Geodesic deviation, Raychaudhuri equation, Newtonian limit, and tidal forces in Weyl-type f(Q,T) gravity
Jin-Zhao Yang, Shahab Shahidi, Tiberiu Harko, Shi-Dong Liang
aa r X i v : . [ g r- q c ] J a n Geodesic deviation, Raychaudhuri equation, Newtonian limit, and tidal forces inWeyl-type f ( Q, T ) gravity Jin-Zhao Yang, ∗ Shahab Shahidi, † Tiberiu Harko,
3, 4, 1, ‡ and Shi-Dong Liang
1, 5, § School of Physics, Sun Yat-Sen University, Xingang Road, Guangzhou 510275, P. R. China, School of Physics, Damghan University, Damghan, 41167-36716, Iran Astronomical Observatory, 19 Ciresilor Street, Cluj-Napoca 400487, Romania, Department of Physics, Babes-Bolyai University,Kogalniceanu Street, Cluj-Napoca 400084, Romania State Key Laboratory of Optoelectronic Material and Technology,and Guangdong Province Key Laboratory of Display Material and Technology,Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China. (Dated: February 2, 2021)We consider the geodesic deviation equation, describing the relative accelerations of nearby par-ticles, and the Raychaudhuri equation, giving the evolution of the kinematical quantities associatedwith deformations (expansion, shear and rotation) in the Weyl-type f ( Q, T ) gravity, in which thenon-metricity Q is represented in the standard Weyl form, fully determined by the Weyl vector,while T represents the trace of the matter energy–momentum tensor. The effects of the Weyl geom-etry and of the extra force induced by the non-metricity–matter coupling are explicitly taken intoaccount. The Newtonian limit of the theory is investigated, and the generalized Poisson equation,containing correction terms coming from the Weyl geometry, and from the geometry matter cou-pling, is derived. As a physical application of the geodesic deviation equation the modifications ofthe tidal forces, due to the non-metricity–matter coupling, are obtained in the weak-field approxi-mation. The tidal motion of test particles is directly influenced by the gradients of the extra force,and of the Weyl vector. As a concrete astrophysical example we obtain the expression of the Rochelimit (the orbital distance at which a satellite begins to be tidally torn apart by the body it orbits)in the Weyl-type f ( Q, T ) gravity.
PACS numbers: 03.75.Kk, 11.27.+d, 98.80.Cq, 04.20.-q, 04.25.D-, 95.35.+d
CONTENTS
I. Introduction 2II. Geometrical preliminaries and the basics of theWeyl-type f ( Q, T ) gravity theory 4A. Quick start for Weyl geometry 4B. The Weyl-type f ( Q, T ) gravity theory 5III. Geodesic deviation equation in f ( Q, T ) theory 6A. The extra force in Weyl-type f ( Q, T ) theory 6B. The geodesic deviation equation 7IV. Generalized Raychaudhuri equation in Weyl-type f ( Q, T ) theory 8V. Weak field approximation, Newtonian andPost-Newtonian li,its, tidal force and Rocheradius in Weyl-type f ( Q, T ) gravity theory 9A. The weak field approximation 9B. Post-Newtonian analysis 10C. Tidal forces in Weyl type f ( Q, T ) gravitytheory 11 ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] D. The Roche radius in Weyl type f ( Q, T )gravity 12VI. Discussions and final remarks 13Acknowledgments 14References 15A. Calculational details for some basic results 161. Obtaining the expression of the scalarnonmetricity Q I. INTRODUCTION
The twentieth century has seen the birth of GeneralRelativity (GR) and of Quantum Mechanics (QM), whichare considered as the most two successful theories de-scribing the nature and properties of the physical world,on scale ranging from the microscopic to the cosmologi-cal one. General Relativity, a geometrical theory of grav-ity, one of the fundamental interactions that shape theUniverse, has provided an excellent description of theobservational data [1], and has led to new insights intothe problems of space and time, and their relation withthe physical Universe. The recent detection [2] of thegravitational waves has again proved the existence of theexcellent correspondence between experimental data andthe theoretical predictions of GR in a range extendingfrom weak to strong gravitational fields. The introduc-tion of Riemann geometry into GR has provided a power-ful mathematical framework to describe the properties ofthe gravitational field. However, despite its remarkablesuccess, a number of recent observational results haveraised some questions about the absolute validity of stan-dard GR, which may still present some limitations espe-cially on astrophysical scales exceeding the Solar Systemone. The two fundamental problems facing present daygravitational theories, and, in particular General Relativ-ity, are the dark energy and the dark matter problems,respectively. For recent reviews on the dark energy anddark matter problems see [3–13]. To fix the theoreticalimperfections of standard General Relativity, two impor-tant physical components, the dark matter and the darkenergy, are introduced in the cosmological scenario in arather ad hoc way, with the major goal of explaining andadapting the standard gravity model to describe realisticphysical situations, related to the motion of massive par-ticles around galaxies, and to solve the problem of theaccelerated expansionary state of Universe. We may callthe approach based on the introduction of two new phys-ical components in the overall matter/energy balance ofthe Universe as the dark components model [14].However, a second approach to gravitational phenom-ena is also possible, and it is called the dark gravity ap-proach [14]. In this approach one assumes that both darkmatter and dark energy can be explained by changing thenature of the gravitational force. Many modified theoryof gravity, going beyond the standard GR model havebeen proposed in order to build a fundamental frame-work for the explanation mysterious dark matter anddark energy [15], and to give a solution of the presentobservational-theoretical contradictions and conflicts. Inthe framework of the Riemannian geometry one can nat-urally generalize the Einstein-Hilbert action by substi-tuting the Ricci scalar R with an arbitrary functions f ( R ). This leads to the f ( R ) modified theory of grav-ity [16]-[23]. There are two approaches in f ( R ) gravity,namely, the metric formulation, in which the metric isconsidered as the only dynamical variable, and the Pala-tini formulation, in which the connection is considered as another independent variable, beside the metric tensor.One can find detailed discussions of the metric formu-lation of f ( R ) gravity in [24]-[29], and for the Palatiniformulation in [30]-[32], respectively. The most obviousdrawbacks of f ( R ) theory is that the scalar field in thePalatini formulation is not dynamical, which implies thatno new degrees of freedom can be introduced, resultinginto the existence of infinite tidal forces that generally arephysically impossible [33]. On the other hand the extrafreedom introduced from the metric formulation wouldlead to contradiction with the observational results ob-tained in the Solar System [34, 35].To allow for the generation of long-range forces andsimultaneously passing the Solar system test, in [36]-[40]a new approach to gravitational effects was proposed. Inthis theory, called Hybrid Metric-Palatini Gravity, theEinstein-Hilbert action is supplemented with a correc-tion term inspired by the Palatini formulation. Anotherinteresting and important modification of gravity is theinclusion of a non-minimal coupling of geometry and mat-ter into the action [41]-[48], by using arbitrary functionsof the scalar curvature and Lagrangian density of matter(in the f ( R, L m ) gravity theory [47]), or by considering agravitational Lagrangian of the form f ( R, T ) [48], where T is the trace of the matter energy-momentum tensor.In these classes of theories the covariant derivative ofthe energy-momentum tensor is always non-zero, whichimplies a non-geodesic motion of test particles, and theappearance of an extra force. For a recent review of somemodified gravity theories in Riemann geometry one canrefer to [14] and [49], respectively.The standard GR theory is formulated in Riemann ge-ometry. Hence, an alternative avenue for searching fora generalized description of gravity is the extensions ofthe geometrical framework on which GR is based. In anattempt to unify gravity and electromagnetism H. Weylintroduced in 1918 a generalization of the Riemann ge-ometry [50]. In the Weyl geometry both the orienta-tion and the length of vectors are allowed to vary underparallel transport, while in Riemann geometry only thevariation of the orientation is allowed. Weyl geometryrepresents a completely consistent generalization of Rie-mannian geometry. In modern language, the vector fieldintroduced by Weyl, which generates a new componentin the connection, which in Weyl geometry is no longermetric-compatible, is actually the dilatation gauge vec-tor. If the vector is the gradient of some function, ascale transformation of the form ˜ g µν = σ g µν , where σ is the scale (conformal) factor, can be applied to cancelthe Weyl vector. In this case the Weyl geometry