PoS(QFTHEP 2013)092 sa.it http://pos.sis ] to compare our theory 3,5 ], we obtain spherically symmetric solution in the tetrad 1,2 1 Speaker [email protected] [email protected] In this paper, following C. Møller [ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeCopyright the Creative Licence. of the author(s) owned by the under terms Commons

 The XXI International Workshop High Energy Physics International Field Theory Workshop The XXI and Quantum 30, 2013 June 23 – June Petersburg Russia Area, Saint 1 Møller gravity. We use this solution and the so-called PPN-formalism [ and theory on the with constants. to restrictions experiment improve Sergey Keyzerov Sergey State Institute University of Physics, Moscow Nuclear Lomonosov Skobeltsyn M.V. Leninskie GSP-1, 119991, Russian Federation 1(2), gory, Moscow E-mail: Edward Edward Rakhmetov State Institute University of Moscow Physics, Nuclear Lomonosov Skobeltsyn M.V. Leninskie GSP-1, 119991, Russian Federation Moscow 1(2), gory, E-mail: Spherically symmetric solution solution symmetric Spherically Møllergravity in PoS(QFTHEP 2013)092 (2.1) (2.2) (2.5) (2.6) (2.4) (2.3) Edward Rakhmetov It is convenient to ]. 1 spherically symmetric organized as follows. ]. The Møller gravity theory

1 which can be obtained in the }

{ indices, which run from 0 to 3, and the . ] diag In § 3, we obtain the 3 [ and to confront this theory with Solar System 2 . The tensor stress for this vector fields (denoted this theory in 1978 [

( ) 1,1,1,1 δ αβ ≡ − (vielbein) , is implied put forward a spherically symmetric solution, . brackets are frame

orthonormal tetrad: orthonormal µ

the

an µ the gravity theory under study

the extended theories of theextended gravitytheories of a brief review of the Møller gravity. ]. For example, the three classical tests of General Relativity are based on the g 4 is, as usually:as is, µ [] ) ()()()()() 2 2 2 1 1 1 In § 4, we gain restrictions on the theory parameters, using PPN-formalism and data of f f f f f ()() µν 0 ∂ α ≡ α ∂ 3 ∑ α= ()()()()()() α µν µ ν It was C. Møller, who first In this paper we consider Save for this little introduction and a short conclusion, this paper is Spherically symmetric solutions are very important to understand the slow-motion and µ ()()()()()() () α α ≡ f f f f f f ≡ δ αα α α ≡ α α ()() f α ≡ ∂ α αβ= ∂α µµβ+ ∂β µµα+ ∂µ βαµ α β ≡ δ αβ Møller gravity review Møller Introduction 2 2 2 2 2 4 1 1 1 1 1 1 ()() ()()()()()()()

µν λ µνλ ()()() µν µ ν µ ν f g g g g g g g g R f f f − µνα νµβ− µνα µνβ+ αµν βµν + ∂µ αβµ− αβµ ννµ− βαµ ννµ+ C g C Where Where scalars: three different tensor the stressdifferent build inwe can ways,from indicesContacting We can obtain the Riche tensor from the stress tensor: thestress from obtain canWe the tensor Riche by Where the antisymmetrization over the indices in the square brackets is implied. The coordinate it below: asis presented turned indices into be(vielbein) frame can indices is constructed from is constructed Here the indices in summation theover indices repeated is a metric theory, in which the metric themetric metric tensor in which is theory, a In § 2, we give solution. experiments Solar System 2. interesting generalization of General Relativity suggested by C. Møller [ use in such analyses so-called PPN-formalism, specially built tooflimit weak-field study the slow-motion and 1. weak-field limit of experiments [ andsolution symmetriclimit. the Schwarzschildweak-field spherically Spherically symmetric solution… symmetric Spherically PoS(QFTHEP 2013)092 3.1) (2.7) (2.8) 3.2) 2.12) (2.10) ( ]: of the 4 (2.11) is equal to 0. ′ 2 k , Edward Rakhmetov ′ 1 k ) ( ) (

