6.3 Geometric existence and uniqueness

The results mentioned so far concerning the well-posed IBVP for Einstein’s field equations in the tetrad formulation of [187Jump To The Next Citation Point], in the metric formulation with harmonic coordinates described in Section 6.1, or in the linearized BSSN formulation described in Section 6.2 allow one, from the PDE point of view, to construct unique solutions on a manifold of the form M = [0,T] × Σ, given appropriate initial and boundary data. However, since general relativity is a diffeomorphism invariant theory, one needs to pose the IBVP from a geometric perspective. In particular, the following questions arise, which, for simplicity, we only formulate for the vacuum case:

These geometric existence and uniqueness problems have been solved in the context of the Cauchy problem without boundaries; see [127] and Section 4.1.3. However, when boundaries are present, several new difficulties appear as pointed out in [186Jump To The Next Citation Point]; see also [187Jump To The Next Citation Point, 184]:

  1. It is a priori not clear what the boundary data ψ should represent geometrically. Unlike the case of the initial surface, where the data represents the first and second fundamental forms of Σ0 as a spatial surface embedded in the constructed spacetime (M, g), it is less clear what the geometric meaning of ψ should be since it is restricted by the characteristic structure of the evolution equations, as discussed in Section 5.
  2. The boundary data (q ,q ,q ,q ) K L Q QQ in the boundary conditions (6.2View Equation, 6.3View Equation, 6.4View Equation, 6.5View Equation) for the harmonic formulation and the boundary data Gij in the boundary condition (6.14View Equation) for the BSSN formulation ultimately depend on the specific choice of a future-directed time-like vector field T at the boundary surface 𝒯. Together with the unit outward normal N to 𝒯, this vector defines the preferred null directions K = T + N and L = T − N, which are used to construct the boundary-adapted null tetrad in the harmonic case and the projection operators μ μ μ μ Π ν = δ ν + T Tν − N N ν and μν μ ν 1 μν P αβ = Π αΠ β − 2Π Πα β in the BSSN one. Although it is tempting to define T as the unit, future-directed time-like vector tangent to 𝒯, which is orthogonal to the cross sections ∂Σt, this definition would depend on the particular foliation Σ t the formulation is based on, and so the resulting vector T would be gauge-dependent. A similar issue arises in the tetrad formulation of [187Jump To The Next Citation Point].
  3. When addressing the geometric uniqueness issue, an interesting question is whether or not it is possible to determine from the data sets (f1,q1) and (f2,q2) alone if they are equivalent in the sense that their solutions u 1 and u 2 induce the same geometric data (h,k, ψ). Therefore, the question is whether or not one can identify equivalent data sets by considering only transformations on the initial and boundary surfaces Σ0 and 𝒯, without knowing the solutions u1 and u2.

Although a complete answer to these questions remains a difficult task, there has been some recent progress towards their understanding. In [186Jump To The Next Citation Point] a method was proposed to geometrically single out a preferred time direction T at the boundary surface 𝒯. This is done by considering the trace-free part of the second fundamental form, and proving that under certain conditions, which are stable under perturbations, the corresponding linear map on the tangent space possesses a unique time-like eigenvector. Together with the unit outward normal vector N, the vector field T defines a distinguished adapted null tetrad at the boundary, from which geometrically meaningful boundary data could be defined. For instance, the complex Weyl scalar Ψ0 can then be defined as the contraction Ψ0 = C αβγδK αQ βK γQ δ of the Weyl tensor Cαβγδ associated to the metric gμν along the null vectors K and Q, and the definition is unique up to the usual spin rotational freedom Q ↦→ eiφQ, and therefore, the Weyl scalar Ψ 0 is a good candidate for forming part of the boundary data ψ.

In [355Jump To The Next Citation Point] it was suggested that the unique specification of a vector field T may not be a fundamental problem, but rather the manifestation of the inability to specify a non-incoming radiation condition correctly. In the linearized case, for example, setting the Weyl scalar Ψ0 to zero computed from the boundary-adapted tetrad is transparent to gravitational plane waves traveling along the specific null direction K = T + N, see Example 32, but it induces spurious reflections for outgoing plane waves traveling in other null direction. Therefore, a genuine non-incoming radiation condition should be, in fact, independent of any specific null or time-like direction at the boundary, and can only depend on the normal vector N. This is indeed the case for much simpler systems like the scalar wave equation on a Minkowski background [153], where perfectly absorbing boundary conditions are formulated as a nonlocal condition, which is independent of a preferred time direction at the boundary.

Aside from controlling the incoming gravitational degrees of freedom, the boundary data ψ should also comprise information related to the geometric evolution of the boundary surface. In [187Jump To The Next Citation Point] this was achieved by specifying the mean curvature of 𝒯 as part of the boundary data. In the harmonic formulation described in Section 6.1 this information is presumably contained in the functions qK, qL and qQ, but their geometric interpretation is not clear.

