Raychaudhuri Equation in Space-Times with Torsion

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Raychaudhuri equation in space-times with torsion

Paulo Luz1, 2, ∗ and Vincenzo Vitagliano2, † 1Centro de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal 2Centro Multidisciplinar de Astrofísica - CENTRA, Departamento de Física, Instituto Superior Técnico - IST, Universidade de Lisboa - UL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal Given a space-time with non-vanishing torsion, we discuss the equation for the evolution of the separation vector between infinitesimally close curves in a congruence. We show that the presence of a torsion field leads in general to tangent and orthogonal effects to the congruence; in particular, the presence of a completely generic torsion field contributes to a relative acceleration between test particles. We derive, for the first time in the literature, the Raychaudhuri equation for a congruence of time-like and null curves in a space-time with the most generic torsion field.

I. INTRODUCTION tal issues as in the case of the derivation of the (modified) Einstein equations as equations of state for a (non) equi- The appearance of singularities in a physical the- librium spacetime thermodynamics [7–9]. ory ineluctably marks the pillars of Hercules of In this paper we generalize, for the first time in the lit- that model. General Relativity, from this point of erature, the Raychaudhuri construction to a congruence view, is not an exception: it did not take a long of curves embedded in a space-time with a non-trivial, time for the community of relativists to realize that completely generic torsion tensor field. The idea of gen- even the two simplest space-time solutions of Ein- eralizing Einstein General Relativity to non-Riemannian stein field equations, the Schwarzschild metric and the geometries has been now around for a while. One of Friedmann–Lemaître–Robertson–Walker (FLRW) met- the prototypical example of this kind of proposals is the ric, harbour a gravitational singularity, that is, a break- Einstein–Cartan–Sciama–Kibble (ECSK) theory. The down of the space-time structure itself! While in the ECSK theory is characterized by assuming an indepen- early 50’s Amal Kumar Raychaudhuri was forcedly work- dent connection (using the so-called Palatini approach to ing on the properties of electronic energy bands in met- find two sets of independent field equations) and further als, he got interested in the debate around the nature requiring the anti-symmetric part of the connection to be of gravitational singularities and the generic features of in general non-vanishing, defining a tensor field dubbed Einstein’s theory of General Relativity (GR). Motivated torsion tensor field; note, however, that the compatibil- by cosmology, Raychaudhuri started from the idea that ity of the connection with the metric is still imposed, a singularity is nothing more than an artifact of the sym- that is, the covariant derivative (defined with the inde- metries of the matter distribution; in his seminal paper pendent connection) of the metric tensor field along any [1] he then proposed a model of time-dependent universe space-time curve is null. without assuming the cosmological principle and its implications on homogeneity and isotropy. The analysis of Theories of gravity with non-vanishing torsion have the flows kinematic that he carried out resulted in the been extensively considered in the literature; the atten- renown Raychaudhuri equation1 for the evolution of the tion toward these models has been initially catalyzed cosmological expansion in a given background. It was by the relatively simple scheme they provide to account Raychaudhuri himself that pointed out the relation of for nontrivial quantum effects in a gravitational environ- his work with the existence of singularities, and it should ment, achieved through a direct coupling between the not come as a surprise that Penrose and Hawking were intrinsic spin of matter and the torsion field [10–18]; late inspired by Raychaudhuri’s paper to define the