is calledintegrable, and the length of vectors will be unchangedunder a parallel transport along closed paths. For a de-tailed discussion of Weyl geometry see [51–53].In an important mathematical and physical develop-ment E. Cartan introduced the anti-symmetric part ofthe connection, known as torsion, into the gravity the-ory, thus formulating an extension of GR [54], whichis known as the Einstein-Cartan theory. The Weyl ge-ometry can be immediately generalized by including thetorsion, which leads to the Weyl-Cartan geometry, anda corresponding geometric theory of gravitation [55–66].For a review of geometrical properties and physical appli-cations of Riemann-Cartan and Weyl-Cartan spacetimesone can refer to [67]. Another important mathemati-cal development with important physical implications isrelated to the work by R. Weitzenb¨ock [68], who devel-oped a geometry with torsion and zero Riemann curva-ture. Since curvature is zero, Weizenb¨ock spaces possessthe interesting property of distant parallelism, known asteleparallelism. The teleparallel approach substitutes themetric tensor, which plays a central role as a basic physi-cal variable in gravitational theories, with a set of tetradvectors. In this approach, the torsion is generated bytetrad fields and it describes the gravitational field en-tirely, once the torsion is properly chosen to eliminatethe curvature. This is the so-called teleparallel equiva-lent of general relativity (TEGR) [69–71], which is alsoknown as f ( ˜ T ) theory, where ˜ T is the trace of torsiontensor. In f ( ˜ T ) theory, torsion completely compensatesthe curvature, and the spacetime becomes flat. Weyl-Cartan-Weizenb¨ock gravity (WCW) was introduced asan extension of the teleparallel gravity models in [72]. Inthis approach, the Weizenb¨ock condition of the vanishingof the total curvature is implemented in a Weyl-Cartanspacetime. Moreover, in [73] the Weizenb¨ock conditionof the exact compensation of torsion and curvature wasintroduced in the action via the Lagrange multiplier ap-proach, in Riemann-Cartan spacetime. For a review ofteleparallel gravity theory see [74].An interesting theory, geometrically equivalent to GR,which is also known as symmetric teleparallel gravity, wasintroduced in 1999 [75]. In this geometric approach thenonmetricity Q of a Weyl geometry represents the basicgeometrical variable. This approach was further devel-oped as f ( Q ) gravity theory, also named as nonmetricgravity [76]. Various physical and geometrical propertiesof symmetric teleparallel gravity have been analyzed in[77–96].An important extension of f ( Q ) theory has been ob-tained in [97] by including a nonminimal curvature-matter coupling into the gravitational Lagrangian, with L = f ( Q ) + f ( Q ) L m , where f and f are arbitraryfunctions of the nonmetricity Q . Similarly to the non-minimal couplings between curvature and matter in Rie-mannian geometry [47, 48], the coupling leads to the non-conservation of the matter energy-momentum tensor aswell, which leads to the existence of a new term in thegeodesic equation, which can be interpreted as an extra-force. A Bayesian statistical analysis using redshift spacedistortions data was performed to test a model of f ( Q )gravity in [98]. The cosmological background evolution issimilar to the ΛCDM one, but differences arise in the per-turbations. The best fit parameters indicate that the σ f ( Q ) gravity model was extended to include a non- minimal coupling in the Lagrangian in [99], in which thegravitational action L is given by an arbitrary function f of the nonmetricity Q and of the trace of the matterenergy-momentum tensor T , with L = f ( Q, T ). Severalcosmological applications of the theory were consideredby chosing some simple functional forms of the function f ( Q, T ), corresponding to additive expressions of f ( Q, T )of the form f ( Q, T ) = αQ + βT , f ( Q, T ) = αQ n +1+ βT ,and f ( Q, T ) = − αQ − βT , respectively, where α , β and n are constants. The Hubble function, the deceleration pa-rameter, and the matter energy density were obtained ineach case as a function of the redshift by using analyticaland numerical techniques. For all considered cases theUniverse experiences an accelerating expansion, endingwith a de Sitter type evolution. Gravitational baryogen-esis in f ( Q, T ) gravity was considered in [100], and it wasfound that f ( Q, T ) gravity can contribute significantlyto this phenomenon. The various cosmological param-eters in Friedmann-Lemaitre-Robertson-walker (FLRW)geometry have been obtained in [101] for different choicesof the function f ( Q, T ) in terms of the scale-factor andredshift z by constraining the energy-conservation law.The observational constraints on the model have beenobtained by fitting the model parameters using the avail-able data sets like Hubble data sets H ( z ), Joint LightCurve Analysis (JLA) data sets and union 2.1 compila-tion of SNe Ia data sets. The various energy conditionsfor cosmological models in f ( Q, T ) gravity were studiedin [102]. The equation of state parameter w = − f ( Q, T ) models the null, weak, and dominantenergy conditions are obeyed, while the strong energyconditions are violated during the present accelerated ex-pansion. The late time cosmology in f ( Q, T ) gravity wasinvestigated in [103]. Constraints on the model parame-ters were imposed from the updated 57 points of Hubbledata sets, and 580 points of union 2.1 compilation super-novae data sets. The performed analysis did show that f ( Q, T ) gravity represents a promising approach for ex-plaining the current cosmic acceleration, and it can pro-vide a consistent solution to the dark energy problem.A particular type of f ( Q, T ) model was considered in[104], in which the scalar non-metricity Q µν of the space-time was expressed in its standard Weyl form, and there-fore it is fully determined by a vector field w µ . The fieldequations of the theory have been obtained under theassumption of the vanishing of the total scalar curva-ture, a condition which was added into the gravitationalaction via a Lagrange multiplier. The cosmological im-plications of the theory were also investigated for a flat,homogeneous and isotropic geometry, and the generalizedFriedmann equations were obtained. Several cosmolog-ical models were investigated by adopting some simplefunctional forms of the function f ( Q, T ), and the predic-tions of the theory have been compared with the standardΛCDM model.The main goal of the present paper is to investigatesome fundamental properties of motion of test particlesin the Weyl type f ( Q, T ) gravity theory, introduced in[104]. We derive the geodesic deviation equation and theRaychaudhuri equation in the Weyl geometry and in thepresence of the extra-force that describes the effects ofthe nonmetricity-matter coupling. As compared to stan-dard general relativity a number of new terms do appearin both equations, indicating the existence of a complexdynamics resulting from the intricate interplay of the ge-ometrical and matter factors. The weak field limit of thetheory is also considered, and the Poisson equation, con-taining the corrections arriving from the Weyl geometryand nonmetricity-matter coupling are determined. As aphysical application of the obtained results we considerthe tidal force problem in Weyl type f ( Q, T ) gravity, anda generalization of the Roche limit is obtained, under thesimplifying assumption that the center of mass of thetwo-body system coincides with the geometrical centerof the massive object with mass M .The present paper is organized as follows. In Section IIwe review the basics of the Weyl geometry, and we in-troduce the Weyl type f ( Q, T ) gravity theory, and itsfield equations. The Geodesic deviation equation is de-rived in Section III, while the Raychaudhuri equation inWeyl geometry and in the presence of the extra force in-duced by the nonmetricity-matter coupling is obtainedin Section IV. In Section V we consider the weak fieldlimit of the theory, and the generalized Poisson equationfor the gravitational potential, containing the correctionsdue to the geometric and coupling effects, is obtained.The properties of the tidal forces, as well as the expres-sion of the Roche limit in Weyl geometry and in the pres-ence of extra force, are also investigated. We discuss andconclude our results in Section VI. Some mathematicalresults used in the derivation of the main relations of thepaper are summarized and detailed in the Appendix A.
II. GEOMETRICAL PRELIMINARIES ANDTHE BASICS OF THE WEYL-TYPE f ( Q, T ) GRAVITY THEORY
In the present Section we briefly review the fundamen-tals of the Weyl geometry, and of the Weyl-type f ( Q, T )gravity.
A. Quick start for Weyl geometry
