Traversable wormholes in Chern-Simons modified gravity
aa r X i v : . [ g r- q c ] J a n Traversable wormholes in Chern-Simons modified gravity
J. R. Nascimento, ∗ A. Yu. Petrov, † and P. J. Porf´ırio ‡ Departamento de F´ısica, Universidade Federal da Para´ıbaCaixa Postal 5008, 58051-970, Jo˜ao Pessoa, Para´ıba, Brazil
In this paper, we examine the existence of traversable wormhole solutions within theChern-Simons modified gravity. We find a non-trivial solution in the theory with dynamicalChern-Simons coefficient in the absence of matter sources. This result displays a situationopposite to GR where the matter sources violating the energy conditions are required.
I. INTRODUCTION
In recent years, the search for modified gravity theories as an alternative to General Relativityhas strongly intensified. The main reasons for this consist in the facts that, first, the GR doesnot allow for a consistent quantum description of gravity due to its non-renormalizability, second,it fails to explain the cosmic acceleration. In this context, various alternative gravity models areconsidered, such as f ( R ) gravity, f ( R, Q ) gravity, Horava-Lifshitz gravity models, theories includingtorsion and nonmetricity and many other examples. Among these models, an important role isplayed by the four-dimensional Chern-Simons (CS) modified gravity [1], whose action looks likesum of the four-dimensional gravitational CS term with the usual GR action, so that it involves aChern-Pontryagin density (third derivatives of the metric background) coupled to a pseudoscalarfield ϑ called the CS coefficient. In particular, this theory represents itself as a first known exampleof CPT-breaking (and, for a special form of the CS coefficient, Lorentz-breaking) gravity model.Additionally, the CS modified gravity also turns out to be relevant in other physical contextssuch as: string theory (after dimensional reduction process) [2], loop quantum gravity and particlephysics, see f.e. [3] and references therein. Further extension of CS modified gravity called thedynamical CS modified gravity included a nontrivial dynamics for ϑ as well (for a general reviewon CS modified gravity, both in dynamical and non-dynamical cases, see f.e. [3]).As it is known, consistency of any modified gravity theory is verified through checking of con-sistency of known metric solutions in this theory. In [4], a wide class of spherically and axially ∗ jroberto@fisica.ufpb.br † petrov@fisica.ufpb.br ‡ pporfi[email protected] symmetric solutions of equations of motion in CS modified gravity was analyzed. In [5], it wasshown that to achieve order-by-order perturbative consistency of the Kerr metric within this theory,it should be modified by the ϑ -dependent terms, and in [6], the consistency of G¨odel-type solutionsin this theory was verified, with the causality conditions were analyzed.Among other interesting gravity solutions, the wormholes are of a special importance. Their keyfeature consists in the fact that in certain cases they allow for trajectories connecting two causallydisconnected points. Another remarkable feature about the wormholes is the possibility of timetravel, namely, it is possible for an observer in such geometries to travel faster than light. It isworth pointing out that around each point of this geometry the local Lorentz invariance is fulfilled,but globally this is not hold. It was shown in [7] the procedure how to build a time machine fromwormholes. Many interesting results related to wormholes can be found in [7, 8]. However, up tonow, the consistency of wormhole solutions within the CS modified gravity never was studied. Thisis the aim of this paper.The structure of this work looks like follows. In the section 2, we give a general descriptionof geometry of traversable wormholes. In the section 3, we present a brief review on CS modifiedgravity. And in the section 4, we verify the consistency of wormholes within the dynamical CSmodified gravity and discuss the related energy conditions. Finally, in the section 5 we present ourconclusions. In the Appendix A we present an explicit form of components of the Cotton tensorand the energy-momentum tensor of the ϑ field. II. TRAVERSABLE WORMHOLE GEOMETRIES
