Dark Gravitational Field on Riemannian and Sasaki Spacetime
aa r X i v : . [ g r- q c ] J a n Dark Gravitational Field on Riemannian and Sasaki Spacetime
Panayiotis Stavrinos ∗ and Christos Savvopoulos † Department of Mathematics, National and Kapodistrian University of Athens, Athens, 15784, Greece Department of Physics, National and Kapodistrian University of Athens, Athens, 15784, Greece
The aim of this paper is to provide the geometrical structure of a gravitational field that includesthe addition of dark matter in the framework of a Riemannian and a Riemann–Sasaki spacetime. Bymeans of the classical Riemannian geometric methods we arrive at modified geodesic equations, tidalforces, and Einstein and Raychaudhuri equations to account for extra dark gravity. We further examinean application of this approach in cosmology. Moreover, a possible extension of this model on thetangent bundle is studied in order to examine the behavior of dark matter in a unified geometricmodel of gravity with more degrees of freedom. Particular emphasis shall be laid on the problem ofthe geodesic motion under the influence of dark matter.
Keywords: Riemannian geometry; Sasaki metric; dark matter; geodesics; deviation of geodesics; Einstein equations;Raychaudhuri equations
I. INTRODUCTION
Recent advances in the theoretical and observational field of cosmology have shown the significance of the existenceof dark matter and dark energy [1–25]. There is considerable evidence that most of the mass in the universe is neitherin the luminous matter in galaxies, nor in the radiation detected so far. Mass can be detected by its gravitationalinfluence even if it cannot be seen directly [26]. Evidence for the existence of dark matter comes for example fromthe study of gravitational lensing, cosmic microwave background radiation (CMB) or the rotational curves of spiralgalaxies [27, 28].According to observational results, dark matter plays a dominant role in the field of evolution and accelerationof the universe. There are many pieces of evidence that the visible matter and detectable radiation comprise only asmall fraction of the mass in the universe, perhaps as little as a few percent [26]. Therefore the study of dark matteris central for cosmology. A geometric model for gravity with dark matter based on a spacetime manifold endowedwith a Riemannian metric is necessary, because of the existence of contributions to gravity, curvature, tidal forces etc.due to dark matter whose gravitational interaction could be described using a Riemmanian geometric frameworksimilar to that of ordinary matter as shall be assumed throughout this work.These contributions can be observed in the velocities of nearby trajectories (geodesics) of particles (or clustersthereof, e.g., planets), in the orbital motions of member galaxies in galaxy clusters or in the relative change of velocityof the deviation vector which shall now be altered to account for the existence of extra “dark” gravity. The abovementioned combined with observational data give rise to a question regarding whether or not one accepts theassumption that test bodies move on geodesics of a given inertial metric of the base spacetime [29, 30]; in other words,whether or not the geodesics remain the same during the evolution of the universe with the addition of dark matter.In a more general geometric framework, the tangent bundle of a spacetime manifold, can extend the limits ofa “unified” gravitational field in more than four dimensions. The development of the metric geometry of tangentbundles first began with the introduction of the natural Sasaki metric, in the fundamental paper of Sasaki publishedin 1958 [31], which we shall assume for a further geometric model with dark matter. This choice proves crucial asthe tangent bundle allows for a gravitational field with more degrees of freedom. Dark matter could cause extragravitational influence on all scales, which can be illustrated well in extra dimensions [32–36]. A geometric frame thatpotentially materializes this concept is, therefore, that of Sasaki, where the underlying metric structure is Riemannian.This work is organized as follows: in Section II, we examine the gravitational field with the addition of extradark gravity in a Riemannian spacetime setting. In particular, in Section II A we provide the geodesic equationsand their deviation, in Section II B we study the Einstein equations and in Section II C the Raychaudhuri equation.In addition, in Section II D we derive the Friedmann equations and the continuity equation as an application usingthe F ( R ) gravity model for dark matter. In Section III, we extend our study of the dark and ordinary gravitationalfield on the tangent bundle of a Sasaki spacetime. In Section III A we derive the equations of geodesic deviation on ∗ Electronic address: [email protected] † Electronic address: [email protected] the Sasaki tangent bundle, and in Section III B we provide the foundations for an extension of dark gravity on thetangent bundle. Finally, in Section IV we summarize the results of this work and in Appendix A we present somefurther geometric results.
II. DARK GRAVITY IN THE RIEMANNIAN SPACETIME
Let us consider a (pseudo-)Riemannian 4-dimensional spacetime M containing both matter and dark matterequipped with a metric g . For the purposes of our study, an additive relation for the metric tensor is assumed, suchthat the contributions to the metric of ordinary and dark matter can be viewed separately.Let ds = g ij ( x ) dx i dx j (1)be the metric of this 4-dimensional spacetime, where we assume that the unified metric g ij ( x ) = g ( O ) ij ( x ) + g ( D ) ij ( x ) (2)where g ( O ) ij ( x ) is the sectoral metric of ordinary matter and g ( D ) ij ( x ) that of dark matter .Due to the geometry of the space that has been chosen, we have to use the unified metric tensor, g ij , for suchoperations as raising, lowering and contracting indices of tensors. Since, in our study, we deal with tensors relatedto ordinary or dark matter’s spacetime, we shall attempt to relate such concepts within the framework of the unifiedgeometric space. From a physical perspective, a unified framework of gravity which includes the gravitationalinteraction of ordinary and dark matter is necessary in order to describe the gravitational effects of the large scaleuniverse structures (e.g., to explain the rotation of galaxies or the motions of clusters). A. Geodesics and Tidal Forces