In the differential geometry of the Riemann spaces theconnection describes the properties of the parallel trans-port, the fundamental concept used for the characteri-zation of the various mathematical aspects associated tothe geometrical objects in curved space-times. In orderto keep the tensor properties of the differentiation op-eration acting on a vector, some new terms, containingthe connection, must be introduced to assure the ten-sorial nature of the differentials of a vector. Hence, inRiemann geometry the covariant derivatives of the con- travariant and covariant vectors are given by [105–107] ∇ ν A µ = ∂ ν A µ + Γ µλν A λ , ∇ ν A µ = ∂ ν A µ − Γ λµν A λ , (1)In Riemann spaces, the connection is of Christoffeltype, it is compatible to the metric g µν , so that ∇ λ g µν =0, and it is given byΓ λµν = 12 g λσ (cid:18) ∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν (cid:19) . (2)With this connection the parallel transportation in aRiemann space will preserve the length of the vectors,and only the orientation of the vector is changed. In1918 H. Weyl proposed a new geometry [50] by introduc-ing a connection with the property that under paralleltransportation both the orientation and the magnitudeof a vector change. The connection in Weyl geometryis no longer metric compatible. Moreover, a new vectorfield, known as Weyl vector field, is introduced, allowingto write the Weyl connection as [52, 53],˜Γ λµν = Γ λµν + g µν w λ − δ λµ w ν − δ λν w µ , (3)where in the following the tilde indicates the quantitiesdefined in the Weyl geometry. With the use of the Weylconnection one can immediately obtain the fundamentalresult that the covariant derivative associated with theconnection ˜Γ λµν , when applied on the metric tensor, givesa non-zero value,˜ ∇ λ g µν = 2 w λ g µν , ˜ ∇ λ g µν = − w λ g µν . (4)In the standard representation of the Weyl geometrythe variation of the length of a vector under paralleltransport is given by δl = l w µ δx µ , (5)where l is the length of the vector before transporta-tion. Thus the variation of the length of a vector undertransport along a closed loop is, δl = l W µν δs µν , (6)where δs µν is the area surrounded by the loop, and W µν is given by W µν = ˜ ∇ ν w µ − ˜ ∇ µ w ν = ∇ ν w µ − ∇ µ w ν . (7)In Weyl geometry one can define a curvature tensor inthe same way as in Riemann geometry, with the use ofthe intrinsic connection,2 ˜ ∇ [ µ ˜ ∇ ν ] A λ = ˜ R σλνµ A σ , (8)2 ˜ ∇ [ µ ˜ ∇ ν ] A λ = − ˜ R λσνµ A σ . (9)The explicit expressions of the curvature tensor inWeyl geometry can be obtained as˜ R µνλσ = R µνλσ + g µν W λσ + 2 ∇ λ w [ µ g ν ] σ + 2 ∇ σ w [ ν g µ ] λ +2 w λ w [ µ g ν ] σ + 2 w σ w [ ν g µ ] λ − w α w α g λ [ µ g ν ] σ . (10)By contracting the first and the third indices of the cur-vature tensor, we find˜ R νσ = R νσ + 2 w ν w σ + 2 ∇ σ w ν + W νσ + g νσ ( ∇ α w α − w α w α ) . (11)Finally, contracting the remaining two indices one canobtain the scalar curvature as˜ R = ˜ R µµ = R + 6( ∇ µ w µ − w µ w µ ) . (12)In order to develop some physical applications we in-troduce two types of nonmetricities as follows, Q λµν = − w λ g µν = − ˜ ∇ λ g µν , (13) Q λµν = − w λ g µν = ˜ ∇ λ g µν . (14)Moreover, the scalar nonmetricity Q is defined as Q = − g µν (cid:0) L αβµ L βνα − L αβα L βµν (cid:1) , (15)where L λµν is given by L λµν = 12 g λσ ( Q µσν + Q νσµ − Q σµν ) . (16)Hence we can rewrite the connection in Weyl geometryas ˜Γ λµν = Γ λµν + L λµν . (17)With the use of Eqs. (13) and (16), we obtain for thescalar nonmetricity the expression (see Appendix A 1 forthe calculational details), Q = 6 w . (18)The explicit expressions of the covariant derivatives forsome vector and tensor quantities in Weyl geometry aregiven in Appendix A 2. B. The Weyl-type f ( Q, T ) gravity theory A particular model of the general f ( Q, T ) theory wasintroduced in [104], and it is based on the following ac-tion, S = Z d x √− g (cid:20) κ f ( Q, T ) − W µν W µν − m w µ w µ + λ ˜ R + L m (cid:21) , (19)where we have denoted κ = 1 / πG , and m is the massof the particle associated to the vector field w µ . More-over, with L m we have denoted the ordinary matter ac-tion. In the action (19), the second and third terms arethe kinetic energy and the mass terms associated to theWeyl-type vector field, respectively. As for the first term,giving the gravitational Lagrangian, it is taken as an ar-bitrary function of the nonmetricity Q and of the trace T of the matter energy-momentum tensor. Moreover, in the present approach, we assume the van-ishing of the total scalar curvature, and we impose in thetotal action the Weizenb¨ock condition with the help ofthe Lagrange multiplier λ . We impose this condition inorder to follow the essential ideas of the teleparallel ap-proach to gravity. The action (19) generalizes the GRequivalent symmetric teleparallel gravity theory, or f ( Q )theory, by introducing a matter-geometry coupling, in asimilar manner as in the f ( R, T ) theory, and adding anextra matter distribution for the particles associated withthe Weyl vector field. However, it should be mentionedthat the present theory is different from f ( R, T ) theorydue to the presence of a boundary term, and hence it canbe seen as a generalized equivalent theory of GR, in thesense discussed in [88].In the case where f ( Q, T ) = Q , and if we have a van-ishing mass m = 0, the theory reduces to the symmetrictelleparallel equivalent to GR, as is fully developed in[76].By first varying the total action (19) with respect tothe vector field w µ , we obtain the generalized Proca equa-tion that describes the evolution of the vector field w µ inthe Weyl geometry, ∇ ν W µν − (cid:0) m + 12 κ f Q + 12 λ (cid:1) w µ = 6 ∇ µ λ, (20)where f Q = ∂f ( Q, T ) /∂Q . Next, we vary the action withrespect to the metric tensor, and with the use of theWeizenb¨ock condition, that is, after omitting the termsthat explicitly contain ˜ R , we obtain the generalized fieldequation of the Weyl-type f ( Q, T ) gravity as follows,12 ( T µν + S µν ) − κ f T ( T µν + Θ µν )= − κ f g µν + 6 κ f Q w µ w ν + λ ( R µν − w µ w ν + 3 g µν ∇ ρ w ρ ) +3 g µν w ρ ∇ ρ λ − w ( µ ∇ ν ) λ + g µν (cid:3) λ − ∇ µ ∇ ν λ, (21)where in the field equations we have denoted by f T thepartial derivative of f with respect to T , f T = ∂f /∂T ,and where we have introduced the energy-momentumtensor S µν associated to the Weyl type vector field, de-fined according to S µν = − g µν W αβ W αβ + W µρ W ρν − m g µν w ρ w ρ + m w µ w ν . (22)As customary, the energy-momentum tensor T µν of theordinary matter is defined as T µν = − √− g δ ( √− g L m ) δg µν . (23)The tensor Θ µν is obtained from the variation of theenergy-momentum tensor with respect to the metric. Byadopting the assumption according to which the ordinarymatter Lagrangian density L m depends only on the met-ric tensor components, and not on their derivatives, weobtain for Θ µν the simple expression,Θ µν = g αβ δT αβ δg µν = g µν L m − T µν . (24)By contracting the indices µ and ν in Eq. (21), weobtain the scalar S , given by S = S µµ = − m w µ w µ . (25)The trace of the matter energy-momentum tensor T canbe obtained from the equation,12 ( T + S ) − κ f T ( T + Θ) = − κ f + 6 κ f Q w µ w µ + λ ( R − w µ w µ + 12 ∇ µ w µ ) + 6 w µ ∇ µ λ + 3 (cid:3) λ (26)By taking the covariant divergence of Eq.(21), and us-ing the constraint ˜ R = 0, we obtain for the divergenceof the ordinary matter energy-momentum tensor the ex-pression˜ ∇ µ T µν = ∇ µ T µν − w µ T µν + w ν T = κ κ f T × (cid:20) ∇ ν ( f T L m ) − f T ∇ ν T − T µν ∇ µ f T (cid:21) − w µ T µν + w ν T. (27)Hence, as one can see from the above equation, theordinary matter energy-momentum tensor is not con-served in the Weyl-type f ( Q, T ) theory. From a physicalpoint of view the nonconservation of the matter energy-momentum tensor can be interpreted as indicating thepresence of an extra-force, acting on massive test par-ticles, and making the motion nongeodesic. It shouldbe noted that in the special case f T = 0, the energy-momentum tensor becomes conserved, as one can be eas-ily seem from Eq. (21). In this case the term in theaction generated by the non-minimal coupling betweenmatter and geometry does vanish, and the theory reducesto f ( Q ) coincident gravity [76].There is a special case where the denominator ofEq. (27) vanishes identically, and therefore the field equa-tions Eq. ((21)) are not valid. This particular case cor-responds to 1 + 2 κ f T = 0, giving immediately f T = − / κ , and f ( Q, T ) = − T / κ + C ( Q ), where C ( Q ) isan arbitrary function depending only on the scalar no-metricity. For this form of f ( Q, T ) the gravitational fieldequations Eqs. (21) become,12 ( S µν + g µν L m )= − κ C ( Q ) g µν + T κ g µν − κ C Q w µ w ν + λ ( R µν − w µ w ν + 3 g µν ∇ ρ w ρ ) +3 g µν w ρ ∇ ρ λ − w ( µ ∇ ν ) λ + g µν (cid:3) λ − ∇ µ ∇ ν λ, (28) where C Q = ∂C ( Q ) /∂Q . The conservation equation ofthe matter energy-momentum tensor reduces to the sim-ple form ∇ ν ( T − L m ) = 0 . (29) III. GEODESIC DEVIATION EQUATION IN f ( Q, T ) THEORY
As a first step in our further investigations of the geo-metric and physical properties of the Weyl type f ( Q, T )gravity, in the present Section we derive the geodesic de-viation equation in this theory, and obtain its form bytaking into account the presence of the extra-force gen-erated by the nonconservation of the energy-momentumtensor.
