Einstein’s field equations with a cosmological constant Λ for a Lorentzian metric g of signature (1,3) on a smooth manifold M take the form
Ric(g)+Λg= 0. (2.1)
The correct generalization to the case ofn+1 dimensions is Ein(g)=Λg, which is equiv-alent to
Ric+ 2Λ n−1
g= 0;
by a slight abuse of terminology and for the sake of brevity, we will however refer to (2.1) as the Einstein vacuum equations also in the general case. Given a globally hyperbolic solution (M, g) and a spacelike hypersurface Σ0⊂M, the negative definite Riemannian metric h on Σ0 induced by g and the second fundamental form k(X, Y)=h∇XY, Ni, X, Y∈TΣ0, of Σ0satisfy the constraint equations
Rh+(trhk)2−|k|2h= (1−n)Λ, δhk+dtrhk= 0,
(2.2)
whereRhis the scalar curvature ofh, and (δhr)µ=−rµν;ν is the divergence of the sym-metric 2-tensorr. We recall that, given a unit normal vector fieldN on Σ0, the constraint equations are equivalent to the equations
Eing(N, N) =12(n−1)Λ, Eing(N, X) = 0, X∈TΣ0, (2.3) for the Einstein tensor Eing=GgRic(g), where
Ggr=r−12(trgr)g. (2.4)
Conversely, given an initial data set (Σ0, h, k), where Σ0 is a smooth 3-manifold,h is a negative definite Riemannian metric on Σ0andkis a symmetric 2-tensor on Σ0, one can consider the non-characteristic initial value problem for the Einstein equation (2.1), which asks for a Lorentzian 4-manifold (M, g) and an embedding Σ0,!M such that h andkare, respectively, the induced metric and the second fundamental form of Σ0inM. We refer to the survey of Bartnik–Isenberg [8] for a detailed discussion of the constraint equations; see also§11.3.
As explained in the introduction, solving the initial value problem is non-trivial be-cause of the lack of hyperbolicity of Einstein’s equations due to their diffeomorphism invariance. However, as first shown by Choquet-Bruhat [59], the initial value problem admits a local solution, provided (h, k) are sufficiently regular, and the solution is unique up to diffeomorphisms in this sense; Choquet-Bruhat–Geroch [21] then proved the exis-tence of amaximal globally hyperbolic development of the initial data. (Sbierski [125]
recently gave a proof of this fact which avoids the use of Zorn’s lemma.)
We now explain the method of DeTurck [48] for solving the initial value problem in some detail; we follow the presentation of Graham and Lee [65]. Given the initial data set (Σ0, h, k), we defineM=Rx0×Σ0and embed Σ0,!{x0=0}⊂M; the task is to find a Lorentzian metricgonM near Σ0solving the Einstein equation and inducing the initial data (h, k) on Σ0. Choose a smooth non-degenerate background metric g0, which can have arbitrary signature. We then define the gauge 1-form
Υ(g) :=g(g0)−1δgGgg0∈ C∞(M, T∗M), (2.5) viewingg(g0)−1as a bundle automorphism ofT∗M. As a non-linear differential operator acting ong∈C∞(M, S2T∗M), the operator Υ(g) is of first order.
Remark 2.1. A simple calculation in local coordinates gives
Υ(g)µ=gµgνλ(Γ(g)νλ−Γ(g0)νλ). (2.6)
Thus, Υ(g)=0 if and only if the pointwise identity map Id: (M, g)!(M, g0) is a wave map. Now, given any local solutiongof the initial value problem for Einstein’s equations, we can solve the wave map equationg,g0φ=0 for φ: (M, g)!(M, g0) with initial data φ|Σ0=IdΣ0 andDφ|Σ0=IdTΣ0. Indeed, recalling that
(g,g0φ)k=gµν ∂µ∂νφk−Γ(g)λµν∂λφk+Γ(g0)kij∂µφi∂νφj
, φ(x) = (φk(xµ)), we see thatg,g0φ=0 is a semi-linear wave equation, hence a solution φ is guaranteed to exist locally near Σ0, and φ is a diffeomorphism of a small neighborhood U of Σ0 ontoφ(U); let us restrict the domain ofφ to such a neighborhoodU. Then gΥ:=φ∗g is well defined onφ(U), andφ: (U, g)!(φ(U), gΥ) is an isometry; hence we conclude that Id: (φ(U), gΥ)!(φ(U), g0) is a wave map, so Υ(gΥ)=0. Moreover, by our choice of initial conditions forφ, the metricgΥ induces the given initial data (h, k) pointwise on Σ0.
