Space-time deformations as extended conformal transformations
aa r X i v : . [ g r- q c ] D ec Space-time deformations as extended conformal transformations
S. Capozziello a,b and C. Stornaiolo b ∗ a Dipartimento di Scienze Fisiche, Universit`a di Napoli ”Federico II”,and b INFN, Sez. di Napoli, Compl. Univ. di Monte S. Angelo,Edificio G, Via Cinthia, I-80126 - Napoli, Italy (Dated: November 2, 2018)A definition of space-time metric deformations on an n -dimensional manifold is given. We showthat such deformations can be regarded as extended conformal transformations. In particular, theirfeatures can be related to the perturbation theory giving a natural picture by which gravitationalwaves are described by small deformations of the metric. As further result, deformations can berelated to approximate Killing vectors (approximate symmetries) by which it is possible to pa-rameterize the deformed region of a given manifold. The perspectives and some possible physicalapplications of such an approach are discussed. PACS numbers: 02.40.-k, 04.50.+h, 04.60.-mKeywords: gauge symmetries; conformal transformations; perturbations; gravitational waves
I. INTRODUCTION
The issue to consider a general way to deform the space-time metrics is not new. It has been posed in different waysand is related to several physical problems ranging from the spontaneous symmetry breaking of unification theoriesup to gravitational waves, considered as space-time perturbations. In cosmology, for example, one faces the problemto describe an observationally lumpy universe at small scales which becomes isotropic and homogeneous at very largescales according to the Cosmological Principle. In this context, it is crucial to find a way to connect background andlocally perturbed metrics [1]. For example, McVittie [2] considered a metric which behaves as a Schwarzschild oneat short ranges and as a Friedman-Lemaitre-Robertson-Walker metric at very large scales. Gautreau [3] calculatedthe metric generated by a Schwarzschild mass embedded in a Friedman cosmological fluid trying to address the sameproblem. On the other hand, the post-newtonian parameterization, as a standard, can be considered as a deformationof a background, asymptotically flat Minkowski metric.In general, the deformation problem has been explicitly posed by Coll and collaborators [4, 5, 10] who conjecturedthe possibility to obtain any metric from the deformation of a space-time with constant curvature. The problemwas solved only for 3-dimensional spaces but a straightforward extension should be to achieve the same result forspace-times of any dimension.In principle, new exact solutions of the Einstein field equations can be obtained by studying perturbations. Inparticular, dealing with perturbations as Lorentz matrices of scalar fields Φ AC reveals particularly useful. Firstly theytransform as scalars with respect the coordinate transformations. Secondly, they are dimensionless and, in each point,the matrix Φ AC behaves as the element of a group. As we shall see below, such an approach can be related to theconformal transformations giving an ”extended” interpretation and a straightforward physical meaning of them (see[7, 8] and references therein for a comprehensive review). Furthermore scalar fields related to space-time deformationshave a straightforward physical interpretation which could contribute to explain several fundamental issues as theHiggs mechanism in unification theories, the inflation in cosmology and other pictures where scalar fields play afundamental role in dynamics.In this paper, we are going to discuss the properties of the deforming matrices Φ AC and we will derive, from theEinstein equations, the field equations for them, showing how them can parameterize the deformed metrics, accordingto the boundary and initial conditions and to the energy-momentum tensor.The layout of the paper is the following. In Sec.II, we define the space-time perturbations in the framework ofthe metric formalism giving the notion of first and second deformation matrices. Sec.III is devoted to the mainproperties of deformations. In particular, we discuss how deformation matrices can be split in their trace, tracelessand skew parts. We derive the contributions of deformation to the geodesic equation and, starting from the curvatureRiemann tensor, the general equation of deformations. In Sec.IV we discuss the notion of linear perturbations underthe standard of deformations. In particular, we recast the equation of gravitational waves and the transverse traceless ∗ Electronic address: [email protected], [email protected] gauge under the standard of deformations. Sec.V is devoted to discuss the action of deformations on the Killingvectors. The result consists in achieving a notion of approximate symmetry. Discussion and conclusions are given inSec.VI. In Appendix, we discuss in details how deformations act on affine connections.
