whereQ(W˜)is a term quadric in ˜W andL(W˜)a term linear in ˜W6. In Lorentzian signature the above system provides a hyperbolic system of equations, which is satisfied by the Weyl tensor ˜W.
A special property in4dimensions is the equivalence of the Bianchi equationB(∇W) =0 and the contracted Bianchi equation div W= 07. The latter is equivalent to divw =0 by Equation (1.111) and that in principle is why imposing divw=0 in the conformal field equations suffices to get an hyperbolic system. The advantage of using divw= 0 in place ofB(∇˜W˜) =0 is that it is regular whereσvanishes. Using a spin frame formalism, H. Friedrich was able to split the conformal field equations into a system of symmetric hyperbolic evolution equations and a set of constraint equations [Fri81a, Fri81b, Fri82, Fri83]. He pointed out that in higher dimension usage of the full Bianchi equations or a conformal analogue may be necessary to get hyperbolic regular PDEs [Fri02]. An ansatz for using the full Bianchi equations will be presented in the next section.
4.3 conformal wave equations
Related to the conformal field equations there have been several results concerning uniqueness and existence issues. Some have been mentioned in the introduction of this thesis. A particular interest is the Cauchy problem with initial data given on a characteristic cone that also acts as conformal infinity. A recent result is the construction of a system of quasilinear wave equations, which corresponds to the conformal field equations [Pae13]. Existence of solutions can be proven if the characteristic initial data at conformal infinity satisfy some smoothness conditions [CP13].
This section will summarise the construction and will provide an ansatz to generalise it to higher even dimensions.
The conformal field equations written in the form
(ii) ∇dσ = −σP−ρg
(iii) ∇ρ = gradgσyP
(iv) R∇(X,Y)Z = σn−3w(X,Y)Z+ (P?g)(X,Y,Z,·)]g (vi) (∇P) (X,Y,Z)−(∇P) (Y,X,Z) = −σn−4dσ(w(X,Y)Z)
(vii) (divgw)(Z,X,Y) = 0.
are the starting point of the construction. The first of the conformal field equations has been used to remove the unknownςin the remaining equations. A second ingredient of the construction is usage of a generalised wave-map gauge (compare section4.1). Considers two metricsgand ˆg onMwith Levi-Civita connections∇=∇gand D=∇gˆ. Thegeneralised wave-gauge vectorthen is defined as
H=trgM −V,
whereM=∇ −Dis the potential andVis a vector field that may explicitly depend onx∈ M and the unknownsu. The explicit set of unknowns that is denoted byudepends on the system under consideration. The reduced Ricci tensor in a generalised wave-map gauge is defined as
Ric(H)(X,Y):=Ric(X,Y)−1
2(g(X,DYH) +g(Y,DXH)). (4.20)
6 an explicit formula is given for example in [Fri02]
7 To see this property one considers the(2, 2)-Weyl tensor as mapMW : Γ(Λ2T∗M) → Γ(Λ2T∗M)on antisymmetric tensor fields. In four dimensions the Weyl tensor decomposes into a self-dual part W+and an anti-self-dual part W−, with∗W±=±W±. Now letd∇:Γ ΛpT∗M⊗Λ2T∗M
→Γ Λp+1T∗M⊗Λ2T∗M
be the exterior differential associated to the Levi-Civita connection ofg[Bes08, (1.12)]. Then locally one has
B(∇W) =d∇W=− ∗ ◦δ∇◦ ∗W=− ∗δ∇W+−δ∇W− ,
whereδ∇is the divergence. Linear independence ofδ∇W+ andδ∇W+ bijectivity of the Hodge dual then gives the equivalence.
80 Chapter 4: Characteristic Cauchy Problem
As mentioned beforeVwill be allowed to depend on gbut not on its higher derivatives. Hence the second term still removes second-order derivatives ofg, which would prevent the principal part of Ric from being a wave-like equation ong. The operator
(H):=∆∇+Ric]−DH−(n−2)P]−Jid (4.21) so still is a reduced wave operator (compare definition4.1.12 for the notation) and may entail a hyperbolic equation even if g is part of the unknowns. J is considered to be an arbitrary map, which locally can be prescribed to the system and will later coincide with the trace of P.
