5.3 special coordinates
5.3.4. Coordinates Prescribing the Conformal Factor and Null Pregeodesics Originating at Vertices
The results of the last section will now be used to construct special coordinates for a neigh-bourhood of vertices ofΣ. In those coordinates, null geodesics originating from the vertex are mapped to straight lines, while at the same timeσis a homogeneous polynomial of degree2. In particular it has the same form that is provided by the Morse lemma for a neighbourhood of the vertex.
withs=±1. Without loss of generality we assume the neighbourhoodUto be a normal neigh-bourhood. By Lemma5.1.3, the framen
∂i:= ∂
∂ϕi
o
is orthogonal at the vertex. The base vectors
128 Chapter 5: almost einstein structures with vanishing almost scalar curvature
do not have to be of length 1 or−1. Nevertheless the frame provides a natural identification I :Rn 3 Y 7→∑iYi∂iTpM. We denote the geodesic coordinates that are defined by the inverse smooth coordinate transformation, which maps Morse to geodesic coordinates in p. Then by construction ζ is cone-preserving, ζ(0) = ϕ˜◦ϕ−1(0) = 0 and for canonic base vectors eµ ∈ Rn we have dζ0(eµ) = dϕ˜p◦ dϕ−1
0(eµ) = dϕ˜p (∂µ)p = dϕ˜p (∂˜µ)p = eµ. In partic-ular dζ0 = id and so ζ is a cone-preserving diffeomorphism close to the identity. By Propo-sition5.3.37, at least locally there exists a special C1-smooth cone-preserving diffeomorphism F = F0,F
: ϕ(U) ⊃ U → R×Rn−1, close to the identity, which extends kζ(y)kkyk ζ(y) away from the cone C∩ U in the suitable way described before. We now define new coordinates ϑ:Up=ϕ−1(U)→Rnby
The coordinate mapϑinherits the smoothness ofF, since ϕis smooth. This gives the first point.
The second point is a consequence of F being close to the identity, i.e. dF0 = id and hence
Hence γ has to be a null pregeodesic originating in p ∈ M. For the last point we use that F preserves cylinders and{t} ×Rn−1-hypersurfaces (Proposition5.3.37(ii)) and get
−ϑ0(y)2+ϑ1(y)2+· · ·+ϑn−1(y)2= −F0◦ϕ(y)2+kF◦ϕ(y)k2n−1 (5.25)
= −ϕ0(y)2+ϕ1(y), . . . ,ϕn−1(y)2 (5.26)
=σ(y). (5.27)
We will now summarise the substance of the last sections in a theorem.
5.3 special coordinates 129
Theorem 5.3.38. Let p ∈ Σd be a vertex of Σ, then for a neighbourhood U of p there exists special coordinatesϑ:U→Rnsuch thatσhas the form
with s = ±1. These coordinates have the smoothness ofσ except in p where they are of class C1. In addition a curve of the form
γ(t) =ϑ−1(tV) is a null pregeodesic for all(06=V)∈RnwithkVk21,n−1=0.
Coordinates Adapted to the Null Cone
The potential advantage of the coordinates in the last theorem is a reduction in the number of variables in the partial differential equation arising from the Cauchy problem defined by the almost Einstein equations. The defining function is prescribed in these coordinates and only the metric and its derivatives are considered to be unknowns to the system. Also the null direction along the cone is provided by straight lines in these coordinates.
The method of introducing special coordinates is used in general relativity to handle parts of the characteristic Cauchy problem for the vacuum field equations and to handle the conformal wave equations with initial data at a characteristic cone [Ren90, CBCMG11b, CP13]. We will par-tially review the method here and point out the modifications that have to be made to prescribe the conformal factor in such coordinates. A basic idea is to start with geodesic coordinatesϕat the vertex p of the characteristic cone. If those coordinates are based on an orthogonal frame, this guarantees that the geodesic null cone inpis mapped to the Minkowskian null cone inRn. On the other hand, geodesics originating at the vertex are affinely parametrised in the sense that γ(t) = ϕ−1(tV) for someV ∈ TpM. We will drop that last requirement in order to have the this wayΣlocally is given by the equation for the Minkowskian null cone. One can now define coordinatesx:U ⊃U→R×R+0 ×Rn−2for a subset of the local future-directed causal cone in Sn−2 → Rn−2 are local spherical coordinates. The coordinates x are singular along the line x1=0 and in particular at the vertex p. Null geodesics originating in phavex0=0,xA=const.
