Conformal scalar fields, isotropic singularities and conformal cyclic cosmologies
aa r X i v : . [ g r- q c ] D ec Conformal scalar fields, isotropic singularitiesand conformal cyclic cosmologies
Christian L¨ubbe ∗ Department of Mathematics, University College London, London, UKGraduate School of Mathematical Sciences, University of Tokyo, Tokyo, JapanNovember 2, 2018
Abstract
We analyse spacetimes with a conformal scalar field source, a cosmological constant and aquartic self-interaction term for the scalar field. We also consider additional matter contentsin the form of Maxwell and Yang-Mills fields or radiation fluids. Existence theorems forweakly asymptotically flat spacetimes are given. We give a generalisation of Bekenstein’sresult [Ann. Phys. 82, 535 (1974)] and use it to derive existence theorems for spacetimesthat contain an isotropic singularity. The results are combined to suggest a mathematicalsetup for Penrose’s CCC scenario using a conformal scalar field cosmology.
Penrose’s work on conformal geometry and general relativity [21], [22] (see also [26] ) has becomean established approach to study the asymptotic structure of spacetimes. The central idea is tostudy a physical spacetime ( ˜
M , ˜ g ) and its asymptotic structure in terms of a conformally relatedspacetime ( M, g ), with g = θ ˜ g . The Einstein field equations are not satisfied in ( M, g ) and hence(
M, g ) is typically referred to as the unphysical spacetime. The conformal boundary of ( ˜
M , ˜ g ) isgiven by the set where the conformal factor θ vanishes. The conformal approach allows one tostudy ( ˜ M , ˜ g ) in terms of the properties of fields at the conformal boundary.The treatment of isotropic singularities [27], also known as conformally compactifiable singu-larities [1] or conformal gauge singularities [14], is similar in spirit to the conformal approach forthe asymptotic structure. Again the physical spacetimes ( ˜ M , ˜ g ) is conformally embedded intoa larger unphysical spacetime ( M, g ), with g = θ ˜ g . However this time the conformal factor θ diverges as one approaches the singularity. In [1], [2], [28] isotropic singularities with differentmatter models were studied to investigate the Weyl tensor hypothesis proposed by Penrose [23],[24], [25]. Penrose argued that at the big bang the gavitational entropy should have been very lowand that the Weyl tensor at the big bang should have vanished (strong Weyl curvature hypothe-sis) or at least been non-singular (weak Weyl curvature hypothesis). In the case of a perfect fluidwith p = ( γ − ρ , Anguige and Tod showed in [1] that an initially vanishing Weyl tensor impliesthat the spacetime is globally conformally flat. In the case of massless Vlasov the same authors[2] showed that there exist spacetimes for which the Weyl tensor vanishes at the isotropic singu-larity but which are not conformally flat. In [1], [2] a vanishing cosmological constant was used,while the case with a de Sitter-like cosmological constant was analysed for spatially homogeneousspacetimes in [29].In [24], [25] Penrose outlined details of his recent proposal of conformal cyclic cosmologies (inthe following CCC). At the centre of the CCC-proposal lies the idea that spacetimes (termedaeons) with a de Sitter-like cosmological constant form a successive chain. Two consecutive aeonsare joint in a bridging spacetime [31] by identifying the future null infinity of one aeon with ∗ E-mail address: [email protected], [email protected] I imposes a vanishing Weyl tensor, while the Weyltensor may be non-zero at the isotropic singularity. Thus the setting in the future aeon appearsto require specific fine tuning. The results of [1], [2] highlight that for a chosen matter model weneed to check whether there exists a sufficiently large family of solutions satisfying the fine-tuningor whether one is automatically reduced to the conformally flat case. The existence of explicitpairs of physical spacetimes satisfying the CCC-proposal which are not conformally flat has beenrecently shown in [31]. Observe that, despite the use of an unphysical ’bridging metric’ to formulate the concepts ofconformal boundary and isotropic singularity, the CCC-scenario requires two physical spacetimesthat are conformally related. A similar conformal relationship between two physical spacetimeswas already observed by Bekenstein in [3]. He showed that a spacetime ( ˜
M , ˜ g ) with an ordinaryscalar field, an electromagnetic field and a radiation fluid is conformally related to a spacetime( ˆ M , ˆ g ) containing a conformal scalar field, an electromagnetic field and a radiation fluid. Itwas also shown that if the spacetime ( ˜ M , ˜ g ) contains only an ordinary scalar field then it isconformally related to two spacetimes ( ˆ M , ˆ g ) and ( ˇ M , ˇ g ) each containing a conformal scalar field.Unlike ( M, g ), the manifolds ( ˜
M , ˜ g ), ( ˆ M , ˆ g ) and ( ˇ M , ˇ g ) are all physical solutions in their ownright, using different matter models. Bekenstein’s spacetimes have the advantage that there is aclear mathematical procedure that generates a new physical spacetime from a given one. Sincethe work in [3] is only concerned with Λ = 0 we will extend the results to include a non-zerocosmological constant. Moreover, we will investigate how such a result could provide new ideasfor generating the new aeon from the previous one. Our results can be summarised as follows
Main Theorem:
Let Σ be a compact spacelike hypersurface. Suppose on Σ we are given initialdata at null infinity for the CEFE with a de Sitter-like cosmological constant whose matter modelis a conformal scalar field ˆ φ with quartic self-interaction term minimally coupled to Einstein-Maxwell-Yang-Mills and radiation fluids. Then the following hold:1. There exists a weakly asymptotically flat spacetime ( ˆ M , ˆ g ) with a conformal scalar fieldminimally coupled to Einstein-Maxwell-Yang-Mills and radiation fluids.2. There exists a second solution ( ˇ M , ˇ g ) with the same matter models for which the valuesof the cosmological constant and the coefficient of the quartic self-interaction term areinterchanged. If the unphysical scalar field ˆΦ = ˆ φ/θ vanishes exactly on Σ then ( ˇ M , ˇ g ) hasan isotropic singularity at Σ that satisfies the strong Weyl curvature hypothesis.3. ( ˆ M , ˆ g ) and ( ˇ M , ˇ g ) can be interpreted as consecutive aeons of Penrose’s conformal cycliccosmology.For the detailed assumptions, precise formulations and technical details the reader is referredto the main text. We start by setting up the necessary geometry and notation in Section 2. In particular, weextend Bekenstein’s results [3] on the duality of conformal scalar field spacetimes to include acosmological constant and a quartic self-interaction term. In Section 3 we discuss the existenceof weakly asymptotically flat spacetimes containting a conformal scalar field coupled with Yang-Mills fields and radiation fluids. Using the duality between conformal scalar field spacetimes wederive existence results for spacetimes contain an isotropic singularity and whose matter is givenby a conformal scalar field, Einstein-Maxwell-Yang-Mills and radiation fluids. In Section 4 weshow that these spacetimes can be joined to form consecutive aeons of the CCC-scenario and New conformally flat solutions were found in [19].