con- however this interest has been extended to the fact that ditions under which their singularity theorem holds [2,3]. torsion appears naturally and inevitably in the low en- Since its inception, the Raychaudhuri equation and its ergy limit of super string theories, any theory of gravity subsequent generalizations (see e.g. the nice review [4]) that considers twistors and in higher dimensional Kaluza- have found application not only, as already mentioned, Klein theories [19]. On top of that, gravitational theo- in terms of singularity theorems but also in a vast range ries with torsion exhibit a unique nature in comparison of different physical contexts, from the study of gravita- with other modified theories of gravity: in general, unless tional lensing [5], to crack formation in spherical astro- some particular further assumptions are imposed, the ef- physical objects [6], and finally to some more fundamen- fects of a non-null torsion field cannot be recast in an effective energy-momentum tensor, in other words, the torsion field cannot be seen, in general, as an extra matter field in a Lorentzian torsion free manifold. ECSK ∗ [email protected] theory is a particular exception to this property, where † [email protected] torsion can in fact be seen as the effect of a matter field 1 Sometimes referred to as the Landau–Raychaudhuri equation. in a Lorentzian manifold, usually called spin-torsion. Yet 2

ECSK theory by no means represents the most generic Having in mind the general affine connection, the com- case of a theory equipped with a torsion field (see for ex- mutator between two vectors can be expressed in terms ample [19, 20]) and, although this circumstance is widely of the torsion-full covariant derivative as recognized, it is intriguing to note that most of the work [u, v]γ = uα∇ vγ − vα∇ uγ − 2S γ uαvβ . (6) done on gravitational theories with non-vanishing torsion α α αβ only considers the ECSK setup: for instance, the very im- This last equation and the definition of the Riemann ten- portant question of whether the presence of torsion can γ sor associated with Cαβ, avoid the formation of singularities resulting from gravi- ρ ρ ρ ρ σ ρ σ tational collapse has only been answered for the case of a Rαβγ = ∂βCαγ − ∂αCβγ + CβσCαγ − CασCβγ . (7) spin-torsion field [11–13]. In this paper we then start fill- lead to a modified version of the relation between the ing such a crucial gap in the literature: we will focus on curvature and the commutator of two covariant derivative the study of the effects of the most generic torsion field in the case of non vanishing torsion, on the kinematics of test particles and derive the Ray- ρ ρ chaudhuri equation for a congruence of null and time-like Rαβγ wρ = [∇α, ∇β] wγ + 2Sαβ ∇ρwγ . (8) curves in the spacetime. The Riemann tensor for a non-symmetric connection The paper is organized as follows: in Sec.II we intro- does not have all the usual symmetries. However, from duce the basic definitions and set the conventions that (7) we see that will be used throughout the article; in Sec.III we derive ρ ρ the evolution equation of the separation vector between Rαβγ = −Rβαγ , (9) test particles in space-times with torsion, define the kinematical quantities of a congruence of curves and derive and, using the symmetries of the contorsion tensor, (4) the Raychaudhuri equation, both for a time-like and null and (5), congruence of curves; finally, we summarize the main re- Rαβγρ = −Rαβργ . (10) sults and sort out our conclusions in Sec.IV. So, the skew symmetry of the Riemann tensor is still ver- ified. To define the Ricci tensor in terms of the Riemann II. CONVENTIONS AND NOTATIONS tensor we will adopt the convention [21, 22] γ Rαβ = Rαγβ , (11) Due to the wide variety of different conventions used in literature, let us start by gently introducing the basic which, using (7), can be expressed in terms of the con- definitions and setting the conventions that will be used nection coefficients as throughout the article. Introduce a covariant derivative, γ γ ρ γ ρ γ Rαβ = ∂γ Cαβ − ∂αCγβ + CαβCγρ − CγβCαρ . (12) β β β σ ∇αU = ∂αU + CασU , (1) The Ricci scalar is defined as R = gαβR . (13) constrained to be metric compatible, ∇αgβγ = 0, but αβ γ otherwise with a completely generic connection Cαβ. The Note that in the case of a general affine metric-compatible anti-symmetric part of the connection define a tensor connection it is still possible to take an independent con- which is called the torsion tensor traction of the Riemann tensor defining a 2-rank tensor, γδ γ γ 1 γ γ Reαβ = g gβRαγδ ; however a further contraction with Sαβ ≡ C = C − C . (2) [αβ] 2 αβ βα the metric results in Re = −R, that is the Ricci scalar is unequivocally defined. Using such definition, it is possible to split the connec- A further important remark is about the notation we tion into an appropriate combination of the torsion tensor γ will be using to describe the matter sector. For a gen- plus the usual metric Christoffel symbols Γαβ, eral Riemann–Cartan theory of gravity, the matter La-

γ γ γ γ γ grangian can couple (non-)minimally to the torsion ten- Cαβ = Γαβ + Sαβ + S αβ − Sβ α . (3) sor which introduce new degrees of freedom in the problem. While the stress-energy tensor is still deﬁned as The sum of the three torsion pieces on the right hand usual as the variation of the matter action with respect side of last equation is frequently dubbed in literature as to the metric, we need to introduce a new object, the in- the contorsion tensor, K γ ≡ S γ + Sγ − S γ ; αβ αβ αβ β α trinsic hypermomentum, deﬁned as the variation of the using the anti-symmetry of the torsion tensor, it is an action with respect to the independent connection, easy task to verify straightforwardly the two symmetries of contorsion, 1 δSMatter ρ √ ∆µν ≡ ρ . (14) −g δΓµν Kαβγ = −Kαγβ , (4) Such quantity encapsulates all the information of the mi- croscopic structure of the particle, i.e. intrinsic spin, dila- γ γ K[αβ] = Sαβ . (5) ton charge and intrinsic shear. 3

III. RAYCHAUDHURI EQUATION For infinitesimally close curves, the evolution of the separation vector along the fiducial curve is entirely described A. The separation vector and its evolution by the tensor field Bαβ. Let us emphasize that in the derivation of (19) and (20) we have not specified the type of the tangent vector to the fiducial curve, U α, hence, this Introduced the basic definitions and identities we are α now in the position to generalize the Raychaudhuri equa- equations are equally valid for the case of U being time- tion for the case of an N-dimensional space-time with like, space-like or light-like, with the fiducial curve being non-null torsion. either a geodesic or not. The notion of separation (sometimes deviation) vector Let us now verify some physical implications of (19) and (20) in the case of the presence a non-vanishing tor- between two infinitesimally close curves is quite intuitive: α define a congruence of curves, not necessarily geodesics, sion tensor. The derivative along c of the quantity nαU such that each curve of the congruence is parameterized reads by an affine parameter λ. Consider a second congruence, D (n U α) α = n aβ + 2S U σU αnγ , (21) this time of geodesics, parametrised by an affine parame- dλ β σγα ter t, such that each geodesic intersects a curve of the first congruence at one and only one point of the space-time. where we have defined the acceleration vector appearing for non geodesics fiducial curves as Given two curves in the first congruence, c1 and c2, and a geodesic of the second congruence, γ, let the two points aα ≡ U γ ∇ U α . (22) p and q be the intersection points of γ with, respectively, γ c1 and c2, with c1 (λ0) = γ (t0) = p. Let us now assume The expression (21) represents the failure of the separa- that the point q is in a small enough neighbourhood of tion vector nα and the tangent vector U α to stay orthog- ∂γ α the point p such that q = γ (t0 + δt) ≈ p + δt. If n onal to each other, that is, if at a given point n and ∂t α t0 U are orthogonal to each other, a general non-null tor- is the tangent vector to the geodesic γ in p, then α sion, Sαβγ , or a non-null acceleration, a , will spoil the preservation of such orthogonality along the curve. Note ∂γ n ≡ δt = q − p (15) that a torsion field, with no further imposed symmetry, ∂t α t0 will lead to effects parallel to the direction of U (second term on the right hand side of (21)), contributing to a gives also a meaningful notion of the separation between relative acceleration between two initially infinitesimally the curves c and c . 