A simple characterization of doubly twisted spacetimes
aa r X i v : . [ g r- q c ] J a n A SIMPLE CHARACTERIZATIONOF DOUBLY TWISTED SPACETIMES
CARLO ALBERTO MANTICA AND LUCA GUIDO MOLINARI
Abstract.
In this note we characterize 1+n doubly twisted spacetimes interms of ‘doubly torqued’ vector fields. They extend Bang-Yen Chen’s charac-terization of twisted and generalized Robertson-Walker spacetimes with torquedand concircular vector fields. The result is a simple classification of 1+ndoubly-twisted, doubly-warped, twisted and generalized Robertson-Walker space-times. Introduction
Several interesting Lorentzian metrics have a block-diagonal form, with timelabelling a foliation with spacelike hypersurfaces. They include doubly-twisted,doubly-warped, twisted, warped, and Robertson-Walker spacetimes [1]. The Frobe-nius theorem characterizes the vector fields u i that are hypersurface orthogonal, forwhich there exist functions λ and f such that, locally, λu i = ∇ i f ([2], p.19). Thisestablishes a dual description of such spacetimes: the special form of the metricallows explicit evaluations, the one in terms of the vector field is covariant. Whilephysicists conceive the vector field as a congruence of timelike trajectories, geome-ters prefer other vectors, as is here illustrated.Doubly twisted spacetimes were introduced (and named ‘conformally separable’)by Kentaro Yano in 1940: ds = − b ( q , t ) dt + a ( t, q ) g ∗ µν ( q ) dq µ dq ν (1)He showed that the metric structure is necessary and sufficient for the hypersur-faces to be totally umbilical [3]. The spacetime is doubly warped if b only dependson q and a only depends on time t .Ferrando, Presilla and Morales [4] proved that doubly twisted spacetimes are co-variantly characterized by the existence of a timelike unit, shear and vorticity freevector field: u i u i = − ∇ i u j = ϕ ( u i u j + g ij ) − u i ˙ u j (2)where ˙ u j = u k ∇ k u j is the acceleration, and ˙ u j u j = 0.In 1979 Bang-Yen Chen introduced twisted spacetimes, eq.(1) with b = 1 [5].Years later he characterized them through the existence of a timelike torqued vectorfield [6]: ∇ i τ j = κg ij + α i τ j , α i τ i = 0(3) Mathematics Subject Classification.
Primary 83C20, Secondary 53B30.
Key words and phrases.
Twisted spacetime, warped spacetime, torqued vector. where κ is a scalar field. We gave the equivalent description in terms of torse-forming time-like unit vectors, eq.(2) with ˙ u i = 0, and obtained the form of theRicci tensor [7], and unicity of the vector, up to special cases [8].Generalized Robertson-Walker (GRW) spacetimes were introduced in 1995 byAl´ıas, Romero and S´anchez [9, 10]: ds = − dt + a ( t ) g ∗ µν ( q ) dq µ dq ν (4)Bang-Yen Chen characterized them through the existence of a timelike concircularvector field ∇ i τ j = κg ij [11, 12] (the statement can be weakened, see Prop.2.3).We gave the alternative characterization (2) with ˙ u i = 0 and ∇ i ϕ = − u i ˙ ϕ , andproved the useful property for the Weyl tensor [13]: u m C jklm = 0 if and only if ∇ m C jklm = 0. If the Weyl tensor is zero, the spacetime is Robertson-Walker.All these cases constitute a rich family of manifolds which are mostly studiedby geometers. They also appear in physics, as inhomogeneous extensions of theRobertson-Walker metric. In fact, the Einstein equations with a source of imper-fect fluid with shear-free and irrotational velocity, lead to doubly-twisted metrics[14, 15, 16, 4, 17, 18]. The Stephani universes fall in this class [2, 19]. The require-ment of geodesic flow specialises the metric to twisted, with interesting applicationsdiscussed by Coley and McManus [20]. Doubly warped and GRW (or warped) man-ifolds have an ample geometric literature [21, 22, 23, 24, 12].In this note we present a simple characterization that includes all such space-times, and discuss some properties of doubly torqued vectors.2. Another characterization
Theorem 2.1.