The conventions and notations in this article are those of [14]. We briefly summarise the mostimportant ones. Throughout we work in n = 4 dimensions and the metrics have signature(+ − −− ). The bundles of scalar conformal densities of conformal weight w is denoted ε [ w ].They can be used to define more general conformal densities by ε A [ w ] := ε A ⊗ ε [ w ], where A is some general bundle index. ε [1] is the bundle of conformal scales. Let ˆ σ ∈ ε [1] denote thephysical conformal scale. Then the physical metric is given by ˆ g ij = ˆ σ − g ij , where g ij ∈ ε ij [2]is the conformal metric. The associated Levi-Civita connection is denoted by ˆ ∇ and satisfiesˆ ∇ i ˆ σ = 0. We will denote the unphysical metric by g and the associated conformal scale and Levi-Civita connection by ν and ∇ . Our curvature conventions are ( ∇ i ∇ j − ∇ j ∇ i ) v k = R kij l v l and R jl = R kkj l . The Schouten tensor for a Levi-Civita connection is given by L ij = ( R ij − Rg ij ).When working with trace-free matter models we will use the rescaled energy-momentum tensor T ij = ˆ θ − ˆ T ij , which also satisfies ∇ i T ij [10]. The associated conformal density is given by T ij = ˆ σ − ˆ T ij ∈ ε ij [ − Let ˆ ϕ ∈ ε [ −
1] be the conformal density representing the conformally scalar field. Observe thatˆ φ := ˆ σ ˆ ϕ is the representation of ˆ ϕ in the conformal scale ˆ σ and thus the value of the conformalscalar field in the physical spacetime.The energy momentum tensor for a conformal scalar field [6], [20] can be given by ǫ ˆ T ij = 4 ˆ ∇ i ˆ φ ˆ ∇ j ˆ φ − ˆ g ij ˆ ∇ k ˆ φ ˆ ∇ k ˆ φ − φ ˆ ∇ i ˆ ∇ j ˆ φ + 2 ˆ φ ˆ L ij − α ˆ φ ˆ g ij (1)where ǫ = ±
1. We refer to the case ǫ = 1 as the (attractive) conformal scalar field and to ǫ = − φ satisfies the inhomogenous conformallyinvariant wave equation ( ˆ L = ˆ L ij ˆ g ij ) ˆ (cid:3) ˆ φ − ˆ L ˆ φ = − α ˆ φ (2)implies that ˆ T ij is trace-free and divergence free. Note that the coefficients of the quartic inter-action term in (1) and of the cubic in (2) have to coincide. The related Einstein field equationwith a cosmological constant ˆΛ is given byˆ G ij + ˆΛˆ g ij = ˆ T ij . (3)We observe that ˆ G ij = 2( ˜ L ij − ˆ L ˆ g ij ) and set ˆΛ = 6ˆ λ for later convenience. As ˆ T ij is tracefree wehave ˆ L = 4 λ so that (3) can be written as2 ˆ L ij − λ ˆ g ij = ˆ T ij (4)and (2) takes the form ˆ (cid:3) ˆ φ − λ ˆ φ = − α ˆ φ . This can be interpreted in terms of a potential V = V ( ˆ φ ) and written asˆ (cid:3) ˆ φ + ∂V∂ ˆ φ = 0 with V − V = α ˆ φ − λα ˆ φ ! = α ˆ φ − ˆ λα ! − ˆ λ α .
3f the spacetime is de Sitter-like, i.e. ˆ λ <
0, then for α > φ = 0 and for α < α = 0 the potentialreduces to V = − λ ˆ φ .Isolating ˆ L ij from (4), substituting into (1) and isolating ˆ T ij allows one to find the commonlyused expression for the physical energy-momentum tensor in terms of ˆ φ, ˆ λ, α onlyˆ T ij = ( ǫ − ˆ φ ) − h ∇ i ˆ φ ˆ ∇ j ˆ φ − ˆ g ij ˆ ∇ k ˆ φ ˆ ∇ k ˆ φ − φ ˆ ∇ i ˆ ∇ j ˆ φ + 2ˆ λ ˆ φ ˆ g ij − α ˆ φ ˆ g ij i . (5)For ǫ = 1 (5) holds as long as ˆ φ = 1. When ˆ φ ∈ ( − ,
1) the factor at the front is positive, whereaswhen | ˆ φ | > ǫ = − φ = 0, which describes vacuum. For a constant scalar fieldˆ φ = ˆ φ = 0 equation (2) gives ˆ φ = ˆ λα which in turn implies that T ij = 0 and thus also describesvacuum. Since we are not interested in vacuum solutions we will throughout assume that ˆ ∇ i ˆ φ = 0or ˆ ∇ i ˆ ∇ j ˆ φ = 0 when ˆ φ = 0 We say a connection is compatible with [ g ] if ∇ i g jk = 0 and ∇ is torsion-free. For a metricˆ g ∈ [ g ] the Levi-Civita connection ˆ ∇ = ∇ (ˆ g ) is compatible with [ g ], as are all the general Weylconnections ∇ .In the following we work in the language of conformal densities as this saves us carryingexplicit conformal factors through all the calculations. For ˆ ϕ ∈ ε [ −
1] the conformally invariantwave equation has the form (cid:3) ˆ ϕ − L ˆ ϕ = − α ˆ ϕ , (6)where ∇ is any connection compatible with [ g ], (cid:3) = g ij ∇ i ∇ j and L = L ij g ij is the conformaltrace of the Schouten tensor of ∇ .Given a conformal density ˆ ϕ ∈ ε [ − ∇ compatible with [ g ] and its Schoutentensor L ij as well as a parameter α we define the following conformal density D ij [ ˆ ϕ, ∇ , α ] := 4 ∇ i ˆ ϕ ∇ j ˆ ϕ − g ij ∇ k ˆ ϕ ∇ k ˆ ϕ − ϕ ∇ i ∇ j ˆ ϕ + 2 ˆ ϕ L ij − α ˆ ϕ g ij ∈ ε ij [ −
2] (7)We observe the following properties of D ij [ ˆ ϕ, ∇ , α ]1. D ij [ C ˆ ϕ, ∇ , α ] = C D ij [ ˆ ϕ, ∇ , α/C ]2. If ∇ and ˆ ∇ are both compatible with [ g ] then D ij [ ˆ ϕ, ∇ , α ] = D ij [ ˆ ϕ, ˆ ∇ , α ] . Thus we can simply write the tensor density as D ij [ ˆ ϕ, α ].3. Taking the trace we have g ij D ij [ ˆ ϕ, α ] = − ϕ (cid:0) (cid:3) ˆ ϕ − L ˆ ϕ + 4 α ˆ ϕ (cid:1) .Thus if ˆ ϕ satisfies (6) then D ij [ ˆ ϕ, α ] is trace-free. Moreover, (6) implies that D ij [ ˆ ϕ, α ] isdivergence-free.4. If ˆ σ denotes the physical scale, ˆ φ = ˆ σ ˆ ϕ and ˆ ∇ satisfies ˆ ∇ i ˆ σ = 0 then we have ˆ σ D ij [ ˆ ϕ, α ] = ǫ ˆ T ij . (8)5. Let ˇ ϕ := ˆ σ − then ˇ ϕ satisfies D ij [ ˇ ϕ, ˆ λ ] = 2 ˇ ϕ ˆ L ij − λ ˇ ϕ g ij = 2 ˇ ϕ ( ˆ L ij − ˆ λ ˆ g ij ) (9)and (cid:3) ˇ ϕ − L ˇ ϕ = ˆ (cid:3) ˇ ϕ − ˆL ˇ ϕ = − λ ˇ ϕ Setting C = ± i in 1.) switches between the cases ǫ = 1 and ǫ = −
1. However, it also transforms a real scalarfield into a purely imaginary one. σ D ij [ ˇ ϕ, ˆ λ ] = 2( ˆ L ij − ˆ λ ˆ g ij ) = ˆ T ij = ǫ ˆ σ D ij [ ˆ ϕ, α ]Thus the Einstein field equation can be recast into the conformally invariant equation D ij [ ˇ ϕ, ˆ λ ] = ǫ D ij [ ˆ ϕ, α ] . (10)The important fact to observe is that in (10) we can interpret either side as the Einstein tensorand the other side as the energy-momentum tensor. The two corresponding physical metrics aregiven by ˆ g ij = ˆ σ − g ij and ˇ g ij = ˇ σ − g ij . Since ˇ ϕ = ˆ σ − and ˆ ϕ = ˇ σ − we have ˆ φ = ˆ ϕ ˆ σ = ˇ σ − ˆ σ and ˇ φ = ˇ ϕ ˇ σ = ˆ σ − ˇ σ = ˆ φ − . Hence the physical metrics are related byˇ g ij = ˆΩ ˆ g ij with ˆΩ = ˆ φ and ˆ g ij = ˇΩ ˇ g ij with ˇΩ = ˇ φ. (11)The existence of two dual solutions to the conformal scalar Einstein field equations was alreadyobserved in Theorem 2 of [3], where ˆ λ = 0 = α was assumed. The above observations lead to astraight forward generalisation of Bekenstein’s result and can be summarised as follows Theorem 1.