II. GENERALITIES ON SPACE-TIME DEFORMATIONS
In order to start our considerations, let us take into account a metric g on a space-time manifold M . Such a metricis assumed to be an exact solution of the Einstein field equations. We can decompose it by a co-tetrad field ω A ( x ) g = η AB ω A ω B . (1)Let us define now a new tetrad field e ω = Φ AC ( x ) ω C , with Φ AC ( x ) a matrix of scalar fields. Finally we introduce aspace-time f M with the metric e g defined in the following way e g = η AB Φ AC Φ BD ω C ω D = γ CD ( x ) ω C ω D , (2)where also γ CD ( x ) is a matrix of fields which are scalars with respect to the coordinate transformations.If Φ AC ( x ) is a Lorentz matrix in any point of M , then e g ≡ g (3)otherwise we say that e g is a deformation of g and f M is a deformed M . If all the functions of Φ AC ( x ) are continuous,then there is a one - to - one correspondence between the points of M and the points of f M .In particular, if ξ is a Killing vector for g on M , the corresponding vector e ξ on f M could not necessarily be a Killingvector.A particular subset of these deformation matrices is given byΦ AC ( x ) = Ω( x ) δ AC . (4)which define conformal transformations of the metric, e g = Ω ( x ) g . (5)In this sense, the deformations defined by Eq. (2) can be regarded as a generalization of the conformal transforma-tions.We call the matrices Φ AC ( x ) first deformation matrices , while we can refer to γ CD ( x ) = η AB Φ AC ( x )Φ BD ( x ) . (6)as the second deformation matrices , which, as seen above, are also matrices of scalar fields. They generalize theMinkowski matrix η AB with constant elements in the definition of the metric. A further restriction on the matricesΦ AC comes from the theorem proved by Riemann by which an n -dimensional metric has n ( n − / III. PROPERTIES OF DEFORMING MATRICES
Let us take into account a four dimensional space-time with Lorentzian signature. A family of matrices Φ AC ( x )such that Φ AC ( x ) ∈ GL (4) ∀ x, (7)are defined on such a space-time.These functions are not necessarily continuous and can connect space-times with different topologies. A singularscalar field introduces a deformed manifold f M with a space-time singularity.As it is well known, the Lorentz matrices Λ AC leave the Minkowski metric invariant and then g = η EF Λ EA Λ F B Φ AC Φ BD ω C ω D = η AB Φ AC Φ BD ω C ω D . (8)It follows that Φ AC give rise to right cosets of the Lorentz group, i.e. they are the elements of the quotient group GL (4 , R ) /SO (3 , AC by a Lorentz matrix induces a different defor-mation matrix.The inverse deformed metric is e g ab = η CD Φ − AC Φ − BD e aA e bB (9)where Φ − AC Φ CB = Φ AC Φ − CB = δ AB .Let us decompose now the matrix Φ AB = η AC Φ CB in its symmetric and antisymmetric partsΦ AB = Φ ( AB ) + Φ [ AB ] = Ω η AB + Θ AB + ϕ AB (10)where Ω = Φ AA , Θ AB is the traceless symmetric part and ϕ AB is the skew symmetric part of the first deformationmatrix respectively. Then standard conformal transformations are nothing else but deformations with Θ AB = ϕ AB = 0[9].Finding the inverse matrix Φ − AC in terms of Ω, Θ AB and ϕ AB is not immediate, but as above, it can be split inthe three terms Φ − AC = αδ AC + Ψ AC + Σ AC (11)where α , Ψ AC and Σ AC are respectively the trace, the traceless symmetric part and the antisymmetric part of theinverse deformation matrix. The second deformation matrix, from the above decomposition, takes the form γ AB = η CD (Ω δ CA + Θ CA + ϕ CA )(Ω δ DB + Θ DB + ϕ DB ) (12)and then γ AB = Ω η AB + 2Ω Θ AB + η CD Θ CA Θ DB + η CD (Θ CA ϕ DB + ϕ CA Θ DB ) + η CD ϕ CA ϕ DB . (13)In general, the deformed metric can be split as ˜ g ab = Ω g ab + γ ab (14)where γ ab = (cid:0)