and are parametrised by x1. In contrast to the construction in [CBCMG11b], where the latter coordinate component is an affine parameter, this is not the case here, as we did not start with geodesic coordinates. Nevertheless the coordinate derivative∂x1 is a null vector tangent to null geodesics originating inp. Moreover along the null coneCp(U)it is perpendicular to the tangent vectors∂xA forA >1, since{x1, . . . ,xn−1}parametrisesCp(U)⊃ x0−1
(0). And hence along the cone one hasg11 = g1A = 0 forA ∈ {2, . . . ,n−1}. The cone itself and in particularΣare characterised by the equationx0 =0. Using the notation of the latter paper, along the cone the metric then is of type
130 Chapter 5: almost einstein structures with vanishing almost scalar curvature
whereν0=g01,ν:M→Rn−1is defined byνA :=g0Afor A≥2 and ˜gare the componentsgAB with A,B≥2. The inverse matrix, denoted bygµνand computed along the cone is
gµν=|Cp(U) The notation is partially motivated by the fact that along the cone ˜g−1is the inverse matrix of ˜g, ν0= 1
ν0 andν−1=g˜−1·ν. The added benefit of the coordinates that have just been constructed is the form of the conformal factor and its derivatives
σ=−x02
For further calculations it is helpful to evaluate Γ0 along the cone. Using Einstein notation, it reduces to the sumg01 ∂igj1+∂jgi1−∂1gij
gij. The componentsg1k with k ≥ 1 vanish along the cone. Hence tangent derivatives∂lg1kwithk,l≥1 will also vanish. This gives
Γ0=|Cp(U) 1
5.3 special coordinates 131
As we started with an almost Einstein structure(M,g,σ), in the coordinatesxthe almost Einstein equation along the null cone (Hessgσ+ρg=0) reads
Prescribing the metricg on the cone clearly fixes the tangent derivatives L∂
xig along the cone fori≥ 1. In addition a couple of transversal derivatives∂x0gij are fixed by the almost Einstein equation and the components of ˜g have to fulfil some constraint equations. First consider the termdx0⊗dx1. The almost Einstein equation implies
0 =|Cp(U) 1− x
On the one hand, provided ˜g is prescribed on the cone, this fixes the transversal derivative of g11on the cone. On the other hand, substituting this equation to the almost Einstein equation in adapted coordinates gives
The coefficients of the terms dx1⊗dxi for i ≥ 1 vanish, since g1i is constant along the cone and so all its tangent derivatives will vanish. The remaining equations then give an ordinary differential equation to ˜g that has to be fulfilled for almost Einstein metrics on the cone in the specific coordinates withG(x˜)positive definite and symmetric. In fact all solutions to the differential-algebraic Equa-tion (5.28) are of this form (see appendix D).
Considering the asymptotic behaviour as x1 = r goes to zero gives further restrictions to the form of ˜g. The coordinatesϑ appeared as a transformation of Morse coordinates ϕin the
132 Chapter 5: almost einstein structures with vanishing almost scalar curvature
preliminaries of Theorem 5.3.38. Conversely one may also write them as transformation of geodesic coordinates by observing
ϑ=F◦ζ−1◦ϕ˜ =: ˜F◦ϕ,˜
which then defines the coordinate transformation ˜F. Both transformations F and ζ are cone-preserving diffeomorphisms close to the identity. In particular we will need the properties F(y) = y+o(kxk) and ζ(x) = x+o(kxk) such that we also have ˜F(x) = x+o(kxk) or dF˜(x) = id+O(kxk) for the Jacobian. We denote with gϑij and gϕij˜ the metrics in different co-ordinates. Thengϑ = dF˜−1t
·gϕ˜·dF˜−1. A property of geodesic coordinates is the asymptotic behaviour gijϕ˜(x) = κηij+O(kxk2) with the Minkowski form η and a constant κ ∈ R, as we have not started with normalised coordinates. By using the expansion fordF˜ this also provides gϑij(x) =κηij+O(kxk2). Following [CBCMG11b, section4.5.] this gives rise to the asymptotic
˜
g(x) =|Cp(U)κ x12
ΩSn−2(x) +O x12
in coordinates x, where ΩSn−2 is the round met-ric in the sphemet-rical coordinatesxA. The decomposition ˜g(x) =|Cp(U)f(x)·G(x2, . . . ,xn−1)then requiresG=ΩSn−2 and f(x) =κ x12
1+O x12 .