1 2 close particles. Let us consider a coordinate neighbourhood that con- The analysis of (21) leads to the conclusion that the tains the points p and q, such that p = {xα}, q = {x0α} = tensor B describing the behaviour of the separation {xα + nα}, with αβ vector will have, for the case of a generic torsion tensor, ∂xα a non-null component tangential to the fiducial curve c. nα = δt , (16) Without loss of generality, it is then possible to write ∂t Bαβ in terms of two components, one orthogonal and α and let U be the tangent vector to the curve c1 (from the other parallel to c: given a projector hαβ onto the here on we will drop the index 1) at p, hypersurface orthogonal to the curve c at a given point, we can write ∂xα U α = . (17) ∂λ Bαβ = B⊥αβ + Bkαβ , (23) In order to find the general expression for the evolution with α of the separation vector, n , we will start by computing B ≡ hγ hσB , (24) the Lie derivative of n over the tangent vector U and vice ⊥αβ α β γσ versa. From the definition of the Lie derivative and using Bkαβ ≡ Bαβ − B⊥αβ . (25) (16) and (17) we find that In analogy to the General Relativity case, we want to define the kinematical quantities identifying expansion, Ln U = LU n = 0 . (18) shear, and vorticity - θ, σαβ and ωαβ, respectively - of neighbouring curves of the congruence. These quantities Using (6) and (18) it is possible to derive an equation will only depend on the orthogonal part of the tensor for the change of the separation vector along the fiducial B , B , so that, defining the expansion, shear and curve c, αβ ⊥αβ vorticity as β α α β γ U ∇βn = Bβ n , (19) θ = B⊥γ , (26) hαβ where σαβ = B⊥(αβ) − γ θ , (27) h γ α α α γ Bβ = ∇βU + 2Sγβ U . (20) ωαβ = B⊥[αβ] , (28) 4

2 B⊥αβ can be decomposed into [24] (see also [25, 26] for early single-pole approximation descriptions). A quite interesting result is that single- hαβ pole particles without intrinsic hypermomentum follow, B⊥αβ = γ θ + σαβ + ωαβ . (29) h γ as in general relativity, geodesics of the metric connec- β (g) α tion, v ∇β v = 0, no matter what is the underlying Before we continue and derive the Raychaudhuri equa- theory of gravity. tion let us emphasize that the definitions of the kinemat- Things become much more complicated for particles ical quantities given by (26)-(28) are always valid when- with non-vanishing intrinsic hypermomentum. We will ever the tensor Bαβ is related with the variation of the focus on the case of Riemann-Cartan space-time, since separation vector between curves of a congruence by an in our setup nonmetricity is trivially zero. We will also equation such as (19). A formal proof for this assertion consider particles endowed with only mass and intrinsic can be seen in [23]. spin but we neglect dilaton charge and intrinsic shear for simplifying reason. This means that the hypermomentum tensor reduces (for a single-pole particle [24]) ρ ρ B. Raychaudhuri equation for a congruence of to ∆[µν] = τµν v , where τµν is the anti-symmetric spin time-like curves density tensor. In this case the equations of motion for a particle read The results in the previous section are quite general κ κ 2 κ β σ and valid for curves of any kind; however, the procedure v ∇κvα = − vαv ∇κ ln m − v ∇κ v v ∇στβα + that defines the projector h strictly depends on the m αβ µ β ν σ specific family of curves considered. Once the projector + Sαβ v (mvµ + 2v v ∇στνµ) + is assigned, (29) will give an actual expression in terms 1 + R ν vβτ µ ≡ a , of the tangent vector and the torsion tensor. 