Wormholes have initially been introduced by Einstein and Rosen in seminal paper [9]. Thiswormhole (Einstein-Rosen bridge) has constructed from a particular extension of maximally ex-tended Schwarzschild solution of General Relativity. It covers two asymptotically flat spaces con-nected by a “bridge” whose spatial section possesses a non-trivial topology. Another importantfeature is that Einstein-Rosen wormhole presents event horizons, thus an observer traveling alongwormhole could not cross it, in other words, it is forbidden traveling from one flat space to theother. However, there must be conditions which prevent the emergence of the event horizons. Suchconditions are so-called of traversability conditions and the wormholes that satisfied such conditionsare known by traversable wormholes.In this section we briefly discuss the most important features of the traversable wormholes. Itsline element in spherical coordinates, following [10], is given by ds = − e r ) dt + dr − b ( r ) /r + r ( dθ + sin θdφ ) . (1)Note that this line element has spherical symmetry and is asymptotically flat. Furthermore, theradial coordinate r is non-monotonic, it may be split into two patches: the first one covers the rangewhere r decreases from + ∞ to r , where r (represents the throat of the wormhole) is the minimumof r , and the second one covers the range where r increases from r to + ∞ . The functions Φ( r )and b ( r ) stands for the gravitational redshift and shape function of the wormhole, respectively.In general, both functions must fulfill some requirements (traversability conditions): in order toavoid the presence of horizons it is necessary that e Φ( r ) is positive everywhere or, identically, Φ( r )must be finite and essentially real. Concerning the shape function we can show using embeddingarguments: it must satisfy the flaring-out condition at the throat, i.e.,( b − b ′ r )2 b > , (2)where the prime means derivative with respect r , leads to b ′ ( r ) < b ( r ) = r , as a conse-quence the proper distance will be minimal at the throat [10].In GR, such a traversable wormhole geometries requires matter sources violating the null energycondition (NEC), see f.e. [7, 10, 11]. This violation is represented by the inequality T ( m ) µν k µ k ν < , (3)where k µ represents any null vector and T ( m ) µν is the energy-momentum tensor of the matter sources.Indeed, the NEC is obtained from the famous Raychaudhuri equation whose form for null geodesicslooks like dθdτ = − θ − σ µν σ µν + ω µν ω µν − R µν k µ k ν , (4)where θ is the expansion, σ µν denotes the shear, ω µν represents the vorticity all them related tothe null geodesics congruence defined by the tangent null vector k µ that emerges the traversablewormhole at one side and going out at the other one. As shown in [7, 12], the vorticity is identicallyzero for a traversable wormhole. In addition, at the throat the expansion θ is also zero, satisfyingthe condition dθdτ ≥
0, thus Eq. (4) reduces to inequality R µν k µ k µ ≤ , (5)implying that null geodesics will defocus so that the null convergence condition [13] does not hold.By means of the Einstein field equations, the direct consequence of Eq. (5) can be shown to looklike T ( m ) µν k µ k ν ≤ , (6)therefore, the NEC is either violated or on the imminent of being violated [14]. We note thatactually our model represents itself as a gravity coupled to a scalar (either ghost or not). Thecoupling of the usual general relativity to the scalar matter has been studied in great details in[15] for a non-ghost matter, and in [16] for a ghost matter. As we see further, our theory involvesadditional non-minimal coupling responsible for the Chern-Simons term. It was argued in [17, 18]that these solutions display a wormhole-like behavior. III. CHERN-SIMONS MODIFIED GRAVITY