Important to the following consideration is the observation that the modification of the Laplacian on the right-hand side can be rewritten to
Ric(X,Y)−g(DXH,Y)−(n−2)P(X,Y)−Jg(X,Y) = Ric(H)(X,Y) +1
2(g(X,DYH)−g(Y,DXH))−(n−2)P(X,Y)−Jg(X,Y). (4.22) It is important due to the following observation. Let g and P be solutions to an arbitrary set of PDEs and assume that for that solution g is a Lorentzian metric and P its Schouten tensor.
Assume furtherg to have vanishing wave gauge vector with respect toD. Then(H)coincides with∆∇and solutions to equations involving(H) will also be solutions to the same equations, where the reduced operator is replaced by the Laplacian∆∇. Let for exampleTbe a tensor field that is a solution to an equation of type(H)T =. . . , where(H)is an operator inD. Then the previous assumptions assure thatTalso is a solution to∆∇. Moreover in place of considering a equation that correspond to Ric[g]is suffice to impose an equation on Ric(H)[g]and to assure that a solutionghas vanishing wave-gauge vector. This is quite important, since equations involving Ric(H)are better behaved in terms of D-derivatives ofg.
Consider the unknownu = (g, P,w,σ,ρ). By differentiating the conformal field equations, a system of Laplace-type equations can be derived [Pae13]
∆∇(P,w,σ,ρ) =F(x,u,∇u)
Ric= (n−2)P+Jg. (4.23)
This system then is replaced by the reduced system
(H)(P,w,σ,ρ) =F(x,u,∇u)
Ric(H)[g] = (n−2)P+Jg. (4.24) The latter system is calledconformal wave equations. The name is a reference to the fact that the reduced Ricci Ric(H)[g] and (H) have leading order term trgDDg = ∆g,D. Solutions to the conformal field equations with vanishing wave-gauge vector H will by constructions also be solutions to the conformal wave equations. A first result then is that provided a set of conditions is fulfilled by the initial data, then solutions to the conformal wave equations, implying the initial data on the initial characteristic set, will also be solutions to the conformal field equations [Pae13, Theorem3.7]. Assuming the initial set is a null cone with respect to the metricgto be constructed and the zero-set ofσ, the second goal is to provide a set of constraint equations to the initial data in a certain wave-map gauge, which guarantees that solutions to the conformal wave equations are also solutions to the conformal field equations and vice versa ([Pae13, Theorem5.1]). An important tool to derive the constraint equations is the usage of adapted null coordinates that provide a way to derive a hierarchical system of algebraic and ordinary differential equations along null rays originating from the vertex of the cone.
The conformal field equations eliminate the degeneracy of the almost Einstein equations by imposing new unknowns and new equations to the system. In particular the unknownw of a solution is the rescaled Weyl tensor. Consequently solutions to the conformal field equations will have vanishing Weyl tensor on the conformal boundary and moreover its asymptotic be-haviour has to be such thatσ3−nW has a sufficiently smooth extension to the boundary. This
4.3 conformal wave equations 81
requirements are quite strong restrictions to the set of possible solutions to the almost Einstein equations.
A second type of wave equations can be derived, where the requirement of vanishing Weyl curvature is weakened. To keep the equivalence between solutions to the wave equations and solutions to the conformal field equations, one needs stronger restrictions to the initial data.
The ansatz is to replace the unknowns (g, P,w,σ,ρ) by a new set of unknowns(g, P, C, W,σ,ρ) and to impose a set of Laplace-type equations on them. W is considered to have the algebraic properties of the Weyl tensor and C is considered to have the algebraic properties of the Cotton tensor. Construction of the system of wave-equations is sketched in the following.