Let ( ˆ
M , ˆ g ) denote a solution of the Einstein field equations with cosmological con-stant ˆΛ = 6ˆ λ and conformal scalar field ˆ φ satisfying (1) with coefficient α . Then there exist a dualsolution ( ˇ M , ˇ g ) with scalar field ˇ φ = ˆ φ − and ˇ g ij = ˆ φ ˆ g ij . Moreover, the role of the cosmologicalconstant ˆ λ and the coefficient of the quartic self interaction term α have been swapped. In Appendix D of [4] the authors gave conformal scalar field spacetimes obtained from staticvacuum spacetimes. In these examples ˆΛ = 0 and α = 0. Equ. (D74) of [4] givesd s = 14 (cid:0) W β ± W − β (cid:1) (cid:2) W α d t − W − α h ij d x i d x j (cid:3) (12) φ = C ∓ W β ± W β = − C W β ∓ W − β W β ± W − β . (13)The top sign gives a solution of type A, denoted (ˆ g, ˆ φ ) here, and the bottom sign a solution oftype B, denoted (ˇ g, ˇ φ ). It is straight forward to check that ˇ g = ˆ φ ˆ g and ˇ φ = ˆ φ − so that solutionsof types A and B are dual to each other. Moreover, when W = e U = 1, i.e. U = 0 we have ˆ φ = 0and ˇ φ = ∞ .A spherically symmetric example is given by equ. (D74) of [4]ˆ g = (cid:16) − m ¯ r (cid:17) d t − (cid:16) − m ¯ r (cid:17) − d¯ r − ¯ r d σ (14)ˆ φ = C m ¯ r − m (15)which is the metric of extremal Reissner-Nordstr¨om black hole. Defining the isotropic radialcoordinate R = ¯ r − m we get ˆ φ = C mR . Clearly ˆ φ = 0 at R = ∞ and ˆ φ = ∞ at R = 0. Observethat the conformal rescaling ˇ g = ˆ φ ˆ g recovers the discrete conformal isometry of the extremalReissner-Nordstr¨om spacetime. This conformal isometry i : ˆ g → ˇ g maps the spacetime onto itselfby i : R → ρ = m R . In particular, the horizon R = 0 is mapped to null infinity R = ∞ andvice versa — see [5], [18] for further details. So in a sense, the metric (14) is its own dual. Ashighlighted in [4] the geometry of the metric (14) is regular at the horizon R = 0 despite theblow-up of ˆ φ . This highlights that the blow-up of ˆ φ need not imply a singular geometry. This is similar to equations (17) and (22) in [20] .3 Conformal scalar fields coupled to other matter models Let us consider the case where the conformal scalar field is coupled to other trace-free mat-ter models, like Einstein-Maxwell-Yang-Mills, radiation fluids, null fluids or massless Vlasov.Throughout this article we will assume that this coupling to other matter is minimal, i.e. theenergy momentum tensor for each individual matter component is divergence-free.We will be particularly interested in Einstein-Maxwell-Yang-Mills and radiation fluids as wewill want to use the conformal Einstein field equations (CEFE) for both models, see [10] and [17],to prove the existence of asymptotically flat solutions. We remark that our results generalise tocombinations of multiple Yang-Mills fields, radiation fluids and conformal scalar fields, however inthe interest of clarity we will only discuss the case of at most one component of each matter type.We would also like to highlight that the results concerning dualities of spacetimes generalise tomore general trace-free models, like null fluids or massless Vlasov. However, a central part of ourwork will rely on the CEFE and the related existence and uniqueness results. At this moment intime there are no CEFE available for these matter models that allow us to formulate an IVP atconformal infinity and evolve the conformal spacetime, which is the main reason why models likenull fluids or massless Vlasov have been excluded from our analysis.Recall that the physical and unphysical energy momentum tensors ˆ T ij and T ij are given byˆ T [ EMY M ] ij = F α il F α lj − g ij F α kl F α kl = ˆ θ T [ EMY M ] ij (16)ˆ T [ rad ] ij = 43 ˆ ρ ˆ u i ˆ u j −
13 ˆ ρ ˆ g ij = ˆ θ T [ rad ] ij . (17)For an observer’s time direction ˆ v i in ( ˆ M , ˆ g ) the associated electric and magnetic (Yang-Mills)fields are given by ˆ E αi = F α ij ˆ v j and ˆ B αi = ∗ F α ij ˆ v j . Both covectors have conformal weight − E αi = ˆ θ − ˆ E αi . The fluid density ˆ ρ and fluid velocity ˆ u i have conformal weights − − ρ = ˆ θ − ˆ ρ and u i = ˆ θ − ˆ u i . In [10] and [17] it was shown that near the conformal boundary ofasymptotically simple spacetimes the unphysical variables E αi , B αi , ρ, u i are bounded in ( M, g ).Thus the physical variables ˆ E αi , ˆ B αi , ˆ ρ, ˆ u i vanish at conformal infinity and ( ˆ M , ˆ g ) satisfies thecosmic no-hair conjecture.We now consider a spacetime ( ˆ M , ˆ g ) with cosmological constant ˆ λ satisfyingˆ G ij + 6ˆ λ ˆ g ij = ˆ T ij = ˆ T [ ˆ ϕ ] ij + ˆ T [ M ] ij , (18)where ˆ T ij has been split into the conformal scalar field part ˆ T [ ˆ ϕ ] ij and all other trace-free mattercomponents ˆ T [ M ] ij . Due to our assumption of minimal coupling we have ˆ ∇ i ˆ T ij = 0.From a mathematical point of view one can also consider an Einstein equation of the formˆ G ij + 6ˆ λ ˆ g ij = − ˇ T ij . (19)We will refer to such matter M as repulsive matter, e.g. repulsive Einstein-Maxwell or a repulsiveradiation fluid. From a physical standpoint such matter is considered unrealistic. However, aswe will see below, it will be useful to consider repulsive matter mathematically. Unless otherwisespecified we shall always assume that we are dealing with standard or attractive matter and use(18). We will refer to the case ǫ = − φ , ˆ T [ ˆ ϕ ] ij can be locally negative for ǫ = 1as well as locally positive for ǫ = − T [ M ] ij = ˆ σ − ˆ T [ M ] ij . Using (8), (9) for ˆ T [ ˆ ϕ ] ij we can rewrite (18) in terms of conformaldensities to give the following generalisation of (10): D ij [ ˇ ϕ, ˆ λ ] = ǫ D ij [ ˆ ϕ, α ] + T [ M ] ij . (20)As before we consider switching the roles of D ij [ ˇ ϕ, ˆ λ ] and D ij [ ˆ ϕ, α ]. Setting ˇ T [ M ] ij = ˇ σ T [ M ] ij ,the Einstein equation in ( ˇ M , ˇ g ij ) takes the formˇ G ij + 6 α ˇ g ij = ˇ T ij = ˇ T [ ˇ ϕ ] ij − ǫ ˇ T [ M ] ij . (21)6e can see that the additional matter models in ( ˆ M , ˆ g ij ) and ( ˇ M , ˇ g ij ) are identical, e.g. aYang-Mills field will lead again to a Yang-Mills field.For an attractive conformal scalar field ( ǫ = 1) the additional matter M changes its charac-teristic from attractive to repulsive (and vice versa) in the dual solution. This problem is dueto the minus sign in (21) and was already observed in [3], where the resulting dual solutionswere considered as unphysical. However if we start from a solution whose additional matter ˆ M is repulsive then the dual solution will contain standard matter ˇ M . We will exploit this aspectlater to obtain new solutions with attractive matter.Note that for the repulsive conformal scalar field ( ǫ = −
1) the problem disappears. Bothˇ T [ M ] ij and ˆ T [ M ] ij are either attractive or repulsive. Thus for ǫ = − M , ˆ g ) and ( ˇ M , ˇ g ), which both describe a repulsive conformal scalar field coupled to attractivetrace-free matter.In summary we have Theorem 2.