In the current section we present a brief review of the four-dimensional Chern-Simons (CS)modified gravity. Its action can be cast into the form [19, 20] S CS = 12 κ Z d x √− gR + α Z d x √− g (cid:18) ϑ ∗ RR (cid:19) − ˜ β Z d x √− g (cid:18) ∂ µ ϑ∂ µ ϑ (cid:19) + S mat , (7)where κ = 8 πG , ϑ is a (pseudo)-scalar field, α and ˜ β are coupling constants with a non-zero massdimension, and S mat is the matter action. Let us make substitutions α = 1 / κ and ˜ β = β/κ , thusthe dimension of ϑ is length squared, [ ϑ ] = L , whereas β has the dimension of inverse fourth powerof length, [ β ] = L − . Regarding to second term in the action, the ∗ RR is a topological term (itdoes not contribute for the equations of motion, if ϑ = const ) called the Chern-Pontryagin termand defined by ∗ RR ≡ ∗ R µ γσν R νµγσ = 12 ǫ γστη √− g R µ ντη R νµγσ , (8)where ǫ γστη is the Levi-Civita symbol and ∗ R µ γσν is the dual Riemann tensor. In more mathemat-ical language, the Chern-Pontryagin term in Eq.(7) can be rewritten as a second order polynomialof the two-form curvature, T r ( R ∧ R ) [21], such a quantity is proportional to the well-known secondChern class which, in turn, is related to Chern-Simons form Ω by means of a total derivative [22], T r ( R ∧ R ) = d Ω , (9)where Ω = T r (cid:18) ω ∧ dω + 23 ω ∧ ω ∧ ω (cid:19) and ω is one-form connection. In a more physical notation,the former equation becomes ∗ RR = 2 ∇ µ K µ , (10)where Γ λνσ represents the Levi-Civita connection coefficients, and K µ = ǫ µναβ √− g (cid:18) Γ λνσ ∂ α Γ σβλ + Γ λνσ Γ σαθ Γ θβλ (cid:19) is the topological current. Therefore, by integrating by parts the second term inEq. (7) we get Z d x √− g (cid:18) ϑ ∗ RR (cid:19) = − Z d x √− g v µ K µ , (11)where v µ = ∂ µ ϑ . When the ϑ becomes a constant, the modified theory reduces to the usual GeneralRelativity.The modified field equations are obtained varying to the action with respect to the metric andscalar field. Doing this, we arrive at: G µν + C µν = κT ( m ) µν + T ( ϑ ) µν ; β (cid:3) ϑ = − ∗ RR, (12)where (cid:3) ≡ √− g ∂ µ ( √− gg µν ∂ ν ) is for the covariant d’Alembertian operator. The energy-momentum tensor is defined as follows: first, the T ( m ) µν = − √− g (cid:18) δ L m δg µν (cid:19) , (13)describes the energy-momentum tensor of the matter sources and T ( ϑ ) µν = β (cid:20) ( ∂ µ ϑ )( ∂ ν ϑ ) − g µν ( ∂ λ ϑ )( ∂ λ ϑ ) (cid:21) , (14)represents the energy-momentum tensor of the contributions of ϑ . Finally, the variation of thesecond term in Eq. (7) with respect to the metric gives rise to the Cotton tensor C µν explicitlywritten as C µν = − (cid:20) v σ (cid:18) ǫ σµαβ √− g ∇ α R νβ + ǫ σναβ √− g ∇ α R µβ (cid:19) + v στ ( ∗ R τµσν + ∗ R τνσµ ) (cid:21) , (15)where v στ = ∇ σ v τ .The theory may be considered within two approaches depending on β coupling, namely: thefirst one is for non-dynamical framework (non-dynamical CS, NCS theory) that implies β = 0 sothat the kinetic term is ruled out, the second one is for the dynamical framework (dynamical CS,DCS theory), in this case β = 0 involves the non-zero kinetic term. IV. TRAVERSABLE WORMHOLES IN CS MODIFIED GRAVITYA. Vacuum solution
In this section we examine the possibility of the existence of traversable wormhole vacuumsolutions both in non-dynamical and dynamical frameworks.To study the equations of motion in our theory, for convenience, we will use the Cartan formal-ism. Within its methodology, for the traversable wormhole manifolds given by (1) we can define alocal Lorentz (orthonormal) co-frame such that θ (0) = e Φ( r ) dt ; θ (1) = (1 − b ( r ) /r ) − / dr ; θ (2) = rdθ ; θ (3) = r sin θdϕ, (16)where ds = η AB θ A θ B , with η AB = diag ( − , +1 , +1 , +1) being the Minkowski metric. It wasshown in [1] that in the Schwarzschild case one has ∗ RR = 0, and in [20], the same result wasshown to hold for all spherically symmetric geometries. Taking into account this result, the fieldequations in the co-frame (16), in the absence of matter, take the form G AB + C AB = T ( ϑ ) AB , (17) β (cid:3) ϑ = 0 , (18)where we have used natural units, κ = 1. The Ricci tensor and the Einstein tensor in the co-frame(16) are diagonal. Furthermore, for the sake of simplicity, we will take Φ ′ = 0 which leads toa constant redshift function (zero tidal force); such a condition avoids possible higher derivativeterms, as well as other complications in the field equations. Carrying out these simplifications, wehave G (0)(0) = b ′ ( r ) r ; G (1)(1) = − b ( r ) r ; G (2)(2) = G (3)(3) = − (cid:18) b ′ ( r ) r − b ( r ) r (cid:19) . (19)The non-vanishing components of C AB and T ( ϑ ) AB are described in Appendix A. B. Non-Dynamical framework
This case is covered by taking β = 0 in the field equations implying that T ( ϑ ) AB is ruled out.Hence, the Eqs. (17,18) reduce to G AB + C AB = 0 . (20)Note that the former equation decouples for spherically symmetric metrics as shown in [4], namely,it can be rewritten as follows R AB = 0 , (21) C AB = 0 , (22)where the first equation is identical to the vacuum Einstein equation.As a result of Birkhoff theorem, the field equations (22) do not describe wormholes. Hence, weconclude that the CS modified gravity in the absence of matter, in non-dynamical framework, doesnot support wormhole-like geometries independently of the CS field form. C. Dynamical framework
Differently from the non-dynamical case, the dynamical framework ( β = 0) is a more richapproach because the CS field is treated as a dynamical one. As a consequence, the field equationsmay dramatically change allowing wormhole vacuum solutions in contrast to NCS theory. In fact,it is reasonable to expect that, because in this case, the energy-momentum tensor of usual matteris zero, but the dynamics of CS scalar field adds up a new contribution to the right-hand side ofusual Einstein equations, so, the space-time now possesses an extra amount of energy (scalar hair).Before proceeding with the components of Eq. (17), let us focus on Eq. (18). It representsitself as a massless Klein-Gordon equation, and in the spaces (1) it can be solved via separation ofvariables. More precisely, within the classification used in [24–26] such geometries describe locallyrotational space-time (LRS), and their Killing vectors yield the Lie algebra T ⊕ so (3), where T represents the Killing vector ∂ t due to the fact that Eq. (1) is static while so (3) is associatedto other three Killing vectors with one of them being ∂ φ , implying rotational symmetry. Addingall this to the fact of linearity of (17), we conclude that it can be solved through separation ofvariables. As shown in [27], the invariant (maximally symmetric) solutions of Eq. (18) have theform ϑ = ϑ ( x l ) e γx j , where j labels the coordinates of the symmetries of (1), and l labels othercoordinates.In particular, as the CS coefficient is a real pseudo-scalar field, it can be written as ϑ ( t, r, θ, φ ) = A ( r ) Y ( θ ) Re (cid:18) e i ( mφ + ωt ) (cid:19) . (23)Substituting it in Klein-Gordon equation, we get (cid:20) θ ) ddθ (cid:18) sin( θ ) ddθ (cid:19) − (cid:18) m sin( θ ) (cid:19) (cid:21) Y ( θ ) = − l ( l + 1) Y ( θ ) , (24)( r − br ) A ′′ − (cid:18) b − r + 12 rb ′ (cid:19) A ′ + r ω A = l ( l + 1) A, (25)where the prime stands for derivative with respect to r and l ( l + 1) is the separation constant( l = 0 , , ... ). The Eq. (24) is nothing more as the well known associated Legendre equationwhose solutions are given by associated Legendre polynomials. On the other hand, the latterequation imposes the relation between A ( r ) and b ( r ).Returning to the modified field equations (17), we may simplify the solution that comes fromKlein-Gordon equation. For this, one must note that the off-diagonal field equations decouple,and, then, the field equations reduce to a set of partial differential equations (PDEs) which mustbe solved for non-trivial ϑ . Accordingly, we have G AB = T ( ϑ ) AB , A=B (26) C AB = T ( ϑ ) AB , A = B (27) (cid:3) ϑ = 0 . (28)It is clear that wormhole solutions in DCS theory are