Junction conditions in extended Teleparallel gravities
Alvaro de la Cruz-Dombriz, Peter K. S. Dunsby, Diego Saez-Gomez
aa r X i v : . [ g r- q c ] J a n Prepared for submission to JCAP
Junction conditions in extendedTeleparallel gravities ´Alvaro de la Cruz-Dombriz a Peter K. S. Dunsby b,c , DiegoS´aez-G´omez b,d a Departamento de F´ısica Te´orica I, Ciudad Universitaria, Universidad Complutense deMadrid, E-28040 Madrid, Spain. b Astrophysics, Cosmology and Gravity Centre (ACGC), Department of Mathematics andApplied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, SouthAfrica. c South African Astronomical Observatory, Observatory 7925, Cape Town, South Africa d Fisika Teorikoaren eta Zientziaren Historia Saila, Zientzia eta Teknologia Fakultatea,Euskal Herriko Unibertsitatea, 644 Posta Kutxatila, 48080 Bilbao, Spain.
Abstract.
In the context of extended Teleparallel gravity theories, we address the issueof junction conditions required to guarantee the correct matching of different regions ofspacetime. In the absence of shells/branes, these conditions turn out to be more restrictivethan their counterparts in General Relativity as in other extended theories of gravity. Infact, the general junction conditions on the matching hypersurfaces depend on the underlyingtheory and a new condition on the induced tetrads in order to avoid delta-like distributionsin the field equations. This result imposes strict consequences on the viability of standardsolutions such as the Einstein-Straus-like construction. We find that the continuity of thescalar torsion is required in order to recover the usual General Relativity results. dombriz [at] fis.ucm.es peter.dunsby [at] uct.ac.za diego.saezgomez [at] uct.ac.za ontents f ( T ) gravity 33 ADM decomposition in Teleparallel gravity 54 Junction conditions 65 The Einstein-Straus reconstruction 96 Conclusions 13 Teleparallel gravity is a gravitational theory for the translation group, associating a Minkows-kian tangent space to every point of the spacetime. Teleparallel gravity parallelly transportsthe so-called vierbeins/tetrads field, providing the name of the theory ( c.f. [1] for a compre-hensive review). In order to achieve this, the theory is constructed in terms of the so-calledWeitzenb¨ock connection instead of the Levi-Civita connection. Unlike the Levi-Civita con-nection, the Weitzenb¨ock connection is not commutative under the exchange of the lowerindices, which not only induces a non-zero torsion, but also a zero Ricci scalar. Thus, thecovariant action for Teleparalell gravity is constructed in terms of the scalar torsion T . Arelevant feature of this theory lies in the fact that every solution of General Relativity (GR)is also a solution for Teleparallel gravity. Moreover, some extensions of Teleparallel gravityin the cosmological context have attracted some interest over the last few years. This is dueto the fact that Dark Energy may be described within the framework of these theories ( c.f. [2]). In analogy of higher-order theories of gravity, such as f ( R ) theories [3], Teleparallel grav-ity has been extended by constructing gravitational Lagrangians in terms of more complexfunctions of the torsion scalar. They have been referred to as f ( T ) theories. When introduc-ing extra terms in the action, the aforementioned equivalence between GR and Teleparallelgravity now does not remain between f ( T ) and f ( R ) theories. However, the f ( T ) gravita-tional field equations are second order whereas modifications of General Relativity, such as f ( R ) theories, are usually higher order. This results for instance in interesting differences,because no extra gravitational waves modes (in comparison to GR) appear [4]. Furthermore,the inflationary scenario has been studied in these theories [5], as well as the non-invarianceof the theory under local Lorentz transformations [6]. In addition, some other extensionsof Teleparallel gravity have also been proposed, such as conformally invariant actions inTeleparallel gravity [7].As it is widely known, extensions of Teleparallel gravity are not invariant under localLorentz transformations (see Ref. [6]). Thus, the field equations will be sensitive to the choiceof tetrads and consequently, the determination of the correct tetrad fields, leading to a metrictensor with some desirable symmetries, has attracted some attention in the last few years.The key-point lies in the requirement of a correct parallelisation of spacetime, i.e., the correct– 1 –hoice of tetrads, in order to guarantee a solution of the field equations while recovering thedesired metric. In fact, the choice of some tetrads may impose restrictions in the form of the f ( T ) theories able to satisfy the field equations. Authors in [8] have