2 αβµ ν α Let us then start by considering the generalized Ray- (32) chaudhuri equation for a congruence of time-like curves. In this case we will impose that the fiducial curve is α σ κ v ∇ατµν − 2v[µv v ∇κτσ|ν] = 0 , (33) parametrised by the proper time, τ, and, in order to avoid confusion with the general case, we will label its tangent α α Notably, this equation is independent by the specific vector as v , with vαv = −1. The operator projecting choice of the gravitational part in the action. It is impor- onto the hypersurface orthogonal to vα is given by tant to stress that for a non-spinning particle, τµν van- ishes, the second of equations (33) is trivially satisfied hαβ ≡ gαβ + vαvβ , (30) while the first one recovers the equation of the geodesics β (g) α of the metric, v ∇β v = 0. Note that the rest mass, and fulfills the following conditions that is the projection of the particle 4-momentum on the α rest frame, is guaranteed to be constant along the con- hαβv = 0 , gruence only in the case of non-spinning particle. γ hαhγβ = hαβ , (31) Using equations (30)-(33) in equations (24) and (25) σ hσ =N − 1 , we find

ρ ρ σ where N is, again, the dimension of the space-time. B⊥αβ = ∇αvβ + 2Sραβv + 2Sρασv v vβ + vαaβ ,(34) In order to calculate the two components, orthogonal ρ σ Bkαβ = −2Sρασv v vβ − vαaβ . (35) and parallel, of Bαβ another ingredient is necessary: we must find an expression for the trajectories along which For the case of a congruence of time-like curves in a N- free particles move. In a general non-Riemannian man- dimensional space-time (29) can be written as ifold, it is not an easy task to determine such physical curves. The statement usually claimed in the litera- 1 ture that particles follow geodesics, either of the Levi- B⊥αβ = hαβθ + σαβ + ωαβ , (36) N − 1 Civita generic connection or of the metric connection, turns out to be rather unsatisfactory and naïve. In a where (31) was used. manifold equipped with a completely generic affine con- Taking the variation of the tensor B along the fidu- nection, the particle trajectories can be correctly deter- ⊥αβ cial curve, and remembering the expression (8) for the mined starting from the equations of motion, with the latter obtained themselves from the conservation laws for canonical energy-momentum and hypermomentum. A comprehensive treatment for the calculation of the propagation equations of single-pole and pole-dipole particles 2 We warn about a slight different notation in [24]: for example in metric-affine theories of gravity has been developed in the definition of the torsion tensor is there twice our definition. 5

Riemann tensor, we find spin, Refs. [14, 28] consider the simplifying assumption γ γ on the torsion tensor Sαβ = Sαβv , with S(αβ) = 0 and DB⊥αβ γ ρ γ S vα = 0; in Ref. [12] instead torsion is constrained =v ∇γ B⊥αβ = Rγαβρv v + ∇αaβ− αβ γ γ σ dτ to be Sαβ = ηαβσ v v S , where ηαβσ is the com- γ γ ρ − (∇αv )(∇γ vβ) − 2v Sγα ∇ρvβ+ (37) pletely anti-symmetric Levi-Civita tensor. However, in γ ρ γ + 2v ∇γ (Sραβv ) + v ∇γ (vαaβ) + these special cases the extra symmetries imposed on the + 2vγ ∇ (S vρvσv ) , torsion tensor imply that the second term on the right γ ρασ β hand side of (20) is null, reducing the problem to the Shear and vorticity are, respectively, traceless and anti- torsion free case. symmetric, so that, contracting α and β on both sides of previous equations, we obtain C. Raychaudhuri equation for a congruence of null Dθ curves = − R vρvγ + ∇ aα − (∇ vγ )(∇ vα) dτ γρ α α γ γ ρ α γ ρ − 2v Sγα ∇ρv + 2v ∇γ (Sρv ) , (38) Let us now derive the Raychaudhuri equation for a congruence of null curves. As in the previous subsection, α where Sρ ≡ Sρα . Taking into account that to avoid any confusion, we will re-label the tangent vector α α to the fiducial curve and call it k , so that k kα = 0. B Bβα = ∇ vβ (∇ vα) + 2vρS α∇ vβ+ ⊥αβ ⊥ α β ρβ α In this case, unfortunately, if ehαβ is the projector onto β ρ α β α ρ γ + 2Sρα v ∇βv + 4Sρα Sγβ v v + the hypersurface that is orthogonal to the fiducial null curve, it cannot be naively defined by (30), since it would + 4S vαvγ aβ , αβγ not be orthogonal to kα (it would be h kα = k 6= 0). (39) eαβ β The way out of this problem is through the introduction and using (36) we can write (38) as of an