With (δg∗u)µν=12(uµ;ν+uν;µ) denoting the symmetric gradient of a 1-form u, the non-linear differential operator
PDT(g) := Ric(g)+Λg−δ∗gΥ(g) (2.7) is hyperbolic. Indeed, following [65,§3], the linearizations of the various terms are given by
DgRic(r) =12gr−δg∗δgGgr+Rg(r), (2.8) where (gr)µν=−rµν; and
Rg(r)µν=rλ(Rg)µνλ+12(Ric(g)µλrλν+Ric(g)νλrλµ), (2.9) and
DgΥ(r) =−δgGgr+C(r)−D(r), (2.10) where
Cµν =12((g0)−1)λ(g0µλ;ν+gνλ;µ0 −g0µν;λ), D=gµνCµν , C(r)=gλCµνλ rµν, D(r)=Dλrλ,
so DgPDT(r) is equal to the principally scalar wave operator 12gr plus lower-order terms. Therefore, one can solve the Cauchy problem for the quasilinear hyperbolic system PDT(g)=0, where the Cauchy data
γ0(g) := (g|Σ0,L∂
x0g|Σ0) = (g0, g1),
g0, g1∈C∞(Σ0, S2TΣ∗
0M), are arbitrary. Moreover, we saw in Remark 2.1 that every solution of the Einstein vacuum equations can be realized as a solution ofPDT(g)=0 by putting the solution into the wave map gauge Υ(g)=0. (On the other hand, a solution of PDT(g)=0 withgeneral Cauchy data (g0, g1) will have no relationship with the Einstein equation!)
We can now explain how to solve the initial value problem for (2.1). Given an initial data set (Σ0, h, k), so h, k∈C∞(Σ0, S2T∗Σ0) (in local coordinates, 3×3 matrices), one first constructs g0, g1∈C∞(Σ0, S2TΣ∗0M) (in local coordinates, 4×4 matrices) with the following properties:
• g0is of Lorentzian signature;
• the data on Σ0 induced by a metric ¯g withγ0(¯g)=(g0, g1) are equal to (h, k);
• Υ(¯g)|Σ0=0 as an element ofC∞(Σ0, TΣ∗
0M).
Note here that the metric induced on Σ0(which we want to be equal toh) only depends ong0, while the second fundamental form and gauge 1-form (which we want to bekand Υ(¯g), respectively) only depend on (g0, g1); in other words, any metricg with the same Cauchy data (g0, g1) induces the given initial data on Σ0and satisfies Υ(g)=0 at Σ0. We refer the reader to [135,§18.9] for the construction of the Cauchy data, and also to§3.6 for a detailed discussion in the context of the black hole stability problem.
Next, one solves thegauged Einstein equation
PDT(g) = 0, γ0(g) = (g0, g1), (2.11) locally near Σ0. ApplyingGgto this equation and using the constraint equations (2.3), we conclude that (Ggδ∗gΥ(g))(N, N)=0, where N is a unit normal to Σ0⊂M with respect to the solution metric g, and (Ggδ∗gΥ(g))(N, X)=0 for all X∈TΣ0. It is easy to see [135, §18.8] that Υ(g)|Σ0=0 and these equations together imply L∂x0Υ(g)|Σ0=0. The final insight is that the second Bianchi identity,(3) written as δgGgRic(g)=0 for any metricg, implies a hyperbolic evolution equation for Υ(g), which we call the (unmodified) constraint propagation equation, to wit
CPg Υ(g) = 0, (2.12)
whereCPg is the (unmodified)constraint propagation operator defined as
CPg := 2δgGgδg∗. (2.13)
The notation is justified: one easily verifies CPg u=gu−Ric(g)(u,·), with g being the tensor Laplacian on 1-forms. The terminology is motivated by the following fact:
(3) The second Bianchi identity is in fact in a consequence of the diffeomorphism invariance Ric(φ∗g)=φ∗Ric(g) of the Ricci tensor; see [23,§3.2].
given a solutiong of (2.11) with arbitrary initial data, and a spacelike surface Σ1 (with unit normal N) at which Υ(g)|Σ1=0, the constraint equations (2.3) are equivalent to (Υ(g)Σ1,LNΥ(g)|Σ1)=0; it is in this sense that (2.12) governs the propagation of the constraints.