A. The extra force in Weyl-type f ( Q, T ) theory As we have already mentioned, due to the presence ofnonmetricity, in Weyl geometry the length of a vector isno longer preserved under parallel transport. However,we define the four-velocity, as usual, according to theexpression u µ = dx µ /dξ , where ξ is the affine parameter,which forms a tangent bundle of congruences of timelikecurves. But in a Weyl geometry we need to normalizethe four-velocity according to [108], u µ u µ = g µν u µ u ν = − ℓ , ℓ = ℓ ( x α ) , (30)where ℓ ( x α ) is an arbitrary function of space and timecoordinates. With this normalization, even if we set theproper time as dτ = − g µν dx µ dx ν , ξ does not necessarilycoincide with τ [108], and by chain-rule of differentiationwe obtain the relation dτ /dξ = ± ℓ .The presence of nonmetricity, as well as its specificproperties related to the background spacetime stronglyaffects the nature of the hypersurfaces orthogonal to thetimelike u µ velocity field. Therefore, in Weyl geometry,a generalized projection tensor operator needs to be in-troduced by the definition [108] h µν = g µν + 1 ℓ u µ u ν . (31)The generalized projection operator has the usual proper-ties h µν = h νµ , and h µν h µν = 3, respectively. The mixedtensor form of the projection operator can be obtainedby raising one of its indices, and thus h µλ h λν = h νµ = δ νµ + 1 ℓ u µ u ν . (32)For mathematical convenience, here onwards we de-note the temporal derivative by ′ , to indicate covariantdifferentiation with respect to ξ , and the spatial deriva-tive D µ for a generalized tensor in Weyl geometry, re-spectively, as follows [108], T β ...β n ′ α ...α m = u µ ˜ ∇ µ T β ...β n α ...α m , (33) D µ T β ...β n α ...α m = h λµ h γ α . . . h γ m α m h β δ . . . h β n δ n ˜ ∇ λ T δ ...δ n γ ...γ m . (34)Due to the presence of nonmetricity in Weyl geometrythere exist two types of four-acceleration, denoted by A µ and a µ , respectively, and defined according to A µ = u µ ′ = u ν ˜ ∇ ν u µ = u ν ∇ ν u µ − ℓ w µ − w ν u ν u µ , (35)and a µ = u ′ µ = u ν ˜ ∇ ν u µ = u ν ∇ ν u µ − ℓ w µ , (36)respectively. The two accelerations are related by theimportant equation, A µ = a µ + Q νλµ u ν u λ . (37)From these two types of acceleration we can obtainsome pure geometrical relations. Multiplying the twoacceleration vectors by u µ , we obtain A µ u µ = −
12 ( ℓ ) ′ + 12 Q µνλ u µ u ν u λ , (38) a µ u µ = −
12 ( ℓ ) ′ − Q µνλ u µ u ν u λ . (39)For the sake of clarity we mention that ( ℓ ) ′ = u µ ˜ ∇ µ ( ℓ ).With the above definitions, the geodesic equation inWeyl geometry takes the following form, where generallythe extra force f µ gives the supplementary accelerationinduced by the nonminimal curvature-matter coupling, u µ ′ = u ν ˜ ∇ ν u µ = d x µ dξ + ˜Γ µνλ u ν u λ = f µ . (40)The energy-momentum tensor of a perfect fluid can bedefined in a Weyl spacetime according to [108] T µν = ( p + ρ ) ℓ u µ u ν + pg µν . (41)The four-velocity u µ of the perfect fluid must be ξ -parameterized according to the definition of velocity inWeyl geometry, and it obeys the normalization condition(30). The details of the derivation of the generalizationof the perfect fluid model in Weyl geometry can be foundin Appendix A 3.Now multiplying with the projection operator the co-variant divergence of the energy-momentum tensor in theWeyl-type f ( Q, T ) theory, we obtain the important corre-spondence between the matter energy-momentum tensor T µν and the four-acceleration A µ as h ρν ˜ ∇ µ T µν = p + ρℓ ( A ρ − Q µνρ u µ u ν ) − p + ρ ℓ (cid:20) ( ℓ ) ′ + Q µνλ u µ u ν u λ (cid:21) u ρ + h ρν ˜ ∇ ν p + h νρ Q µµν p. (42)With the use of Eqs.(35) and (40), from Eq. (42) wecan obtain the expression of the extra force in Weyl-type f ( Q, T ) gravity as follows (for the calculational detailssee Appendix A 4), f ρ = u µ ′ = ℓ p + ρ h ρν ˜ ∇ µ ( T µν − pg µν ) +12 ℓ (cid:20) ( ℓ ) ′ + Q µνλ u µ u ν u λ (cid:21) u ρ + Q µνρ u µ u ν . (43)By substituting the expression of the nonmetricity ten-sor in the above equation, we obtain for the extra forcethe final expression f ρ = ℓ p + ρ h ρν ˜ ∇ µ ( T µν − pg µν ) + ( ℓ ) ′ ℓ u ρ − w µ u µ u ρ . (44) B. The geodesic deviation equation
Consider now a one-parameter congruence of curves x µ ( ξ ; σ ) satisfying the geodesic equation for the parame-ter ξ in Weyl geometry for each σ = σ = constant. Weintroduce the four-vectors U µ = ∂x µ ( ξ ; σ ) ∂σ , η µ = U µ δσ. (45)The second temporal derivative of U µ can be expressedas U µ ′′ = u ν ˜ ∇ ν ( u α ˜ ∇ α U µ ) = u ν ˜ ∇ ν ( U α ˜ ∇ α u µ )= ( ˜ ∇ ν ˜ ∇ α u µ ) U α u ν + ( ˜ ∇ ν U α )( ˜ ∇ α u µ ) u ν . (46)Using the definition of curvature tensor, we have U µ ′′ = ( − ˜ R µβαν u β + ˜ ∇ α ˜ ∇ ν u µ ) U α u ν + U ν ( ˜ ∇ ν u α )( ˜ ∇ α u µ )= − ˜ R µβαν U α u β u ν + U α ˜ ∇ α ( u ν ˜ ∇ ν u µ ) . (47)Hence we obtain the geodesic deviation equation inWeyl geometry in the presence of an extra force as, U µ ′′ = − ˜ R µναβ U α u β u ν + U α ˜ ∇ α f µ . (48)By multiplying with δσ both sides of the above equa-tion gives η µ ′′ = − ˜ R µναβ η α u β u ν + η α ˜ ∇ α f µ . (49)Eq. (49) represents the geodesic deviation equation inWeyl geometry in the presence of an extra force gener-ated by the coupling between nonmetricity and matter,as follows from the mathematical and physical structureof f ( Q, T ) gravity. In the special case when f T = 0,the energy-momentum tensor becomes conserved, andEq. (49) reduces to the geodesic deviation equation inWeyl geometry, as considered in [108]. Note that in thiscase the present theory becomes equivalent to the coinci-dent gravity theory [76]. In the more general case wherethe Weyl-vector also vanishes, the space-time becomesflat and the right hand side of Eq. (49) identically van-ishes. IV. GENERALIZED RAYCHAUDHURIEQUATION IN WEYL-TYPE f ( Q, T ) THEORY
In the present Section we derive the Raychaudhuriequation in the Weyl-type f ( Q, T ) theory, and in thepresence of an extra force. Our approach basically fol-lows, and generalizes, the similar analysis performed in[108], in which the presence of the extra-force has notbeen taken into account.We begin our investigation by decomposing the covari-ant derivative of the four-velocity u µ into its temporaland spatial components, according to [108],˜ ∇ ν u µ = D ν u µ − ℓ ( u µ ξ ν + a µ u ν ) − ℓ ( u λ a λ ) u µ u ν , (50)where D ν u µ = h βν h λµ ˜ ∇ β u λ and ξ µ = u ν ˜ ∇ µ u ν = u ν ∇ µ u ν − ℓ w µ . We can see that by construction wehave the relation ξ µ u µ = a µ u µ .Similarly, the projected covariant derivative can be de-composed into D ν u µ = 13 (cid:18) θ + 1 ℓ a α u α (cid:19) h µν + σ µν + ω µν , (51)where θ = g µν ˜ ∇ ν u µ = D µ u µ − ℓ a µ u µ = ∇ µ u µ − w µ u µ , (52)is the ”volume” scalar in Weyl geometry. We can alsoconstruct the scalar D µ u µ = h µν ˜ ∇ ν u µ .In a Weyl geometry and in the presence of nonmetricitythe shear tensor is defined according to σ µν = D h ν u µ i = h β ( ν h λµ ) ˜ ∇ β u λ − (cid:18) θ + 1 ℓ a α u α (cid:19) h µν , (53)while the vorticity tensor is introduced based on the def-inition ω µν = D [ ν u µ ] = 12 ( D ν u µ − D µ u ν ) . (54)The shear tensor is symmetric and trace-free due toits construction. As for the vorticity tensor, it is natu-rally anti-symmetric. Thus, based on these mathemat-ical properties, we find that the following relations arealways satisfied, σ µµ = 0 = ω µµ , σ µν u ν = 0 = ω µν u ν , σ µν h µν = 0 = ω µν h µν and σ µν ω µν = 0.From Eqs. (50) and (51) we obtain the explicit expres-sion of the covariant derivative of u µ as˜ ∇ ν u µ = 13 (cid:18) θ + 1 ℓ a α u α (cid:19) h µν + σ µν + ω µν − ℓ ( u µ ξ ν + a µ u ν ) − ℓ ( u α a α ) u µ u ν , (55)while for the covariant derivative of u µ we find˜ ∇ ν u µ = 13 (cid:18) θ + 1 ℓ a α u α (cid:19) h µν − ℓ ( u µ ξ ν + a µ u ν ) − ℓ ( u α a α ) u µ u ν + Q νµα u α + σ µν + ω µν . (56) Raychaudhuri’s equation is a purely geometrical rela-tion, and it immediately follows from a set of fundamen-tal geometric relations, known as Ricci’s identities. In aWeyl spacetime and in the presence of nonmetricity, thedefinition of the curvature tensor is,( ˜ ∇ µ ˜ ∇ ν − ˜ ∇ ν ˜ ∇ µ ) u λ = ˜ R βλνµ u β . (57)We multiply both sides of Eq.