In this section, we review the form of the vacuum Einstein equations (2) written with respect to a natural null frame attached to a local double null foliation of a Lorentzian manifold. See Christodoulou [11] for a detailed exposition. It is these equations which we shall formally linearise in§5.1to obtain the equations of linearised gravity. The reader not interested in the derivation of the linearised system can skip immediately to§4.
An outline of the current section is as follows: We begin in§3.1with preliminaries, defining the notion of double null gauge and associated notation. Ricci coefficients and curvature components are then defined in§3.2. Finally, the vacuum Einstein equations are presented in§3.3.
3.1. Preliminaries
Let (M,g) be a (3+1)-dimensional Lorentzian manifold.
3.1.1. Local double null gauge
In a neighbourhood of any pointp∈M, we can introduce local coordinatesu,v,θ1and θ2such that the metric is expressed in “canonical double-null form”:
g=−4Ω2du dv+/g
CD(dθC−bCdv)(dθD−bDdv) (70)
for a spacetime function Ω:M!R, anSu,v-tangent vectorbA and a symmetric Su,v -tangent covariant symmetric 2-tensor/gAB. Here,Su,v denotes the 2-dimensional (Rie-mannian with metric/gAB) manifold arising as the intersection of the hypersurfaces of constantuandv.
3.1.2. Normalised frames
We can define a normalised null frame associated with the above coordinates as follows.
We define
e3=Ω−1∂u, e4=Ω−1(∂v+bA∂θA), eA=∂θA forA= 1,2, (71) for which we note the relations
g(e3, e4) =−2, g(e3,eA) = 0, g(e4,eA) = 0, g(eA,eB) =/gAB.
In particular, {e1,e2} constitutes a (local) coordinate frame field (not necessarily or-thonormal) of the orthogonal complement of the span of e3 and e4 (i.e. in the tangent space of the submanifoldSu,v). In view of the above relations, we shall refer to the null frameN={e3,e4,e1,e2} as beingnormalised.
3.1.3.Su,v-tensor algebra
In §3.2, we will express the Ricci coefficients and curvature components of the metric (70) with respect to the null frame (71). These objects will then becomeSu,v-tangent tensors, orSu,v-tensors for short (see [11]). Two types of such Su,v-tensors will play a particularly important role: 1-forms ξ and symmetric 2-tensors θ, the latter being defined as satisfying θAB=θBA in any coordinate patch. A traceless symmetric Su,v
2-tensorθ satisfies in addition/gABθAB=0.
Let ξ and ξ˜be arbitrary Su,v 1-forms, and θ and θ˜be arbitrary symmetric Su,v
2-tensors.
We denote by ?ξand ?θthe Hodge-dual (on (Su,v,/g)) ofξandθ, respectively, [11], and denote byθ] the tensor obtained fromθ by raising an index withg./
We define the contractions
(ξ,ξ) :=˜ /gABξAξ˜B and (θ,θ) :=˜ g/ABg/CDθACθ˜BD,
and|ξ|2=(ξ,ξ) and|θ|2=(θ,θ). We denote byθ]·ξthe 1-formθABξB arising from the contraction withg./
We finally define the 2-tensorsθ×θ˜andξ⊗bξ, and the scalar˜ θ∧θ˜via (θ×θ)˜BC:=g/ADθABθ˜DC,
(ξ⊗bξ)˜AB:=ξAξ˜B+ξBξ˜A−/gCDξCξ˜D/gAB, θ∧θ˜:=/εAB/gCDθACθ˜BD,
where/εAB denotes the components of the volume form associated with/gonSu,v. Note thatξ⊗bξ˜is a symmetric traceless 2-tensor.
3.1.4. Su,v-projected Lie and covariant derivates
We define the derivative operatorsD andD to act on an Su,v-tensor φas the projec-tion ontoSu,v of the Lie-derivative of φin the direction ofΩe3 and Ωe4, respectively.
We hence have the following relations between the projected Lie-derivatives D andD, and the Su,v-projected spacetime covariant derivatives ∇/3=∇/e3 and ∇/4=∇/e4 in the directione3 ande4, respectively:
Df=Ω∇/4f on functionsf, Dξ=Ω∇/4ξ+Ωχ]·ξ on 1-formsξ,
Dθ=Ω∇/4θ+Ωχ×θ+Ωθ×χ on symmetric 2-tensorsθ,
(72)
and similarly for∇/3, replacingχbyχandD byD. See [11] for details.
3.1.5. Angular operators on Su,v
We employ the following notation (adapted from [14]) for operators on the manifolds Su,v.
Letξbe an arbitrary 1-form andθ be an arbitrary symmetric traceless 2-tensor on Su,v.
• ∇/ denotes the covariant derivative associated with the metricg/
AB onSu,v.
• D/1 takesξinto the pair of functions (div/ ξ,curl/ ξ), wherediv/ ξ=g/AB∇/AξB and curl/ ξ=/εAB∇/AξB.
• D/?1, the formal(14) L2-adjoint of D/1, takes any pair of scalars % and σ into the Su,v-1-form−∇/A%+/εAB∇/Bσ.
• D/2 takesθ into theSu,v-1-form (div/ θ)C=g/AB∇/AθBC.
• D/?2, the formal L2 adjoint of D/2, takes ξ into the symmetric traceless 2-tensor (D/?2ξ)AB=−12(∇/BξA+∇/AξB−(div/ ξ)/gAB).
(14) In our application, the surfacesSu,v will be compact topological spheres and this will indeed define an adjoint on appropriate spaces.
3.2. Ricci coefficients and curvature components
We now define the Ricci coefficients and curvature components associated with the metric (70) with respect to the normalised null frameN={e3,e4,e1,e2}.