auxiliary null vector field, ξα, such that [29] Dθ 1 kαξ = −1 , (41) = − R vρvγ − θ2 + σ σαβ + ω ωβα + α γρ αβ αβ α dτ N − 1 ξαξ = 0 . (42) β ρ 1 α α α Using (41) and (42) we can now properly introduce a + 2Sρα v hβ θ + σβ + ωβ N − 1 projector onto the hypersurface orthogonal to both kα α γ ρ β ρ α α + ∇αa + 2v ∇γ (Sρv ) + 2Sρα v vβa . and ξ as (40) ehαβ = gαβ + kαξβ + ξαkβ , (43) This equation represents the generalization of the Ray- chaudhuri equation for a time-like congruence of curves satisfying the following properties in the presence a generic torsion field. Here, we would h kα =h ξα = 0 , like to make a couple of comments: first of all, this equa- eαβ eαβ σ tion has been obtained using only geometrical arguments, eh αehσβ =ehαβ , (44) plus the canonical energy-momentum conservation equa- hσ =N − 2 . tion to define the equations of motion of the particles; eσ this means that the result is independent by the spe- As in the time-like case, the extra ingredient to be cific geometrical theory that we are choosing: once that taken into account is the effective particles trajectories. the theory has been assigned, then the Ricci tensor can An important caveat is here due: particles that are mov- be related with the energy-momentum tensor accordingly ing along null curves are massless particle; the most nat- with the (modified) Einstein field equations. Secondly, it ural candidates in the Standard Model are then photons. in interesting to stress that the extra-force responsible However, the minimal coupling procedure to general- of the acceleration term reported in the first line of (40) ize the electromagnetic field to non-Riemannian environ- is of purely geometric origin, related to the extra cou- ments preserve photons by having a non-vanishing intrin- pling of the intrinsic spin (viz. intrinsic hypermomentum sic hypermomentum; hence, photons follow the geodesics in the most general setup) with post-Riemannian struc- determined by the Christoffel symbols, tures. Note also that our result differs from previous kα∇(g)kβ = 0 . (45) versions of the torsion-full Raychaudhuri equation avail- α able in literature. More concretely, refs. [4, 19, 27] did As a side note, let us stress that some theories, such as the not take properly into account the relation between the Standard Model Extension [30], allow for a non-minimal tensor Bαβ and the evolution of the separation vector be- coupling of the electromagnetic field with geometry (and tween infinitesimally close curves of the congruence, as- eventually other fields), which means a non trivial intrin- suming that the expression of the tensor Bαβ is a priori sic hypermomentum: anyway, in this case the extra oper- the same as in the case of null torsion. Few quite specific ators will also introduce some effective mass for the pho- models for torsion accidentally result anyway in the cor- ton (the simplest case being the explicitly massive Proca rect expression: retracing the properties of the intrinsic field) and a fortiori they could not follow null curves. 6

In the context of the Standard Model, neutrinos de- Tracing Eq. (46) and computing its covariant deriva- serve a separate mention; the recent confirmation of the tive along the fiducial null curve we find phenomenon of neutrino oscillations [31, 32] directly imply that neutrinos might be massive [33], avoiding them Dθ α β 1 2 αβ βα = −Rαβk k − θ + σαβσ + ωαβω to follow null paths3. In spite of that, let us take into ac- dλ N − 2 count the behaviour of the Standard Model massless neu- h˜ α + 2S βkρ β θ + σ α + ω α + 2kγ ∇ (S kρ) trino, viz. a single-pole, massless Dirac particle. Dirac ρα N − 2 β β γ ρ particles have a completely anti-symmetric hypermomen- α γ ρ βγ α α2 tum, which implies, in the single-pole approximation, a + 2∇α (S γρk k ) + 4 h Sαβγ k − Sαk vanishing intrinsic spin density tensor, τµν = 0 [34, 35]. αβ µ h δ ρ δi Back to (33), this would mean that the particles follow + 4Sµ k Sδβαk − hβSδραk metric geodesics, as in the non-spinning case, and as for + Sα kµ hνβ − gνβ 4B + 6B photons. µν e kαβ kβα It is finally possible to use (41)-(45) to find the or- α δβ δβ − 2Bkδ Bkβα eh − g , thogonal and tangential components of the tensor Bαβ that defines the dynamics of the separation vector of the (50) null congruence of geodesics; a straightforward calcula- which represents the Raychaudhuri equation for a con- tion gives gruence of null geodesics in the presence of a torsion tensor field. Here the literature is more sparse: ref. [36] de- rives the Raychaudhuri equation for a null congruence γ γ σ γ σ of curves in the particular case of a completely anti- B⊥αβ = ∇αkβ − 2Sαγβk − 2Sαγσk k ξβ + 2Sγσβkαk ξ γ σ ρ γ γ symmetric torsion, missing in any case the correct defi- + 2Sγρσkαk k ξβξ + kαξ ∇γ kβ + kβξ ∇αkγ nition of the Bαβ tensor (although without affecting the γ σ γ σ − 2Sαγσkβk ξ + kαkβξ ξ ∇σkγ final result). γ σ γ ρ σ + 2Sβγσk k ξα + 2Sγσρkαkβk ξ ξ Looking at Eq. (50) it is not clear that the Raychaud- + 2k ξ ξσS kγ kρ (46) huri equation for a null congruence is independent of the β α σγρ choice of the auxiliary null vector ξα that was introduced to define the projector onto the orthogonal hypersurface B = −2S kρk ξσ − k ξρ∇ k + 2S kρk ξσ to the congruence, Eq. (43): we have terms that depend kαβ ρσβ α β α ρ αρσ β αβ ρ σ γ ρ σ on the projector eh and Bkαβ, which themselves depend − 2Sρσγ k kαkβξ ξ + 2Sαρσk k ξβ on ξα. However, since the expansion θ is a scalar quan- ρ σ γ σ γ ρ − 2Sργσk k kαξ ξβ − 2ξ ξαkβSσγρk k tity, its rate of variation along the congruence will be γ σ ρ − 2ξαSβγσk k − kαξ ∇ρkβ related to the the rate of variation of θe by ρ σ − kαkβξ ξ ∇σkρ (47) Dθ µ µ µ µ (g) Deθe = k ∇µθ = k ∂µθ = k ∂µθe = k ∇µ θe = . Similarly to what was done in (29), we can relate the dλ dλ (51) component B⊥αβ with the expansion, shear and vorticity D µ (g) of the congruence of the null curves, where dλe = k ∇µ represents the covariant derivative along the fiducial curve where only the metric connection is considered. Computing the derivative, we recover the ehαβ B = θ + σ + ω , (48) usual general relativity expression for the Raychaudhuri ⊥αβ N − 2 αβ αβ equation governing the evolution of the expansion: that in combination with (46) provides an expression for the expansion of the congruence of null geodesics in terms α α Deθe α β 1 2 αβ βα of k , ξ and Sαβγ , given by = −Reαβk k − θe + σαβσ + ωαβω , dλ N − 2 e e e e (52) θ = ∇ kγ + 2S kγ = ∇(g)kγ ≡ θ . (49) γ γ γ e where the tilded quantities are calculated with the Christoffel connection, and Note that also in this case, as in general relativity, the scalar expansion θ does not depend on the auxiliary vec- θe =∇(g)kµ , tor ξµ chosen to define a proper projection operator to µ treat the null curves congruence case. (g) γ (g) γ (g) σeµν =∇(µ kν) + ξ k(µ∇γ kν) + ξ k(ν ∇µ) kγ (53) γ σ (g) (g) µ + kµkν ξ ξ ∇σ kγ − ∇µ k , (g) γ (g) γ (g) ωeµν =∇[µ kν] + k[µξ ∇γ kν] + ξ k[ν ∇µ] kγ . 3 For the sake of completeness, we want to mention that also in this case there is still room for the (unlikely) scenario with one Note that, in spite of the explicit appearance of the aux- out of three neutrino species exactly massless. iliary vector ξµ in the two expressions for σeµν and ωeµν , 7 the total expression (52) does not depend on ξµ because Knowing how the kinematical decomposition of the it is possible to find that geodesics congruence is influenced by the torsion tensor allows the possibility to test models equipped with a 1 2 1 nontrivial Riemannian connection through the study of σ σµν = − θ˜2 + θ˜ + ∇(g)kν · ∇(g)kµ eµν e N − 2 N − 2 2 µ ν the motion of test particles. Let us expand a bit on this 1 point: the matter source appearing on the right hand side + ∇(g)kν · ∇(g)µk , 2 µ ν of the equation of motion for the torsion tensor (what is 1 1 usually dubbed hypermomentum, obtained from the vari- ω ωνµ = ∇(g)kν · ∇(g)kµ − ∇(g)kν · ∇(g)µk , eµν e 2 µ ν 2 µ ν ation of the classical matter action with respect to the (54) independent connection) depends on the specific coupling showing that Eq. (52) is independent of the auxiliary null between matter and torsion itself: different models will vector ξµ. Since Eq. (52) is equivalent to