claimed that once goodtetrads have been chosen the functional form of f ( T ) must not be restricted. Literaturedevoted to this subtlety in Teleparallel gravity has mainly focused on standard solutionssuch as Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universes [9] or vacuum static andspherically symmetric (Schwarzschild) solution [10]. Considerable attention has been devotedto the Schwarzschild solution because of its importance [11–15]. The use of naive tetrad fieldswhich were expected to represent a Schwarzschild solution were shown to be inconsistent withthis solution [10], [12]-[14]. Nevertheless, a set of tetrads (the so-called Schwarzschild frame ),able to cover this solution can be obtained by performing a convenient boost (or rotation [8])of the isotropic set of tetrads [10]. Moreover, in the context of f ( T ) theories the existenceof vacuum solution with both vanishing R and T leads to f ( T ) = T + O ( T ), for which theSchwarzschild tetrads lead to vacuum solutions. In addition Birkhoff’s theorem has beenaddressed after several attempts [8, 15, 16] leading to Schwarzschild-de Sitter solution withapplications in 3-dimensional scenarios [10].Cosmological solutions in Teleparallel theories have also been given extensive coveragein the literature [2]. Analogously to the subtleties about the correct choice of tetrads in staticscenarios, the use of diagonal tetrad in spherical coordinates in f ( T ) theories, constrains thegravitational Lagrangian, since f T T = 0 is a necessary condition to fulfill the field equations.In other words, the f ( T ) Teleparallel gravity would reduce to GR. Nevertheless, rotation ofthese tetrads shows that no constraints on the f ( T ) action appear and consequently, generalclasses of f ( T ) models are allowed, in principle in principle allowed [8]. The fact that rotationssuffice to find adequate tetrads providing either spherically symmetric and static solutionsor homogeneous and isotropic solutions, opens up the possibility of considering more generallocal Lorentz transformations guaranteeing the existence of well-behaved tetrads.The application of junction conditions is a key issue in every gravitational theory aimingto describe satisfactorily, among other things, stellar configurations or collapsing bodies.Since these conditions guarantee a smooth transition through the matching hypersurfacebetween different spacetime regions, their violation poses serious shortcomings in any theoryunder consideration [17–19]. Previous work dealing with other types of extended theoriesof gravity demonstrated that additional junction conditions with respect to standard GRconditions are required. A case in point is f ( R ) gravity, where the scalar (Ricci) curvaturehas to be continuous across the matching hypersurface when only jumps in the matter sectorare permitted ( c.f. [20–22]). This result implies that matched solutions in Einstein gravitywill not necessarily exist in the context of fourth-order f ( R ) gravity due to the presence ofthese additional junction conditions. With respect to Teleparallel gravity, i.e., f ( T ) = T ,this theory does recover the standard GR results, and consequently this fact demonstratesthat the usual Israel junction conditions [17] are satisfied in this case independently of thetetrads, as the theory remains invariant under local Lorentz transformations. Nevertheless,this is not the case in f ( T ) theories with f ( T ) = T , where the matching between twodifferent spacetime regions provides junction conditions different from those in GR. In thispaper we shall demonstrate that the presence of a new junction condition, which is absent inTeleparallel gravity, plays a crucial role in the possibility of matching two different spacetimeregions. In the absence of branes, i.e., just considering the case of two different regionsof spacetime described by different metrics, the new conditions are f ( T ) model-dependentand require the continuity of the induced tetrad. It is not straightforward to combine both– 2 –equirements in order to recover usual classical solutions. However, to illustrate this difficulty,we attempt to construct the Einstein-Straus model, where an empty spherically symmetricregion is matched to a general FLRW metric [23]. Unlike the case of f ( R ) gravity where thecondition on the Ricci scalar does not allow one to reconstruct the Einstein-Straus model ( c.f. [21]), we shall show how this model may be reconstructed in f ( T ) gravity by an adequateprocedure.The paper is organised as follows: in Section 2 a brief introduction to f ( T ) theoriesis given. Section 3 deals with the Arnowitt-Deser-Misner (ADM) decomposition in thesetheories. This decomposition, when a trivial redefinition of coordinates is