First, we find that the Chistoffel symbols of first kind are: Γ ijk = Γ ( O ) ijk + Γ ( D ) ijk (3)where, Γ ( O ) ijk = (cid:18) ∂ g ( O ) ij ∂ x k + ∂ g ( O ) ik ∂ x j − ∂ g ( O ) jk ∂ x i (cid:19) (4) Γ ( D ) ijk = (cid:18) ∂ g ( D ) ij ∂ x k + ∂ g ( D ) ik ∂ x j − ∂ g ( D ) jk ∂ x i (cid:19) (5)are the Christoffel symbols of first kind of a space occupied exclusively by ordinary or dark matter, respectively .The geodesics will then be given by the well-known equation, which needs to be modified to account for theexistence of dark matter as d x i dt + ( Γ ( O ) ijk + Γ ( D ) ijk + γ ijk ) dx j dt dx k dt = (6)since the Christoffel symbols of second kind shall have the following form: Γ ijk = Γ ( O ) ijk + Γ ( D ) ijk + γ ijk (7) For all three metric tensors a metric signature ( − ,+,+,+) shall be assumed in their respective space. One must be careful that Γ ( O ) ijk and Γ ( D ) ijk do not function as Christoffel symbols for the unified space ( M , g ) . where t is an affine parameter and Γ ( O ) ijk = g ( O ) il (cid:18) ∂ g ( O ) lj ∂ x k + ∂ g ( O ) lk ∂ x j − ∂ g ( O ) jk ∂ x l (cid:19) (8) Γ ( D ) ijk = g ( D ) il (cid:18) ∂ g ( D ) lj ∂ x k + ∂ g ( D ) lk ∂ x j − ∂ g ( D ) jk ∂ x l (cid:19) (9)are the Christoffel symbols of ordinary and dark matter’s metric contribution, respectively, and γ ijk = Γ ijk − ( Γ ( O ) ijk + Γ ( D ) ijk ) (10)is the interaction part of the Christoffel symbols since it could be shown that γ ijk represents the interaction betweenthe ordinary and dark matter gravitational potentials and their respective intensities .In view of relation (6), one can see that for a test particle m i moving along geodesics in the spacetime of our modelthe following relation holds: m i (cid:18) d x i dt + Γ ( O ) ijk dx j dt dx k dt (cid:19) = − m i (cid:18) Γ ( D ) ijk + γ ijk (cid:19) dx j dt dx k dt = F i ( D ) (11)Seemingly, the left-hand side of Equation (11) does not give a geodesic in the Riemannian sense since the right-handside does not vanish unless we consider dark matter to be absent. The dark gravitational field and its interaction withordinary matter influences the geodesics in a dominant way and give rise to a dark pseudo-force field F i ( D ) which canbe interpreted as a gravitational source for inertial force fields of interaction by an observer who does not take intoaccount the existence of dark matter in his considerations. The addition of dark matter also influences the curvaturetensor of the unified space, R abcd , which assumes the form shown in the following relation: R abcd = R ( O ) abcd + R ( D ) abcd + r abcd (12)where, R ( O ) abcd = ∂ Γ ( O ) abd ∂ x c − ∂ Γ ( O ) abc ∂ x d + Γ ( O ) ace Γ ( O ) ebd − Γ ( O ) ade Γ ( O ) ebc (13)is ordinary matter’s Riemann tensor, R ( D ) abcd = ∂ Γ ( D ) abd ∂ x c − ∂ Γ ( D ) abc ∂ x d + Γ ( D ) ace Γ ( D ) ebd − Γ ( D ) ade Γ ( D ) ebc (14)is dark matter’s curvature tensor and, r abcd = ∂γ abd ∂ x c − ∂γ abc ∂ x d + Γ ( O ) ae [ c Γ ( D ) ed ] b + Γ ( D ) ae [ c Γ ( O ) ed ] b + Γ ( O ) ae [ c γ ed ] b + Γ ( D ) ae [ c γ ed ] b + γ ae [ c Γ ( O ) ed ] b + γ ae [ c Γ ( D ) ed ] b + γ ae [ c γ ed ] b (15)is the interaction part of the Riemann tensor , .Similarly, the equation of geodesic deviation needs to be modified, as well. It will then take the following form: δ t ( δ t W a ) = − ( R ( O ) abcd + R ( D ) abcd + r abcd ) V b W c V d (16) These γ ijk symbols could be explicitly calculated using [37]. The lower-indices [ cd ] denote an antisymmetrization (similarly, indices between parentheses shall denote symmetrization). One must be careful that for the unified space ( M , g ) there is but one curvature tensor; the unified R abcd . where, δ t W a = ∂ W a ∂ t + ( Γ ( O ) abc + Γ ( D ) abc + γ abc ) W b V c (17) V i = ∂ t x i , W i = ∂ s x i and x is ( t ) is a two-parameter geodesic family with t being the affine parameter and s the selectorparameter [26].In view of relations (12) and (16), the extra terms of the product of the Christoffel symbols of ordinary and darkmatter represent the coupling of the intensities of their respective gravitational fields, i.e., an interaction whichhas been instigated by γ ijk and propagated by the curvature tensor in the deviation of geodesics. This shows thatthe interaction between ordinary and dark matter has been incorporated in the geometry of the spacetime due tocurvature being an intrisic geometric property of space. Thus, the interaction of the gravitational field intensities isnow being manifested in the Deviation Equation (16) and therefore in the tidal forces. This means that the geodesicswe understand are deviated by the unified curvature including both dark and ordinary matter, instead of the geodesicmotion in the ordinary spacetime as we have already noticed in rel. (11). B. Einstein Equations
We shall now present the Riemann curvature tensor in its covariant form since it shall prove useful later on.By lowering the upper index in Equation (12) we get: R abcd = R ( O ) abcd + R ( D ) abcd + ̺ abcd (18)where, R ( O ) abcd = g ( O ) ae R ( O ) ebcd (19)is ordinary matter’s covariant Riemann tensor, R ( D ) abcd = g ( D ) ae R ( D ) ebcd (20)is dark matter’s covariant curvature tensor and, ̺ abcd = g ( D ) ae R ( O ) ebcd + g ( O ) ae R ( D ) ebcd + g ae r ebcd (21)is the interaction part of the tensor.By virtue of Equation (18), we can now find the Ricci curvature: R ab = R ( O ) ab + R ( D ) ab + ξ ab (22)where, R ( O ) ab = g ( O ) cd R ( O ) cadb (23)is ordinary matter’s Ricci curvature tensor, R ( D ) ab = g ( D ) cd R ( D ) cadb (24)is dark matter’s Ricci curvature tensor and, ξ ab = R ab − ( R ( O ) ab + R ( D ) ab ) (25)is the interaction part of the Ricci tensor. For such operations we must always use the unified metric.