Finally, we obtain a reduced form for the metric in coordinatesxadapted to the null cone
gµν=|Cp(U)
g00 ν0 ν ν0 0 0 νt 0 fΩSn−2
, (5.29)
whereg00−κ,ν0+κ,νand f are at least of orderO x12
(see [CBCMG11b, section4.5.] for the first components).
6 O U T L O O K
We will now summarise open problems that emerged while writing the thesis.
A first challenge is a more detailed attempt to generalise the conformal wave equations to higher even dimensions as it is only sketched in section4.3. By introducing powers∆kP,∆kC,
∆kW and their derivatives as new variables to the system, we would like to have a more explicit expression of the obstruction tensor in terms of the new set of unknowns. The next step would be to provide initial data on the characteristic cone for that set of unknowns.
The adapted coordinates constructed in the latter section may provide a tool for the analysis of such an initial data problem. But at the moment it is only a tool that lacks an application, since the author is not an expert in partial differential equations. The matter of whether the advantage of eliminating the conformal factor that is gained by the disadvantage of loosing smoothness at the vertex really is of some use will have to be examined in future works.
On the way to the construction of adapted coordinates we got a lifting result for diffeomor-phisms on the sphere and on the compact cylinder that are close to the identity map (Proposition 5.3.21and5.3.31). The problem may be generalised in the following way. Let Mbe a compact manifold with a Lie groupGacting on it. Then consider a localCm-diffeomorphism f :M→M, which admits a mapg: M→gsuch that f(x) =exp(g(x))·x. The mapgis not even assumed to be continuous. The question is, what additional requirements on M, G and f suffice to en-sure thatg inherits the smoothness of f. A simple counterexample can be found for M = S1, G = SO(2), g = R. The local smooth diffeomorphism f : eiϕ 7→ e2ϕ cannot be written as f(x) =exp(g(x))·xwith a smooth mapg, as f(eiϕ) =eiϕ·eiϕwould up to a constant 2πkimply g eiϕ
=ϕ. Such a map cannot even be continuous.
Main considerations of the thesis start with almost Einstein structures instead of conformally compactified Einstein manifolds. Such structures correspond to parallel tractors in the tractor bundle over a conformal structure. The existence of an almost Einstein structure can then equiva-lently be treated as existence problem for parallel tractors. Parallel sections in the tractor bundle on the other hand can be approached via examination of the holonomy of the tractor connec-tion. There actually is a lot of interest in conformal holonomy in Lorentzian signature. We have not followed this path at this time but it may lead to new examples of conformally compact-ified Einstein manifolds. Closely related to that issue is the property of the singularity set to decompose into a set of isolated point and null hypersurfaces. This corresponds to the curved orbit decomposition [ ˇCGH14] of an almost Einstein structure. Now vice versa it would be in-teresting to identify the requirements of a manifoldMn without boundary that guarantee that it at least topologically admit such a disjoint decomposition, In particular a decomposition into open sets ˚M,(n−1)-dimensional submanifoldsΣcand isolated pointsΣdsuch that in addition
∂M˚ =Σc∪Σd,∂Σc=Σdand close to isolated points the unionΣc∪Σdshould have the topology of a double cone. Next question then would be what is needed to assure that this decomposition is compatible with a metric in the sense that it locally givesΣc∪Σdthe causal structure of a null cone or null hypersurface.