Assumegis a metric,∇its Levi-Civita connection, P the Schouten tensor, C the Cotton tensor and W the Weyl tensor. Then by Equation (1.52),(g, P, C, W)satisfies the equation
∆∇W=−(∆∇P+HessgJ)?g+F[W, C,∇C, P,g], (4.25) where F[W, C,∇C, P]is a rational algebraic term depending only on W, C, ∇C and P. For the Cotton tensor Equation (1.53) provides a similar condition
(∆∇C)(X,Y,Z) = ∇∆∇P+HessgJ
(Y,Z,X)
−∇∆∇P+HessgJ
(Z,Y,X) +F[W, C, P,∇P,g]. ( 4.26) The dependence on J and its derivatives is partially suppressed on the right-hand side. An equation involving the Laplace of the Schouten tensor then in even dimensionnis provided by the obstruction tensor as for example given in [GH05]8
O[g] =∆∇n2−2
∆∇P+HessgJ
+Fn−1, (4.27)
The term F refers to derivatives of the metric of order less or equal ton−1. Vanishing of the obstruction tensor gives an equation that necessarily has to be fulfilled to get an almost Einstein manifold.
Now the remaining Laplace-type equations for σ, ρ and the metric g are provided by the demand that(M,g,σ)has to be an almost Einstein structure. The definition givesnρ=∆σ−Jσ and hence an equation toσ. Equation (1.122) provides an equation toρand Ric[g] = (n−2)P+ Jgcorresponds to an equation ong. There are two problems remaining, if one is interested in hyperbolic equations on the unknowns (g, P, C, W,σ,ρ). The first is that the equations on the Cotton and Weyl tensors do involve second- and third-order derivatives of the Schouten tensor on the right-hand side and the Obstruction tensor gives only a Laplace-type equation on the Schouten tensor in 4 dimensions. The second challenge is that ifg is considered to be part of the unknowns, even the Laplace operator with respect to∇on the left-hand side is not diagonal in its leading order term. The second problem can be solved by replacing the Laplace operator by the reduced d’Alembertian introduced in Equation (4.21) and by using the reduced Ricci given in Equation (4.20). The emerging Laplacian∆g,Dwith respect to some connectionDthen is diagonal in its leading order term. Part of this method is to impose vanishing of the wave-gauge vectorHon the initial data. It then has to be shown that vanishing of His propagated to full solution, i.e. that solutions to the reduced system with such initial data automatically have vanishing wave-gauge vector.
The first problem only appears in even dimensionn>4. In 4dimensions, the Bach tensor is the obstruction tensor and for example given in Equation (1.45). Its vanishing implies that
∆∇P+HessgJ=F[P,g]
holds for almost Einstein structures, whereF[P,g]only depends on the Schouten tensor and the metric. At first sight, this gives an equation to the Schouten tensor. To get rid of the Hessian of
8 If compared to the first appearance of the obstruction tensor in Equation (3.5), this time the dimension ofMisn. This explains the different exponents.
82 Chapter 4: Characteristic Cauchy Problem
J, one uses that at least locally Jmay be prescribed to an almost Einstein structure by exploiting conformal covariance. So by fixingJ, HessgJ just becomes an inhomogenity on the right-hand side, which will coincide with the trace of P for solutions to the final system. The Laplacian on the left-hand side then is replaced by the reduced d’Alembertian. Moreover this equation provides a method to substitute the second- and third-order derivatives on the right-hand side of the equations for W and C by at most first-order derivatives ofgand P. As a matter of fact, the new derivatives of P can be replaced by C due to the antisymmetrisation that is involved [Pae13]. Finally inn=4 dimensions this gives a reduced system
(H)(P, C, W,σ,ρ) =F[x,u,Du]
Ric(H)= (n−2)P+Jg, (4.28) where the set of unknowns isu= (g, P, C, W,σ,ρ). The system is referred to asalternative system of conformal wave equations. [Pae13, Theorem6.4] then states that provided the initial data fulfil a set of constraint equations on the initial surface9, then solutions to the reduced system will also solve the system
∆∇(P, C, W,σ,ρ) =F[x,u,∇u]
Ric[g] = (n−2)P+Jg. (4.29) Moreover it is shown that this theorem also applies to initial data on a characteristic cone, which by construction is the zero set ofσ.