Using the form of the Ricci tensor given in Equation (22), we can find the Ricci scalar: R = R ( O ) + R ( D ) + κ (26)where, R ( O ) = g ( O ) ab R ( O ) ab (27)is ordinary matter’s Ricci scalar, R ( D ) = g ( D ) ab R ( D ) ab (28)is dark matter’s Ricci scalar and, κ = R − ( R ( O ) + R ( D ) ) (29)is the interaction part of the Ricci scalar.The actions of the sectoral ordinary and dark matter gravitational fields and matter are, respectively: S O = Z (cid:18) π R ( O ) + L M O (cid:19)q − g ( O ) d x (30)and S D = Z (cid:18) π R ( D ) + L M D (cid:19)q − g ( D ) d x (31)where L M O and L M D describe the sectoral ordinary and dark matter fields, g ( O ) = det g ( O ) ij and g ( D ) = det g ( D ) ij , M O and M D are the ordinary and dark matter masses, respectively . It must be noted that with the addition of darkmatter to our model, one must consider the action of the unified gravitational field and of the total matter , whichshall be S = Z (cid:18) π R + L M (cid:19)p − g d x (32)where L M describes the unified matter fields, g = det g ij and the total matter is M = M O + M D . By varying thisaction and by following the standard procedure of deriving the field equations we obtain the following unifiedEinstein equations: R ab + Λ g ab − Rg ab = π T ab (33)where Λ is the cosmological constant and T ab is the unified energy-momentum tensor. Using Equations (2), (22) and(26) in (33), we obtain the Einstein equation in the following modified form given below : (cid:0) R ( O ) ab + Λ g ( O ) ab − R ( O ) g ( O ) ab (cid:1) + (cid:0) R ( D ) ab + Λ g ( D ) ab − R ( D ) g ( D ) ab (cid:1) + (cid:0) ξ ab − κ g ab − R ( O ) g ( D ) ab − R ( D ) g ( O ) ab (cid:1) = π GT ab (34)where, R ( O ) ab + Λ g ( O ) ab − R ( O ) g ( O ) ab = π GT ( O ) ab (35) We will refrain from using specific Lagrangians neither for the ordinary nor for the dark matter sector due to the existence of a plethora ofpotential Lagrangians for ordinary matter and a possible need for a complicated Lagrangian in order to effectively reproduce the dark sectorphenomenology [38]. One can clearly see that S = S O + S D . We assume c = . R ( D ) ab + Λ g ( D ) ab − R ( D ) g ( D ) ab = π GT ( D ) ab (36)and ξ ab − κ g ab − R ( O ) g ( D ) ab − R ( D ) g ( O ) ab = π G τ ab (37)One can clearly observe that Equation (35) is the Einstein equation for ordinary matter, with T ( O ) ab being the energy-momentum tensor of ordinary matter. The corresponding relation for dark matter is also true as can be seen fromEquation (36), where T ( D ) ab plays the role of dark matter’s energy-momentum tensor.On the other hand, Equation (37) is a relation containing terms that hint towards gravitational interaction betweenordinary and dark matter. The tensorial object τ ab acts as the interaction part of the unified energy-momentum tensorsince T ab = T ( O ) ab + T ( D ) ab + τ ab (38)As we shall see, the Einstein Equations (34) yield interesting results pertaining to the gravitational interaction ofdark and ordinary matter. By taking the trace of Equation (34) we obtain the following result: κ = Λ − π T − R ( O ) − R ( D ) (39)where T is the trace of the unified energy-momentum tensor. C. Raychaudhuri Equation
The Raychaudhuri equation which constitutes an extension of the geodesic deviation equation, plays a significantrole in relativity theory and cosmology due to its connection with singularities [39–42]. It governs the behavior andevolution of a family of test particles moving in world lines with given certain variables ( θ , σ , ω ), where θ measuresthe rate of change of the cross-sectional area enclosing a family of geodesics , the shear σ measures anisotropy, and ω shows a rotation. The form of the equation depends on the geometry of the spacetime; if a geodesic motion of afamily of test particles converges, the expansion θ is negative, and diverges when θ is positive.We shall henceforth consider that both ordinary and dark matter can be considered as perfect fluids, each in-dividually and together as a unified matter perfect fluid comprising the sum of ordinary and dark matter perfectfluids [43–45]. The stress-energy tensor for the perfect fluid distribution of the unified matter (ordinary and dark)with pressure p and energy density ρ falling under its own gravity with 4-velocity u a is then known to be: T ab = ( ρ + p ) u a u b + pg ab (40)However, the unified fluid is composed of two-perfect fluids [46, 47] the stress-energy tensor and shall thereforeassume the following form [48, 49]: T ab = ( ρ ( O ) + p ( O ) ) u ( O ) a u ( O ) b + p ( O ) g ab + ( ρ ( D ) + p ( D ) ) u ( D ) a u ( D ) b + p ( D ) g ab (41)where p ( O ) , p ( D ) is the pressure of the ordinary and dark matter fluids with energy density ρ ( O ) and ρ ( D ) and4-velocity u ( O ) a and u ( D ) a , respectively.Using our assumption (2) Equation (41) becomes: T ab = ( ρ ( O ) + p ( O ) ) u ( O ) a u ( O ) b + p ( O ) g ( O ) ab + ( ρ ( D ) + p ( D ) ) u ( D ) a u ( D ) b + p ( D ) g ( D ) ab + p ( O ) g ( D ) ab + p ( D ) g ( O ) ab (42)One can therefore see that for a perfect fluid and in accordance with Equation (38) T ( O ) ab = ( ρ ( O ) + p ( O ) ) u ( O ) a u ( O ) b + p ( O ) g ( O ) ab (43) T ( D ) ab = ( ρ ( D ) + p ( D ) ) u ( D ) a u ( D ) b + p ( D ) g ( D ) ab (44) τ ab = p ( O ) g ( D ) ab + p ( D ) g ( O ) ab (45) The same apply to any curves in general.