(57) with g λν u µ , thus ob-taining g λν u µ ˜ ∇ µ ˜ ∇ ν u λ − g λν u µ ˜ ∇ ν ˜ ∇ µ u λ = − ˜ R βλµν u β u µ g λν . (58)The first term on the left hand side of Eq. (58) can beevaluated as g λν u µ ˜ ∇ µ ˜ ∇ ν u λ = θ ′ − (cid:18) θ + 1 ℓ a α u α (cid:19) u µ Q µ − u µ σ νλ Q µνλ − ℓ (cid:18) θ − ℓ a α u α (cid:19) u µ u ν u λ Q µνλ +1 ℓ ( ξ λ + a λ ) u µ u ν Q µνλ , (59)while the second term on the left hand side of Eq. (58)evaluates to g λν u µ ˜ ∇ ν ˜ ∇ µ u λ = − θ + ˜ ∇ µ a µ − ℓ θa α u α +23 ℓ ( a α u α ) − (cid:18) θ + 1 ℓ a α u α (cid:19) u µ Q ννµ + 2 ℓ a α ξ α − ℓ (cid:18) θ − ℓ a α u α (cid:19) u µ u ν u λ Q µνλ +1 ℓ ( u µ ξ ν + a µ u ν ) u λ Q µνλ − σ µν σ µν + ω µν ω µν − ( σ µν + ω µν ) u λ Q µνλ . (60)Hence in Weyl geometry the left hand side of Eq. (58)finally reads, g λν u µ ˜ ∇ µ ˜ ∇ ν u λ − g λν u µ ˜ ∇ ν ˜ ∇ µ u λ = θ ′ + 13 θ − ˜ ∇ µ a µ + 2 θ ℓ a α u α − ℓ ( a α u α ) − ℓ a α ξ α + σ µν σ µν − ω µν ω µν + ( σ µν + w µν ) u λ Q µνλ +13 (cid:18) θ + 1 ℓ a α u α (cid:19) u µ Q ννµ − (cid:18) θ + 1 ℓ a α u α (cid:19) u µ Q µ − u µ σ νλ Q µνλ + 1 ℓ ( u µ u ν a λ − a µ u ν u λ ) Q µνλ = (cid:20) θ − ( ℓ ) ′ ℓ − w µ u µ (cid:21) ′ + 13 (cid:20) θ − ( ℓ ) ′ ℓ − w µ u µ (cid:21) + σ µν σ µν − ω µν ω µν − ∇ µ f µ + 2 w µ f µ + 1 ℓ f µ ∇ µ ℓ + (cid:20) ( ℓ ) ′ ℓ − w µ u µ (cid:21) ′ − (cid:20) ( ℓ ) ′ ℓ − w µ u µ (cid:21) . (61)The right hand side of the above equation can be eval-uated as˜ R βλµν u β u µ g λν = R µν u µ u ν + u µ u ν ˜ ∇ ν Q ααµ − u µ u ν ˜ ∇ α Q µνα − Q αβµ Q βαν u µ u ν = R µν u µ u ν + 2 u µ u ν ∇ µ w ν +2( w µ u µ ) − ℓ ∇ µ w µ + 2 ℓ w µ w µ . (62)With the use of Eqs.(61) and (62) we finally obtainthe Raychaudhuri equation in Weyl geometry, and in thepresence of an extra force, as, (cid:20) θ − w µ u µ (cid:21) ′ = − (cid:20) θ − ( ℓ ) ′ ℓ − w µ u µ (cid:21) + (cid:20) ( ℓ ) ′ ℓ − w µ u µ (cid:21) − R µν u µ u ν − σ µν σ µν + ω µν ω µν + ∇ µ f µ − ℓ f µ ∇ µ ℓ − w µ f µ − w µ u µ ) − u µ u ν ∇ µ w ν + ℓ ∇ µ w µ − ℓ w µ w µ . (63)It is worth mentioning again that the effect of thenon-minimal matter-geometry coupling f ( Q, T ) will en-ter into the Raychaudhury equation (63) through the ex-pression of the extra force, given by Eq. (44). As wehave already mentioned earlier, in the case of minimalmatter-geometry coupling f T = 0, the above equationreduces to the generalized Raychaudhuri equation in thecoincidence gravity theory [76]. As is well-known, thecoincidence gravity is a generalization of the symmet-ric teleparallel gravity [76]. In this sense, Eq. (63) with f T = 0 could be considered as the Raychaudhuri equa-tion of the generalized symmetric teleparallel equivalentto GR.The special case with zero extra force of the Raychaud-huri equation has been obtained and discussed in [108].The first three lines in Eq. (63) have analogue formsto the similar terms in modified gravity theories withgeometry-matter coupling formulated in Riemann geom-etry [109]. The Raychaudhuri and optical equations fornull geodesic congruences with torsion were investigatedin [110]. V. WEAK FIELD APPROXIMATION,NEWTONIAN AND POST-NEWTONIAN LI,ITS,TIDAL FORCE AND ROCHE RADIUS INWEYL-TYPE f ( Q, T ) GRAVITY THEORY
In the present Section, based on the previous mathe-matical results, we will consider the weak field limit ofthe Weyl-type f ( Q, T ) gravity theory. The generalizedPoisson equation, as well as the expressions of the tidalforce tensor are derived. As an astrophysical applicationof the obtained results we consider the modifications tothe Roche limit that are induced by the assumption ofthe presence of nonmetricity in the space-time geometry.
A. The weak field approximation
If one considers the physical situation in which themotion of the test particles is slow, and that the gravi-tational field intensity created by the material particlesis comparably weak, one could easily compare general-ized metric theories of gravity with each other, with theexperimental observations, as well as with Newtoniangravity. In this case, the first order approximation isadequately accurate to compare the theoretical predic-tions of the gravitational theories with past, present andfuture Solar System observations. Generally, this kindof approximation of gravitational theories, the so-calledpost-Newtonian limit, is valid within the near-zone ofthe system, corresponding to a spherical region with sizesmaller than one gravitational wavelength.As a first step in our investigation of the physical prop-erties of the f ( Q, T ) gravity theory, in the following weinvestigate the linear approximation of the metric field,or the weak field approximation, by assuming the decom-position of the metric tensor as g µν = η µν + H µν , | H µν | ≪ , (64)where η µν = diag( − , , ,
1) is the metric tensor in theMinkowski spacetime. In Weyl geometry, by keeping onlythe first order of H µν , the connection and curvature ten-sor can be respectively formed from their Riemann part[106], to which we add the non-Riemannian (Weyl) com-ponents of the connection˜Γ λµν = 12 ( ∂ ν H µλ + ∂ µ H νλ − ∂ λ H µν ) + g µν w λ − g λµ w ν − g λν w µ , (65)and of the curvature tensor,˜ R αµβν = 12 ( ∂ β ∂ µ H να + ∂ ν ∂ α H µβ − ∂ β ∂ α H µν − ∂ ν ∂ µ H αβ )+2 ∇ β w [ α g µ ] ν + 2 ∇ ν w [ µ g α ] β + 2 w β w [ α g µ ] ν +2 w ν w [ µ g α ] β − w λ w λ g β [ α g µ ] ν + g αµ W βν , (66)respectively. In the following the Latin letters denotethe spatial components ( i = 1 , ,
3) of the tensors, and wewill use
Greek letters to indicate both spatial and temporalcomponents ( µ = 0 , , , H µν is also a static field,one can obtain the explicit weak-field expressions of theChristoffel part in curvature tensor as [106, 107],Γ µ = − g µν ∂ ν g ≈ − η µµ ∂ µ H , (67)Γ = − η ∂ H = 0 , Γ i = − η ii ∂ i H . (68)From Eq. (66) we obtain the component of R γ β , R γ β = − ∂ β ∂ γ H , (69)0and R µν = 12 × (cid:0) ∂ µ ∂ α H αν + ∂ ν ∂ α H αµ − ∂ ν ∂ µ H αα − ∂ α ∂ α H µν (cid:1) . (70)For the details of the calculations of the Riemann andRicci tensors in the weak field approximation see Ap-pendix A 7. B. Post-Newtonian analysis
In this Section we will discuss in detail the Newtonianlimit of f ( Q, T ) gravity, by assuming that the perfectfluid filling the space-time is nonrelativistic. First of all,we will chose a reference frame where the motion of thefluid is static. Moreover, we assume that matter is in theform or pressureless dust with the property ρ ≫ p , andthat the velocity of particles is small as compared to thespeed of light.Hence we can approximate the four-velocity of the fluidas u µ = ( ℓ, u i ), and u µ = ( − ℓ, u i ), respectively, and wekeep only the first order terms in u i . In the frameworkof these approximations the matter energy-momentumtensor T µν reads, T µν = ρℓ u µ u ν . (71)Assuming that λ = λ + δλ and ℓ = 1 + δℓ , where δλ and δℓ are perturbation variables, from Eq.(5) andEq.(20), we obtain the first-order perturbation of δλ and δℓ as δx µ ∂ µ λ = δλ = − (2 κ f Q + 2 λ ) w µ δx µ = − (2 κ f Q + 2 λ ) δℓ, (72)Using the above equation together with equation (71),one finds that the only non-vanishing component of theenerg-momentum tensor is T = ρ .It should be noted that in the Newtonian limit, onedecompose the tensor H µν asΦ = − H , Ψ = − H ii . (73)where Φ and Ψ are Newtonian potentials. Also we as-sume that the Weyl vector can be written as ω µ = (0 , ω i )where ω i ∼ O ( ǫ ).From these definitions, one can see that the tensor S µν in (22) becomes second order in perturbation variablesand does not contribute to the Newtonian limit of thetheory.Considering the metric field equations (21) at back-ground level, it follows that the background value of theLagrange multiplier λ should be constant, and also thecondition f (0) = 0, where by (0) we mean the backgroundvalue, must hold. By expanding the field equation (21) up to first orderin the perturbation parameters, and by using the back-ground constraints derived above, one can obtain the (00)and ( ii ) components of the field equations as2 λ ∆Φ − δλ = 6 λ ∇ i ω i + (1 + κ f (0) T ) ρ, (74) λ (∆Φ − − δλ = 9 λ ∇ i ω i − κ f (0) T ρ. (75)Also, from the off-diagonal components of the metricfield equation (21), we obtain, δλ = λ (Ψ − Φ) . (76)In order to close the system of dynamical equations, oneshould also consider the constraint equation ˜ R = 0, andalso the vector field equation (20) to first order in per-turbations. The constraint equation becomes∆Φ − ∇ i ω i , (77)and the divergence of the vector field equation becomes6 λ (∆Φ − ∆Ψ)= 13 (∆Φ − (cid:16) m + 12 λ + 12 κ f (0) Q (cid:17) . (78)Now, by substituting ∇ i ω i and ∆ δλ from equations(77) and (76) in the ( ii ) component of the metric fieldequation (75), one can see that f (0) T ≈