Let ( ˆ
M , ˆ g ) denote a solution of the Einstein field equations with cosmological con-stant ˆΛ = 6ˆ λ and a conformal scalar field ˆ φ with α as coefficient of the self-interaction term in (1) coupled to additional trace-free matter ˆ M . Then the following hold:1. There exists a dual solution ( ˇ M , ˇ g ) with ˇ g ij = ˆ φ ˆ g ij with cosmological constant A = 6 α anda conformal scalar field ˇ φ with ˆ λ as coefficient of the self-interaction term in (1) .2. The conformal scalar field models have the same parameter ǫ .3. The additional matter ˇ M is of the same type as ˆ M .4. If ǫ = 1 then repulsive matter ˆ M gives attractive matter ˇ M , and vice versa.5. If ǫ = − then the additional matter components ˆ M and ˇ M are either both attractive orboth repulsive. Corollary 1.
The dual of an attractive conformal scalar field ( ǫ = 1) coupled to a repulsiveYang-Mills field and a repulsive radiation fluid is an attractive conformal scalar field coupled toan attractive Yang-Mills field and an attractive radiation fluid. Corollary 2.
The dual of a repulsive conformal scalar field ( ǫ = − coupled to a Yang-Millsfield and a radiation fluid is an repulsive conformal scalar field coupled to a Yang-Mills field anda radiation fluid. Remark: We require ˆ T [ M ] ij to be trace-free, so that ˇ T [ M ] ij will be divergence free as well. ( ˆ M , ˆ g ) and ( ˇ M , ˇ g ) and the unphysical manifold ( M, g ) Considering ( ˆ
M , ˆ g ) as our physical solution we can see that as long as ˆ φ = 0 there is no problemin finding the dual solution ( ˇ M , ˇ g ). In the case ˆ φ = 0 we have ˇ φ = ∞ and the conformal factorrelating ˆ g and ˇ g will vanish, see (24) and (27) below. Similarly, we are interested in the asympoticbehaviour of both spacetimes, i.e the neighbourhood of the set where ˆ σ vanishes.In the following we will assume that we can choose a conformal scale ν ∈ ε [1] such that g ij = ν − g ij is regular throughout M . The conformal scales ˆ σ and ˇ σ provide a conformalembedding of the physical solutions ( ˆ M , ˆ g ) and ( ˇ M , ˇ g ) into ( M, g ). Thus we will consider (
M, g )as our unphysical spacetime for both solutions. The corresponding Levi-Civita connections will bedenoted ˆ ∇ , ˇ ∇ , ∇ respectively. Analogous notation will be used for quantities related to quantitiesassociated with a particular choice of conformal scale or metric.The three metrics g, ˆ g and ˇ g are defined in terms of the conformal metric g by g ij = ν − g ij , ˆ g ij = ˆ σ − g ij , ˇ g ij . = ˇ σ − g ij (22)7e define the conformal factorsˆ θ = ˆ σν − , ˇ θ = ˇ σν − , ˆΩ = ˆ σ ˇ σ − = ˆ θ ˇ θ , ˇΩ = ˇ σ ˆ σ − = ˇ θ ˆ θ . (23)Hence the following conformal relationships hold between the metrics g, ˆ g and ˇ gg ij = ˆ θ ˆ g ij , g ij = ˇ θ ˇ g ij ˇ g ij = ˆΩ ˆ g ij , ˆ g ij = ˇΩ ˇ g ij . (24)The physical scalar fields ˆ φ, ˇ φ and the unphysical scalar fields ˆΦ , ˇΦ are the realisations of theconformal densities ˆ ϕ and ˇ ϕ in the conformal scales ˆ σ, ˇ σ and ν . They are given byˆ φ = ˆ σ ˆ ϕ, ˇ φ = ˇ σ ˇ ϕ, ˆΦ = ν ˆ ϕ, ˇΦ = ν ˇ ϕ. (25)Recall that the conformal scales ˆ σ, ˇ σ and the conformal scalar fields ˆ ϕ, ˇ ϕ are related byˇ ϕ = ˆ σ − , ˇ σ = ˆ ϕ − . (26)Using (23) and (25) we find the following relationships between ˆ φ, ˇ φ, ˆΦ and ˇΦˆΦ = 1ˇ θ , ˇΦ = 1ˆ θ , ˆ φ = ˆ θ ˆΦ = ˆΩ , ˇ φ = ˇ θ ˇΦ = ˇΩ , ˇ φ = ˆ φ − . (27)The energy-momentum tensors in (18) and (21) are related byˇ T ij = ˆΩ − ˆ T ij = ˆ φ − ˆ T ij , ˆ T ij = ˇΩ − ˇ T ij = ˇ φ − ˇ T ij . (28)We recall that a spacetimes ( ˆ M , ˆ g ) is said to be weakly asymptotically flat if there exists aconformally related spacetime ( M, g ) with g = ˆ θ ˆ g satisfying i) ˆ θ | ˆ M > I = { ˆ θ = 0 } one has ∇ i ˆ θ = 0. Note minor exceptions to ii) exists when ˆ λ = 0 as there exist isolated pointswhere ˆ θ = 0 and ∇ i ˆ θ = 0, but ∇ i ∇ j ˆ θ = 0.A spacetime ( ˇ M , ˇ g ) has an isotropic singularity if there exists a conformally related spacetime( M, g ) with ˇ g = Ξ g , ˇ M ⊂ M and a spacelike hypersurface Σ where i) Ξ = 0, ii) g is a regularmetric, iii) ˇ g is singular. The hypersurface Σ is interpreted as the singularity of ( ˇ M , ˇ g ). Here wemean regular in the sense that the curvature is sufficiently smooth and thus finite at Σ, whilesingular means that part of the curvature blows up as we approach Σ (strictly speaking Σ isnot part of ( ˇ M , ˇ g )). Since g is regular at Σ the Weyl tensor C kij l must be regular at Σ. Thusthe singular behaviour of ˇ g must arise from a blow up of the Ricci tensor ˇ R ij , respectively theSchouten tensor ˇ L ij or the energy momentum tensor ˇ T ij .For a conformal scalar field spacetime the divergence of the scalar field on its own is insufficientfor an isotropic singularity. For example at the horizon the extremal Reissner-Nordstr¨om blackhole Ω = φ = ∞ , while the spacetime curvature is regular there. Hence the horizon cannot beinterpreted as a curvature singularity, or even an isotropic singularity. This example illustratesthat for an isotropic singularity we require ˇ T ij to diverge.We proceed by analysing the effects of ˆ φ vanishing or diverging on a hypersurface Σ. Lemma 1.
Suppose ˆ T ij is given by (1) . Suppose there exists a hypersurface Σ in ( ˆ M , ˆ g ) where ˆ φ vanishes smoothly but ˆ T ij = 0 . Then in the dual spacetime ( ˇ M , ˇ g ) ˇ φ and ˇ T ij diverge on Σ andif Σ is a spacelike hypersurface then it represents an isotropic singularity.Proof. Set g = ˆ g and Ξ = ˆΩ = ˆ φ . Then g is regular at Σ and Ξ = 0 there. If Σ is spacelike theconditions for an isotropic singularity are satisfied. Equations (27) and (28) imply that ˇ φ and ˇ T ij diverge at Σ. Since the Weyl tensor is finite Σ represents an isotropic singularity. Lemma 2.