We further define the following four operators acting on a vector, e.g., u a (or similarly on a tensor) of ( M , g ) ,as follows: u a | k = ∂ u a ∂ x k + u b Γ ( O ) abk (46)is the covariant derivative with respect to the ordinary matter subspace, in other words, the covariant derivative of aspace comprised of purely ordinary matter .Similarly, u a || k = ∂ u a ∂ x k + u b Γ ( D ) abk (47)is the covariant derivative w.r.t. the dark matter subspace . u a \ k = ∂ u a ∂ x k − u b γ abk (48)will be an operation that resembles a covariant derivative but uses the interaction symbols, γ abc . Finally, the unifiedspace covariant derivative is given by : u a ; k = ∂ u a ∂ x k + u b Γ abk = u a | k + u a || k − u a \ k (49)Using this covariant derivative in the rest system of a perfect matter fluid (40) the conservation of the energy-momentum tensor is considered T ab ; b = (50)Assuming that the velocity u corresponding to the total matter fluid (ordinary and dark) is given by u = u ( O ) + u ( D ) ,the Raychaudhuri equations shall be [39]: ˙ θ = − ( σ − ω ) − θ − R ab u a u b + ˙ u a ; a (51)where σ = σ ab σ ab , ω = ω ab ω ab .The vorticity tensor, ω ab = u [ a ; b ] − ˙ u [ a u b ] , is given by ω ab = ω ( O ) ab + ω ( D ) ab − w ab (52)where ω ( O ) ab = u ( O )[ a | b ] − ˙ u ( O )[ a u ( O ) b ] (53) ω ( D ) ab = u ( D )[ a || b ] − ˙ u ( D )[ a u ( D ) b ] (54)are the vorticity tensors of ordinary and dark matter respectively, and w ab = u [ a \ b ] − u ( O )[ a || b ] − u ( D )[ a | b ] + ˙ u ( O )[ a u ( D ) b ] + ˙ u ( D )[ a u ( O ) b ] (55)is the interaction part of the vorticity tensor. This is true only if the vector (or tensor) acted upon belongs to the space with metric g ( O ) . This is true only if the vector (or tensor) acted upon belongs to the space with metric g ( D ) . This operation does not constitute a covariant derivative as the symbols γ abc are not proper Christoffel symbols and there is no correspondinggeometric space. One must be careful that only u a ; k is the covariant derivative of u a ∈ ( M , g ) ; all other operations defined before represent arbitrary operators inthe framework of the unified space and can only be treated otherwise if we restrict our study in the corresponding subspaces. The unified expansion scalar, θ = u k ; k assumes the form θ = θ ( O ) + θ ( D ) − ϑ (56)where θ ( O ) = u k | k (57) θ ( D ) = u k || k (58)are the expansion scalars of ordinary and dark matter and ϑ = u ( O ) k || k + u ( D ) k | k − u k \ k (59)is the interaction part of the expansion scalar.The projection tensor, h ab = g ab − u a u b , has the following form: h ab = h ( O ) ab + h ( D ) ab − u ( O )( a u ( D ) b ) (60)where h ( O ) ab = g ( O ) ab − u ( O ) a u ( O ) b , (61) h ( D ) ab = g ( D ) ab − u ( D ) a u ( D ) b (62)are the ordinary and dark matter projection tensors.The unified shear tensor, σ ab = u ( a ; b ) − θ h ab − ˙ u ( a u b ) , shall be given by σ ab = σ ( O ) ab + σ ( D ) ab − s ab (63)where σ ( O ) ab = u ( O )( a | b ) − θ ( O ) h ( O ) ab − ˙ u ( O )( a u ( O ) b ) (64) σ ( D ) ab = u ( D )( a || b ) − θ ( D ) h ( D ) ab − ˙ u ( D )( a u ( D ) b ) (65)are the shear tensors of ordinary matter and dark matter fluid respectively, and s ab = u ( a \ b ) − u ( O )( a || b ) − u ( D )( a | b ) + ˙ u ( O )( a u ( D ) b ) + ˙ u ( D )( a u ( O ) b ) ++ (cid:18) θ ( O ) ( h ( D ) ab − u ( O )( a u ( D ) b ) ) + θ ( D ) ( h ( O ) ab − u ( O )( a u ( D ) b ) ) − ϑ h ab (cid:19) (66)is the interaction part of the shear tensor. Finally, using the Einstein Equation (34) and in conjunction with relations(43)–(45) we can find that, apart from the unified Raychaudhuri scalar, R ab u a u b = π ( ρ + p − Λ π ) , which includesordinary and dark matter as well as their interaction, the following are also true: R ( O ) ab u ( O ) a u ( O ) b = π (cid:18) ρ ( O ) + p ( O ) − Λ π (cid:19) (67)is the Raychaudhuri scalar of ordinary matter and, R ( D ) ab u ( D ) a u ( D ) b = π (cid:18) ρ ( D ) + p ( D ) − Λ π (cid:19) (68)is the dark matter Raychaudhuri scalar.By vitrue of relations (51), (52), (56) and (63) the expansion θ , the shear σ and the vorticity ω , which in our modelinclude contributions from dark matter and its interaction with ordinary matter, extend the Raychaudhuri equationwhich now takes into account the existence of extra mass in the form of dark matter. Since the Raychaudhuri equationplays a dominant role in the evolution of the universe, Equation (51) gives rise to a potential need to differentiate theexisting considerations of singularities. D. Conformal Dark FLRW-Metric Structure
In this section we consider an application of our model in cosmology. In particular, we shall use a FLRW metricstructure for the ordinary matter sector and we will assume a conformal relation for the unified spacetime containingboth ordinary and dark matter. Using a conformal factor F ( R ( O ) ) for the unified metric, we shall derive modifiedFriedmann equations and the continuity equation for a model which is described by F ( R ) gravity [50, 51]. We havechosen such an assumption for the metric of the unified space because studies in the conformal structure in thefields of General Relativity and Cosmology have given rise to viable theories related to dark matter and dark energy(e.g., [50–56]) and this framework has been seen to play a significant role as the angles in the light-cone structure arepreserved [52]. Indeed, we shall further choose a particular class of F ( R ( O ) ) [53] and we shall see that the metric ofthe dark matter sector is also conformal with the ordinary matter FLRW metric structure; namely we assume that F ( R ( O ) ) = R ( O ) + f ( R ( O ) ) (69)where F ( R ( O ) ) and f ( R ( O ) ) are functions of the Ricci scalar of the ordinary matter sector R ( O ) . The function F ( R ( O ) ) is the conformal factor of the unified metric and f ( R ( O ) ) serves as the conformal factor of the dark metric. This isseen as a direct consequence of the assumption (2) as g ab = F ′ ( R ( O ) ) g ( O ) ab = ( + f ′ ( R ( O ) )) g ( O ) ab = g ( O ) ab + f ′ ( R ( O ) ) g ( O ) ab (70)where F ′ ( R ( O ) ) = dF ( R ( O ) ) dR ( O ) and f ′ ( R ( O ) ) = d f ( R ( O ) ) dR ( O ) .Alternatively one could start from the assumption that the dark metric is conformal with the ordinary FLRW metricwith conformal factor f ( R ( O ) ) [52]; namely g ( D ) ab = f ′ ( R ( O ) ) g ( O ) ab (71)and using the assumption (2) we find that g ab = g ( O ) ab + g ( D ) ab = ( + f ′ ( R ( O ) )) g ( O ) ab (72)Therefore the unified metric is also conformal with the ordinary FLRW metric if we assume a conformal factor F ( R ( O ) ) of the form given in (69).Using this previous assumption in conjunction with relations (7), (22), (26) and (38) we can calculate the Christoffelsymbols, the Ricci curvature as well as the energy momentum tensor . From the Einstein Equation (34) we thenderive the following modified Friedmann equations (cid:18) ˙ aa (cid:19) = π G ρ F ′ − κ a + Φ ( t ) (73) ¨ aa = − π G ( ρ + p ) F ′ + Φ ( t ) (74)where a is the scale factor, κ = ±