performed, willenable us to provide the key equations of our analysis in a transparent way once the spacetimefoliation is performed in the correct coordinates. Then, the junction conditions for f ( T )gravity are derived in Section 4. Section 5 is devoted to the reconstruction of the Einstein-Straus model within these theories. We conclude in Section 6 with a discussion about possibleapplications of our results.Throughout the paper we shall follow the following conventions: D µ will representthe covariant derivative with respect to the Levi-Civita connection Γ αµν and • D µ holds forthe covariant derivative in terms of the Weitzenb¨ock connection • Γ αµν . Greek indexes suchas µ, ν... refer to spacetime indexes, the latin letters a, b, c... refer to the tetrads indexesassociated with the tangent space, while the latin letters i, j, k... express indexes of a 3-hypersurface of the spacetime. f ( T ) gravity Teleparallel gravity can be described as an orthonormal basis of tetrads e a ( x µ ) defined inthe tangent space at every point x µ on the spacetime. Then, the metric can be expressed interms of the tetrads as follows d s = g µν d x µ d x ν = η ab θ a θ b , (2.1)d x µ = e µa θ a , θ a = e aµ d x µ , (2.2)where η ij = diag(+1 , − , − , −
1) and e µa e aν = δ µν or e µa e b µ = δ ba . The tetrads e aµ representthe dynamic fields of the theory.Furthermore, in Teleparallel gravities, the connection describing the covariant deriva-tives of tensors is given by the Weitzenb¨ock connection instead of the Levi-Civita connection: • Γ αµν = e αa ∂ ν e aµ = − e aµ ∂ ν e αa , (2.3)which leads to a vanishing scalar curvature but non-zero torsion. Then, the torsion tensor T αµν is defined as the antisymmetric part of the connection (2.3): T αµν = • Γ ανµ − • Γ αµν = e αa (cid:0) ∂ µ e aν − ∂ ν e aµ (cid:1) , (2.4)while the contortion tensor K µνα is defined as the difference between the Weitzenb¨ock andthe Levi-Civita connections: K αµν = • Γ αµν − Γ αµν = 12 (cid:0) T αµ ν + T αν µ − T αµν (cid:1) , (2.5)– 3 –r equivalently, K µνα = −
12 ( T µνα − T νµα − T µνα ) . (2.6)Furthermore, in order to construct the gravitational action, a tensor S µνα is defined, by usingthe torsion and contortion tensors: S µνα = 12 (cid:16) K µνα + δ µα T βνβ − δ να T βµβ (cid:17) , (2.7)whereas the torsion scalar is constructed contracting the torsion tensor (2.4) and the tensor(2.7) as follows, T = T αµν S µνα , (2.8)Alternatively, the torsion scalar can also be expressed in terms of the torsion tensor: T = 14 T λµν T µνλ + 12 T λµν T νµλ − T ρ µρ T νµν . (2.9)Then, the well-known action for Teleparallel gravity is given by [1] S G = 12 κ Z e T d x . (2.10)In analogy of f ( R ) gravity, this action can be further generalised into more general functionsof scalar torsion: S G = 12 κ Z e f ( T )d x . (2.11)By assuming a matter action S m = R e L m d x with L m being the matter Lagrangian, thefield equations can be obtained by varying the action (2.11) with respect to the tetrads,yielding S νρµ ∂ ρ T f
T T + h e − e iµ ∂ ρ ( ee αi S νρα ) + T αλµ S νλα i f T + 14 δ νµ f = κ T νµ , (2.12)where T νµ = e νa e δ L m δe µa holds for the matter energy-momentum tensor, f T = d f ( T ) / d T and f T T = d f ( T ) / d T . In addition, by using the contortion tensor (2.5), the relation betweenthe Ricci scalar and the torsion scalar yields, R = − T − D µ T νµν . (2.13)Thus, it is straightforward to check that Teleparallel gravity is equivalent to GR, since thecovariant derivative in (2.13) can be removed in the action (2.10). Furthermore, the relation(2.13) enables us to rewrite the field equations (2.12) in the more usual covariant description, f T (cid:18) R µν − g µν R (cid:19) + 12 g µν ( f ( T ) − f T T ) + 2 f T T S νµρ D ρ T = κ T µν , (2.14)where R µν and R the Ricci tensor and Ricci scalar, respectively. Let us stress at this stagethat by setting f ( T ) = T , the field equations (2.14) reduce to the standard Einstein fieldequations. – 4 – ADM decomposition in Teleparallel gravity
In order to determine the junction conditions between different regions of a spacetime forthe theory studied in this paper, it is convenient to foliate the spacetime in hypersurfacesorthogonal to the direction where the potential discontinuity may in principle lie. Althoughthe relevant split to address the junction problem will be of the form 3 + 1 where the firstthree coordinates refers to one temporal and two spatial coordinates – the ones describingthe two-dimensional hypersurfaces – and the last one refers to the orthogonal coordinatewith respect to the boundary. This split can be understood as totally analogous to ADMdecomposition [24] up to signs issues, so we decided to illustrate the ADM decomposition inthe frame of f ( T ) theories, a tool