0. Now, solvingthe remaining equations (74) and (78) for the Newtonianpotentials, we find ∆Φ = 12 G eff ρ, (79)∆Ψ = 12 γG eff ρ, (80)where we have defined the generalized Newtonian gravi-tational constant G eff as G eff = 2 λ + 1 m + 12 κ f (0) Q ! , (81)and the PPN γ parameter as γ = m + 12 κ f (0) Q − λ m + 12 κ f (0) Q + 3 λ ) . (82)One can see from its definition that the generalizedNewton gravitational constant depends on the derivativeof the function f ( Q, T ) with respect to the trace of theenergy-momentum tensor T , on the mass of the Weylvector field, and on the background value of the Lagrangemultiplier.It is worth mentioning that the value γ = 1, whichcorresponds to the GR result, and which is confirmed by1the observations at the Solar System level, occurs when f (0) Q = − λ κ − m / κ . In this case we obtain λ =1 / G , where G is the Newtonian constant.The present analysis does show that the effects of thematter geometry couplings f ( Q, T ) do appear already inthe first-order perturbation of the theory. This is in factdifferent from the generalized teleparallel theory, wherethe first-order perturbation analysis does not reveal theextra degrees of freedom, and one should take into ac-count higher order perturbation analysis [117]. However,in order to explore the detailed dependence of the func-tion f ( Q, T ) on the trace of the energy-momentum tensor T , one should consider higher order perturbation analysisof the model.The extra force (44), can be decomposed into anenergy-momentum related component, and a geometryrelated component, f ρ ≈ h ρν ˜ ∇ µ T µν ρ − w µ u µ u ρ + ( ℓ ) ′ ℓ u ρ = F ρ − w µ u µ u ρ + ( ℓ ) ′ ℓ u ρ , (83)where we have defined the tensor F ρ ≡ h ρν ˜ ∇ µ T µν ρ , (84)which can be written explicitly with the use of Eq. (27)as, F ρ = − ℓ h ρν w µ T µν ρ + ℓ h ρν w ν Tρ − κ ℓ h ρν (1 + 2 κ f T ) ρ × (cid:20) ∇ ν ( ρf T ) + f T ∇ ν T + 2 T µν ∇ µ f T (cid:21) , (85)where u µ F µ = 0 should be always fulfilled. Using the def-inition of the energy-momentum tensor and also expand-ing the above expression to first order in perturbationvariables, one obtains F ρ = − ℓ ω ρ + ℓu ρ ω − κ ℓ κ f T h ρν ∇ ν f T , (86)where we have assumed that the energy density variesslowly in this limit. At first order in perturbationsone obtains, F = 0. Also note the second term inEq. (86) is non-vanishing only in the case of ρ = 0.Since in the following calculations we will use only the i components, we will omit this term. In the partic-ular case of 1 + 2 κ f T = 0, f T = − / κ , we findlim f T →− / κ ∇ ν f T / (cid:0) κ f T (cid:1) = 0, and thus we stillobtain for the extra-force the non-trivial expression F ρ = − ℓ ω ρ + ℓu ρ ω .The equation describing a world line in Weyl geometrywith extra force reads, u ν ∂ ν u µ + Γ µνσ u σ u ν = F µ + w ν u ν u µ − w µ u ν u ν + ( ℓ ) ′ ℓ u µ . (87) By adopting the linear approximation and the Newtonianlimit, the spatial component in Eq. (87) becomes d x i dτ = − ℓ∂ i Φ + ℓw i + F i ℓ + ( ℓ ) ′ ℓ u i . (88)This equation represents the generalization of the New-tonian equation of motion in the Weyl-type f ( Q, T ) grav-ity theory.
C. Tidal forces in Weyl type f ( Q, T ) gravity theory In a Weyl geometries, the properties of parallel trans-port of a vector along a geodesic line are preserved withthe important exception of the magnitude of a vectorchanging after transport. Similarly to Riemannian ge-ometry, in the Weyl geometry one can always find a setof tangent spaces generated by the four velocity u µ forevery point along a world line, where their axes (withthe same index) remain parallel, under parallel transportalong the world line. It should be noted that these fourvelocities are not necessarily normalized to one in everytangent space. We have already defined u µ u µ = − ℓ , andin this case we will not take ℓ to one. Here onward wewill consider geodesic reference frame in which all connec-tion components vanish, indicating that ˜Γ λµν = 0 pointwisely.We at first go back to Eq. (66) to evaluate the spa-tial total curvature tensor ˜ R i j , which is of significantimportant in the following part. Hence we obtain˜ R i j = R i j + ∇ j w i g + ∇ w g ij + w j w i g + w g ij − w g ji g ≈ R i j + η ∂ j w i = R i j − ∂ j w i , (89)where Γ λµν = − g µν w λ + δ λµ w ν + δ λν w µ is obtained fromthe condition of zero total connection. Raising the index i , we have ˜ R i j = R i j − ∂ j w i . (90)Hence the geodesic deviation equation Eq. (49) willbecome, u α ∂ α ( u β ∂ β η µ ) = − ˜ R µναβ η α u β u ν + η α ∂ α f µ (91)If we consider the linear approximation, the Newtonianlimit, and the zero total connection for this system, andwe also use η = 0 to indicate that the accelerations ofthe particles are compared at equal times, and f = 0,respectively, indicating that the thermodynamic param-eters of the matter do not depend on time, we obtain ℓ d η i dt = − ℓ ˜ R i j η j + η j ∂ j F i . (92)Now we write the explicit expression of the curvaturetensor in Weyl geometry by using Eq. (90) as, d η i dt = − ( R i j − ∂ j w i ) η j + 1 ℓ η j ∂ j F i , (93)2and hence we can reformulate Eq. (93) by introducing anewly defined tidal force vector F i , and the tidal matrix K ij , which has been modified by the matter-curvaturecoupling and Weyl geometry, d η i dt = F i = K ij η j , (94)with the explicit expression of K ij given by K ij = − R i j + ∂ j w i + 1 ℓ ∂ j F i , (95)where F i is the component of extra force, and has beendefined in Eq.(86). The contraction of the tidal matrixgives, K = K ii = ∂F i ∂η i = − R + ∂ i w i + 1 ℓ ∂ i F i , (96)and by using Eq. (69), (70, and (73), we obtain K ij = − ∂ Φ ∂x i x j + ∂ j w i + 1 ℓ ∂ j F i , (97)and the scalar K reads, K = − ∆Φ + ∂ i w i + 1 ℓ ∂ i F i , (98)Substituting the expression (86) for F i , one obtains K ij = − ∂ Φ ∂x i x j − κ ∂ j (cid:18) ∂ i f T κ f T (cid:19) , (99)and K = − ∆Φ − κ (1 + 2 κ f T ) × h (1 + 2 κ f T ) (cid:3) f T − κ ∂ i f T ∂ i f T i , (100)respectively. D. The Roche radius in Weyl type f ( Q, T ) gravity In Newtonian gravity, the spherical potential of a givenparticle with mass M is given byΦ( r ) = − M πr , (101)where we have assumed that 8 πG = 1. In a frameof reference with the x axis passing through the parti-cle’s position, indicating that the particle is located at( x = r, y = 0 , z = 0), the Newtonian tidal force tensor τ ij will be diagonal, and has only the following nonzerocomponents [109], τ ij = − ∂ i ∂ j Φ = diag (cid:18) M πr , − M πr , − M πr (cid:19) . (102) The components of the Newtonian tidal force F i can bewritten as F x = 2 M ∆ m x πr , F y = − M ∆ m y πr , F z = − M ∆ m z πr , (103)respectively [107].The Roche limit, an important astrophysical and astro-nomical concept, is defined as the closest distance r Roche that a cosmic object, having mass m , radius R m , anddensity ρ m , respectively, can come near a massive star ofmass M , radius R M and density ρ M , respectively, with-out being torn apart by the tidal gravity of the star. Inthe following we consider a simplified case with M ≫ m ,a condition which allows us to set the center of mass inthe geometrical center of the mass M .We consider a small object of mass ∆ m located at thesurface of the small body of mass m . The gravitationforce from the small mass acting on ∆ m is given by F G = m ∆ m πR m , (104)while the tidal force from the big massive body acting on∆ m is obtained as F = 2 M ∆ mR m πr , (105)where r is the distance between the centers of the twocelestial objects, and we have neglected the differencesin the distances between ∆ m and M , and ∆ M and m ,respectively. The Roche limit is reached if the two forcesacting on ∆ m are equal, F G = F . Thus we obtain theRoche limit in Newtonian gravity r Roche as r Roche = R m (cid:18) Mm (cid:19) = 2 R M (cid:18) ρ M ρ m (cid:19) . (106)With the use of Eq. (88) we obtain the modificationof the gravitational force in Weyl-type f ( Q, T ) gravity,which can be represented as F total = F gravity + F geometry + F Extraforce = F gravity − M ℓ πR m + ℓw r + F r ℓ + ( ℓ ) ′ ℓ u r , (107)where the index r indicates the radial components, F gravity is the Newtonian gravitational force, F geometry gives by modifications from geometry components be-yond Riemann geometry, and F Extraforce is the compo-nent generated by the geometry-matter coupling.Thus, in Weyl geometry, by using Eq. (97), the Rochelimit r Roche is obtained as " M πr Roche + ∂ r w r + 2 w r + 1 ℓ ∂ r F r R m = m πR m − M ℓ πR m + ℓw r + F r ℓ + ( ℓ ) ′ ℓ u r , (108)3and the vectors containing index r (no summation upon r ) must be evaluated in the coordinate system in whichthe Newtonian tidal tensor is diagonal. Considering thatthe gravitational effects due to the coupling between mat-ter and curvature are small as compared to the Newto-nian ones, we have r Roche ≈ R m (cid:18) Mm (cid:19) × " M ℓ m + 8 πR m m (cid:18) ∂ r w r + 1 ℓ ∂ r F r (cid:19) − πR m m (cid:18) ℓw r + F r ℓ + ( ℓ ) ′ ℓ u r (cid:19) . (109)Substituting F r from Eq. (86) one finally obtains r Roche ≈ R m (cid:18) Mm (cid:19) × " M ℓ m + 16 πκ R m m ∂ r (cid:18) h rr ∂ r f T κ f T (cid:19) +8 πR m m (cid:18) κ ℓ κ f T h rr ∂ r f T − ( ℓ ) ′ ℓ u r (cid:19) . (110) VI. DISCUSSIONS AND FINAL REMARKS