as α ( µνλ νµλ ) , if we used just the metric tensor without µ 2 } () ) L g µ µν

3 ) and 1 (

L ( ′′′′

sin β ( The symmetric gives: The part αγ

α βγ ( ) ≡ 3 L f f } ) (

separately. separately. βαγ {

gives us General Relativity. As we can see, in the Møller gravity theory µ µ µν , , are arbitrary dimensional constants. dimensional are arbitrary pq 2 1 µν µν αβγ

′′ 3 α α 1 1 2 2 ) 0 k α ≡ S S 2 , () () 2 δ δ e g () 2 g g L f f − k µναβ α βµν αβ µνλ µνλ δ µ δ µ   ÷  ′′′′′ 1 2 2 1 2 , 3 1 2 1 2 [ ] 1 g g ) αβγ γ 1 1 r r are equal to zero, the antisymmetric part vanishes. The symmetric part k 4 2 2 2 2 ( ()() 0

{ , e γ α 2 2 4 2 3 0 ′ 2 0 1 1 2 2 3 3 ∫ X 2 2 0   ÷ ÷  k =− − +Λ++∇ −− + k = − ∇ − ∇ − − = αβγ κ ( α α ′′′′′ = Let us write the required spherically symmetric metric using isotropic coordinates symmetric coordinates [ write metric spherically using the Let usrequired isotropic 1 2 2 1 2 1 2 = − + θ + θ ϕ X ∫ , kkff kff gkff kff ′ µ ≡ µ ≡ = αβ αβαβ αβµ µναβ αβ αβµ µαβ µναβ 1 2 2 2 0 ≡ 2 2 2 2 2 2 2 ()()[] () ( ) 0 [][][] = Λ + + + = + + + Spherically symmetric solution Spherically symmetric k ( αβ 1 X k k f k f k k f f X kRgR kkf kkff X X g L f f S R k L k L gdx S k k L k L k L gdx ds e dt e dr r d d + + − + + = Then the metric tensor is: metric thetensor Then under the rotations of the frame, unlike in General Relativity. frame,unlikeof in General the under rotations the 3. If equations of motion of General than inRelativity, framevectors,the additional restrictions because on more we have antisymmetric part of the equations of motion. And now the action of the theory is not invariant The antisymmetric part of the equations of motionantisymmetricof theequations part is The of we can write the symmetric and the antisymmetric parts of the equations of motion We could not write the action with the terms generalization metric ofgravity. a Thus, we the have vielbein formalism. As we can see, Møller gravity theory coincides with General Relativity, if theaction of to with respect Denoting the variation Using (2.5) we can obtain more usual expression for Einstein-Møller action (up to a complete divergence): (2.9) Then, in general case, the simplest action quadratic in the partial derivatives simplest the in the quadratic partial case, is: action in general Then, where Spherically symmetric solution… symmetric Spherically PoS(QFTHEP 2013)092 ) 3.6) 3.5) 3.7) 3.3) ( ( 3.14 ( 3.4) ( 3.13) 3.12) ( ( 3.11) ( ( 3.9) 3.10) ( 3.8) ( ( are equal to ′ 2

k ) ( , ′ ) Edward Rakhmetov 1 2 k

)  ) ξ

( + −α ) (

( ) ) ) 2 2 = − r sin cos sin sin q q r , − θ ϕ − ξ after some calculations we obtain three − θ ϕ

) 1 2 1 ()()   ÷ ÷ ÷  ( as well. as g a e g a = − θ ϕ , ( q 4 ( ) ξ ) ) (3) cos sin   ÷ ÷

g r ξ α , , ′′ 1 2 1 − ξ ( k k − ξ + ) 2 0 1 3 ( = 1 3 1 2 q q 2 2 2 ( = + ( ξ ≡ + = 1 − ξ + ξ − ξ r r ()() ( 1 2 1 2 , , θ ϕ g a e g a (