A Lorentzian spacetime is doubly-twisted if and only if it admits atimelike vector field, which we name ‘doubly torqued’: ∇ i τ j = κg ij + α i τ j + τ i β j (5) with α i τ i = 0 , β i τ i = 0 , and nκ = ∇ i τ i .Proof. We prove the equivalence of (5) with (2).Let N = √− τ i τ i , and introduce the vector u i = τ i /N . Evaluate: ∇ i N = − τ j ∇ i τ j = − κτ i + 2 α i N . Then: ∇ i N = − κu i + α i N . Next: N ∇ i u j = ∇ i τ j − u j ∇ i N = κg ij + N α i u j + N u i β j + κu i u j − N α i u j Therefore: ∇ i u j = ( κ/N )( u i u j + g ij ) + u i β j . Contraction with u i shows that β j = − ˙ u j , and eq.(2) is obtained.Given (2), the corresponding metric is (1). Define β j = − ˙ u j and τ i = Su i , where S is to be found. Multiply eq.(2) by S : ∇ i ( Su j ) − u j ∇ i S = ϕS ( u i u j + g ij ) + Su i β j This is eq.(5) with the vector α i = ( ∇ i S ) /S + ϕu i .We must impose the condition: 0 = α i τ i = u i ∇ i S − Sϕ i.e. ϕ = u i ∇ i log S .In the frame (1) it is u = − b , u µ = 0, and the condition becomes ϕ = ( ∂ t S ) / ( Sb ).With the Christoffel symbols in appendix we obtain ˙ u = u ( ∂ t − Γ ) u = 0 and˙ u µ = − u Γ µ u = b µ /b . The time component of (2), ∇ u = ϕ ( u + g ), gives ϕ = ( ∂ t a ) / ( ab ). Therefore S = a up to a constant factor. (cid:3) SIMPLE CHARACTERIZATION OF DOUBLY TWISTED SPACETIMES 3
There is some freedom in the choice of the doubly torqued vector: multiplicationof eq.(5) by a function λ gives an equation for a vector λτ i that is orthogonal tothe hypersurfaces and ∇ i ( λτ j ) = ( λκ ) g ij + ( α i + ∂ i λ/λ )( λτ j ) + ( λτ i ) β j . It is doublytorqued provided that: τ i ∂ i λ = 0(6)We show the relation of the special vectors α i and β i with the scale functions a and b of the metric (1).In the coordinate frame ( t, q ), the vectors are τ i = ( τ , α i = (0 , α µ ) and β i =(0 , β µ ). The equations (5) are: ∂ τ − Γ τ = − κb , ∂ µ τ − Γ µ τ = τ α µ , − Γ µ τ = β µ τ and − Γ µν τ = κa g ∗ µν . They can be rewritten as follows: β µ = − ∂ µ log b (7) ∂ t τ = τ ∂ t log( ab )(8) ∂ µ τ = τ ( α µ − β µ )(9) κb = − ∂ t ( τ /b )(10)The second equation is integrated: τ ( t, q ) = c ( q )( ab )( t, q ), where c is an arbitraryfunction. Then, the first and third equation give α µ = ∂ µ log( ca ).In coordinates ( t, q ) the condition (6) is τ ∂ t λ = 0 and implies that λ does notdepend on time t . The transformation ( τ , α µ , β µ , κ ) → ( λτ , α µ + ∂ µ λ/λ, β µ , λκ )leaves the above equations unchanged. This freedom is used to put c ( q ) = ± τ ( t, q ) = − ( ab )( t, q ) (if τ > α µ = ∂∂q µ log a, β µ = − ∂∂q µ log b, κ = 1 b ∂a∂t (11)and establish a simple relation of the vectors with the metric. An interestinginvariant is: τ k τ k = − b τ = − a (12)For a doubly warped 1 + n metric, the scale functions b does not depend on timeand a does not depend on q . In this case, the analysis shows that α i is either zeroor a gradient orthogonal to τ , that can be absorbed by a rescaling of τ .We obtain the characterization: Theorem 2.2. A n spacetime is doubly warped if and only if there is a timelikevector such that ∇ i τ j = κg ij + τ i β j with τ i β i = 0 , and β i is closed.Proof. In a doubly warped spacetime, b ( q ) and a ( t ) are given and specialize eq.(5).Eq.(11) gives: α µ = 0 i.e. α i = 0. Eq.(7) with ∂ t b = 0 implies that β i is closed.If α i = 0 in (5), then eq.(9), 0 = ∂ µ τ /τ − ∂ µ log b , has solution τ ( t, q ) = F ( t ) b ( t, q ). The result in (8) gives: ∂ t log a = ∂ t F/F , so that a = a ( t ). β i closedbecomes β µ = − ∂ µ r ( q ). Eq.(7) gives b ( t, q ) = exp[ r ( q ) + s ( t )]. The metric ds = − e r ( q ) e s ( t ) dt + a ( t ) g ∗ µν dq µ dq ν is doubly warped with a redefinition of time. (cid:3) In twisted 1+n spacetimes, a depends on t and q , and b = 1. Then β i = 0and we recover Chen’s result (3). In a GRW spacetime the scale function a onlydepends on t , so that also the vector α i is zero. C. A. MANTICA AND L. G. MOLINARI
Proposition 2.3.