2Ω Θ AB + η CD Θ CA Θ DB + η CD (Θ CA ϕ DB + ϕ CA Θ DB )+ η CD ϕ CA ϕ DB (cid:1) ω Aa ω Bb (15)In particular, if Θ AB = 0, the deformed metric simplifies to e g ab = Ω g ab + η CD ϕ CA ϕ DB ω Aa ω Bb (16)and, if Ω = 1, the deformation of a metric consists in adding to the background metric a tensor γ ab . We have toremember that all these quantities are not independent as, by the theorem mentioned in [5], they have to form atmost six independent functions in a four dimensional space-time.Similarly the controvariant deformed metric can be always decomposed in the following way e g ab = α g ab + λ ab (17)Let us find the relation between γ ab and λ ab . By using f g ab f g bc = δ ca , we obtain α Ω δ ca + α γ ca + Ω λ ca + γ ab λ bc = δ ca (18)if the deformations are conformal transformations, we have α = Ω − , so assuming such a condition, one obtain thefollowing matrix equation α γ ca + Ω λ ca + γ ab λ bc = 0 , (19)and ( δ ba + Ω − γ ba ) λ cb = − Ω − γ ca (20)and finally λ cb = − Ω − ( δ + Ω − γ ) − ab γ ca (21)where ( δ + Ω − γ ) − is the inverse tensor of ( δ ba + Ω − γ ba ).To each matrix Φ AB , we can associate a (1,1)-tensor φ ab defined by φ ab = Φ AB ω Bb e aA (22)such that e g ab = g cd φ ca φ db (23)which can be decomposed as in Eq.(16). Vice-versa from a (1,1)-tensor φ ab , we can define a matrix of scalar fields as φ AB = φ ab ω Aa e bB . (24)The Levi Civita connection corresponding to the metric (14) is related to the original connection by the relation(see the Appendix for details) e Γ cab = Γ cab + C cab (25)(see [9]), where C cab = 2 e g cd g d ( a ∇ b ) Ω − g ab e g cd ∇ d Ω + 12 e g cd ( ∇ a γ db + ∇ b γ ad − ∇ d γ ab ) . (26)Therefore, in a deformed space-time, the connection deformation acts like a force that deviates the test particlesfrom the geodesic motion in the unperturbed space-time. As a matter of fact the geodesic equation for the deformedspace-time d x c dλ + ˜Γ cab dx a dλ dx b dλ = 0 (27)becomes d x c dλ + Γ cab dx a dλ dx b dλ = − C cab dx a dλ dx b dλ . (28)The deformed Riemann curvature tensor is then e R dabc = R dabc + ∇ b C dac − ∇ a C dbc + C eac C dbe − C ebc C dae , (29)while the deformed Ricci tensor obtained by contraction is e R ab = R ab + ∇ d C dab − ∇ a C ddb + C eab C dde − C edb C dae (30)and the curvature scalar e R = e g ab e R ab = e g ab R ab + e g ab (cid:2) ∇ d C dab − ∇ a C ddb + C eab C dde − C edb C dae (cid:3) (31)From the above curvature quantities, we obtain finally the equations for the deformations. In the vacuum case, wesimply have e R ab = R ab + ∇ d C dab − ∇ a C ddb + C eab C dde − C edb C dae = 0 (32)where R ab must be regarded as a known function. In presence of matter, we consider the equation R ab + ∇ d C dab − ∇ a C ddb + C eab C dde − C edb C dae = e T ab − e g ab e T (33)we are assuming, for the sake of simplicity 8 πG = c = 1. This last equation can be improved by considering theEinstein field equations R ab = T ab − g ab T (34)and then ∇ d C dab − ∇ a C ddb + C eab C dde − C edb C dae = e T ab − e g ab e T − (cid:18) T ab − g ab T (cid:19) (35)is the most general equation for deformations. IV. METRIC DEFORMATIONS AS PERTURBATIONS AND GRAVITATIONAL WAVES
Metric deformations can be used to describe perturbations. To this aim we can simply consider the deformationsΦ AB = δ AB + ϕ AB (36)with | ϕ AB | ≪ , (37)together with their derivatives | ∂ϕ AB | ≪ . (38)With this approximation, immediately we find the inverse relation(Φ − ) AB ≃ δ AB − ϕ AB . (39)As a remarkable example, we have that gravitational waves are generally described, in linear approximation, asperturbations of the Minkowski metric g ab = η ab + γ ab . (40)In our case, we can extend in a covariant way such an approximation. If ϕ AB is an antisymmetric matrix, we have e g ab = g ab + γ ab (41)where the first order terms in ϕ AB vanish and γ ab is of second order γ ab = η AB ϕ AC ϕ BD ω Ca ω Db . (42)Consequently e g ab = g ab + γ ab (43)where γ ab = η AB ( ϕ − ) CA ( ϕ − ) DB e aC e bD . (44)Let us consider the background metric g ab , solution of the Einstein equations in the vacuum R ab = 0 . (45)We obtain the equation of perturbations considering only the linear terms in Eq.