Abandoning the metricity conditions, and includingnonminimal curvature-matter couplings are some promis-ing ways to modify standard general relativity, and toexplain the major challenges present day gravity theoriesface. A possible geometric avenue for the generalizationof general relativity is represented by the so-called sym-metric teleparallel gravity theory [75], and by its exten-sions [76, 97, 98]. In particular, the role of matter andof the geometry-matter couplings have been analyzed in[99] and [104], respectively. In the present paper we haveextended the previous analyses of the Weyl type f ( Q, T )gravity, a particular version of the general f ( Q, T ) typetheories, by developing some basic theoretical tools thatwould allow not only to further investigations of the fun-damental geometrical and physical properties of thesegravity theories, but can also open the possibility of theirobservational testing.More exactly, from the fundamental point of view ofthe analysis of the Weyl type f ( Q, T ) theories, we haveobtained two of the basic equations of the gravitationalphysics, namely, the geodesic deviation equation, and theRaychaudhuri equation, respectively. The geodesic de-viation equation geodesic describes the way objects ap-proach or recede from one another when moving underthe influence of a spatially varying gravitational field.One of the important applications of the geodesic devia-tion equation is in the study of the tidal forces, which in modified theories of gravity acquire some extra terms dueto the presence of the new terms that modify the gravi-tational interaction. Hence the geodesic equation can beused to observationally test the Weyl type f ( Q, T ) grav-ity model through the observations of the effects of thetides produced by an extended mass distribution. Tidaleffects play an important role in the eccentric inspirallingneutron star binaries [111]. The neutron stars can bemodelled as a compressible ellipsoid, which can deformnonlinearly due to tidal forces, while the orbit evolutioncan usually be described with the post-Newtonian the-ory. In general, the tidal interaction can accelerate theinspiral, and cause orbital frequency and phase shifts.Tidal interactions have an essential effect on the star for-mation in galaxies, since tidal perturbations induced byclose companions increase the gas accretion rates [112].By using gravitational wave detector networks one canconstrain the equation of state of binary neutron-stars,and extract their redshifts through the imprints of tidaleffects in the gravitational waveforms [113]. The exis-tence of light, fundamental bosonic fields is an attractivepossibility that can be tested via black hole observations.The effect of a tidal field caused by a companion star orblack hole on the evolution of superradiant scalar-fieldstates around spinning black holes can test the existenceof light bosonic fields [114]. For large tidal fields thescalar condensates are disrupted, and the impact of tidescan be relevant for known black-hole systems such as theone at the center of our galaxy or the Cygnus X-1 system.The companion of Cygnus X-1 will disrupt possible scalarstructures around the black hole for large gravitationalcouplings. Tidal effects in massless scalar-tensor theorieswere considered in [115], where a new class of scalar-typetidal Love numbers. It turns out that in a system domi-nated by dipolar emission, tidal effects may be detectableby LISA or third generation gravitational wave detectors.Another astrophysical situations in which the effects ofthe tides are of major importance are perturbations of theOort cloud by the galactic field, globular clusters evolv-ing under the influence of the galactic mass distribution,and galactic encounters [116]. As we have seen in ouranalysis of the F ( q, t ) gravity, the curvature-matter cou-pling significantly modifies the nature of the tidal forces,as well as the equation of motion in the Newtonian limit.Therefore, the comparison of the theoretical predictionsof the Weyl type f ( Q, T ) gravity about the modificationsof the tidal forces with the observational evidences, com-ing from a large class of astrophysical phenomena couldgive, at least in principle, some insights into the funda-mental aspects of the gravitational interaction, and itsgeometric description.We have also obtained the generalization of the Pois-son equation, describing the properties of the gravita-tional potential. The Poisson equation, and its solution,is an important tool in the investigation of many gravi-tational effects involving small velocities and low matterdensities. The modifications of the gravitational poten-tial, and the new terms appearing in the equation may4provide a theoretical explanation for the observed dy-namics of the particles moving on circular orbits aroundgalaxies. These observations are usually explained bypostulating the existence of dark matter, a mysteriousmajor component of the Universe, which has not beendetected yet. Hence the novel geometric effects inducedby the Weyl-type f ( Q, T ) gravity may provide a geomet-ric explanation for the galactic dynamics of test particleswithout having to resort to the dark matter hypothesis.In present paper we have obtained the equation of mo-tion of the particles in the Weyl-type f ( Q, T ) gravity, andwe have discussed it in detail. Note that the extra forcehas two components of different origins, coming from thematter distribution and from the geometrical propertiesof the space-time. The geometry provides an extra de-gree of freedom, with the nonminimal curvature-mattercoupling also generating more degrees of freedom for thegravitational interaction. Hence these extra degrees offreedom contribute with new terms to the extra force,the tidal force, and the Roche limit. These extra termsmay have observational (and even experimental) effects,which can be used to test the theoretical gravity modelwe have investigated in this paper.The Raychaudhuri equation is of major importancein the investigation of the space-time singularities, andin construction of cosmological models. For the sake ofcompleteness we briefly mention some cosmological ap-plications of our results. Let’s consider a flat Friedmann-Lemaitre-Robertson-Walker Universe, with metric givenby ds = − dt + a ( t ) δ ij dx i dx j , where a ( t ) is the scalefactor. We take ℓ = 1, and we adopt a co-moving ref-erence system, with u µ = (1 , , , ′ = u µ ˜ ∇ µ = u ˜ ∇ , and, whenapplied on a scalar, we have ′ = d/dτ = d/dt . We alsointroduce the Hubble function H = (1 /a ( t )) da ( t ) /dt , de-scribing the rate of change of a ( t ) with respect to time.We consider a general model for the cosmological non-metricity, which was introduced in [118], and accordingto which Q λµν = A ( t ) u λ h µν + B ( t ) h λ ( µ u ν ) + C ( t ) u λ u µ u ν ,where A , B , C are time-dependent functions representingthe behaviour of non-Riemannian degrees of freedom and h µν is the projection tensor previously defined. With theWeyl vector defined in Eq. (13), only one extra degreesof freedom is added by the nonmetricity, By assuming A = − C , and by taking B = 0, we obtain for the cosmo-logical nonmetricity the expression Q λµν = Au λ g µν = − w λ g µν , (111)giving for the Weyl vector the simple expression w λ = − A u λ (112)By taking advantage of Eq.(83), we obtain for the extraforce the expression, f ρ = F ρ − A u ρ . (113) We also notice that under conditions that we took for-merly in this approach, from a geometrical point of viewwe have, f ρ = u µ ˜ ∇ µ u ρ = − A u ρ , (114)leading to F ρ = 0. This condition is a direct consequenceof the use of the co-moving reference frame. In the labo-ratory reference system the extra-force does not vanish.If we keep the terms containing F ρ , the Raychaudhuriequation in the Weyl spacetime, given by Eq. (63), cannow be written as (see Appendix A 8 for the calculationaldetails),3 H ′ + 3 H = 3 A ′ A H − ˜ R µν u µ u ν + ˜ ∇ µ F µ , (115)and it can also be reformulated into a equation of a ( t ), a ′′ a = A ′ Aa ′ a −
13 ˜ R µν u µ u ν + ˜ ∇ µ F µ , (116)which represents the generalized cosmic accelerationequation. Considering the fact that F ρ = 0 in theadopted coordinate system, and after substituting theexpression of ˜ R µν u µ u ν , one can see that the above equa-tion is identically satisfied. The most general form of theacceleration equation in the presence of torsion and non-metricity has been obtained, in a Friedmann-lemaitre-Robertson-Walker geometry, with the help of the gen-eralized Raychaudhuri equation with torsion and non-metricity, in [118]. Cosmological Hyperfluids, repre-senting fluids with intrinsic hypermomentum that in-duce spacetime torsion and non-metricity, were studiedin [119], where the most general form of the Friedmannequations with torsion and non-metricity were also ob-tained.To conclude, the present investigation of some funda-mental aspects of the Weyl type f ( Q, T ) gravity opensfurther possibilities for the theoretical, observational andeven experimental study of the alternative purely geo-metrical theories of gravity in the presence of the couplingbetween geometry and matter. Moreover, the results ob-tained in the present paper may also be relevant for otherclasses of modified gravity theories.