− ξ − − ξ + ξ , ) ) 1 1 2 1  sin sin sin cos = = ) r = − θ ϕ ξ D B 0 ) the equations of motion,   ÷ ÷ ÷  , , + = − θ ϕ ) ) ( ) 0 ( 2 2 2 ( q ( ( r r r g a ) 2 , (2) cos cos −γ ( ( g r − ξ ( ) )

3.4) and , , + ξ+ ξ + ξ + ξ + ξ ( 1 2 1 CDAB = frame in 3-dimensional space. flatin 3-dimensional frame ( ) 2 2 2 2 ) 1 1 1 − − . 0 ) − ξ r r 2(1 2 ) 1 2 4(1 2 ) − ξ θ ′′ 1 3 1 2 is the Schwarzschild radius. istheSchwarzschild 0 1 3.1)- (0) 0 1 1 3 ( ( k k ( 0 ξ 0 1 0 1 g e r 2 r   ÷ ÷ r r r , cos 2 1 2 2 1 γ ξ 1 1 2 2 1 − − − ( (    ÷ ÷ ÷  − − − ξ + − ξ + ξ ξ ≡ + 1 / 1 / 1 / 1 / r r r 1 1 1 ( r r r = 2 1 2 1 ln 1 / ln 1 / = − θ   ÷      ÷  2 2 2 2 4 + ln 1 / ln 1 / − ξ + − ξ + ξ Up to the second order in inverse radius we obtain for the space-time metric thespace-timefor inversewe in obtain radius Up to the order second Using 0 = − − − − − + + orthonormal A r r B r r q 1 2 1 1 2 1 = − + − + + + + rrr r rrrr rr r C r r D r r ,,,,,,,,, 2 2 2 2 2 2 2 2 2 = − = − rrr r r r r rrrr r r rrr r r rr r = 2 1 1 2 2 1 ,,,,,,,,,,,,,,,,,, (1) sin (0) α+α+ α=ξα−γ−γ−αγ− γ 0 A g e g r γ = − + − ds dt dx dy dz rds r r r r r dt r r r r dx dy dz 2 4 2 2 2 4 α = − + − α +α+ α + γ α+ α + γ+γ+γ=ξ−α+γ−αγ− γ = −ξ α + γ + α α C It is convenient to use this expression in the so-called PPN-formalism to with experiment compare our theory 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 in the case when but zero separately, And And where sphericallythe lookslike: Thus symmetric metric Obviously, this system is overdetermined. Nevertheless, it has a solution in the form of a hasNevertheless, the form solution in thisit isoverdetermined. Obviously, system Where Where Where is is an theframe: andparameterizing functions the metric two for equations Where Where Spherically symmetric solution… symmetric Spherically the frame: for anconvenient ansatz to write It is PoS(QFTHEP 2013)092 ] 5,6 4.2) [ ( ], but

4.1) ( 1 γ 4.4) 4.5) 4.3) ( ( ( PPN-parameters Edward Rakhmetov for Møller theory: gravityMøller for γ

5 − and ]. It is sufficiently accurate to the β . Recently tests of ] 3,4 3 [

parameters 5

of the theory parameters has to be very small to be ) ′′

1 2 Time delay tests give the best estimation for k k the spherically symmetric solution coincides with ]. 2 ( 1.1 1.2) 10 ′′ 1 2 5 General Relativity

k k [ ′′ 5 2 1 2 spherically symmetricmatter distributionsphericallyof ξ ≡ + = − ± × − 1 2 ... k k ( 2 ξ ≡ + are the well-known Eddington–Robertson–Schiff parameters

= − + γ δ + γ ξ ≡ + 4.2), PPN- we obtain 4.2), ij ij ( g U ]. In this approximation the space-time metric predicted by nearly , 5 and

