use the weak-field limit. This approximation is known as the post-Newtonian limit [3,4]. It is sufficiently accurate to the most solar-system tests [5]. In this approximation the space-time metric predicted by nearly every metric theory of gravity has the same structure. The space-time metric can be written as an expansion about the Minkowski metric in terms of dimensionless gravitational potentials of varying degrees of smallness: = − + β2 + = −( + γ) δ + PoS(QFTHEP 2013)092 g00 1 2 U 2 U ..