which will enable us to advance in Section 4. In fact, atthe beginning of Section 4 we provide the required change to go from one split (ADM) tothe other (3 + 1). Once these changes are introduced, both the discontinuities and Dirac-likedistributions appearing in the field equations can be more easily analysed using Gaussiancoordinates.Let us then consider the usual ADM formalism and foliate the spacetime into hypersur-faces of constant time, such that the metric can be written as followsd s = N d t − h ij (cid:0) d x i + N i d t (cid:1) (cid:0) d x j + N j d t (cid:1) . (3.1)Then the Ricci scalar can be rewritten in terms of the above metric: R = Θ − Θ ij Θ ij + R (3) − √ hN ∂ ( √ h Θ) + 2 N (cid:0) Θ N i − h ij N ,j (cid:1) ,i , (3.2)where Θ ij is the extrinsic curvature defined byΘ ij = n µ ; ν ∂x µ dx i ∂x ν dx j , (3.3)and n µ is the normal vector to the hypersurface. Then, the extrinsic curvature on a hyper-surface of constant time t yieldsΘ ij = − N (cid:16) ˙ h ij − D i N j − D j N i (cid:17) . (3.4)Here both dot and ∂ denote derivative with respect to time t . In the case of f ( T ) theories,the set of tetrads which gives the metric (3.1), is not unique and each choice may lead todifferent solutions. Consequently, for simplicity the following set of tetrads will be assumed[25, 26]: e µ = ( N, ) , e aµ = ( N a , h ai ) , (3.5)where N a = h ai N i and h ai is the set of spatial components of the tetrads. To simplifythe notation, let us use i = 1 , ,
3, which refer to the spatial indexes of the spacetime ofthe hypersurface of constant time, while we denote now a = 1 , , h ij = η ab h ai h b j , (3.6)where again a, b = 1 , , η ab . Then, the torsion components aregiven by, T j = ∂ j NN , T ij = • D j N i − N i N ∂ j N − h ia ∂ h aj , T ijk = (3) T ijk . (3.7)– 5 –he expression of the torsion scalar can be easily obtained by using the relation (2.13) andthe expression of the Ricci scalar (3.2): T = Θ ij Θ ij − Θ − R (3) + 2 √ hN ∂ ( √ h Θ) − N (cid:0) Θ N i − h ij N ,j (cid:1) ,i − D µ T νµν . (3.8)Alternatively and in analogy with the extrinsic curvature (3.4), we may define the extrinsictorsion as follows, • Θ ij = − N (cid:16) ˙ h ij − • D i N j − • D j N i (cid:17) , (3.9)which is related to the extrinsic curvature (3.4) by • Θ ij = Θ ij − N k N ( T ijk + T jik ) , (3.10)whereas the scalar torsion can be rewritten in terms of the extrinsic torsion (3.9): T = • Θ ij • Θ ij − (cid:18) • Θ + N k N T iik (cid:19) + T (3) + N k N l N T ijl ( T ijk + T jik ) + 2 N k N T ijk • Θ ij + 2 N (cid:20) √− h ∂ (cid:18) √− h N i N T kik (cid:19) + D j (cid:18) N i N j N ∂ i N (cid:19) − T iji ∂ j N (cid:21) . (3.11)In the next section, we use this formalism for foliating the spacetime into hypersurfaceswhich act as matching boundaries between different regions of the spacetime. Thus theADM formalism has allowed us to express the geometrical sector of the f ( T ) field equationsin terms of geometrical quantities defined on the boundary hypersurface. Let us now examine the junction conditions for a general f ( T ) action. Here we consider thecase where only jumps in the matter sector are allowed, so that there are no shells/braneslocated at the boundary, only discontinuities in the energy-momentum tensor. In other words,the matter side of the equations does not contain any delta function in the distribution, whichforbids any delta function within the gravitational sector of the field equations. In order toobtain the junction conditions, we can express the metric tensors in a Gaussian-normal frameadapted to describe the aforementioned scenario with two regions:d s = d y + γ ∗ ij d x i d x j . (4.1)For this particular choice of the coordinates, the boundary between the two different space-time regions can be located at y = 0, where γ ∗ ij is the induced metric or first fundamentalform on the matching hypersurface and each region of the spacetime induces a particularmetric. In comparison with the ADM decomposition, where the spacetime is foliated byconstant time hypersurfaces, here we have not assumed any particular foliation of the space-time, keeping y as a general coordinate. Then, by following the definition (3.3), the extrinsiccurvature is given by, Θ ij = 12 ∂ y γ ij . (4.2)Note that the choice of Gaussian coordinates (4.1) excludes crossing terms between the y and x i coordinates in the line element (4.1), in such a way that a 3 + 1 decomposition can– 6 –e followed from the previous ADM decomposition by setting N i = 0 and replacing thecoordinate t by the unknown coordinate y , while the first fundamental form γ ij = − h ij , asthe negative sign is removed in order to keep the hypersurfaces of constant y as general aspossible, with no particular time-like or space-like nature. Then, the set