In order to illustrate some of the issues related to the geometric existence and uniqueness problem in a simpler context, in what follows we analyze the IBVP for linearized gravitational waves propagating on a Minkowski background. Before analyzing this case, however, we make two remarks. First, it should be noted [186] that the geometric uniqueness problem, especially an understanding of point (iii), also has practical interest, since in long term evolutions it is possible that the gauge threatens to break down at some point, requiring a redefinition. The second remark concerns the formulation of the Einstein IBVP in generalized harmonic coordinates, described in Sections 4.1 and 6.1, where general covariance was maintained by introducing a background metric ˚gμν on the manifold M. IBVPs based on this approach have been formulated in [369Jump To The Next Citation Point] and [264Jump To The Next Citation Point] and further developed in [434] and [433]. However, one has to emphasize that this approach does not automatically solve the geometric existence and uniqueness problems described here: although it is true that the IBVP is invariant with respect to any diffeomorphism ϕ : M → M, which acts on the dynamical and the background metric at the same time, the question of the dependency of the solution on the background metric remains.

6.3.1 Geometric existence and uniqueness in the linearized case

Here we analyze some of the geometric existence and uniqueness issues of the IBVP for Einstein’s field equations in the much simpler setting of linearized gravity on Minkowski space, where the vacuum field equations reduce to

− ∇ μ∇ μh αβ − ∇α ∇βh + 2∇ μ∇ (αh β)μ = 0, (6.15 )
where hαβ denotes the first variation of the metric, αβ h := η h αβ its trace with respect to the Minkowski background metric η αβ, and ∇μ is the covariant derivative with respect to ηαβ. An infinitesimal coordinate transformation parametrized by a vector field ξμ induces the transformation
hαβ ↦→ &tidle;hαβ = h αβ + 2∇ (αξβ), (6.16 )
where β ξα := η αβξ.

Let us consider the linearized Cauchy problem without boundaries first, where initial data is specified at the initial surface Σ0 = {0} × ℝ3. The initial data is specified geometrically by the first and second fundamental forms of Σ 0, which, in the linearized case, are represented by a pair (h (0),k(0)) ij ij of covariant symmetric tensor fields on Σ0. We assume (0) (0) (h ij ,kij ) to be smooth and to satisfy the linearized Hamiltonian and momentum constraints

ijrs (0) ijrs (0) G ∂i∂jhrs = 0, G ∂jkrs = 0, (6.17 )
where Gijrs := δi(rδs)j − δijδrs. A solution hαβ of Eq. (6.15View Equation) with the induced data corresponding to (0) (0) (hij ,kij ) up to a gauge transformation (6.16View Equation) satisfies
h | = h (0)+ 2 ∂ X , ∂ h − 2 ∂ h || = − 2(k(0)+ ∂ ∂ f ), (6.18 ) ijΣ0 ij (i j) t ij (i j)0Σ0 ij ij
where Xj = ξj and f = ξ0 are smooth and represent the initial gauge freedom. Then, one has:

Theorem 9. The initial-value problem (6.15View Equation, 6.18View Equation) possesses a smooth solution hαβ, which is unique up to an infinitesimal coordinate transformation &tidle; h αβ = hαβ + 2∇ (αξβ) generated by a vector field ξα.

Proof. We first show the existence of a solution in the linearized harmonic gauge μ 1 Cβ = ∇ h βμ − 2∇ βh = 0, for which Eq. (6.15View Equation) reduces to the system of wave equations μ ∇ ∇ μh αβ = 0. The initial data, (h αβ|Σ0 , ∂thαβ|Σ0), for this system is chosen such that hij| = h (0) Σ0 ij, | ∂thij| = 2∂(ihj)0| − 2k (0) Σ0 Σ0 ij and ij (0) ∂th00|Σ0 = 2δ kij, i (0) 1 kl (0) 1 ∂th0j|Σ0 = ∂(h ij − 2δijδ hkl ) + 2 ∂jh00|Σ0, where (0) (0) (hij ,kij ) satisfy the constraint equations (6.17View Equation) and where the initial data for h00 and h0j is chosen smooth but otherwise arbitrary. This choice implies the satisfaction of Eq. (6.18View Equation) with Xj = 0 and f = 0 and the initial conditions C | = 0 β Σ0 and ∂ C | = 0 t β Σ0 on the constraint fields C β. Therefore, solving the wave equation μ ∇ ∇ μhαβ = 0 with such data, we obtain a solution of the linearized Einstein equations (6.15View Equation) in the harmonic gauge with initial data satisfying (6.18View Equation) with Xj = 0 and f = 0. This shows geometric existence for the linearized harmonic formulation.