Suppose ˇ T ij is given by (1) . Suppose there exists a hypersurface Σ in ( ˇ M , ˇ g ) where ˇ φ diverges, ( ∂ i ˇ φ ) ˇ φ − = 0 and ˇ T ij is finite. Then the dual spacetime ( ˆ M , ˆ g ) is weakly asymptoticallyflat, the hypersurface Σ represents part of conformal infinity and ˆ φ and ˆ T ij vanishes on Σ . Note that our convention differs from [31]. This is also referred to as a conformally compactifiable singularity [1] or as a conformal gauge singularity [14].
M , ˆ g ) ( M, g )Σ = I + Σ = I − ( ˇ M , ˇ g )Σ = IS DualityFigure 1:
The conformal duality:
The bridging spacetime (
M, g ) is depicted (twice). Thephysical spacetimes ( ˆ
M , ˆ g ), a weakly asymptotically flat spacetime, and ( ˇ M , ˇ g ), a spacetime withan isotropic singularity (IS), can be conformally embedded into ( M, g ). The surface Σ representsnull infinity, respectively the isotropic singularity (zigzagged line). The strength of each (physical)energy-momentum tensor is indicated by the strength of the shading. Note the upper halves alsodescribes two dual physical spacetimes.
Proof.
The vanishing of ˆ φ on Σ follows directly from (27). Since the curvature in ( ˇ M , ˇ g ) is regularat Σ find that ˇ T ij is bounded there and hence ˆ T ij = ˇΩ − ˇ T ij = ˆ φ ˇ T ij vanishes at Σ as well.Since i) ˇ g ij = ˆΩ ˆ g ij = ˆ φ ˆ g ij , ii) ˆΩ = ˆ φ = 0 on Σ with ˆΩ = 0 on ˇ M \ Σ iii) ˆ ∇ i ˆΩ = ∂ i ( ˇ φ − ) =( ∂ i ˇ φ ) ˇ φ − = 0 on Σ and iv) Σ is a regular hypersurface in ( ˇ M , ˇ g ) where the metric ˇ g ij is regular,it follows that Σ satisfies the definition of conformal infinity for ( ˆ M , ˆ g ).Remark: The nature of the conformal boundary of ( ˆ M , ˆ g ) depends on the cosmological con-stant ˆ λ , i.e. the coefficient of the quartic self-interaction term for ˇ φ .For the CCC-scenario we would like Σ to describe both conformal infinity and the isotropicsingularity. However, neither conformal infinity nor the isotropic singularity are part of theirphysical spacetime, while the above lemmas assume explicitly that Σ is a regular hypersurface inthe respective physical spacetime. Below we will show that working with a bridging spacetime( M, g ) and making suitable assumptions, Σ can describe conformal infinity for ( ˆ
M , ˆ g ) and theisotropic singularity for ( ˇ M , ˇ g ) simultaneously. Lemma 3.
Suppose ( ˆ
M , ˆ g ) is a weakly asymptotically flat spacetime with a conformal scalar field ˆ φ and energy momentum tensor (1) . Suppose ( ˆ M , ˆ g ) is conformally embedded into ( M, g ) andthe conformal boundary of ( ˆ M , ˆ g ) is described by a spacelike hypersurface Σ in ( M, g ) . Supposethat ˆΦ = ˆ φ ˆ θ − vanishes on Σ , while T ij = ˆ θ − ˆ T ij is finite, but non-vanishing there. Then in thedual spacetime ( ˇ M , ˇ g ) the hypersurface Σ represents an isotropic singularity. (See Figure 1)Proof. Combining (24) and (27) we find ˇ g ij = ˇ θ − g ij = ˆΦ g ij . Since T ij is non-zero we find thatˆ T ij = ˆ θ T ij vanishes while ˇ T ij = ˇ θ T ij diverges. Furthermore ( M, g ) has regular Weyl curvatureand hence (
M, g ) and ( ˇ
M , ˇ g ) satisfy the requirements for Σ to represent an isotropic singularityfor ( ˇ M , ˇ g ).Remark: In Lemma 1 and 3 the requirement that Σ be spacelike has been chosen here in orderto fit the definition of isotropic singularities in [11], which uses the level set Σ = { T = 0 } of atime function T . The condition can be dropped if one allows for isotropic singularities along nullor timelike hypersurfaces. Lemma 4.
The above three lemmas hold if the conformal scalar field is coupled to trace-freematter, in particular Yang-Mills fields and radiation fluids. roof. Recall that ˆ T ij = ˆ T [ ϕ ] ij + ˆ T [ M ij implies T ij = T [ ϕ ] ij + T [ M ij where each part rescales individuallyby Ξ − . In particular, recall that the fields E αi , B αi , ρ, u i have conformal weights − , − , − , − M, g ) are finite.Lemma 1: Again set ν = ˆ σ , so that ˆ θ = 1 and ( M, g ) = ( ˆ
M , ˆ g ). It follows that ˆΩ = ˆ φ = ˇ θ − =ˇ φ − . Thus, ˆΩ = 0 and ˇ φ = ∞ at Σ. Since ˆ T ij = 0, (28) implies that ˇ T ij diverges while the Weyltensor is finite. Hence Σ represents an isotropic singularity. In particular, the fields E αi , B αi , ρ, u i ,which have same finite values in ( M, g ) and ( ˆ
M , ˆ g ), rescale to ˇ E αi = ˆΩ − E αi , ˇ B αi = ˆΩ − B αi , ˇ ρ =ˆΩ − ρ . Thus any field that does not vanish in ( M, g ) = ( ˆ
M , ˆ g ) will diverge in ( ˇ M , ˇ g ).Lemma 2: The proof goes directly through as before with ˆ T ij , ˆ E i , ˆ B i , ˆ ρ each vanishing at Σ.Lemma 3: Once more ˆ T ij = ˆ θ T ij vanishes while ˇ T ij = ˇ θ T ij diverges at Σ, which hencerepresents an isotropic singularity in ( ˇ M , ˇ g ).Remark: Lemma 4 works for ǫ = ± In this section we will prove the existence of spacetimes with a conformal scalar field that areeither weakly asymptotically flat or contain an isotropic singularity. Our approach uses theconformal Einstein field equations (CEFE). As Friedrich showed, a key feature of the CEFE forvacuum [8], [9] or trace-free matter [10] is that a solution of the CEFE on (
M, g ) implies a solutionof the related Einstein field equations on each connected component of the subset { Θ = 0 } of( M, g ). The method has been extended to spacetimes containing Einstein-Maxwell-Yang-Mills[10], radiation fluid [17] or conformal scalar field [12]. In particular, one can formulate a Cauchyproblem for each of these CEFE. In order to prove existence and uniqueness for the CEFE oneshows that the CEFE can be transformed into a first order symmetric hyperbolic system andthat the constraints are propogated. As we will discuss in the next section the method can bedirectly extended to spacetimes containing a combination of the above mentioned matter fieldsin a minimally coupled form. In particular, one can prove the following theorem
Theorem 3.
Let Σ be an initial surface with initial data for the CEFE of a standard confor-mal scalar field ( ǫ = 1) minimally coupled to Einstein-Maxwell-Yang-Mills and a radiation fluid.Suppose that on Σ the initial conformal scalar field satisfies − < Θ ∗ φ ∗ < . Then in a neigh-bourhood U of Σ there exists a solution to the CEFE. In each connected component of U \ { θ = 0 } the solution gives rise to a physical spacetime whose matter model is a conformal scalar fieldminimally coupled to Einstein-Maxwell-Yang-Mills and a radiation fluid. We are free to set Θ ∗ = 1 and φ ∗ = 0 across Σ. Thus we find Corollary 3.