of tetrads describingthe metric (4.1) can be chosen as follows e yµ = (1 , ) , e aµ = ( , γ ai ) , (4.3)which yields γ ij = η ab γ ai γ bj . The non-zero components of the torsion tensor are given by T iyj = γ i a ∂ y γ aj , T ijk = (3) T ijk (4.4)and the contortion tensor and the tensor S µνα are K yi j = 12 γ ik ∂ y γ kj , K ij y = 12 η ab (cid:16) γ ia ∂ y γ jb − γ ja ∂ y γ ib (cid:17) , K ij k = (3) K ij k ,S yjy = 12 T ij i , S yij = 12 (cid:18) γ ik ∂ y γ kj − δ ij γ ka ∂ y γ ak (cid:19) , S ijy = 12 K ij y , S ijk = (3) S ijk . (4.5)Thus, by using the above relations among this decomposition and the ADM decompositionof the previous section, the torsion scalar (3.11) is easily obtained in terms of the extrinsictorsion, T = • Θ ij • Θ ij − • Θ + T (3) , (4.6)where the extrinsic torsion is given by • Θ ij = Θ ij = ∂ y γ ij . Lets remember that the foliationof the spacetime is achieved by hypersurfaces of constant y . Then, the junction conditionsat the hypersurface y = 0 can be obtained by analysing the field equations for a general f ( T ) action through the equations (2.14). The y − y , y − j and i − j components of the fieldequations are f T G yy + 12 ( f − T f T ) + f T T T iki ∂ k T = κ T yy ,f T G yj + f T T (cid:18) γ ik ∂ y γ kj − δ ij γ ka ∂ y γ ak (cid:19) ∂ i T = κ T yj ,f T G ij + 12 γ ij ( f − T f T ) + f T T h S kji ∂ k T + (Θ ij − γ ij Θ) ∂ y T i = κ T ij , (4.7)where G µν = R µν − g µν R is the Einstein tensor whose components are given by [20] G yy = − (cid:16) Θ ij Θ ij − Θ + (3) R (cid:17) ,G yj = −∇ i (cid:0) Θ ij − δ ij Θ (cid:1) ,G ij = ∂ y (Θ ij − γ ij Θ) + 2Θ ki Θ kj − ij + 12 γ ij (cid:16) Θ kl Θ kl + Θ (cid:17) + (3) G ij . (4.8)Hence, the junction conditions can now be analysed on the hypersurface y = 0. It is straight-forward to see that according to the definition of the extrinsic curvature, which contains firstderivatives of the induced metric across the boundary and the expression of the scalar torsion– 7 –4.6), the first fundamental form has to be continuous in order to avoid powers of deltas inthe expression of T . This leads to [ γ ij ] + − = 0 , (4.9)which can also be expressed in terms of the tetrads as follows, (cid:2) η ab γ ai γ aj (cid:3) + − = 0 . (4.10)Nevertheless, note that the above condition does not imply [ γ ai ] + − = 0, as the choice oftetrads is not unique. However, the second equation in (4.7) imposes continuity on theinduced tetrads in order to avoid delta-like distributions in the equations, which leads to thesecond junction condition in f ( T ) gravities (absence in Teleparallel gravity),[ γ ai ] + − = 0 . (4.11)Furthermore, we see in the field equations (4.7) that the ij -component may also containdelta-like distributions through the term, f T ∂ y (Θ ij − γ ij Θ) + f T T (Θ ij − γ ij Θ) ∂ y T = ∂ y [ f T (Θ ij − γ ij Θ)] ∝ [ f T (Θ ij − γ ij Θ)] + − δ ( y ) , (4.12)which gives [ f T (Θ ij − γ ij Θ)] + − = 0 . (4.13)Hence, taking the trace of the above expression, the third junction condition is obtained,namely [ f T Θ ij ] + − = 0 . (4.14)Recall that by assuming f T = 1, i.e., Teleparallel gravity, the usual junction conditions of GRare recovered, where (4.11) becomes irrelevant and (4.14) leads to [Θ ij ] + − = 0. In addition,the third junction condition is model-dependent which means that in general, matchings inGR will not be able to be reconstructed in f ( T ) gravity. Nevertheless, in order to recover thesolutions of GR, we may consider a more restrictive set of conditions that can be obtainedby imposing the continuity on both terms of the left-hand side (l.h.s.) of equation (4.12)separately. This would involve the continuity of the second fundamental form or extrinsiccurvature: [Θ ij ] + − = 0 , (4.15)which consequently implies the continuity of the extrinsic torsion (3.11). Then, provided that f T T = 0, the second term of the l.h.s. of (4.12) contains a divergence of the scalar torsion: ∂ y T = ∂ y T + θ + ( y ) + ∂ y T − θ − ( y ) + [ T ] + − δ ( y ) . (4.16)which leads to the condition [ T ] + − = 0 . (4.17)It is straightforward to check that (4.15) and (4.17) form a subset of the third condition(4.14). In conclusion, the junction conditions for a general f ( T ) Teleparallel theory turnsout to be (4.9), (4.11) and (4.14).Nevertheless, note that the set of tetrads considered here (4.3) is not the most generalone, but just a particular choice. Let us consider now a general set of tetrads that describesthe metric (4.1), g µν = η ab ˜ e aµ ˜ e b ν . (4.18)– 8 –he first fundamental form or induced metric on the hypersurface y = 0 is defined as follows, γ ij = g µν ∂x µ ∂x i ∂x ν ∂x j = η ab ˜ e aµ ˜ e b ν ∂x µ ∂x i ∂x ν ∂x j = η ab ˜ γ ai ˜ γ bj , (4.19)where we have defined the induced tetrads as ˜ γ ai = ˜ e aµ ∂x µ ∂x i . Note that for the set (4.3), theinduced tetrads are given by γ ai , defined in (4.3). Furthermore, as both sets of tetrads leadto the same metric, (4.3) and (4.18) are connected by a Lorentz transformation,˜ e aµ = Λ ab e b µ , (4.20)while the torsion tensor for the general set of tetrads (4.18) can be written as,˜ T λµν = T λµν + ζ λµν , where ζ λµν = Λ ba e λb e c [ ν ∂ µ ] Λ ac . (4.21)Thus the torsion scalar yields˜ T = T + 12 T λµν ζ µνλ + T λµν ζ νµλ − T ρµρ ζ νµν + ζ , (4.22)where ζ = ζ λ µν ζ µνλ + ζ λ µν ζ νµ λ − ζ ρ µρ ζ νµ ν . Then, the crucial term ˜ S yij of the yi -component of the field equations yields˜ S yij = S yij + .... − Λ ba (cid:16) e ib e c j ∂ y Λ ac + γ im γ jk e kb e c m ∂ y Λ ac − δ ij e kb e c k ∂ y Λ ac (cid:17) (4.23)The component S yij contains terms as ∂ y γ aj , which does not provide any delta-like dis-tributions because of the second junction condition (4.11), so in order to avoid delta-likedistributions the Lorentz transformation should be the same on both sides of the bound-ary, [Λ ab ] + − = 0, or in other words, the induced tetrads of the general set { ˜ e aµ } has to becontinuous, [ γ ai ] + − = Λ ab h ˜ γ bi i + − = 0 → [˜ γ ai ] + − = 0 . (4.24)Following the same procedure, it is straightforward to check that the third condition (4.14)has to also be satisfied when assuming the general set of tetrads (4.18).Hence, the novelty of these results with respect to other extended theories of gravitynot only lies in equations (4.11) and (4.14), but also in the possibility of recovering theGR junction conditions from (4.15) and (4.17). In the following section, we illustrate thereconstruction procedure for the Einstein-Straus model in general f ( T ) gravity. In this section let us consider the matching of two spacetime regions within the frameworkof extended Teleparallel gravity. We focus on the so-called Einstein-Straus model [23]. Thisreconstruction consists of an empty region with a point-like mass at the center, surroundedby an expanding FLRW spacetime. The aim here is to match both regions within the generalframework of f ( T ) gravity using the junction conditions found in the previous section. In theoriginal Einstein-Straus construction, the interior solution is described by the Schwarzschildsolution: d s = (cid:18) − Mr (cid:19) d t − (cid:18) − Mr (cid:19) − d r − r dΩ , (5.1)– 9 –here dΩ = d θ + sin θ d φ . On the other hand, the exterior region consists of a FLRWspacetime, whose metric written in the usual comoving coordinates is given by:d˜ s = d˜ t − a (˜ t ) (cid:18) d˜ r − k ˜ r + ˜ r dΩ (cid:19) . (5.2)Here the tilde is used to refer to those quantities defined in the FLRW region of the spacetime.The boundary between both regions is located at r = R in the Schwarzschild spacetimeand at ˜ r = ˜ R in the FLRW spacetime, so that we can proceed to apply the junction conditionsshown in the previous section. To do so, let’s obtain the induced metric on the hypersurface r = R from the Schwarzschild side, which yields γ ij d x i d x j = A − ˙ RA ! d t − R dΩ , (5.3)where the indexes { i, j } refer to coordinates defined on the hypersurface r = R and A = (cid:0) − Mr (cid:1) / . Moreover, the induced metric from the FLRW region gives˜ γ ij d x i d x j = d˜ t − a (˜ t ) ˜ R dΩ . (5.4)According to the first junction condition (4.9), both induced metrics have to coincide, i.e.,[ γ ij ] + − = 0, which gives the following relations R = a (˜ t ) ˜ R and d˜ t = p A − ˙ R A d t , (5.5)leading to γ ij = ˜ γ ij . Let us now apply the second junction condition (4.11). For that purpose,the following choice of tetrads for the Schwarzschild metric (5.1) is preliminarily assumed, e aµ = diag (cid:18) − Mr (cid:19) / , (cid:18) − Mr (cid:19) − / , r, r sin θ ! , (5.6)while the non-null components of the induced tetrads lead to γ at = Aδ at + ˙ RA δ ar , (5.7)According to the second junction condition, γ at = ˜ γ at , so the choice of tetrads in the FLRWregion is determined by such condition. Let’s assume the following set:˜ e aµ = A a ( t ) ˙ R √ (1 − k ˜ r )( A − ˙ R ) ˙ RA a ( t ) q (1 − k ˜ r )(1 − ˙ R A ) ra