ACKNOWLEDGMENTS
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Appendix A: Calculational details for some basicresults
In this Appendix we present explicitly some of the cal-culational details and intermediate steps that were usedin the derivation of the basic results in the main text ofour paper.
1. Obtaining the expression of the scalarnonmetricity Q One can obtain L λµν explicitly in Weyl geometry by itsdefinition Eq.(16), as L νµν = w λ g µν − w µ δ λν − w ν δ λµ . (A1)7Then with the use of Eq.(15) we obtain the scalar non-metricity Q as follows, Q = − g µν (cid:20) (cid:0) w β δ αµ + w µ δ αβ − w α g βµ (cid:1) × (cid:0) w α δ βν + w ν δ βα − w β g να (cid:1) − (cid:0) w β δ αα + w α δ αβ − w α g αβ (cid:1) × (cid:0) w µ δ βν + w ν δ βµ − w β g µν (cid:1) (cid:21) = 6 w µ w µ . (A2)
2. The explicit expressions of the covariantderivative in Weyl geometry
We will present below the explicit expression of thecovariant derivative in Weyl geometry of certain types oftensors. For vectors,˜ ∇ µ A λ = g µν ˜ ∇ ν A λ = g µν ( ∂ ν A λ + ˜Γ λσν A σ )= ∇ µ A λ + w λ A µ − w µ A λ − g µλ w σ A σ , (A3)˜ ∇ µ A λ = g µν ˜ ∇ ν A λ = g µν ( ∂ ν A λ − ˜Γ σλν A σ )= ∇ µ A λ − δ µλ w σ A σ + w µ A λ + w λ A µ . (A4)For second order tensors we obtain,˜ ∇ λ A µν = g λσ ˜ ∇ σ A µν = g λσ ( ∂ σ A µν + ˜Γ µσα A αν + ˜Γ νσα A µα )= ∇ λ A µν + w µ A λν + w ν A µλ − g λµ w α A αν − g λν w α A µα − w λ A µν , (A5)˜ ∇ λ A µν = g λσ ˜ ∇ σ A µν = g λσ ( ∂ σ A µν − ˜Γ ασµ A αν − ˜Γ ασν A µα )= ∇ λ A µν − δ λµ w α A αν − δ λν w α A µα + w µ A λν + w ν A λµ + 2 w λ A µν , (A6)˜ ∇ λ A µν = ˜ ∇ λ ( g σν A µσ ) = Q λσν A µσ + g σν ˜ ∇ λ A µσ = ∇ λ A µν + w µ A λν + w ν A µλ − g λµ w α A αν − δ λν w α A µα − w λ A µν . (A7)
3. The perfect fluid model in Weyl geometry
We should first notice that since the magnitude ofthe four-velocity in Weyl geometry is not preserved, and u µ u µ = − ℓ , the perfect fluid model must be generalizedto include the effects ℓ . Generally, a perfect fluid canbe characterized by its four-velocity u , and the thermo-dynamic quantities - the proper density ρ , the isotropicpressure p , the temperature T , and the specific entropy s ,or the specific enthalpy ω = ( p + ρ ) /n [106] (in this part ω is defined independently from the main body). Here n is the conserved baryon number density, and n does notchange its magnitude during parallel transport. We also introduce the particle number density four-vector n µ , de-fined as n µ = n √− gu µ . (A8)Consequently, n = r g µν n µ n ν ℓ g , (A9)where g is the determinant of the matrix g µν . Next weneed to introduce the matter Lagrangian, which we as-sume as depending only on the energy density scalar thein local rest frame of the fluid, L m = − ρ. (A10)The energy-momentum tensor of the fluid is given by T µν = − ∂ L m /∂g µν + g µν L m under the constraints [106] δs = 0 , δn µ = 0 . (A11)From the thermodynamical relation, (cid:18) ∂ρ∂n (cid:19) = ω, (A12)we immediately obtain δρ = ωδn .Using Eqs. (A8,A9,A12), we obtain δn = 12 n (cid:18) n µ n ν ℓ g δg µν − n µ n ν g µν ℓ g δg (cid:19) = n (cid:18) − u µ u ν ℓ δg µν + u µ u µ ℓ g δg (cid:19) . (A13)By using the basic properties of the metric variation, δg µν = − g µλ g νσ δg λσ , (A14) δgδg µν = − g αµ g βν δgδg αβ = − gg µν , (A15)we obtain δn = n (cid:18) u µ u ν ℓ + g µν (cid:19) δg µν . (A16)The derivative of the matter Lagrangian with respect tothe metric tensor is given by ∂ L m ∂g µν = − nω (cid:18) u µ u ν ℓ + g µν (cid:19) , (A17)and hence we finally obtain the energy-momentum tensorof the perfect fluid model in Weyl geometry as, T µν = p + ρℓ u µ u ν + pg µν . (A18)8
4. The details of the calculation of the extra forcein Weyl geometry
The divergence of the matter energy-momentum tensorin Weyl geometry is obtained as follows. We first obtain˜ ∇ µ T µν = ∇ µ T µν − w µ T µν + w µ T = ˜ ∇ µ p + ˜ ∇ µ ρℓ u µ u ν − p + ρℓ u µ u ν ˜ ∇ µ ℓ + p + ρℓ ( u µ ˜ ∇ µ u ν + u ν ˜ ∇ µ u µ ) + g µν ˜ ∇ µ p + p ˜ ∇ µ g µν = p ′ + ρ ′ ℓ u ν − p + ρℓ ( ℓ ) ′ u ν + p + ρℓ ( a ν + u ν ˜ ∇ µ u µ ) + ˜ ∇ ν p + pQ µµν . (A19)Then after multiplication with the projection tensor wehave h ρν ˜ ∇ µ T µν = h νρ (cid:18) p + ρℓ a ν + ˜ ∇ ν p + pQ µµν (cid:19) = p + ρℓ ( A ρ − Q µνρ u µ u ν ) + p + ρℓ a µ u µ u ρ + h ρν ˜ ∇ ν p + h νρ Q µµν p = p + ρℓ ( A ρ − Q µνρ u µ u ν ) − p + ρ ℓ (cid:20) ( ℓ ) ′ + Q µνλ u µ u ν u λ (cid:21) u ρ + h ρν ˜ ∇ ν p + h νρ Q µµν p. (A20)
5. Calculational details of the derivation of thegeodesic deviation equation
The total derivative in Weyl geometry of the term u ν ˜ ∇ ν U µ is given by, u ν ˜ ∇ ν U µ = u ν ( ∇ ν U µ + w µ U ν − w ν U µ − δ µν w α U α ) , (A21)and U ν ˜ ∇ ν u µ = U ν ( ∇ ν u µ + w µ u ν − w ν u µ − δ µν w α u α ) , (A22)respectively. Note that from the definition ∂U µ /∂λ = ∂u µ /∂σ and w ν U [ µ u ν ] = 0, in Weyl space we obtain u ν ˜ ∇ ν U µ = U ν ˜ ∇ ν u µ .
6. Details of the calculations in the derivation ofthe Raychaudhuri equation
Some important steps an intermediate results in thederivation of the Raychaudhury equation are as follows: θ = g µν ˜ ∇ ν u µ = ∇ µ u µ − w µ u µ , (A23) a µ = u α ∇ α u µ − ℓ w µ = f µ + 2 w ν u ν u µ , (A24) ξ µ = u α ∇ µ u α − ℓ w µ , (A25) ∇ µ ( u α u α ) = −∇ µ ( ℓ ) = 2 u α ∇ µ u α = 2 u α ∇ µ u α , (A26) σ µν = 12 ( ∇ ν u µ + ∇ µ u ν ) − ∇ α u α g µν +23 ℓ u µ u ν u α u β ∇ α u β + 12 ℓ ( u ν u α ∇ α u µ + u µ u α ∇ α u ν + u µ u α ∇ ν u α + u ν u α ∇ µ u α ) − ℓ ( u α u β ∇ α u β g µν + u µ u ν ∇ α u α ) , (A27) ω µν = 12 ( ∇ ν u µ − ∇ µ u ν ) + 12 ℓ ( u ν u α ∇ α u µ − u µ u α ∇ α u ν + u µ u α ∇ ν u α − u ν u α ∇ µ u α ) . (A28)
7. The Riemann and Ricci tensors in the weak-fieldapproximation
In the weak field approximation the Riemann and theRicci tensors can be obtained as follows R α β = g αρ R ρ β = R i l , (A29) R α β = g ρ R ρα β = − R α β , (A30) R lm = R l m + δ ji R iljm = − R l m + R ilim . (A31)
8. Calculations of the cosmological terms in theflat FLRW spacetime
In a flat FLRW geometry the expressions of the rele-vant cosmological quantities can be obtained as follows: ∇ µ w µ = 1 √− g ∂ ( √− gw ) ∂x = − A ′ − A H, (A32) θ = 1 √− g ∂ ( √− gu ) ∂x − A = 3 H − A, (A33) R µν u µ u ν = R ( u ) = − H ′ + H ) , (A34) W µν = ∇ ν w µ − ∇ µ w ν = ∂ ν w µ − ∂ µ w ν = 0 , (A35) σ µν = 0 = ω µν , (A36)Γ µµ = 12 g µν ( g νµ, + g ν ,µ − g µ ,ν )= 12 g µµ ( g µµ, + g µ ,µ − g µ ,µ )= 12 g µµ ∂ g µµ ..