There exist spacetimes with a conformal scalar field minimally coupled to Einstein-Maxwell-Yang-Mills and a radiation fluid for which the conformal scalar field vanishes on a regularspacelike hypersurface Σ . Remark: There is no restriction in Theorem 3 and Corollary 3 on the Einstein-Maxwell-Yang-Mills fields and the radiation fluid. They can be attractive or repulsive.
As outlined above we will use the unphysical momentum tensor T ij = T [ ϕ ] ij + T [ EMY M ] ij + T [ rad ] ij with ∇ i T [ ϕ ] ij = 0, ∇ i T [ EMY M ] ij = 0 and ∇ i T [ rad ] ij = 0 implying ∇ i T ij = 0.The CEFE for Einstein-Maxwell-Yang-Mills, conformal scalar field and radiation fluid werediscussed in detail in [10], [12], [17] respectively. The general details can be found in thesereferences. Here we will only focus on the essential points.The variables used in the CEFE can be split into geometric variables (frame fields, connectioncoefficients, Schouten tensor and Weyl curvature) and matter variables (conformal scalar, Yang-Mills fields, fluid density and fluid velocity). As shown in [10], [12], [17] one needs to includederivatives of the matter variables in order to obtain an overall first order system. In the followingwe will refer to these derivatives as matter fields as well. The prinicipal part of the CEFE for the10eometric variables is identical to the vacuum case and due to the minimal coupling of the mattermodels the principal part for the matter variables is a disjoint combination of the equations forthe individual models.In the case of a conformal scalar field two equations require particular attention. In terms ofthe metric g = ˆ θ ˆ g and its Levi-Civita connection ∇ they take the form ∇ k d kij l = t ijl (29)2 ∇ [ i L j ] l = d kij l d k + ˆ θt ijl (30)where d kij l = ˆ θ − C kij l , d i = ∇ i ˆ θ and t ijl = 2ˆ θ − ˆ ∇ [ i ˆ L j ] l . For trace-free matter models t ijl can beexpressed in terms of T ij and ∇ as follows t ijl = ˆ θ ∇ [ i T j ] l + 3 d [ i T j ] l − g l [ i T j ] k d k . (31)Since (31) is linear in T ij we have t ijk = t [ ϕ ] ijk + t [ EMY M ] ijk + t [ rad ] ijk . The problem that arises in thecase of a spacetime with a single conformal scalar field is that Θ ∇ [ i T j ] l and thus t ijl gives rise to ∇ [ i L j ] l appearing on the RHS of (29) and (30). In this form the CEFE cannot be rendered intoa first order system. As shown in [12], one expands t ijl to give t ijl = 2ˆ θ ˆΦ ∇ [ i L j ] l + l . o . t . . Here lower order terms (l.o.t.) mean those terms which are polynomial combinations of thevariables but contains no explicit derivatives. Now substituting (30) gives t ijl = m ijl := (1 − ˆ θ ˆΦ ) − (cid:16) ˆ θ ˆΦ d kij l d k + l . o . t . (cid:17) . (32)Thus m ijl contains no explicit derivatives and (29) and (30) can be rewritten as ∇ k d kij l = m ijl (33)2 ∇ [ i L j ] l = d kij l d k + ˆ θm ijl (34)leading to the desired form of the CEFE. Observe that t ijl , and hence m ijl , contains ∇ [ i T [ EMY M ] j ] l and ∇ [ i T [ rad ] j ] l . Since the derivatives of the matter fields have been introduced as additionalvariables, one can replace any derivatives in ∇ [ i T [ EMY M ] j ] l and ∇ [ i T [ rad ] j ] l in terms of these additionalvariables. Therefore in the minimally coupled problem analysed here, the tensor m ijl thus containsmatter variables but no explicit derivatives. The remainder of the argument follows the proofin [12]. As in the case of a single conformal scalar field one has to restrict the system to theregion where 1 − ˆ θ ˆΦ > , i.e. | ˆ φ | = | ˆ θ ˆΦ | <
1, in order for the CEFE and their subsequenthyperbolic reduction to form a regular system of PDEs. In the following we restrict ourselves tothe case − < ˆ θ ˆΦ <
1. This still allows us to work in the asymptotic region. The argumentsfor the existence and uniqueness presented in [12] essentially follow [10] and carry through to thesetting given here. This proves Theorem 3. Observe that both attractive and repulsive versionsof Einstein-Maxwell-Yang-Mills and radiation fluid are permitted, since the repulsive cases onlyintroduce a factor of − θ = 0 everywhere on Σ, then Σ is a regular hypersurface and we can use theconformal freedom to work with ˆ θ = 1 everywhere. The CEFE simplify to the problem in thephysical spacetime and the restriction becomes 1 − ˆ φ >
0. The local solution then describes aregular region of the physical spacetime away from the conformal boundary. In [12] the physical variables are denoted with a tilde, while unphysical variables have no marker. The equationsfrom [12] presented her have been adapted to our notation, so that ˜ φ , φ and Ω have been replaced by ˆ φ , ˆΦ and ˆ θ In [12] the conditon is 1 − Ω φ > φ differs by a factor from ˆΦ. .2 The CEFE for a conformal scalar field with ǫ = − The derivation of the CEFE is almost identical for the case ǫ = −
1. The key difference rests onthe fact that in (32) the factor is now (1 + ˆ θ ˆΦ ) − , which is regular for all values of ˆ φ = ˆ θ ˆΦ aslong as neither ˆ θ or ˆΦ diverge. Following through the remainder of the argument one then findsthat the CEFE and their hyperbolic reduction are regular and that for ǫ = − φ = ˆ θ ˆΦ is required. We thus have the following version of Theorem 3 for a conformal scalar fieldwith ǫ = − Lemma 5.
Let Σ be an initial surface with initial data for the CEFE of a conformal scalarfield with ǫ = − , which is minimally coupled to Einstein-Maxwell-Yang-Mills and a radiationfluid. Then a solution to the CEFE exists in a neighbourhood of Σ , which gives rise to a physicalspacetime whose matter model is a conformal scalar field with ǫ = − minimally coupled toEinstein-Maxwell-Yang-Mills and a radiation fluid. For the de Sitter-like case, λ <
0, one can setup a Cauchy problem for the CEFE on the spacelikehypersurface Σ representing I . This is referred to as the initial value problem at null infinity.The solution, which exists in a neighbourhood of I , generates two physical solutions. Onesolution lies to the past and has a future null infinity, the other one lies to the future with apast null infinity. These two solutions are then future (past) geodesically complete and weaklyasymptotically flat.The construction of such spacetimes is discussed below. Let Σ denote the initial surface and { n, e a } form a g -orthonormal frame, where n i is the normal of Σ. In the following the letters a, b, c, d are reserved for spatial indices and take values 1 , ,
3. Let ˆ λ < λ the coefficient of quartic self-interaction term. Recall that d i = ∇ i ˆ θ , so that d = ∇ n ˆ θ . Let h ab denote the 3-metric induced on Σ. Furthermore, d kij l = ˆ θ − C kij l and we let d ab = d a b and d abc = d a bc . Then the rescaled electric and magnetic Weyl tensor are given by E ab = d ab , H ab = d a cd ǫ cdb In order to find the initial values for the CEFE at I we analyse the conformal constraint equationsat I , i.e. ˆ θ = 0 and d a = D a ˆ θ = 0. It was shown in [9], [10] that the conformal constraints at I can be solved. The initial data for the geometric variables is given by d = p − λ/ , s = p − λ/ t, χ ab = − th ab , (35a) L a = D a t, L ab = l ab , H ab = − p − /λB ab (35b) D a E ab = p − λ/ T b . (35c)where t is a smooth real function, h ab is a 3-metric on Σ, E ab is a symmetric trace-free tensor and B ab and y abc are the Bach and the Cotton-York tensor of the 3-metric h ab B cd ǫ dab = − y abc = D b l ac − D a l bc . Observe that L or equivalently L = L kk have not been determined by the initial data. L canbe chosen freely as a gauge-source function for the conformal factor ˆ θ . A convenient choice is aconstant L .For the conformal scalar field, the Einstein-Maxwell-Yang-Mills case and the radiation fluidthe RHS of (35c) is given by T [ φ ]0 b = 4Ξ D b ˆΦ − D b Ξ − χ cb D c ˆΦ + 2 ˆΦ L b , (36a) T [ EMY M ] b = F α bc F α c , (36b) T [ rad ] b = 43 ρ √ − u c u c u b , (36c) ǫ bcd = ǫ bcd is the 3 dimensional volume form of h ab . ∇ n ˆΦ. For the repulsive analogues multiply the RHS by − ∗ = 0, and hence ( D a ˆΦ) ∗ = 0, then (36a) reduces to ( T [ φ ]0 b ) ∗ = 0. If the Maxwellfield at I is either purely electric or purely magnetic then ( T [ EMY M ]0 b ) ∗ = 0. If the radiationfluid is orthogonal to I then ( u b ) ∗ = 0 and ( T [ rad ]0 b ) ∗ = 0.Thus for the Cauchy problem at I the free geometrical data is given by a scalar function t , a 3-metric h ab and a symmetric trace-free tensor E ab satisfying (35c), which itself dependson (36a)-(36c). This geometrical initial data is supplemented with solutions to the conformalconstraint equations for the matter variables. Overall, we then obtain initial data for the Cauchyproblem at null infinity. Using Theorem 3 and Lemma 5 we can thus develop a large class ofweakly asymptotically flat spacetimes with a conformal scalar field coupled to Yang-Mills fieldsand radiation fluids.In particular we find that Corollary 4.