00 0 0 ˜ ra sin θ , (5.8)where we have assumed the change of coordinates (5.5). Then, it is straightforward to showthat the induced tetrads are continuous across the boundary.– 10 –o analyse the third junction condition, we need to obtain the normal vector of thehypersurface from both sides of the boundary. On the vacuum side we have n µ = A p A − ˙ R (cid:16) − ˙ Rδ tµ + δ rµ (cid:17) , (5.9)while on the FLRW side, we obtain ˜ n µ = a − k ˜ R δ ˜ rµ . (5.10)Then, the non-zero components of the extrinsic curvature (3.3) on the Schwarzschildside areΘ tt = 6 A A ,r − A A ,r − A ¨ R A ( A − ˙ R ) / , Θ θθ = s A R A − ˙ R , Θ φφ = s A R A − ˙ R sin θ . (5.11)In the FLRW region, we have˜Θ θθ = a (˜ t ) ˜ R p − k ˜ R , ˜Θ φφ = a (˜ t ) ˜ R p − k ˜ R sin θ . (5.12)In addition, the scalar torsion for the Schwarzschild tetrads (5.6) is given by T = AR (cid:18) A ′ + AR (cid:19) = 2 R , (5.13)while the scalar torsion for the FLRW tetrads (5.8) yields˜ T = 1 R ( A − ˙ R ) (cid:20) A − ˙ R ) − R a ( A − ˙ R )( A k + 3 A ˙ a − k ˙ R )+8 R p − kR /a A ˙ R s − ˙ RA − R p − kR /a A s − ˙ RA ¨ R , (5.14)where we have used (5.5). Let us now apply the third junction condition (4.14), i.e.,[ f T Θ ij ] + − = 0, to this setup. To do so, we may assume a particular action: for simplic-ity let’s take f ( T ) = T m , which leads to the condition˙ R = A " − A − kR /a (cid:18) T ˜ T (cid:19) m − , (5.15)where the usual Einstein-Straus model is recovered when T = ˜ T . This is clearly not thecase, as can be seen by comparing (5.13) and (5.14). In order to recover the Einstein-Strausresult, the continuity on the torsion tensor has to be guaranteed, so that the more restrictiveconditions (4.15) and (4.17) have to be satisfied. However, as f ( T ) gravity is not invariantunder local Lorentz transformations, different sets of tetrads may lead to different solutions(see Ref. [6]). In particular, it is well known that diagonal tetrads in (5.6) do not allow theexistence of Schwarzschild solution for any action different from f ( T ) = T , which obviouslyreduces to Teleparallel gravity or equivalently to GR. Nevertheless, by transforming the orig-inal set of tetrads (5.6) by applying a particular Lorentz transformation, the Schwarzschild– 11 –etric might turn out to be a solution of a more general f ( T ) action [8]. This situationalso appears in the case of non-flat FLRW metrics, as pointed out in Ref. [8, 10]. This isalso the case when matching different spacetime regions and one tries to recover the usualEinstein-Straus model.In order to keep the second junction condition (4.11) valid, the same Lorentz transfor-mation is applied on both sides. Let us consider the transformation: e ′ aµ = Λ ab e b µ . (5.16)As shown in the previous section, the first junction condition remains unaffected by the localLorentz transformation due to the fact that it can be expressed in terms of Lorentz invariantquantities. Also the second junction condition (4.11) remains valid when applying the sametransformation. Nevertheless, the torsion scalar T is not a Lorentz invariant, which may leadto (4.15) and (4.17) by the appropriate transformation, recovering the usual result of GR forthe Einstein-Straus model.We consider the following rotationΛ ab = φ sin θ − (sin φ cos α + cos θ cos φ sin α ) sin φ sin α − cos φ cos θ cos α φ sin θ cos φ cos α − cos θ sin φ sin α − (cos φ sin α + sin φ cos θ cos α )0 cos θ sin θ sin α sin θ cos α , (5.17)where α = α ( t, r ). Then, by keeping α arbitrary, the torsion scalar (5.13) for the Schwarzschildregion evaluated at the hypersurface r = R , turns out to be T ′ = 4 R − M + A ( R − M ) sin α + AR ( R − M ) cos α α r | r = R R ( R − M ) (5.18)By also applying the rotation matrix (5.17) to the FLRW tetrads, the torsion scalar evaluatedat ˜ r = ˜ R = R/a becomes:˜ T ′ = − R ( A − ˙ R ) (cid:26) R (cid:16) A − A ˙ R (cid:17) ˙ a a − RA ˙ R (cid:18) aa sin α + cos αα t (cid:19) − (cid:18) − kR a (cid:19) ˙ R + A (cid:20) − kR /a − (cid:18) sin α + R cos αα r | r = R a (cid:19) q (1 − kR /a )(1 − ˙ R /A ) (cid:21) + 2 A ˙ R (cid:20) − R ˙ A (cid:18) sin α + q (1 − kR /a )(1 − ˙ R /A ) (cid:19) + ˙ R (cid:18) − kR /a + (cid:18) sin α + Ra cos αα r | r = R (cid:19) q (1 − kR /a )(1 − ˙ R /A ) (cid:19)(cid:21) + 2 RA " sin α (cid:18) R ˙ aa + 1 (cid:19) + ¨ R q (1 − kR /a )(1 − ˙ R /A ) + ˙ Ra cos αα t . (5.19)Then, it may be possible to get T ′ = ˜ T ′ for an appropriate function α ( t, r ). Nevertheless, letus be reminded that the expression for α ( t, r ) can not be obtained analytically in general butwould require numerical resources. In case of the existence of such function, the GR junctionconditions are recovered and in combination with (5.5), equation (5.15) leads in this case tothe following well-known result [21]:1 a (cid:18) ∂a∂ ˜ t (cid:19) + ka = 2 M ˜ R a (˜ t ) . (5.20)– 12 –his condition requires that the FLRW side is filled with a pressureless fluid, whose energycoincides with the energy enclosed in the Schwarzschild region. Note however, that whilekeeping T = ˜ T in (5.15), the matching between a FLRW spacetime and the Schwarzschildsolution will lead to different descriptions of the cosmic evolution that will depend on f ( T ),while using the above tetrads and a particular expression for α ( t, r ), it gives rise to (5.20)independently of f ( T ). In this paper, we have obtained for the first time, the junction conditions matching