An interesting question is to what extent this method can be generalised to arbitrary dimen-sion. A method may be implied in even dimensions, where in4 dimensions the Bach tensor is used to get rid of higher-order derivatives of P. In higher even dimension the obstruction tensor has to be used to get a set of equations of Laplace type. This evidently involves powers of the Laplace operator. In particular vanishing of the obstruction tensor gives an equation for∆n2−1P with∆=∆∇for the moment. Letn≥6 be even. To get rid of the terms involving(∆P+HessJ) on the right-hand side of the equations for Cotton and Weyl tensors, one will have to consider higher powers of the Laplacian. In case of Equation (4.25) this reads as
∆n2−1W=−∆n2−2(∆P+HessgJ)?g+F[∇n−4W,∇n−3C,∇n−4P,g], (4.30) where ∇k just denotes the highest derivative of the specific tensor. The situation is little more complicated in case of the equation for the Cotton tensor, since one has to commute the operator with the covariant derivative on the right-hand side. The initial situation is the equation
∆n2−1C
(X,Y,Z) = ∆n2−2(∇(∆P+HessJ))(Y,Z,X)
−∆n2−2(∇(∆P+HessJ))(Z,Y,X) +F[∇n−4W,∇n−4C,∇n−3P,g].
(4.31)
Using (1.55) and Lemma1.1.14 for commuting the Laplacian with the covariant derivative on tensors, the remaining terms will involve only derivatives of P, C and W of highest ordern−4 and hence one is left with an equation of type
∆n2−1C
(X,Y,Z) = ∇∆n2−2∆∇P+HessJ
(Y,Z,X)
−∇∆n2−2
∆∇P+HessJ
(Z,Y,X) +F[∇n−4W,∇n−4C,∇n−3P,g].
(4.32)
Now terms involving∆n2−1P on the right-hand side of Equations (4.30) and (4.32) can be elimi-nated by demanding a vanishing obstruction tensor on almost Einstein manifolds. Vanishing of the obstruction tensor itself gives an equation to∆n2−1P with lower order derivatives of Schouten, Cotton and Weyl tensors on the right-hand side.
9 For example the wave gauge vector, its covariant derivative and 2σρ+kdσk2g(=S[g,σ])have to vanish.
4.3 conformal wave equations 83
The conjecture is that by introducing powers ∆kP, ∆kC and ∆kW as new variables to the system, it can be rewritten as a system of wave-like equations as it has been done in case of a system for the metric and the Anderson-Fefferman-Graham equations in [AC05]. A further analysis of this conjecture is beyond the immediate intention of this thesis and left to a future investigation.
5 A L M O S T E I N S T E I N S T R U C T U R E S W I T H V A N I S H I N G A L M O S T S C A L A R C U R V A T U R E
The main focus of this thesis is on almost Einstein structures in Lorentzian signature(−+· · ·+) with vanishing almost scalar curvature. Throughout this chapter(M,g,σ)will be such a struc-ture with S[g,σ] = 0. If not specified differently, ∇ = ∇g will be the Levi-Civita connection with respect to g. The chapter will start with a short analysis of the differentiability of the defining functionσ. Next, the structure of its zero set Σis considered. The critical points of σ along the zero set will be identified as focal points and vanishing points of the Weyl tensor. The analysis ofΣis completed by a section about the asymptotic behaviour ofσ−2g-geodesics near the conformal boundary Σ. This implies an approach to regain similar results on conformally compactified spacetimes in the context of almost Einstein structures. Finally, a set of special coordinates is constructed, which is adapted to the local topology of the singularity setΣ. The coordinates represent a compromise between Morse coordinates, in which the mapσ has a par-ticularly simple form, and geodesic coordinates, in which radial geodesics have a most simple form. They in particular combine the properties of keeping the simple form ofσin Morse coor-dinates and at the same time ensure a simple form of radial null geodesics. This compromise on the other hand is accompanied by a loss of differentiability at the origin and the loss of a generic affine parametrisation of radial null geodesics. The simple characterisation of radial timelike and spacelike geodesics is also lost. Finally, the form of an almost Einstein metric will be calculated in such coordinates.