As for uniqueness, suppose we had two smooth solutions of Eqs. (6.15View Equation, 6.18View Equation). Then, since the equations are linear, the difference hαβ between these two solutions also satisfies Eqs. (6.15View Equation, 6.18View Equation) with trivial data (0) h ij = 0, (0) k ij = 0. We show that hαβ can be transformed away by means of an infinitesimal gauge transformation (6.16View Equation). For this, define h&tidle;αβ := hαβ + 2∇ (αξβ) where ξβ is required to satisfy the inhomogeneous wave equation

0 = ∇ α&tidle;hαβ − 1∇ β&tidle;h = ∇ αhα β − 1∇ βh + ∇ α∇ αξβ (6.19 ) 2 2
with initial data for ξβ defined by ξ0| = − f Σ0, ξi| = − Xi Σ0, ∂tξ0| = − h00∕2 Σ0, ∂tξi|Σ0 = − h0i + ∂if. Then, by construction, &tidle; hαβ satisfies the harmonic gauge, and it can be verified that | | &tidle;h αβ|| = ∂t&tidle;hαβ|| = 0 Σ0 Σ0. Therefore, &tidle;h αβ is a solution of the wave equation ∇ μ∇ μ&tidle;hαβ = 0 with trivial initial data, and it follows that &tidle;hα β = 0 and that hαβ = − 2∇ (αξβ) is a pure gauge mode. □

It follows from the existence part of the proof that the quantities h | 00 Σ0 and h | 0jΣ0, corresponding to linearized lapse and shift, parametrize pure gauge modes in the linearized harmonic formulation.

Next, we turn to the IBVP on the manifold M = [0,T ] × Σ. Let us first look at the boundary conditions (6.2View Equation6.5View Equation), which, in the linearized case, reduce to

∇K hKK | = qK , ∇K hKL | = qL, ∇K hKQ | = qQ, ∇K hQQ − ∇QhQK | = qQQ.(6.20 ) 𝒯 𝒯 𝒯 𝒯
There is no problem in repeating the geometric existence part of the proof on M imposing these boundary condition, and using the IBVP described in Section 6.1. However, there is a problem when trying to prove the uniqueness part. This is because a gauge transformation (6.16View Equation) induces the following transformations on the boundary data,
2 2 2 &tidle;qK = qK + 2∇ K ξK, &tidle;qL = qL + ∇ KξL + ∇K ∇L ξK, &tidle;qQ = qQ + ∇ K ξQ + ∇K ∇Q ξK, &tidle;qQQ = qQQ + ∇Q (∇K ξQ − ∇Q ξK ),
which overdetermines the vector field ξβ at the boundary. On the other hand, replacing the boundary condition (6.5View Equation) by the specification of the Weyl scalar Ψ0, leads to [286Jump To The Next Citation Point, 369Jump To The Next Citation Point]
2 || ∇ KhQQ + ∇Q (∇QhKK − 2 ∇K hKQ ) 𝒯 = Ψ0. (6.21 )
Since the left-hand side is gauge-invariant, there is no over-determination of ξβ at the boundary any more, and the transformation properties of the remaining boundary data qK, qL and qQ provides a complete set of boundary data for ξK, ξL and ξQ, which may be used in conjunction with the wave equation ∇ μ∇ μξβ = 0 in order to formulate a well-posed IBVP [369Jump To The Next Citation Point]. Provided Ψ0 is smooth and the compatibility conditions are satisfied at the edge S = Σ0 ∩ 𝒯, it follows:

Theorem 10. [355] The IBVP (6.15View Equation, 6.18View Equation, 6.21View Equation) possesses a smooth solution hαβ, which is unique up to an infinitesimal coordinate transformation &tidle;hαβ = hαβ + 2∇ (αξβ) generated by a vector field ξα.

In conclusion, we can say that, in the simple case of linear gravitational waves propagating on a Minkowksi background, we have resolved the issues (i–iii). Correct boundary data is given to the linearized Weyl scalar Ψ0 computed from the boundary-adapted tetrad. To linear order, Ψ0 is invariant with respect to coordinate transformations, and the time-like vector field T appearing in its definition can be defined geometrically by taking the future-directed unit normal to the initial surface Σ0 and parallel transport it along the geodesics orthogonal to Σ0.

Whether or not this result can be generalized to the full nonlinear case is not immediately clear. In our linearized analysis we have imposed no restrictions on the normal component ξN of the vector field generating the infinitesimal coordinate transformation. However, such a restriction is necessary in order to keep the boundary surface fixed under a diffeomorphism. Unfortunately, it does not seem possible to restrict ξN in a natural way with the boundary conditions constructed so far.


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