also solutions in GR by requiring the vanishingof Eq. (27) for some non-trivial ϑ satisfying the other equations, (26) and (28).Now, we shall solve the system of PDEs (27). As a first step, let us substitute the CS coefficientof the form (23) into the non-vanishing components of this system, explicitly discriminated inAppendix A. We get that the non-trivial solutions must fulfill the followings requirements: first,the separation constant should be zero ( l = 0). As a consequence one has m = 0 as well, second, ω = 0. As a result of that, it arrives that the only solution of Eq. (24) is dY ( θ ) dθ = 0. Therefore, thefield equations themselves force the CS coefficient to take the form ϑ ( t, r, θ, φ ) = A ( r ) leading tothe vanishing of all the components of Cotton tensor and non-diagonal components of T ( ϑ ) AB .We may exactly solve the radial equation that reduces to( r − br ) A ′′ − (cid:18) b − r + 12 rb ′ (cid:19) A ′ = 0 , (29)whose solution can be expressed in terms of the shape function, namely, A ( r ) = γ (cid:20) Z r (cid:18) − b ( r ) r (cid:19) − / dr (cid:21) , (30)where γ is an integration constant whose dimension is length cubed. The gradient of A ( r ) is actuallythe vector v µ defined in previous section. Notice that v A = (cid:20) , γr , , (cid:21) , evaluated in Lorentz co-frame (16), represents a space-like vector with a preferred direction on space-time. Therefore, thereis a preferred local Lorentz-frame where v A takes a non-zero value. It is interesting to note the factthat v A does not depends on b ( r ).Following this procedure, remains us to solve the diagonal components of the modified fieldequations. Note that the CS field in the form (30) gives rise to simplest non-trivial components ofenergy-momentum tensor of CS field, obtained in the co-frame (16) in which they do not dependon b ( r ): T ( ϑ )(0)(0) = β γ r ; T ( ϑ )(1)(1) = β γ r ; T ( ϑ )(2)(2) = − β γ r ; T ( ϑ )(3)(3) = − β γ r . (31)In addition, the CS field in the form (30) leads to the vanishing Cotton tensor.Inserting them in the modified field equations, we have a solution that leads to a specific formfor the shape function, b ( r ) = − βγ r , (32)explicitly depending upon γ and β parameters. In particular, the shape function given by Eq. (32)only obeys the flaring-out condition when β is strictly negative. However, such a choice on β leadsto the wrong-sign kinetic term in the action (7) so that the energy of the theory is not boundedfrom below, in other words, perturbations around the vacuum expectation value are unstable.Effectively, our manner to introduce the dynamics of the Chern-Simons coefficient ϑ provides aghost-like dynamics for it (for discussion of different issues related to ghosts, see f.e. [23]). Also, wenote that the use of the exotic matter is necessary for existence of wormholes and other noncausalsolutions, see f.e. [7, 10]. Nevertheless, we should emphasize that the ϑ is not a common matterfield but an ingredient of the CS modified gravity, so, this theory itself contains the possibility ofwormhole solutions, even in a vacuum, since there is no common matter contributions in (27). Inorder to meet the minimum distance condition at the throat, it is necessary which r = − βγ implying b ( r ) = r r . (33)0It just putting the above equation into Eq. (30) one finds an explicit form of CS field, namely, ϑ ( r ) = A ( r ) = π r − r arctan (cid:18)s r r − (cid:19) . (34)So, our main result consists in a possibility to find the shape function b ( r ) from the known CScoefficient ϑ ( r ), or vice versa.It is worth calling attention that the arctan (cid:18)q r r − (cid:19) is a multi-valued function then, takingthis into account and the fact that its argument is strictly positive, its image is inside the interval[0 , π/ ϑ ( r ) = A ( r ) = π r , and goes to zerofar away from it, i.e., lim r →±∞ r arctan (cid:18) q r r − (cid:19) = 0 . (35)For rr − ≪