tworegions of a spacetime within the framework of the so-called f ( T ) theories of gravity, anatural generalisation of Teleparallel gravity. One might have expected that since GeneralRelativity and Teleparallel gravity are equivalent theories, the scenario of equivalence mightbe extrapolated to arbitrary functions of Ricci and torsion scalar, i.e., when comparing f ( R )and f ( T ) solutions. However, we have shown above that f ( R ) and f ( T ) do not constituteequivalent theories as it is obvious from our expression (2.13). Therefore, matching resultsin f ( R ) gravity theories do not provide in principle any hint about the phenomenology in f ( T ) theories.Furthermore, the f ( T ) field equations are not invariant under local Lorentz transfor-mations, which may lead to different solutions while working with different sets of tetrads,an element which is absent in other modified gravities. In this way, we have shown howthe non-invariance of the equations affects the accomplishment of the junction conditions.For instance, the choice of the tetrads plays a crucial role due to the fact that the inducedtetrads on the boundary have to be continuous (4.11). We have illustrated this fact for theEinstein-Straus model, where the matching between a Schwarzschild-vacuum solution anda Robertson-Walker region leads to a different reconstruction from the one obtained whenthis model is studied in pure Einsteinian gravity. In addition, the third junction condition(4.14) depends upon the underlying gravitational theory, which leads to different config-urations of the Einstein-Straus model unless one imposes continuity on the torsion scalar(4.17), which may be achieved whenever a particular rotation is applied (on both sides of theboundary), such that the General Relativity result can be recovered. A natural consequenceis that the Oppenheimer-Snyder collapse scenario [27], which represents the complementarymatching of the Einstein-Straus configuration, i.e., collapsing star to form a black hole, isdirectly addressed since its existence and phenomenology is equivalent to the Einstein-Strausconstruction studied here.In addition, it is interesting to remember that a similar issue (existence of solutionsdepending upon the chosen tetrads) arises when dealing separately with Schwarzschild andFriedmann-Robertson-Walker metrics in the context of f ( T ) theories. This may supportthe idea of the existence of good and bad tetrads, as pointed out in Ref. [8]. In particular,not only the existence of the second junction condition (4.11), absent in Teleparallel gravity,affects directly the choice of tetrads, but also the third one (4.14) in case one wants to recoverthe GR results by imposing continuity on the torsion scalar, which is not invariant underlocal Lorentz transformations.In summary, we have obtained the general junction conditions for every f ( T ) gravity,different from the ones of Teleparallel gravity, and explicitly given by (4.9), (4.11) and (4.14),which implies the continuity of the first fundamental form, the induced tetrads and thequantity f T Θ ij . These conditions depend upon both the f ( T ) model and the choice of the– 13 –etrads. This demonstrates not only the necessity of further study to determine appropriatetetrad fields that describe the junction conditions properly, but also the necessity of analyzingpossible new solutions that arise depending on the f ( T ) Lagrangian. Specifically, the ideathat the Israel equations in the presence of shells/branes could be of universal validity fortheories invariant under diffeomorphisms, as claimed for fourth-order theories of gravity [22]constitutes a natural extension of this investigation. Acknowledgments:
We would like to thank Jos´e M. M. Senovilla and Ra¨ul Verafor useful comments about this project. A.d.l.C.D. acknowledges financial support fromMINECO (Spain) projects FPA2011-27853-C02-01, FIS2011-23000 and Consolider-IngenioMULTIDARK CSD2009-00064. P.K.S.D. thanks the NRF for financial support. D. S.-G.acknowledges the support from the University of the Basque Country, Project ConsoliderCPAN Bo. CSD2007-00042, the NRF financial support from the University of Cape Town(South Africa) and MINECO (Spain) project FIS2010-15640. A.d.l.C.D. thanks the hospital-ity of Kavli Institute for Theoretical Physics China (KITPC), Chinese Academy of Sciences,Beijing (China) for support during the early stages of preparation of this manuscript aswell as the Centre de Cosmologie, Physique des Particules et Ph´enom´enologie CP3, Univer-sit´e catholique de Louvain, Louvain-la-Neuve, Belgium and Theoretical Physics Department,Basque Country University, Spain for assistance with the final steps of this manuscript.
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