(32) and neglecting the contributionsof quadratic terms. We find e R ab = ∇ d C dab − ∇ a C ddb = 0 , (46)and, by the explicit form of C dab , this equation becomes (cid:0) ∇ d ∇ a γ db + ∇ d ∇ b γ da − ∇ d ∇ d γ ab (cid:1) − (cid:0) ∇ a ∇ d γ db + ∇ a ∇ b γ dd − ∇ a ∇ d γ db (cid:1) = 0 . (47)Imposing the transverse traceless gauge on γ ab , i.e. the standard gauge conditions ∇ a γ ab = 0 (48)and γ = γ aa = 0 (49)Eq.(47) reduces to ∇ b ∇ b γ ac − R bacd γ bd = 0 , (50)see also [9]. In our context, this equation is a linearized equation for deformations and it is straightforward toconsider perturbations and, in particular, gravitational waves, as small deformations of the metric. This result canbe immediately translated into the above scalar field matrix equations. Note that such an equation can be applied tothe conformal part of the deformation, when the general decomposition is considered.As an example, let us take into account the deformation matrix equations applied to the Minkowski metric, whenthe deformation matrix assumes the form (36). In this case, the equations (47), become ordinary wave equations for γ ab . Considering the deformation matrices, these equations become, for a tetrad field of constant vectors, ∂ d ∂ d ϕ CA ϕ CB + 2 ∂ d ϕ CA ∂ d ϕ CB + ϕ CA ∂ d ∂ d ϕ CB = 0 . (51)The above gauge conditions are now ϕ AB ϕ BA = 0 (52)and e dD (cid:2) ∂ d ϕ CA ϕ CB + ϕ CA ∂ d ϕ CB (cid:3) = 0 . (53)This result shows that the gravitational waves can be fully recovered starting from the scalar fields which describe thedeformations of the metric. In other words, such scalar fields can assume the meaning of gravitational wave modes. V. APPROXIMATE KILLING VECTORS
Another important issue which can be addressed starting from space-time deformations is related to the symmetries.In particular, they assume a fundamental role in describing when a symmetry is preserved or broken under the actionof a given field. In General Relativity, the Killing vectors are always related to the presence of given space-timesymmetries [9].Let us take an exact solution of the Einstein equations, which satisfies the Killing equation( L ξ g ) ab = 0 (54)where ξ , being the generator of an infinitesimal coordinate transformation, is a Killing vector. If we take a deformationof the metric with the scalar matrix Φ AB = δ AB + ϕ AB (55)with | ϕ AB | ≪ , (56)and ( L ξ e g ) ab = 0 , (57)being ( L ξ e A ) a = 0 , (58)we have ( L ξ ϕ ) AB = ξ a ∂ a ϕ AB = 0 . (59)If there is some region D of the deformed space-time M deformed where | ( L ξ ϕ ) AB | ≪ ξ is an approximate Killing vector on D . In other words, these approximate Killing vectors allow to”control” the space-time symmetries under the action of a given deformation. VI. DISCUSSION AND CONCLUSIONS
In this paper, we have proposed a novel definition of space-time metric deformations parameterizing them in termsof scalar field matrices. The main result is that deformations can be described as extended conformal transformations.This fact gives a straightforward physical interpretation of conformal transformations: conformally related metricscan be seen as the ”background” and the ”perturbed” metrics. In other words, the relations between the Jordanframe and the Einstein frame can be directly interpreted through the action of the deformation matrices contributingto solve the issue of what the true physical frame is [7, 8].Besides, space-time metric deformations can be immediately recast in terms of perturbation theory allowing acompletely covariant approach to the problem of gravitational waves.Results related to those presented here has been proposed in [4, 5]. There it is shown that any metric in a threedimensional manifold can be decomposed in the form e g ab = σ ( x ) h ab + ǫs a s b (61)where h ab is a metric with constant curvature, σ ( x ) is a scalar function, s a is a three-vector and ǫ = ±