In a doubly twisted spacetime with doubly-torqued vector τ i , if β i = ∇ i θ and τ i ∇ i θ = 0 then the spacetime is conformally equivalent to a twistedspacetime.Proof. Consider the conformal map ˆ g ij = e θ g ij . The new Christoffel simbols areˆΓ kij = Γ kij + δ ki ∂ j θ + δ kj ∂ i θ − g ij g kl ∂ l θ The vector ˆ τ i = e θ τ i solves:ˆ ∇ i ˆ τ j = ∇ i ( e θ τ j ) − ˆ τ i ∂ j θ − ˆ τ j ∂ i θ + g ij g kl ˆ τ k ∂ l θ =ˆ τ j ∂ i θ + e θ ( κg ij + α i τ j + τ i ∂ j θ ) − ˆ τ i ∂ j θ − ˆ τ j ∂ i θ + ˆ g ij ˆ g kl ˆ τ k β l =( e − θ κ )ˆ g ij + α i ˆ τ j The absence of β i characterizes a twisted spacetime. (cid:3) A consequence of the proof is the following statement (obvious if regarded onthe side of the metric):
Proposition 2.4.
Conformal transformations ˆ g ij = e θ g ij map doubly twisted todoubly twisted spacetimes.The same conformal transformation, with the condition τ i ∇ i θ = 0 , maps doublywarped to doubly warped spacetimes, or twisted to twisted spacetimes.Proof. Given the vector τ i , consider the vector ˆ τ i = e θ τ i in the new metric. Itsolves the equation of a doubly torqued vector:ˆ ∇ i ˆ τ j = ( κ + ˆ τ k ∂ k θ )ˆ g ij + ( α i + ˆ h ik ∂ k θ )ˆ τ j + ˆ τ i ( β j − ˆ h jk ∂ k θ )where ˆ h ij = ˆ g ij − ˆ τ i ˆ τ j / ˆ τ is a projection.If instead the vector ˆ τ i = e θ τ i is considered, in the new metric it solvesˆ ∇ i ˆ τ j = ( e − θ κ + ˆ τ k ∂ k θ )ˆ g ij + α i ˆ τ j + τ i ( β j − ∂ j θ )If the space is doubly warped ( α i = 0 and β closed) then, with the condition τ i ∂ i θ = 0, the vector ˆ τ i solves the equation for a doubly warped spacetime. Thesame is true for twisted spacetimes ( β i = 0). (cid:3) Conclusion
The introduction of doubly torqued vectors, defined by the equation (5), hasthe virtue of covariantly describing a class of 1 + n spacetimes in simple manner:doubly-twisted ( α = 0, β = 0), doubly-warped ( α = 0, β = 0 closed), twisted( α = 0, β = 0), generalized Robertson-Walker ( α = β = 0).We provide some examples from the physics literature.Stephani universes [19] are conformally flat solutions of the Einstein field equationswith a perfect fluid source. As in the Robertson-Walker space-time, the hypersur-faces orthogonal to the matter world-lines have constant curvature, but now itsvalue k , and even its sign, changes from one hypersurface to another. The lineelement is doubly twisted: ds = − D ( t, x ) dt + R ( t ) V ( t, x ) ( dx + dy + dz )with V = 1 + k x − x ( t ) k , D = F ( t )[ ˙ V /V − ˙ R/R ], k = [ C ( t ) − /F ( t )] R ( t ),arbitrary functions of time C , F , R and x . SIMPLE CHARACTERIZATION OF DOUBLY TWISTED SPACETIMES 5
Another example is the solution by Banerjee et al. [15] for a matter field with shearand vorticity free velocity and heat transfer, in a conformally flat metric: ds = − V ( t, x ) dt + 1 U ( t, x ) ( dx + dy + dz )with U V = A ( t ) k x k + A ( t ) · x + A ( t ), U = B ( t ) k x k + B ( t ) · x + B t , where A , B , A i and B i are arbitrary functions of time.Coley studied the case with no acceleration, V = 1, which makes the above metrictwisted ([20] eq.3.12). The same paper contains an example of GRW spacetime (eq.1.2), which is also Bianchi VI : ds = − dt + X ( t ) ( dx + e − x dy + e x dz )For a discussion and examples of doubly warped metrics, see [21]. Appendix
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