There exists a class of weakly asyptotically flat spacetimes ( ˆ
M , ˆ g ) for which theunphysical conformal scalar field vanishes at null infinity, i.e. ˆ θ ∗ = 0 and ˆΦ ∗ = 0 . We start from initial data at null infinity which describes a conformal scalar field coupled to re-pulsive Einstein-Maxwell-Yang-Mills and a repulsive radiation fluid. Using Theorem 3 we obtainthe corresponding physical solution ( ˆ
M , ˆ g ). By Corollary 1 the dual solution ( ˇ M , ˇ g ) describes aconformal scalar field coupled to attractive Einstein-Maxwell-Yang-Mills and an attractive radi-ation fluid. Applying Lemma 4 we find that ( ˇ M , ˇ g ) has an isotropic singularity. Since the Weyltensor vanishes at the conformal infinity of ( ˆ M , ˆ g ) it must vanish at the isotropic singularity of( ˇ M , ˇ g ). We thus have the following result. Theorem 4.
Let Σ be a spacelike hypersurface. Suppose on Σ we are given initial data at nullinfinity for the CEFE of a conformal scalar field coupled to repulsive Einstein-Maxwell-Yang-Millsand a repulsive radiation fluid. Then there exists a solution ( ˇ M , ˇ g ) to the Einstein field equationswith conformal scalar field coupled to Einstein-Maxwell-Yang-Mills and a radiation fluid thatsatisfies the strong Weyl curvature hypothesis. In other words, ( ˇ M , ˇ g ) has an isotropic singularityat which the Weyl curvature vanishes. The rescaled Weyl tensor need not vanish at Σ and hence the Weyl tensor will be non-zeroaway from I . Hence the above spacetimes provide a new class of spacetimes which satisfy thestrong Weyl curvature hypothesis but are not conformally flat. Therefore Theorem 4 extends theresults in [2]. Moreover, it shows that despite the apparent fine tuning required by the strongWeyl curvature hypothesis there exists a large class of spacetimes satisfying the hypothesis.The work in [1], [2], [29] showed the existence of spacetimes with isotropic singularities satis-fying the weak Weyl curvature hypothesis. Suppose we start with initial data for the CEFE of aconformal scalar field coupled to repulsive Einstein-Maxwell-Yang-Mills and a repulsive radiationfluid on a regular hypersurface of ( ˆ M , ˆ g ), i.e. ˆ θ = 0. In a generic spacetime the Weyl tensor willbe non-zero away from I , so that it is finite and non-vanishing on Σ. Combining Corollary 3with Lemma 1 we get Corollary 5.
There exist solutions ( ˇ
M , ˇ g ) to the Einstein field equations with conformal scalarfield coupled Yang-Mills fields and radiation fluids that have an isotropic singularity at whichthe Weyl curvature does not vanish identically and that hence satisfy the weak Weyl curvaturehypothesis. In [10], [16] and [17] the stability of Einstein-Maxwell-Yang-Mills and of radiation fluid spacetimeswere studied. In particular the stability of de Sitter space in the class of Einstein-Maxwell-Yang-Mills spacetimes and the stability of FLRW in the class of radiation fluids were proven. For theconformal scalar field this aspect of stability was briefly addressed in Theorem 4 of [12]. Since theCEFE can be written as a regular first order symmetric hyperbolic system on can apply theorems13y Kato [13] to prove stability. Thus given a regular (reference or background) solution of theCEFE, one can prove that it is stable under small perturbations.We observe that for conformal scalar fields and Einstein-Maxwell-Yang-Mills we can considervacuum or electro-vacuum spacetimes as a background spacetime. In particular, we can usethe radiative electro-vacuum spacetimes (ˆ λ = 0) as reference solutions for the CEFE and con-sider small perturbations in the family of spacetimes containing conformal scalar fields minimallycoupled to Einstein-Maxwell-Yang-Mills fields. Note that we cannot include perturbations byradiation fluids for these spacetimes since the formulation of the CEFE in [17] require that theunphysical fluid density never vanishes. However we can set FLRW with ˆ λ < k = 1 as ourreference time (see [17]) and study small perturbations with respect to all three matter models.We can thus prove that Theorem 5. a) The region near future null infinity of radiative electromagnetic spacetimes is stable againstsmall perturbations within the class of spacetimes containing conformal scalar fields minimallycoupled to Einstein-Maxwell-Yang-Mills fields.b) The region near future null infinity of de Sitter space is stable against small perturbationswithin the class of spacetimes containing conformal scalar fields minimally coupled to Einstein-Maxwell-Yang-Mills fields.c) The region near future null infinity of the radiation fluid FLRW spacetime with ˆ λ < and k = 1 is stable against small perturbations within the class of spacetimes containing conformalscalar fields minimally coupled to Einstein-Maxwell-Yang-Mills fields and radiation fluids. For the classes of spacetimes in part b) and c) one can consider perturbations of the initialvalues at null infinity. Since the conformal scalar field ˆΦ in the reference solutions vanishes atnull infinity, we can consider the subclass of perturbations for which the initial data at conformalinfinity satisfies ˆΦ ∗ = 0 and Ξ ∗ = ( ∇ n ˆΦ) | Σ = 0. Every spacetime generated by this subclassof initial data is weakly asymptotically flat and has a dual solution which contains an isotropicsingularity. Note that in the reference spacetimes chosen in b) and c) are not part of the abovesubclass since both reference sapcetimes satisfy ˆΦ = 0 everywhere.Remark: All the reference spacetimes used above have a vanishing conformal scalar field.While explicit solutions with conformal scalar field exist, they typically consider either ˆ λ = 0or α = 0. However, it should be possible to generalise some of these solutions under suitablesymmetry assumptions. The CCC-scenario proposes that future conformal infinity of ( ˆ
M , ˆ g ) and the past isotropic sin-gularity of ( ˇ M , ˇ g ) are identified at the spacelike hypersurface Σ. This requires ˆ λ < I + (see remarks above). The results in the previoussections allow us to construct explicit solutions to this scenario.Recall from our discussion above that the solution of the CEFE with initial data at null infinitygives rise one unphysical solution ( M, g ). From (
M, g ) we obtain two separate physical solutions,one to the past and one to the future of Σ. Denote the solution to the past ( ˆ
M , ˆ g ). By Lemma4 the solution to the future has itself a dual solution that contains an isotropic singularity withvanishing Weyl curvature. Denote this dual solution ( ˇ M , ˇ g ). Then the two spacetimes ( ˆ M , ˆ g )and ( ˇ M , ˇ g ) are joint at the surface Σ, which in ( ˆ M , ˆ g ) represents future null infinity and for( ˇ M , ˇ g ) represents the isotropic singularity (Figure 2). The unphysical spacetime ( M, g ) playsthe role of the bridging spacetime. The spacetimes ( ˆ
M , ˆ g ) and ( ˇ M , ˇ g ) satisfy the criteria of theCCC-scenario.In order to guarantee that the next aeon ends in a spacelike future null infinity, we need tomake sure that the coefficient α of the quartic self-interaction term in this aeon, which becomesthe cosmological constant of the next aeon, is negative.14 ˇ M , ˇ g )Σ = IS ( ˆ M , ˆ g )Σ = I + future aeonpast aeon( M, g ) duality transformationˆ g → ˇ g = ˆ φ ˆ g Figure 2:
Proposal for CCC-scenario:
The past aeon is represented by the weakly asymp-totically flat spacetime ( ˆ
M , ˆ g ) for which Σ represents future null infinity. The future aeon isrepresented by the spacetime ( ˇ M , ˇ g ) with its past isotropic singularity. The transition betweenthe two aeons is implemented by rescaling by ˆ φ (duality transformation) at Σ (big bang). Theshading indicates the strength of the (physical) energy-momentum tensor. We should first of all highlight that this proposal is close in its nature to observations made in [30],[31]. In [30] a single radiation fluid in ( ˆ
M , ˆ g ) leads to a radiation fluid coupled to a conformalscalar field in ( ˇ M , ˇ g ). However, the conformal factor is known only implicitly through waveequations. In contrast, our proposal uses a conformal scalar field in both ( ˆ M , ˆ g ) and ( ˇ M , ˇ g ) andthe duality provides a mathematical mechanism for fixing the conformal factor relating the twophysical spacetimes, irrespective of the chosen bridging metric. As seen above this has allowedfor the generation of a large class of spacetimes satisfying the CCC-scenario.If we are dealing with a single conformal scalar field as our matter then we get a satisfactorytransition between ( ˆ M , ˆ g ) and ( ˇ M , ˇ g ) since both spacetimes are described by the same type( ǫ = ±
1) of conformal scalar field. However, if we couple the conformal scalar field with ǫ = 1to Einstein-Maxwell-Yang-Mills and a radiation fluid then one of the two spacetimes containsstandard attractive matter fields while the other sapcetime contains repulsive matter fields. Thisdoes not seem satisfactory in light of the fact that such repulsive matter is considered unphysical.However, one could argue that our own observations of attractive matter fields satisfying thedominant energy condition only give us information about our current aeon. We cannot inferthat the same should hold in the previous or the subsequent aeon (assuming they exist). It isunclear whether the aeon with repulsive matter develops any pecularities or whether gravitationalclumping could be largely overcome by the repulsive matter and the presence of a de Sitter-likecosmological constant, leading to an expanding and potentially fairly homogeneous spacetimewhich is weakly asymptotically flat to the future.If one considers the case ǫ = − M , ˆ g ) and ( ˇ M , ˇ g ) contain standard attractiveEinstein-Maxwell-Yang-Mills fields, radiation fluids and possibly additional attractive conformalscalar fields. Hence the matter contents of subsequent aeons satisfies the same physical properties.The second advantage of ǫ = − φ ) never vanishes. What is unclear for this scenario is whether there existsany physical process which is described by (1) with ǫ = − φ = ± for ǫ = 1Since ˆ φ → − ˆ φ preserves (1), we only need to consider ˆ φ = 1. The factor (1 − ˆ φ ) − diverges inthat case. We observe ˆ φ = 1 = ⇒ ˇ φ = 1 so that the dual solution faces the same problem inthe same location. Some authors refer to this behaviour as a singularity. This viewpoint may bejustified when talking about the CEFE degenerating as a PDE system. For this PDE problem one15as to consider other methods for determining whether a solution can be extended past ˆ φ = 1.However from a spacetime point of view, ˆ φ = 1 does not imply a curvature singularity. The factorin square brackets in (5) may vanish as well so that the quotient can give a finite limit. If themetric is sufficiently regular then its curvature is bounded, in particular the Einstein tensor is.However this implies in turn a finite energy momentum tensor. The Reissner Nordstr¨om blackhole is an illustrative example for this. ˆ φ = 1 at R = m , which is a regular location in the exteriorregion where the curvature and energy momentum tensor are bounded. A natural question to ask is whether the above stability result for the weakly asymptotically flatend leads to a stability result for the dual spacetimes with isotropic singularities. We argue forcaution in this case. Firstly, the reference spacetimes themselves have no dual solution (sinceˆ φ = 0 everywhere) against which to make a comparison. Secondly, the stability proof tells usthat the unphysical spacetimes ( M, g ) are close. So are the weakly asymptotically flat solutions( ˆ
M , ˆ g ) since ˆ θ << M , ˇ g ) small unphysical perturbations are increasinglymagnified in the physical quantities closer to the singularity, since ˇ θ >> θ vanishare close to each other, but they need not coincide. Thus the null infinity of ( ˆ M , ˆ g ) and theisotropic singularity of ( ˇ M , ˇ g ) may no longer coincide. Note that Lemma 1 still assures existenceof the singularity in ( ˇ M , ˇ g ) as long as ˆ T ij = 0. From this point of view our proposal for theCCC-scenario using the duality of the conformal scalar fields appears to require some form offine tuning of initial data away from null infinity to achieve the identification of null infinity andisotropic singularity.Suppose we drop the requirement of the coincidence of null infinity and the isotropic singularityin the bridging spacetime ( M, g ). In this case the vanishing of ˆ φ , on some surface S say, stilltriggers the switch to the dual spacetime, which once more contains an isotropic singularity(Lemma 1). Note that in ( M, g ) the past aeon and the next aeon coexist for some conformal timeuntil the past aeon has reached null infinity. It is unclear how close S should be to null infinity(with respect to the bridging metric g ) to assure that all black holes have already evaporated ?Or what would happen in ( ˇ M , ˇ g ) if S contained trapped surfaces in ( ˆ M , ˆ g )? Recall that theconformal scalar field may violate the dominant energy condition, so that singularity theoremsmay not be applied. We will not pursue these kind of questions here.The focus of this article has been to identify an explicit mathematical formulation that candescribe how the new aeon in the CCC-scenario can be generated from the current aeon. In ourproposal the conformal factor relating these two aeons is determined by the physical conformalfield ˆ φ . It is thus independent of the choice of bridging metric or the reciprocal hypothesis.Indepentently of the discussion of the CCC-scenario, we have extended the work of [3] toinclude cosmological constants and quartic self-interaction terms. As a result we observed thatunder Bekenstein’s duality the cosmological constant and the coefficient of the quartic switch roles.We have shown the existence, uniqueness and stability of weakly asymptotically flat spacetimescontaining a conformal scalar field, Yang-Mills fields and radiation fluids (Theorem 3). Moreover,we have proven the existence of a large class of spacetimes with the above combination of mattermodels that contain an isotropic singularity modelling the big bang. As highlighted above thematter contents can be generalised to other trace-free matter models if suitable formulations ofthe CEFE can be obtained. Acknowledgements
The author would like to thank UCL for Visiting Research Fellowship and JSPS for a Re-search Fellowship. Further, the author would like to acknowledge financial support by the grant Note that [7] argues that black holes will not pop (completely evaporate).
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