1. A relationhas to be imposed between σ and s a and then the metric can be defined, at most, by three independent functions.In a subsequent paper [6], Llosa and Soler showed that (61) can be generalized to arbitrary dimensions by the form e g ab = λ ( x ) g ab + µ ( x ) F ac g cd F db (62)where g ab is a constant curvature metric, F ab is a two-form, λ ( x ) and µ ( x ) are two scalar functions. These results arefully recovered and generalized from our approach as soon as the deformation of a constant metric is considered andsuitable conditions on the tensor Θ AB are imposed.In general, we have shown that, when we turn to the tensor formalism, we can work with arbitrary metrics andarbitrary deforming γ ab tensors. In principle, by arbitrary deformation matrices, not necessarily real, we can passfrom a given metric to any other metric. As an example, a noteworthy result has been achieved by Newman and Janis[11]: They showed that, through a complex coordinate transformation, it is always possible to achieve a Kerr metricfrom a Schwarzschild one. In our language, this means that a space-time deformation allows to pass from a sphericalsymmetry to a cylindrical one. Furthermore, it has been shown [12, 13] that three dimensional black hole solutionscan be found by identifying 3-dimensional anti-de Sitter space on which acts a discrete subgroup of SO (2 , all the tensors defined on the manifold. On the other hand, our definition of metric deformations and deformedmanifolds can be straightforwardly related to the standard notion of perturbations since, in principle, it works on agiven region D of the deformed space-time (see, for example, [14, 15]). VII. APPENDIX
We can calculate the modified connection ˆΓ cab in many alternative ways. Let us introduce the tetrad e A and cotetrad ω B satisfying the orthogonality relation i e A ω B = δ BA (63)and the non-integrability condition (anholonomy) dω A = 12 Ω ABC ω B ∧ ω C . (64)The corresponding connection isΓ ABC = 12 (cid:16) Ω ABC − η AA ′ η BB ′ Ω B ′ A ′ C − η AA ′ η CC ′ Ω C ′ A ′ B (cid:17) (65)If we deform the metric as in (2), we have two alternative ways to write this expression: either writing the “deforma-tion” of the metric in the space of tetrads or “deforming” the tetrad field as in the following expressionˆ g = η AB Φ AC Φ BD ω C ω D = γ AB ω A ω B = η AB ˆ ω A ˆ ω B . (66)In the first case, the contribution of the Christoffel symbols, constructed by the metric γ AB , appearsˆΓ ABC = 12 (cid:16) Ω ABC − γ AA ′ γ BB ′ Ω B ′ A ′ C − γ AA ′ γ CC ′ Ω C ′ A ′ B (cid:17) + 12 γ AA ′ (cid:0) i e C dγ BA ′ − i e B dγ CA ′ − i e A ′ dγ BC (cid:1) (67)In the second case, using (64), we can define the new anholonomy objects ˆ C ABC . d ˆ ω A = 12 ˆΩ ABC ˆ ω B ∧ ˆ ω C . (68)After some calculations, we haveˆΩ ABC = Φ AE Φ − DB Φ − F C Ω EDF + 2Φ AF e aG (cid:16) Φ − G [ B ∂ a Φ − F C ] (cid:17) (69)As we are assuming a constant metric in tetradic space, the deformed connection isˆΓ ABC = 12 (cid:16) ˆΩ ABC − η AA ′ η BB ′ ˆΩ B ′ A ′ C − η AA ′ η CC ′ ˆΩ C ′ A ′ B (cid:17) . (70)Substituting (69) in (70), the final expression of ˆΓ ABC , as a function of Ω
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