β , where , , 2 r 4.1) and 4.1) m ( density of rest mass as measured in a local freely falling frame momentarily + ξ + ξ ), ), 1 (2.1 2.3) 10 4.4) it follows that 4.4) 1 y x ( 1 2 3 γ − = ± × ]. For example, for example, the ].for For 3.14 is the 5 β = , we present the spherically symmetric solution of Møller gravity in an explicit form. , , 3 y y have reached high precision, including the light deflection, the perihelion advance of () − () § 2 ρ γ ρ x y 4.3) and 4.3) 1 1 2 2 ... | | ( In The peculiar moment is that The parameters The only way that one metric theory differs from another is in the numerical values of the The main idea of the parameterized post-Newtonian formalism is that the comparison of + ξ are constructed from the matter variables in imitation of the Newtonian gravitational ∫ = − + β + 1 2

= = ≡ and ]. Conclusions

Spherically symmetric solution and solution PPN-formalism the Spherically symmetric 00 3 g U U γ = β U d r Schwarzschild metric not only in the trivial case, when our theory coincides with General As we can see, the spherically symmetric additional constants solutionof the Einstein-Møller action areappears small, as it was shownnot by Møller [ only in the case,of too. constants also arbitrary in the case when the in agreement with experimental data. This is the only restriction on the theory parameters thespherically experiments. get the solar-system data from can symmetric ofsolution and we 5. Mercury and the Shapiro time delay result with a From As we can see, the combination used to describe the classical tests of coefficients that appear in front of the metric potentials. The PPN-formalism inserts parameters in place of these coefficients. In such waya values dependparameters on the theory studyunder [ ( Comparing potential [ potential Where with the matter. commoving gravitating an expansion about the Minkowski metric in terms of dimensionless gravitational potentials of smallness: of varying degrees that metric theories of gravity with experiment becomes simple when we use the weak-field limit. This approximation is known as the post-Newtonian limit [ most solar-system tests [ every metric theory of gravity has the same structure. The space-time metric can be written as Spherically symmetric solution… symmetric Spherically 4. PoS(QFTHEP 2013)092 as ′′ 1 2 k k 2 0 Living

, ξ ≡ + = Edward Rakhmetov , Cambridge Press, University Mat. Fys. Medd. Vid. Fys. Dan. Mat. , , Springer Science, 2011. (miscellany) Akademie-Verlag, Berlin, 1979 6

A test of general relativity using radio links with the Cassini Cassini general the links with relativityA test using of radio of the theory parameters has to be very small (of the order of to small the has(of order very theorybe ofparameters the Einstein-centinarium, Einstein-centinarium,

Beyond Einstein Gravity ′′ 1 2 k k 2 , 2006. 9 are equal to zero separately), but in the case when Theory physics and experiment in gravitational The Confrontation RelativityExperiment between and General ξ ≡ + ′ 2 Møller, etMøller, al, k , ′ 1 On the Crisis of Gravitation and a Possible Solution of Crisis Gravitation the On

k , n.13, 1978. , Nature, 425, 374–376, 2003. 39 v. Møller Clifford M. Will,Clifford Iess, L. B. P. Bertotti, Tortora, H.-J C. Treder, M. Will,Clifford V.S. Capozziello, Faraoni, C. The authors are grateful to E.E. Boos, I.P. Volobuev, M.N. Smolyakov for valuable The combination The Reviews in Relativity, in Relativity, Reviews spacecraft 1981. Selsk., [5] [6] [4] [2] [3] [1] ) to be in agreement with experimental data. This is the only restriction on the theory -5 NS-3042.2014.2. References Acknowledgements discussions. The work was supported by grant of Russian Ministry of Education and Science well. 10 parameters we can get from the spherically symmetric solution andexperiments. data of the solar-system Spherically symmetric solution… symmetric Spherically Relativity (when