Eq. (50), we find lead to different field equations for the torsion and hence- that the Raychaudhuri equation for a null congruence is forth to different solutions. This means that the eventual independent of the vector field ξµ. contributions of torsion to the various kinematical quan- As a final comment, let us address the following im- tities and their evolution will be dependent on the spe- portant remark: one could be tempted to naively think cific chosen model; the analysis then of the evolution of a that, since (52) does not depend explicitly of the torsion matter fluid in a region of space-time will give a chance tensor, then the evolution of the expansion of the curves to distinguish between different allowed couplings. followed by massless particles does not depend on the One of the most important achievements of this paper presence of a torsion field. This conclusion is not correct is the generalization of Raychaudhuri equation - the since although the torsion tensor does not explicitly ap- equation for the evolution of the expansion of a congru- pear in (52), it does affect the geometry of the space-time ence of curves - for the case of time-like and null curves in a way described by the (modified) Einstein field equa- in an N-dimensional space-time with the most generic tions; the metric solution of the Einstein equations will torsion field. The study of the evolution of the expansion be itself affected by the such modification4, and so will be of time-like and null curves in spacetime is obviously important due to its role in defining the evolution of the corresponding (metric) Ricci tensor Reµν appearing in (52). gravitational collapse and possible formation of singularities. Moreover, as was shown in Ref. [36], the expansion of null light-rays also plays a preponderant role on the definition of the throats of (dynamical) wormholes, IV. CONCLUSIONS namely exotic solutions requiring the violation of energy conditions; a still open question is whether or not the In this paper we derived the equation for the evo- degrees of freedom of a completely generic torsion field lution of the separation vector between infinitesimally can somehow avoid the violation of the null energy close curves of a congruence in space-times with non-null condition of whatever matter present at the dynamical generic torsion field, clarifying some of the ambiguities wormhole’s throat. We firmly believe that the study of lingering in the literature about the role of the torsion such possibilities will be of great interest in the near tensor. We concluded that the presence of a torsion field future. leads in general to tangent and orthogonal effects to the congruence, in particular, the presence of a generic torsion field contributes to a relative acceleration between test particles. This effects happen either for free-falling ACKNOWLEDGMENTS or accelerated particles following time-like, null or space- like curves. The Authors wish to thank José P. S. Lemos for early The evolution equation of the separation vector can discussions on a first version of the paper. We thank be further separated and be used to study the kinemat- FCT-Portugal for financial support through Project No. ical quantities that characterize a congruence of curves, PEst - OE/FIS/UI0099/ 2015. PL thanks IDPASC and namely the expansion, shear and vorticity. We derived, FCT-Portugal for financial support through Grant No. for the first time in the literature, how such kinematical PD/BD/114074/2015. VV is supported by the FCT- quantities depend on a completely generic torsion field. Portugal grant SFRH/BPD/77678/2011.

4 Just as a workable example, consider the simple Einstein–Cartan tion, rewritten in terms of the Christoﬀel connection eliminating case: one of the equations of motion describes algebraically the the torsion tensor, relates the (Christoﬀel) Einstein tensor to a torsion tensor as a function of the spin tensor; the other equa- combined version of the stress-energy tensor, that now takes into account also spin density terms [10]. 8

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