1, we can approximate arctan (cid:18)q r r − (cid:19) ≃ q rr − ϑ shift symmetry (the constant cannot be dropped), so there onemeets the spontaneous breaking down of the shift symmetry. On the other hand, the solution holdsthe symmetry of the metric under transformation r → − r . For a more detailed analysis, we shallmake the following transformation: | x | = s r r − , (36)where the modulus has been used to ensure the symmetry under transformation x → − x . Notethat the minimal value of | x | corresponds to Eq. (36) evaluated at the throat, x ( r ) = 0. Thistransformation is convenient because it introduces a monotonic variable (one chart) covering thewhole wormhole instead of two charts. All the aforementioned features are displayed in Figs. (1,2)as well as the global behavior of x and ϑ ( r ).The energy-momentum tensor of CS field given by Eq. (31) may be reinterpreted through itssplitting into two parts: the first contribution, T ( d ) AB , is identical to a dust-like contribution withnegative energy density, thus violating the null energy condition (which is not unusual since thewormhole solutions require exotic matter, see f.e. [7]), while the second contribution, T ( e ) AB has aform analogous to a source-free circularly symmetric electric or magnetic field, where the explicitform of T ( e ) AB is given by [29]. Thus, we can relate the radius of the throat r with an effectiveelectric charge q through the expression r = q π ; this charge arises from a flux due to a non-trivialtopology of our background. Explicitly, we have T ( ϑ ) AB = T ( d ) AB + T ( e ) AB , (37)1 Figure 1: The figure displays r/r in terms of x . The dashed lines show the behavior asymptotic corre-sponding to Minkoswkian space that implies b ( r ) → ϑ ( r ) → x . We take r = 2. T ( d ) AB = − r r diag(1 , , , , (38)and T ( e ) AB = r r diag(1 , − , , . (39)So, the CS field in the presence of the background metric (1) behaves as a combination of the dust-like matter violating the energy conditions and a free electric charge (as expected such a behavioris identical to that one in GR, see [30]). Such a interpretation arises due to the non-trivial topologyof (1); in [31], this mechanism is referred as charge without charge. D. Non-null redshift
As argued in [20], the condition ∗ RR = 0 does not necessarily imply in vanishing of the Cottontensor, for an explicit example see [6]. Having this in mind, a natural generalization of the metric(1) is the case characterized by a non-constant redshift, g = const . Our aim in this subsection isto inspect the influence of the non-constant redshift on the Cotton tensor. For this purpose, let ususe a more general wormhole metric, see f.e. [32], ds = − A ( x ) dt + 1 B ( x ) dx + r ( x ) ( dθ + sin θdφ ) , (40)where A ( x ), B ( x ) and r ( x ) are functions of the arbitrary radial coordinate x . Similarly to Eq.(1),the metric 40 must satisfy some extra conditions for describing a traversable wormhole geometry,namely, r ( x ) has a global minimum at a point x = x , as mentioned in the Section 1 such a point isreferred by throat, further A ( x ) must be positive definite in order to prevent event horizons aroundthe throat. For sake of simplicity, but without loss of generality, it is natural to make the gaugechoice, A ( x ) = B ( x ).We shall take a similar ansatz to (23) for the CS field, i.e., ϑ ( t, x, θ, φ ) = U ( x ) Y ( θ ). Thenon-vanishing components of the Einstein, Cotton and energy-momentum tensor in the coordinatebasis are explicitly displayed in Appendix B. Thus, the field equations again decouple and take theform G µν = T ( ϑ ) µν , ν = µ (41) C µν = T ( ϑ ) µν , ν = µ (42) (cid:3) ϑ = 0 . (43)3By subtracting the 2 , , U ( x ) ddθ Y ( θ ) ⇒ ddθ Y ( θ ) = 0 , (44)imposing the constraint that non-trivial solutions have the form ϑ ( t, x, θ, φ ) = U ( x ) culminating inthe cancellation of the Cotton tensor and off-diagonal components of T ( ϑ ) µν .Briefly, we have shown in both cases: with and without constant redshift, that the field equationsnaturally decouple for wormhole-like geometries (1,40). The non-trivial solutions require thatthe CS field has an exclusive dependence on the radial coordinate. On the other hand, such arequirement leads to the vanishing of Cotton tensor. We conclude that even starting from DCStheory, the field equations reduce to GR and scalar field ones, and our results are in accordancewith those ones suggested in [4] for the dynamical framework. V. SUMMARY