1, 0 , Φ ( t ) = φ t φ t + φ t | t + φ t | t − ( φ t ) , Φ ( t ) = φ t φ t + φ t | t − φ t | t + ( φ t ) , φ t = ∂ t [ ln [ F ′ ]] , ρ = ρ ( O ) + ρ ( D ) , p = p ( O ) + p ( D ) and the operator “ | ” has been defined in (46).The idea of inflation can be incorporated into the model of Friedmann equations by taking into account the Planckmass m Pl = ( π G ) − [57, 58]. We can then write H = ρ m Pl F ′ − κ a + Φ ( t ) (75) ¨ aa = − ( ρ + p ) m Pl F ′ + Φ ( t ) (76) The velocities are also assumed to follow a conformal relation [54]. H = ˙ a / a is the Hubble parameter. By virtue of the above relations we can see that the additional terms Φ i ( t ) , i =
1, 2 constitute an extension of the Friedmann equations and may contribute to a differentiation to the acceleratedexpansion of the universe which is incorporated in the observations. In a spacetime where dark matter is absent,therefore f ′ = or equivalently F ′ = , it can be easily verified that Φ i ( t ) also vanish and thus, we recover the usualform of the Friedmann equations for a FLRW ordinary matter space.Taking into account that the energy momentum tensor of the ordinary sector is conserved, T ( O ) µ | µ = , we cansee that the energy momentum tensor of the unified two component cosmological fluid is also conserved and usingthe fact that T µ µ = we derive the following continuity equation ˙ ρ + ( + w ) ρ H = (77)where we assume that the total pressure p = w ρ . This result is also compatible with the general conservation relation(50). III. GRAVITY ON THE SASAKI TANGENT BUNDLE
For the study of the tangent bundle we use a metric, g , on a Riemannian manifold ( M , g ) , along with its Levi-Civita connection, to construct a new natural Riemannian metric on the tangent bundle TM , called the Sasaki metric.The geometry of tangent bundles in general and particularly the Sasaki metric find a lot of applications in physics,especially in the study of gravity. This consideration extends the limits of conventional general relativity for amodified gravity approach, and provides the gravitational field with extra degrees of freedom. In particular, a Sasakiextension of spacetime constitutes the minimum metric generalization of spacetime in the framework of a tangentbundle of a four-dimensional spacetime. A. Deviation of Geodesics of a Sasaki Spacetime
In this section, we shall derive the deviation of geodesics of a Sasaki spacetime with metric [31] d σ = g αβ ( x ) dx α dx β + g αβ ( x ) Dy α Dy β (78)where, Dy α = dy α + N αβ ( x , y ) dx β (79) N µα ( x , y ) represents the “pre-Finsler non-linear connection”, g αβ ( x ) is the Riemann metric tensor of the n-dimensionaldifferentiable manifold (here n = 4) and both the horizontal and vertical part of the total metric (78) of the 2n-dimensional tangent bundle. α , β ∈ {
1, 2, ..., n = } .Furthermore, for the purpose of our study, the non-linear connection is assumed to be N µα = Γ µακ y κ (80)The metric (78) may, then be rewritten using the fundamental covariant metric tensor, G ij as d σ = G ij dx i dx j (81)where i , j ∈ {
1, 2, ..., 2 n = } and G αβ = g αβ + g µν Γ µρα Γ νκβ y ρ y κ (82) G α ( n + β ) = Γ µαβ y µ (83)and G ( n + α )( n + β ) = g αβ (84)where Γ µαβ and Γ µρα are Christoffel’s symbols of the first and second kind of M n respectively.1If ¯ Γ ijk and ¯ Γ ijk are Christoffel’s symbols of first and second kind, respectively, that correspond to the Sasaki tangentbundle T ( M n ) as calculated using the fundamental metric, then the geodesic equation is known to be d x i d σ + ¯ Γ ijk dx j d σ dx k d σ = (85)While it is usually more useful to express this equation in terms of quantities of M n , for the present we shall contentourselves with using relation (85) instead, in order to avoid perplexing our equations.If x is ( t ) is a two-parameter geodesic family with t being the affine parameter and s the selector parameter then weshall denote the tangent vectors as V i = ∂ t x i , W i = ∂ s x i so that ∂ s V i = ∂ t W i . Moreover, let X i ( x k ) be a vector fielddefined over a region of the subspace of T ( M n ) defined by the net of x is ( t ) . Then we define the δ − derivatives as: δ X i δ t = X i ; h V h = (cid:16) ∂ X i ∂ x h + ¯ Γ ihk X k (cid:17) V h (86)and δ X i δ s = X i ; h W h = (cid:16) ∂ X i ∂ x h + ¯ Γ ihk X k (cid:17) W h (87)Using the previous result in conjunction with the fact that the Christoffel’s symbols are symmetric w.r.t. theirsuffixes, we get that δ V i δ s = ∂ x i ∂ t ∂ s + ¯ Γ ihk V k W h = δ W i δ t (88)Following the standard procedure of deriving the deviation equations [59], given the previous relations, we findthat δ W i δ t = δ V i δ s δ t = δ V i δ t δ s − K ijhk V j V h W k (89)Finally, using Equation (85) we find the following geodesic deviation equation δ W i δ t = − K ijhk V j V h W k (90)where K ijhk is the curvature tensor of the tangent bundle. An abstract form of the curvature tensor is given by [60, 61].All the components of the curvature tensor K ijhk are given explicitly in Appendix A. B. Dark Gravity on the Tangent Bundle
Experimental research [32–36] suggests that a theory of dark gravity with extra dimensions may be necessaryin order to effectively describe the total mass distribution in the universe. A first step towards such a geometricgravitational theory could be obtained by retaining our assumption (2) and expanding our Riemannian frameworkfrom Section II on the tangent bundle using an underlying Sasaki structure (78) for the total space of ordinary anddark matter.First, taking into account relation (80) and using Equation (7) we find that N µα = N ( O ) µα + N ( D ) µα + ν µα (91)where, N ( O ) µα = Γ ( O ) µακ y κ (92)is the ordinary matter non-linear connection, N ( D ) µα = Γ ( D ) µακ y κ (93)2is dark matter’s non-linear connection, and ν µα = γ µακ y κ (94)is the interaction part of the non-linear connection. ν µα acts as a correlation between the non-linear connectionsof ordinary and dark matter. Thus, in the framework of a Sasaki spacetime, ν µα plays the fundamental role ofinterconnecting the ordinary and dark matter sectors within the line-element. Such a connection between ordinaryand dark matter is absent from the corresponding line-element of the Riemannian spacetime. It can therefore beconcluded that a Sasaki spacetime involves a stronger interaction between the two component matter sectors whichinfluences even the notion of arc-length as we shall also see below.In virtue of relations (79) and (91), we obtain Dy µ = D ( O ) y µ + D ( D ) y µ − δ y µ (95)where, D ( O ) y µ = dy µ + N ( O ) µα ( x , y ) dx α (96) D ( D ) y µ = dy µ + N ( D ) µα ( x , y ) dx α (97)and δ y µ = dy µ − ν µα ( x , y ) dx α (98)In a Sasaki spacetime , the non-linear connection influences the gravitational potential giving rise to extra degreesof freedom in the internal structure of the spacetime in the form of y -dependence. As evidenced by relations (2), (78),(79) and (95), with the addition of dark matter on the tangent bundle of the spacetime , three types of line-elementscan be presented: d σ = d σ O + d σ D + d σ I (99)where d σ O = g ( O ) αβ dx α dx β + g ( O ) αβ D ( O ) y α D ( O ) y β (100)is the line-element of the Sasaki tangent bundle of ordinary matter, d σ D = g ( D ) αβ dx α dx β + g ( D ) αβ D ( D ) y α D ( D ) y β (101)is the line-element of the Sasaki tangent bundle of dark matter, and d σ I = g αβ Dy α Dy β − g ( O ) αβ D ( O ) y α D ( O ) y β − g ( D ) αβ D ( D ) y α D ( D ) y β (102)is the interaction term of the line-element.It can be seen that the non-linear connection, N ( x , y ) , plays a fundamental role in this approach, since it facilitatesthe introduction of three differentials (95) that produce extra interaction terms between ordinary and dark matter.In the base manifold ( M , g ) of Section II, the introduction of dark matter extends the arc-length of the spacetime ofordinary matter in an additive way since one could write relation (1) as ds = ds O + ds D (103)where ds O = g ( O ) αβ dx α dx β (104)is the line-element of an ordinary matter spacetime, and ds D = g ( D ) αβ dx α dx β (105)3is the line-element of a dark matter spacetime. Such “trivial” extension is not possible in the Sasaki tangent bundlebecause of the non-linear connection introducing extra interaction that influences even the arc-length itself.We have seen that the geodesics are provided by the condensed relation (85) which is known to be expanded inthe following two equations [31]: d x µ d σ + Γ µνρ dx ν d σ dx ρ d σ = R µναβ dx ν d σ y α Dy β d σ (106) D y µ d σ = (107)where Γ µνρ and R µναβ are the Christoffel symbols and Riemann curvature tensor, respectively, corresponding to thebase manifold ( M , g ) as discussed in Section II, and Dy µ d σ = dy µ d σ + Γ µνρ y ν dx ρ d σ (108) D y µ d σ = d y µ d σ + dd σ (cid:26) Γ µνρ y ν dx ρ d σ (cid:27) + Γ µαβ Dy α d σ dx β d σ (109)The expansion of the condensed geodesic Equation (85) gave rise to two geodesic equations. The horizontalgeodesic (106) can be physically interpreted as the generalization of the corresponding Riemannian geodesic curveson the tangent bundle. The expanded family of geodesic curves described by (106) shall be further examined below.On the other hand, the physical interpretation of the vertical Equation (107) remains an open question and althoughwe shall continue to provide the vertical equations, we shall limit our present work to the physical study of thehorizontal geodesics.If we compare the Riemannian setting with the Sasaki tangent bundle, we can see from relation (106) that the classof curves that have the geodesic property has now been expanded. On ( TM , G ) we can distinguish two geodesicfamilies the first of which is obtained by lifting a geodesic of ( M , g ) on TM . In that case the expanded form of thegeodesic equations reduces as [31] d x µ d σ + Γ µνρ dx ν d σ dx ρ d σ = (110) Dy µ d σ = In this case, with reference to the base manifold M of TM , an observer shall continue to perceive a state of rest sinceEquation (110) coincides with their notion of a Riemannian geodesic. We will henceforth call an observer limitedto perceiving only a Riemannian manifold occupied by ordinary matter i.e., neglecting dark matter and ignoringthe existence of any higher-dimensional structure, a constrained observer. In essence, “constrained” observers thinkthat they are experiencing the submanifold ( M ( O ) , g ( O ) ) of the base manifold, thereby perceiving anything deviatingfrom their notion of a Riemannian structure as due to external forces or effects. Such interpretation is compatiblewith the idea of an apparent metric as proposed by [52].The second family of geodesics on the Sasaki tangent bundle is comprised by curves that are not obtained bylifting geodesics of M . In that case Equation (106) cannot be further reduced. This causes a constrained observer toperceive apparent pseudo-coupling forces of the gravitational field with the velocity field of TM , preventing themfrom understanding that curve as a geodesic. The r.h.s. part of Equation (106) seemingly disagrees with the notionof a Riemannian geodesic. However, this effect is only due to the observer’s inability to perceive TM .By virtue of relations (95) and (108) we have that: Dy µ d σ = D ( O ) y µ d σ + D ( D ) y µ d σ − δ y µ d σ (111)where D ( O ) y µ d σ = dy µ d σ + Γ ( O ) µνρ y ν dx ρ d σ (112)4 D ( D ) y µ d σ = dy µ d σ + Γ ( D ) µνρ y ν dx ρ d σ (113) δ y µ d σ = dy µ d σ − γ µνρ y ν dx ρ d σ (114)Using Equations (109) and (111) we get: D y µ d σ = D ( O ) y µ d σ + D ( D ) y µ d σ + δ y µ d σ + D ( O ) d σ (cid:26) D ( D ) y µ d σ − δ y µ d σ (cid:27) + D ( D ) d σ (cid:26) D ( O ) y µ d σ − δ y µ d σ (cid:27) − δ d σ (cid:26) D ( O ) y µ d σ + D ( D ) y µ d σ (cid:27) (115)where D ( O ) y µ d σ = d y µ d σ + dd σ (cid:26) Γ ( O ) µνρ y ν dx ρ d σ (cid:27) + Γ ( O ) µαβ D ( O ) y α d σ dx β d σ (116) D ( D ) y µ d σ = d y µ d σ + dd σ (cid:26) Γ ( D ) µνρ y ν dx ρ d σ (cid:27) + Γ ( D ) µαβ D ( D ) y α d σ dx β d σ (117) δ y µ d σ = d y µ d σ + dd σ (cid:26) γ µνρ y ν dx ρ d σ (cid:27) + γ µαβ δ y α d σ dx β d σ (118)From the point of view of an observer limited to the ordinary matter subspace relations (6) and (11) show thatthe geodesic equations in the Riemannian setting are perturbed when dark matter is taken into account. Let usnow assume a test particle moving along geodesics on the tangent bundle in a Sasaki spacetime. In analogy to theRiemannian case the geodesics in higher dimensions will deviate from the previously thought geodesic motion dueto the presence of dark matter and its interaction with ordinary matter. In this case, using our previous results inconjunction with relations (7), (12) and (111) in Equations (106) and (107), the unified form of the correspondinggeodesic equation will be given by the following: d x µ d σ + (cid:0) Γ ( O ) µνρ + Γ ( D ) µνρ + γ µνρ (cid:1) dx ν d σ dx ρ d σ = (cid:0) R ( O ) µναβ + R ( D ) µναβ + r µναβ (cid:1) dx ν d σ y a (cid:18) D ( O ) y b d σ + D ( D ) y b d σ − δ y b d σ (cid:19) (119)and D ( O ) y µ d σ + D ( D ) y µ d σ + δ y µ d σ = − D ( O ) d σ (cid:26) D ( D ) y µ d σ − δ y µ d σ (cid:27) − D ( D ) d σ (cid:26) D ( O ) y µ d σ − δ y µ d σ (cid:27) + δ d σ (cid:26) D ( O ) y µ d σ + D ( D ) y µ d σ (cid:27) (120)As we have discussed, the expanded family of geodesics of a Sasaki spacetime includes curves that can be obtainedby lifting geodesics of the base manifold. In this case, from the perspective of a constrained observer who takesinto account only the ordinary matter sector of the Riemannian space ( M , g ) , the test particle moves along a curvethat deviates from their expected Riemannian geodesics due to the presence of dark pseudo-forces . If however,the test particle at rest happens to be moving along a Sasaki geodesic that cannot be obtained by such a lift, then theconstrained observer will detect, in addition to the dark pseudo-forces, the pseudo-coupling forces given by the r.h.s.part of Equation (119). Therefore, in a Sasaki spacetime a constrained observer shall always observe deviated curvesas is the case with a Riemannian spacetime. The difference between a Riemannian and a Sasaki setting lies in theobservation of the pseudo-coupling forces which may or may not be detected depending on the geodesic.By virtue of relation (A1) it can be shown that K ijhk = K ( O ) ijhk + K ( D ) ijhk + κ ijhk (121)where K ( O ) ijhk , K ( D ) ijhk , are the Sasaki curvature tensors of ordinary and dark matter, respectively, and κ ijhk = K ijhk − K ( O ) ijhk − K ( D ) ijhk . These are the same F i ( D ) as in Equation (11). δ W i δ t = − ( K ( O ) ijhk + K ( D ) ijhk + κ ijhk ) V j V h W k (122)As we have seen from relation (A1), the curvature tensor, K , has been modified in higher dimensions due tothe geometry of the Sasaki tangent bundle. Consequently, we can conclude that the gravitational field has alsobeen influenced by the higher-dimensional metric structure of the spacetime, i.e., the Sasaki tangent bundle, sincegravity is intertwined with the notion of curvature. The gravitational field has been endowed with extra degreesof freedom and this has been incorporated in the tidal forces (122) by means of the generalized curvature tensor.As a result, the deviation of the 4-dimensional free motion of nearby particles (or clusters thereof, e.g., galaxies) (16)can be differentiated from its higher-dimensional counterpart because of the form of the curvature tensor whichis an intrinsic geometric property of the spacetime and therefore independent of the relative observer. Therefore,the resulting dark tidal forces manifest both the addition of extra matter and interaction in the form of dark matteras well as the higher-dimensional geometric structure of the spacetime and the subsequent extra degrees of freedomof the gravitational field.The previous result is of great importance for cosmology. For instance, a positive curvature of a Riemannianspacetime which is connected with converging neighbouring geodesics (Jacobi field) may correspond to a negativecurvature on the Sasaki tangent bundle. Hence, taking into account the higher-dimensional structure could potentiallyreveal that what we initially thought of as convergent is in reality divergent, and vice-versa, due to extra forcesstemming from the geometry of the spacetime. IV. CONCLUDING REMARKS
Motivated by observational results, we laid the foundations for a geometric theory of dark gravity in relation tothe ordinary gravitational field in the framework of a “unified” space. By extending the already existing notions ofgeometry that apply for ordinary matter, we used them in the case of dark gravity. In particular, in this work weexamined the unified gravitational field of ordinary and dark matter in a Riemannian metric framework derivingthe geodesic equations and their deviation, as well as the Einstein and Raychaudhuri equations. Furthermore,by extending our geometric methods on the tangent bundle we presented a unified dark and ordinary gravity on aSasaki spacetime.The geometric methods used throughout this paper gave rise to the concept of dark forces which could play arole in offsetting the centrifugal orbital tendency of the galaxies. In particular, the gravitational influence of darkmatter generated modified geodesic curves that seemingly deviate from the Riemannian geodesic notion due to thepresence of dark pseudo-forces (11) and (119) which appear to dominate the deviated geodesic motion. The extensionof the spacetime on the tangent bundle also caused the potential emergence of the pseudo-coupling forces (119) dueto the inability of the observer to perceive the higher dimensional structure. Both of these cases are deemed aspseudo-forces because they are the product of a constrained observer. The third kind of dark forces however, the darktidal forces (16) and (122), result from an intrinsic geometric property of the spacetime; the dark curvature. Thisshows that the gravitational interaction between ordinary and dark matter has been incorporated in the geometry ofthe spacetime and therefore, dark tidal forces exist independent of the observer.With the introduction of dark matter to the Riemannian spacetime setting, the Einstein and Raychaudhuri equationswere modified to account for the existence of extra matter and thus, for extra gravitational interaction. Theseequations could provide invaluable information concerning the interaction of dark and ordinary matter and henceprovide details about the internal structure of the unified space. The modification of these equations due to darkmatter resulted in modified Friedmann Equations (73)–(76) as examined using a conformal metric structure for thedark matter sector. The study of inflation and the cosmological implications of these relations in the evolutionof the universe as well as the application of other models of dark metric structure could constitute the subject offurther research.Overall, the existence of dark matter calls into question the concept of geodesic motion. Indeed the appearance ofpseudo-force fields dominating the motion of test particles causes a constrained observer to conclude that the motionis not geodesic and that the ordinary matter spacetime has been externally perturbed by dark matter. Nonetheless,a closer look at the overall structure of the spacetime reveals that those apparent forces exist only relative to the pointof view of a particular observer and reaffirms the geodesic property of the curves.Ultimately, we consider that the use of geometric methods in the formulation of a gravitational theory that includesdark matter could form a theoretical basis upon which observational results could be interpreted. The contributionof dark energy in the framework of this theoretical effort remains an even greater question for future research.