For the Ricci coefficients, using the shorthand ∇A=∇eA we define χAB=g(∇Ae4,eB), χAB=g(∇Ae3,eB),
ηA=−12g(∇e3eA,e4), ηA=−12g(∇e4eA,e3), ω=b 12g(∇e4e3,e4), ωb=12g(∇e3e4,e3), ζA=12g(∇Ae4,e3).
(73)
Note that, in view ofΩ−1e3 andΩ−1e4being geodesic vectorfields, all other connection coefficients automatically vanish. It is natural to decomposeχinto its trace-free partχ,b a symmetric tracelessSu,v 2-tensor and its tracetrχ, and similarly χ.(15)
With Rdenoting the Riemann curvature tensor of (70), the null-decomposed cur-vature components are defined as follows:
αAB=R(eA,e4,eB,e4), αAB=R(eA,e3,eB,e3), βA=12R(eA,e4,e3,e4), βA=R(eA,e3,e3,e4),
%=14R(e4,e3,e4,e3), σ=14?R(e4,e3,e4,e3),
(74)
with ?R denoting the Hodge dual on (M,g) ofR. The above objects are Su,v-tensors (functions, vectors, symmetric 2-tensors) on (M,g); cf. [14]. Note also the relations
ωb=∇/3Ω
Ω , ωb=∇/4Ω
Ω , ηA=ζA+∇/AlogΩ, ηA=−ζA+∇/AlogΩ.
3.3. The Einstein equations
If (M,g) satisfies the vacuum Einstein equations
Rµν[g] = 0, (75)
the Ricci coefficients defined in (73) and curvature components (74) satisfy a system of equations, which is presented in this section.
(15) This is of course unrelated to the ˆ in the case of the scalar quantityω, which distinguishes itb from other normalisations; we retain the notationωbto facilitate comparison with [11].
3.3.1. The null structure equations
First, we have the important first variational formulas:(16)
Dg/= 2Ωχ= 2Ωχ+Ω trb χg/ and Dg/= 2Ωχ= 2Ωχ+Ω trb χg./ (76) Second,
∇/3χ+trb χχ−b ωb χb=−α, ∇/4χ+trb χχ−b ωbχb=α, (77)
∇/3(trχ)+12(trχ)2−ωbtrχ=−(χ,b χ),b ∇/4(trχ)+12(trχ)2−ωbtrχ=−(χ,b χ).b (78) Note that the last two equations are the celebrated Raychaudhuri equations. We also have
∇/3χ+b 12trχχ+b ωbχb=−2D/?2η−12trχχ+(ηb ⊗η),b (79)
∇/4χ+b 12trχχ+b ωbχb=−2D/?2η−12trχχ+(ηb ⊗η),b (80)
∇/3(trχ)+12(trχ)(trχ)+ωbtrχ=−(χ,b χ)+2(η,b η)+2%+2div/ η, (81)
∇/4(trχ)+12(trχ)(trχ)+ωbtrχ=−(χ,b χ)+2(η,b η)+2%+2div/ η, (82)
∇/3η=χ]·(η−η)+β, ∇/4η=−χ]·(η−η)−β,
D(Ωω) =b Ω2[2(η,η)−|η|2−%], D(Ωω) =b Ω2[2(η,η)−|η|2−%], Ω2ωb=DΩ, Ω2ωb=DΩ, ηA+ηA= 2∇/AlogΩ,
∂ubA= 2Ω2(ηA−ηA). (83) Finally, we have
curl/ η=−12χ∧χ+σ and curl/ η=12χ∧χ−σ, the Codazzi equations
div/ χb=−1
2χb]·(η−η)+1
4trχ(η−η)+1
2∇/trχ−β
=−1
2χb]·(η−η)−1
2trχη+ 1
2Ω∇(Ω tr/ χ)−β, div/ χb=1
2χb]·(η−η)−1
4trχ(η−η)+1
2∇/trχ+β
=1
2χb]·(η−η)−1
2trχη+ 1
2Ω∇(Ω tr/ χ)+β, and the Gauss equation (Kdenoting the Gauss curvature of the metric/g)
K=−14trχtrχ+12(χ,b χ)−%.b (84)
(16) Note that these formulas are equivalent to the statement that∇/3/g=0=∇/4/g.
3.3.2. The Bianchi equations
We finally turn to the equations satisfied by the curvature components of (M,g), which are the well-known Bianchi equations:
∇/3α+12trχ α+2ωαb =−2D/?2β−3χ%−3b ?χσ+(4η+ζ)b ⊗β,b
∇/4β+2trχ β−ωβb =div/ α+(η]+2ζ])·α,
∇/3β+trχ β+ωβb =D/?1(−%,σ)+3η%+3?ησ+2χb]·β,
∇/4%+32trχ %=div/ β+(2η+ζ,β)−12(χ,b α),
∇/4σ+32trχ σ=−curl/ β−(2η+ζ)∧β+12χ∧α,b
∇/3%+32trχ %=−div/ β−(2η−ζ,β)−12(χ,b α),
∇/3σ+32trχ σ=−curl/ β−(2η−ζ)∧β−12χ∧α,b
∇/4β+trχ β+ωβb =D/?1(%,σ)−3η%+3?ησ+2χb]·β,
∇/3β+2trχ β−ωβb =−div/ α−(η]−2ζ])·α,
∇/4α+12trχ α+2ωαb = 2D/?2β−3χ%+3b ?χσ−(4η−ζ)b ⊗β.b
We note that the vacuum equations (75) further imply that the symmetric tensorsαand α are in addition traceless. The above equations encode the essential hyperbolicity of (75). See [14].