We verified the conditions of persistence of wormhole-like vacuum solutions within CS modifiedgravity. The vacuum solutions in non-dynamical framework are identical to GR ones. Therefore,wormhole geometries are not vacuum solutions of NCS theory.Unlike non-dynamical framework, we have shown that wormhole vacuum solutions are allowed indynamical framework. We realized that the off-diagonal modified field equations in local Lorentzframe lead to a non-trivial setup for CS field ϑ , satisfying its evolution equation. Such a fieldconfiguration implies in vanishing of the Cotton tensor as well as the off-diagonal energy-momentumtensor of ϑ .We found the CS field setup allowing for the wormhole solution. Besides, the flaring-out con-dition presumes β to be essentially negative, as a consequence the kinetic energy associated to thescalar field has a wrong sign, thus it behaves like a ghost field. It turns out to be that this field maybe seen like a combination of ghost-like dust and electromagnetic field, moreover, the non-trivialproperties of the wormhole geometry lead us to a topological charge associated to the throat of thewormhole. This solution setup reduces to the same as GR plus scalar field, as it can be expectedbecause the Cotton tensor disappears, whilst the diagonal components of energy-momentum tensorof ϑ continue to be non-zero.By considering non-zero tidal force we have arrived at the same conclusions as in the case ofzero redshift, i.e., the modified field equations for DCS theory in vacuum always will reduce to GRand scalar field ones, independently of the redshift function. The exotic character of the CS field is4nevertheless natural – indeed, it is well known that noncausal solutions like wormholes, Alcubierrewarp drive etc., require exotic matter, see f.e. [7]. The natural generalization of this study couldconsist in considering a generic situation where a Cotton tensor does not vanish. We plan to carryout this generalization in a forthcoming paper. VI. APPENDIX A. NON-VANISHING COMPONENTS OF THE COTTON ANDENERGY-MOMENTUM TENSORS
In this appendix we write the non-zero components of the Cotton and energy-momentum tensor.For this calculus, we used the
GRtensor program. Accordingly, for the Cotton tensor we get C (0)(2) = − r sin( θ ) (cid:18) − b ( r ) r (cid:19) − / (cid:20) b ( r ) ∂∂φ ϑ ( t, r, θ, φ ) − b ( r ) r ∂ ∂r∂φ ϑ ( t, r, θ, φ ) −− (cid:18) ∂∂φ ϑ ( t, r, θ, φ ) (cid:19) rb ′ ( r ) + r (cid:18) ∂ ∂r∂φ ϑ ( t, r, θ, φ ) (cid:19) b ′ ( r ) ++ (cid:18) ∂∂φ ϑ ( t, r, θ, φ ) (cid:19) r b ′′ ( r ) (cid:21) ; (45) C (0)(3) = − r (cid:18) − b ( r ) r (cid:19) − / (cid:20) b ( r ) ∂∂θ ϑ ( t, r, θ, φ ) − b ( r ) r ∂ ∂r∂θ ϑ ( t, r, θ, φ ) −− (cid:18) ∂∂θ ϑ ( t, r, θ, φ ) (cid:19) rb ′ ( r ) + r (cid:18) ∂ ∂r∂θ ϑ ( t, r, θ, φ ) (cid:19) b ′ ( r ) ++ (cid:18) ∂∂θ ϑ ( t, r, θ, φ ) (cid:19) r b ′′ ( r ) (cid:21) ; (46) C (1)(2) = − r sin ( θ ) (cid:18) ∂ ∂t∂φ ϑ ( t, r, θ, φ ) (cid:19) (cid:0) b ( r ) − rb ′ ( r ) (cid:1) ; (47) C (1)(3) = 14 r (cid:18) ∂ ∂t∂θ ϑ ( t, r, θ, φ ) (cid:19) (cid:0) b ( r ) − rb ′ ( r ) (cid:1) . (48)5The non-vanishing components of T ϑAB are T ϑ (0)(0) = 12 r (sin ( θ )) (cid:20) − (cid:18) ∂∂r ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) b ( r ) + (cid:18) ∂∂φ ϑ ( t, r, θ, φ ) (cid:19) ++ (cid:18) ∂∂t ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) + (cid:18) ∂∂θ ϑ ( t, r, θ, φ ) (cid:19) (sin ( θ )) ++ (cid:18) ∂∂r ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) (cid:21) ; (49) T ϑ (0)(1) = − (cid:18) − b ( r ) r (cid:19) − / (cid:18) ∂∂t ϑ ( t, r, θ, φ ) (cid:19) ∂∂r ϑ ( t, r, θ, φ ) ; (50) T ϑ (0)(2) = − r (cid:18) ∂∂t ϑ ( t, r, θ, φ ) (cid:19) ∂∂θ ϑ ( t, r, θ, φ ) ; (51) T ϑ (0)(3) = − r sin ( θ ) (cid:18) ∂∂t ϑ ( t, r, θ, φ ) (cid:19) ∂∂φ ϑ ( t, r, θ, φ ) ; (52) T ϑ (1)(1) = − r (sin ( θ )) (cid:20) (cid:18) ∂∂r ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) b ( r ) − (cid:18) ∂∂r ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) −− (cid:18) ∂∂t ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) + (cid:18) ∂∂θ ϑ ( t, r, θ, φ ) (cid:19) (sin ( θ )) ++ (cid:18) ∂∂φ ϑ ( t, r, θ, φ ) (cid:19) (cid:21) ; (53) T ϑ (1)(2) = 1 r (cid:18) − b ( r ) r (cid:19) − / (cid:18) ∂∂θ ϑ ( t, r, θ, φ ) (cid:19) ∂∂r ϑ ( t, r, θ, φ ) ; (54) T ϑ (1)(3) = 1 r sin ( θ ) (cid:18) − b ( r ) r (cid:19) − / (cid:18) ∂∂r ϑ ( t, r, θ, φ ) (cid:19) ∂∂φ ϑ ( t, r, θ, φ ) ; (55) T ϑ (2)(2) = − r (sin ( θ )) (cid:20) − (cid:18) ∂∂r ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) b ( r ) − (cid:18) ∂∂θ ϑ ( t, r, θ, φ ) (cid:19) (sin ( θ )) ++ (cid:18) ∂∂r ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) − (cid:18) ∂∂t ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) ++ (cid:18) ∂∂φ ϑ ( t, r, θ, φ ) (cid:19) (cid:21) ; (56) T ϑ (2)(3) = 1 r sin ( θ ) (cid:18) ∂∂θ ϑ ( t, r, θ, φ ) (cid:19) ∂∂φ ϑ ( t, r, θ, φ ) ; (57) T ϑ (3)(3) = 12 r (sin ( θ )) (cid:20) (cid:18) ∂∂r ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) b ( r ) + (cid:18) ∂∂φ ϑ ( t, r, θ, φ ) (cid:19) −− (cid:18) ∂∂r ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) + (cid:18) ∂∂t ϑ ( t, r, θ, φ ) (cid:19) r (sin ( θ )) −− (cid:18) ∂∂θ ϑ ( t, r, θ, φ ) (cid:19) (sin ( θ )) (cid:21) (58)6 VII. APPENDIX B. NON-VANISHING COMPONENTS OF THE EINSTEIN, COTTONAND ENERGY-MOMENTUM TENSORS FOR THE METRIC 40
The non-vanishing components of G µν are G = 2 A ( x ) r ( x ) d dx r ( x ) + (cid:0) ddx A ( x ) (cid:1) r ( x ) ddx r ( x ) − A ( x ) (cid:0) ddx r ( x ) (cid:1) ( r ( x )) , (59) G = (cid:0) ddx A ( x ) (cid:1) r ( x ) ddx r ( x ) − A ( x ) (cid:0) ddx r ( x ) (cid:1) ( r ( x )) , (60) G = G = 12 " (cid:0) ddx A ( x ) (cid:1) ddx r ( x ) + 2 A ( x ) d dx r ( x ) + (cid:16) d dx A ( x ) (cid:17) r ( x ) r ( x ) . (61)The non-vanishing components of C µν are C = 14 ( r ( x )) " sin ( θ ) (cid:18) ddθ Y ( θ ) (cid:19) U ( x ) ( r ( x )) d dx A ( x ) −− U ( x ) (cid:18) ddx A ( x ) (cid:19) r ( x ) d dx r ( x ) − (62) − ddx U ( x ) − A ( x ) U ( x ) r ( x ) d dx r ( x ) − (cid:18) ddx U ( x ) (cid:19) r ( x ) (cid:18) ddx r ( x ) (cid:19) ddx A ( x ) ++ 2 A ( x ) (cid:18) ddx U ( x ) (cid:19) (cid:18) ddx r ( x ) (cid:19) + (cid:18) ddx U ( x ) (cid:19) ( r ( x )) d dx A ( x ) −− A ( x ) (cid:18) ddx U ( x ) (cid:19) r ( x ) d dx r ( x ) + 2 A ( x ) U ( x ) (cid:18) ddx r ( x ) (cid:19) d dx r ( x ) ! ,C = −
14 ( r ( x )) sin ( θ ) " A ( x ) (cid:18) ddθ Y ( θ ) (cid:19) U ( x ) ( r ( x )) d dx A ( x ) −− U ( x ) (cid:18) ddx A ( x ) (cid:19) r ( x ) d dx r ( x ) − (63) − ddx U ( x ) − A ( x ) U ( x ) r ( x ) d dx r ( x ) − (cid:18) ddx U ( x ) (cid:19) r ( x ) (cid:18) ddx r ( x ) (cid:19) ddx A ( x ) ++ 2 A ( x ) (cid:18) ddx U ( x ) (cid:19) (cid:18) ddx r ( x ) (cid:19) + (cid:18) ddx U ( x ) (cid:19) ( r ( x )) d dx A ( x ) −− A ( x ) (cid:18) ddx U ( x ) (cid:19) r ( x ) d dx r ( x ) + 2 A ( x ) U ( x ) (cid:18) ddx r ( x ) (cid:19) d dx r ( x ) ! . T ( ϑ ) µν are T ( ϑ ) 00 = −
12 ( r ( x )) A ( x ) (cid:0) ddx U ( x ) (cid:1) ( Y ( θ )) + ( U ( x )) (cid:0) ddθ Y ( θ ) (cid:1) ( r ( x )) , (64) T ( ϑ ) 11 = 12 ( r ( x )) A ( x ) (cid:0) ddx U ( x ) (cid:1) ( Y ( θ )) − ( U ( x )) (cid:0) ddθ Y ( θ ) (cid:1) ( r ( x )) , (65) T ( ϑ ) 12 = A ( x ) (cid:18) ddx U ( x ) (cid:19) Y ( θ ) U ( x ) ddθ Y ( θ ) (66) T ( ϑ ) 21 = (cid:0) ddx U ( x ) (cid:1) Y ( θ ) U ( x ) ddθ Y ( θ )( r ( x )) (67) T ( ϑ ) 22 = −
12 ( r ( x )) A ( x ) (cid:0) ddx U ( x ) (cid:1) ( Y ( θ )) − ( U ( x )) (cid:0) ddθ Y ( θ ) (cid:1) ( r ( x )) (68) T ( ϑ ) 33 = −
12 ( r ( x )) A ( x ) (cid:0) ddx U ( x ) (cid:1) ( Y ( θ )) + ( U ( x )) (cid:0) ddθ Y ( θ ) (cid:1) ( r ( x )) . (69) Acknowledgments.
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