Conflicts of interest:
The authors declare no conflict of interest.6
Appendix A: The Curvature Tensor of a Sasaki Tangent Bundle
Using the Christoffel symbols ¯ Γ ijk as given by S. Sasaki in relation (7.4) in [31] and after long calculations, it can beseen that the curvature tensor of a n -dimensional Sasaki tangent bundle ( TM , G ) is K i ( n + α )( n + β )( n + γ ) = (A1) K δ ( n + α )( n + β ) γ = ∂∂ x ( n + β ) [ R δγαλ y λ ] + R δǫβµ R ǫγαν y µ y ν K ( n + δ )( n + α )( n + β ) γ = ∂∂ x ( n + β ) [ Γ δαγ − Γ δµλ R λγαν y µ y ν ] + [ Γ δβǫ − Γ δµλ R λǫβν y µ y ν ] R ǫγακ y κ K δ ( n + α ) βγ = ∂∂ x β [ R δγαλ y λ ] − ∂∂ x γ [ R δβαλ y λ ] + [ Γ δβǫ + ( R δǫκµ Γ κνβ + R δβκµ Γ κνǫ ) y µ y ν ] R ǫγαλ y λ − [ Γ δγǫ + ( R δǫκµ Γ κνγ + R δγκµ Γ κνǫ ) y µ y ν ] R ǫβαλ y λ + [ Γ ǫαγ − Γ ǫµκ R κγαν y µ y ν ] R δβǫλ y λ − [ Γ ǫαβ − Γ ǫµκ R κβαν y µ y ν ] R δγǫλ y λ K ( n + δ )( n + α ) βγ = ∂∂ x β [ Γ δαγ − Γ δµλ R λγαν y µ y ν ] − ∂∂ x γ [ Γ δαβ − Γ δµλ R λβαν y µ y ν ]+ R ǫγαξ y ξ [( R δβλǫ + R δǫλβ + ∂ Γ δβǫ ∂ x λ ) y λ + Γ δνκ ( R κǫµη Γ ηλβ + R κβµη Γ ηλǫ ) y λ y µ y ν ]+ [ Γ δǫβ − Γ δµκ R κβǫν y µ y ν ][ Γ ǫαγ − Γ ǫλη R ηγαξ y λ y ξ ] − R ǫβαξ y ξ [( R δγλǫ + R δǫλγ + ∂ Γ δγǫ ∂ x λ ) y λ + Γ δνκ ( R κǫµη Γ ηλγ + R κγµη Γ ηλǫ ) y λ y µ y ν ] − [ Γ δǫγ − Γ δµκ R κγǫν y µ y ν ][ Γ ǫαβ − Γ ǫλη R ηβαξ y λ y ξ ] K δ ( n + α ) β ( n + γ ) = − ∂∂ x n + γ [ R δβαλ y λ ] − R δǫγµ R ǫβαν y µ y ν K ( n + δ )( n + α ) β ( n + γ ) = − ∂∂ x ( n + γ ) [ Γ δαβ − Γ δµλ R λβαν y µ y ν ] − [ Γ δγǫ − Γ δµλ R λǫγν y µ y ν ] R ǫβακ y κ K δαβγ = ∂∂ x β [ Γ δαγ + ( R δγκµ Γ κνα + R δακµ Γ κνγ ) y µ y ν ] − ∂∂ x γ [ Γ δαβ + ( R δβκµ Γ κνα + R δακµ Γ κνβ ) y µ y ν ]+ [ Γ δβǫ + ( R δǫκµ Γ κνβ + R δβκµ Γ κνǫ ) y µ y ν ][ Γ ǫαγ + ( R δγκλ Γ κξα + R δακλ Γ κξγ ) y λ y ξ ] − [ Γ δγǫ + ( R δǫκµ Γ κνγ + R δγκµ Γ κνǫ ) y µ y ν ][ Γ ǫαβ + ( R δβκλ Γ κξα + R δακλ Γ κξβ ) y λ y ξ ]+ R δβǫξ y ξ [( R ǫαλγ + R ǫγλα + ∂ Γ ǫαγ ∂ x λ ) y λ + Γ ǫνκ ( R κγµη Γ ηλα + R καµη Γ ηλγ ) y λ y µ y ν ] − R δγǫξ y ξ [( R ǫαλβ + R ǫβλα + ∂β ǫαβ ∂ x λ ) y λ + β ǫνκ ( R κβµη β ηλα + R καµη β ηλβ ) y λ y µ y ν ] K ( n + δ ) αβγ = ∂∂ x β [( R δαλγ + R δγλα + ∂ Γ δαγ ∂ x λ ) y λ + Γ δνκ ( R κγµη Γ ηλα + R καµη Γ ηλγ ) y λ y µ y ν ] − ∂∂ x γ [( R δαλβ + R δβλα + ∂ Γ δαβ ∂ x λ ) y λ + Γ δνκ ( R κβµη Γ ηλα + R καµη Γ ηλβ ) y λ y µ y ν ]+ [( R δβλǫ + R δǫλβ + ∂ Γ δβǫ ∂ x λ ) y λ + Γ δνκ ( R κǫµη Γ ηλβ + R κβµη Γ ηλǫ ) y λ y µ y ν ] ˙˙ [ Γ ǫαγ + ( R ǫγκξ Γ κρα + R ǫακξ Γ κργ ) y ρ y ξ ] − [( R δγλǫ + R δǫλγ + ∂ Γ δγǫ ∂ x λ ) y λ + Γ δνκ ( R κǫµη Γ ηλγ + R κγµη Γ ηλǫ ) y λ y µ y ν ] ˙˙ [ Γ ǫαβ + ( R ǫβκξ Γ κρα + R ǫακξ Γ κρβ ) y ρ y ξ ]+ [( R ǫαλγ + R ǫγλα + ∂ Γ ǫαγ ∂ x λ ) y λ + Γ ǫνκ ( R κγµη Γ ηλα + R καµη Γ ηλγ ) y λ y µ y ν ][ Γ δǫβ − R κβǫξ Γ δρκ y ρ y ξ ] − [( R ǫαλβ + R ǫβλα + ∂ Γ ǫαβ ∂ x λ ) y λ + Γ ǫνκ ( R κβµη Γ ηλα + R καµη Γ ηλβ ) y λ y µ y ν ][ Γ δǫγ − R κγǫξ Γ δρκ y ρ y ξ ] K δαβ ( n + γ ) = ∂∂ x β [ R δαγλ y λ ] − ∂∂ x ( n + γ ) [ Γ δαβ − ( R δβκµ Γ κλα + R δακµ Γ κλβ ) y λ y µ ]+ R δβǫν y ν [ Γ ǫγα − R καγλ Γ ǫµκ y λ y µ ]+ R ǫαγν y ν [ Γ δβǫ + ( R δǫκµ Γ κλβ + R δβκµ Γ κλǫ ) y λ y µ ] − R δǫγν y ν [ Γ ǫαβ + ( R ǫβκµ Γ κλα + R ǫακµ Γ κλβ ) y λ y µ ] K ( n + δ ) αβ ( n + γ ) = ∂∂ x β [ Γ δαγ − R καγλ Γ δµκ y λ y µ ] − ∂∂ x ( n + γ ) [( R δαλβ + R δβλα + ∂ Γ δαβ ∂ x λ ) y λ + Γ δνκ ( R κβµη Γ ηλα + R καµη Γ ηλβ ) y λ y µ y ν ]+ R ǫαγρ y ρ [( R δβλǫ + R δǫλβ + ∂ Γ δβǫ ∂ x λ ) y λ + Γ δνκ ( R κǫµη Γ ηλβ + R κβµη Γ ηλǫ ) y λ y µ y ν ]+ [ Γ ǫγα − R καγλ Γ ǫµκ y λ y µ ][ Γ δǫβ − R κβǫξ Γ δρκ y ρ y ξ ] − [ Γ ǫαβ + ( R ǫβκν Γ κρα + R ǫακν Γ κρβ ) y ρ y ν ][ Γ δǫγ − R κǫγξ Γ δµκ y µ y ξ ] K δα ( n + β ) γ = ∂∂ x ( n + β ) [ Γ δαγ + ( R δγκµ Γ κλα + R δακµ Γ κλγ ) y λ y µ ] − ∂∂ x γ [ R δαβλ y λ ]+ R δǫβν y ν [ Γ ǫγα + ( R ǫγκµ Γ κλα + R ǫακµ Γ κλγ ) y λ y µ ] − R ǫαβν y ν [ Γ δγǫ + ( R δǫκµ Γ κλγ + R δγκµ Γ κλǫ ) y λ y µ ] − R δγǫν y ν [ Γ ǫβα − R καβλ Γ ǫµκ y λ y µ ] K ( n + δ ) α ( n + β ) γ = ∂∂ x ( n + β ) [( R δαλγ + R δγλα + ∂ Γ δαγ ∂ x λ ) y λ + Γ δνκ ( R κγµη Γ ηλα + R καµη Γ ηλγ ) y λ y µ y ν ] − ∂∂ x γ [ Γ δαβ − R καβλ Γ δµκ y λ y µ ] − R ǫαβρ y ρ [( R δγλǫ + R δǫλγ + ∂ Γ δγǫ ∂ x λ ) y λ + Γ δνκ ( R κǫµη Γ ηλγ + R κγµη Γ ηλǫ ) y λ y µ y ν ] − [ Γ ǫβα − R καβλ Γ ǫµκ y λ y µ ][ Γ δǫγ − R κγǫξ Γ δρκ y ρ y ξ ]+ [ Γ ǫαγ + ( R ǫγκν Γ κρα + R ǫακν Γ κργ ) y ρ y ν ][ Γ δǫβ − R κǫβξ Γ δµκ y µ y ξ ] K δα ( n + β )( n + γ ) = ∂∂ x ( n + β ) [ R δγαλ y λ ] − ∂∂ x ( n + γ ) [ R δβαλ y λ ] + R δǫβν R ǫαγµ y µ y ν − R δǫγν R ǫαβµ y µ y ν K ( n + δ ) α ( n + β ) γ = ∂∂ x ( n + β ) [ Γ δαγ − R καγλ Γ δµκ y λ y µ ] − ∂∂ x ( n + γ ) [ Γ δαβ − R καβλ Γ δµκ y λ y µ ]+ [ Γ ǫαγ + ( R ǫγκν Γ κρα + R ǫακν Γ κργ ) y ρ y ν ][ Γ δǫβ − R κǫβξ Γ δµκ y µ y ξ ] − [ Γ ǫαβ + ( R ǫβκν Γ κρα + R ǫακν Γ κρβ ) y ρ y ν ][ Γ δǫγ − R κǫγξ Γ δµκ y µ y ξ ] where Γ αβγ , R δαβγ are the Christoffel symbols and Riemann curvature tensor of the n -dimensional base manifold ( M , g ) , respectively, and, inkeeping with our previous convention for Section III, the greek indices { α , β , ... } = {
1, 2, · · · n } and the latin indices { a , b , ... } = {
1, 2, · · · n } .It should be noted that if the base manifold is flat, i.e., in an appropriate coordinate system R δαβγ = and Γ αβγ = then ¯ Γ ijk = and K ijhk = . Therefore, a flat base manifold induces a flat Sasaki tangent bundle. In this case the“geodesics” of the flat tangent space are all trivially produced by lifting the “geodesics” of the base manifold. [1] Farnes, J.S. A unifying theory of dark energy and dark matter: Negative masses and matter creation within a modified Λ CDMframework.
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