Now, the uniqueness of solutions of the Cauchy problem for the equation (2.12) implies Υ(g)≡0, and thusg indeed satisfies the Einstein equation Ric(g)+Λg=0 in the wave map gauge Υ(g)=0, andg induces the given initial data on Σ0. This justifies the terminology ‘gauged Einstein equation’ for the equation (2.11), since its solution solves the Einstein equation in the chosen gauge.
2.2. Initial value problems for linearized gravity
Suppose now that we have a smooth familygs,s∈(−ε, ε), of Lorentzian metrics solving the Einstein equation Ric(gs)+Λgs=0 on a fixed (n+1)-dimensional manifoldM, and a hypersurface Σ0⊂M which is spacelike forg=g0. Let
r= d dsgs
s=0
.
Differentiating the equation ats=0 gives thelinearized (ungauged)Einstein equation
Dg(Ric+Λ)(r) = 0. (2.14)
The linearized constraints can be derived as the linearization of (2.2) around the initial data induced byg0, hence they are equations for the linearized metrich0and the linearized second fundamental formk0, withh0, k0∈C∞(Σ0, S2T∗Σ0); alternatively, we can use (2.3):
if Ns is a unit normal field to Σ0 with respect to the metric gs, so Ns also depends smoothly ons, then differentiating Eings−12(n−1)Λgs
(Ns, Ns)=0 ats=0 gives DgEin(r)−12(n−1)Λr
(N, N) = 0, (2.15)
where we used (2.3) to see that the terms coming from differentiating either of theNs
gives 0; on the other hand, using the Einstein equation, the derivative of Eings(Ns, X)=0 ats=0 takes the form
DgEin(r)(N, X)+12(n−1)Λg(N0, X) = 0, N0= d dsNs
s=0
;
nowgs(Ns, X)=0 yields g(N0, X)=−r(N, X) upon differentiation, and hence we arrive at
DgEin(r)−12(n−1)Λr
(N, X) = 0, X∈TΣ0. (2.16)
If moreover each of the metricsgs is in the wave map gauge Υ(gs)=0, with Υ given in (2.5) for a fixed background metric g0, then we also get
DgΥ(r) = 0;
therefore, in this case,rsolves the linearized gauged Einstein equation
Dg(Ric+Λ)(r)−δ∗gDgΥ(r) = 0. (2.17) As in the non-linear setting, one can use (2.17), which is a principally scalar wave equation as discussed after (2.7), to prove the well-posedness of the initial value problem for the linearized Einstein equation: given an initial data set (h0, k0) of symmetric 2-tensors on Σ0
satisfying the linearized constraint equations, one constructs Cauchy data (r0, r1) for the gauged equation (2.17) satisfying the linearized gauge condition at Σ0; solving the Cauchy problem for (2.17) yields a symmetric 2-tensorr. The linearized constraints in the form (2.15)–(2.16) imply thatGgδg∗DgΥ(r)=0 at Σ0, which as before impliesL∂x0DgΥ(r)=0 at Σ0. Finally, since Ric(g)+Λg=0, linearizing the second Bianchi identity ing gives
δgGgDg(Ric+Λ)(r) = 0,
and thus (2.17) implies the evolution equationCPg (DgΥ(r))=0, and henceDgΥ(r)≡0, and we therefore obtain a solutionrof (2.14).
Analogously to the discussion in Remark 2.1, one can put a given solution r of the linearized Einstein equation (2.14), withgsolving Ric(g)+Λg=0, into the linearized gaugeDgΥ(r)=0 by solving a linearized wave map equation. Concretely, the diffeomor-phism invariance of the non-linear equation (2.1) implies thatDg(Ric+Λ)(δg∗ω0)=0 for allω0∈C∞(M, T∗M); indeed,δg∗ω0=12L(ω0)]gis a Lie derivative. Thus, puttingrinto the gaugeDgΥ(r)=0 amounts to findingω0 such that
DgΥ(r+δg∗ω0) = 0 (2.18)
holds, or equivalently
Υgω0= 2DgΥ(r), (2.19)
where we define
Υg =−2DgΥδg∗, (2.20)
which agrees to leading order with the wave operatorgon 1-forms due to (2.10). Thus, one can solve (2.19), with any prescribed initial dataγ0(ω0), and then (2.18) holds. Taking γ0(ω0)=0 ensures that the linearized initial data, i.e. the induced linearized metric and linearized second fundamental form on Σ0, ofr andr+δ∗gω0 coincide.
We remark that, if one chooses the background metricg0in (2.5) to be equal to the metricg around which we linearize, thenDgΥ(r)=−δgGgrby (2.10), and thus
Υg =g+Λ;
see also (2.13).