4.1 The initial value problem

How does one formulate an initial value problem for Einstein’s equations? What should the initial data be? Is there uniqueness in any reasonable sense? These questions can be formulated in the presence of various matter fields, but let us, for the sake of simplicity, restrict our attention to the vacuum case here.

4.1.1 The vacuum equations

The vacuum equations are given by

G = 0,

where

G = Ric − 1-Sg 2

is the Einstein tensor, Ric is the Ricci tensor and S is the scalar curvature of a Lorentz manifold (M, g ). Clearly, Einstein’s vacuum equations are equivalent to

Ric = 0. (5 )
Note that Ric can be thought of as a differential operator acting on the metric. A thorough discussion of the principal symbol of this operator, of the implications of the diffeomorphism invariance of the equations etc…can be found in [35], see also [15]. However, here we shall take a more pedestrian approach. Writing down Ric in local coordinates, we have
1- αβ αβ γδ Rμν = − 2 g ∂α∂βg μν + ∇ (μΓ ν) + g g [Γ αγμΓ βδν + Γ αγμΓ βνδ + Γ αγνΓ βμδ], (6 )
where
1- μν β α Γ αγβ = 2(∂αgβγ + ∂βg αγ − ∂ γgαβ), Γ α = gαβg Γμν, ∇ μΓ ν = ∂μΓ ν − ΓμνΓ α,

and a parenthesis denotes symmetrization, i.e.,

∇ Γ = 1(∇ Γ + ∇ Γ ). (μ ν) 2 μ ν ν μ

The notation ∇μΓ ν is questionable due to the fact that Γ μ are not the components of a covector. Nevertheless, we shall use it. The highest-order derivatives are contained in

− 1-gαβ∂ ∂ g + ∇ Γ . (7 ) 2 α β μν (μ ν)

4.1.2 Formulation, intuition

If the second term in Equation (7View Equation) were not there, Equation (5View Equation) would, in local coordinates, be a nonlinear wave equation, and it would be straightforward to formulate an initial value problem. Furthermore, given a solution to Equation (5View Equation), it is possible to choose local coordinates such that Γ μ vanishes and Equation (5View Equation) takes the form of a nonlinear wave equation. On the other hand, if Equation (5View Equation) were a nonlinear wave equation when expressed with respect to arbitrary coordinates, we would obtain uniqueness for the coordinate expression of the metric. This statement is incompatible with diffeomorphism invariance. Thus, even though Equation (5View Equation) in some respects can be viewed as a hyperbolic differential equation, the geometric aspect of the equation must not be forgotten.

Due to the above observations, it seems natural to expect the right PDE problem to formulate for Einstein’s equations to be the initial value problem. Furthermore, it seems clear that this problem should be given a geometric formulation. Naively, one would expect it to be necessary to specify the metric and the first time derivative of the metric at the initial hypersurface. However, these quantities are not geometric. The induced metric and second fundamental form are, on the other hand, geometric quantities and they contain part of the information one would naively expect to need. Furthermore, they, in the end, turn out to constitute sufficient information. The question arises of what should be required of the initial hypersurface? Since we wish to avoid issues of consistency, we shall require the hypersurface to be such that it has no causal tangent vectors. In other words, we require that it be spacelike (this is, strictly speaking, not necessary; there are formulations in the null case as well; see, e.g., [65]).

4.1.3 Formulation, formal definition

The above discussion suggests the following. The initial data should, at the very minimum, consist of a manifold, say Σ (which should be thought of as the initial hypersurface), a Riemannian metric on Σ, say ρ (which should be thought of as the induced metric on the initial hypersurface), and a symmetric covariant two-tensor on Σ, say κ (which should be thought of as the induced second fundamental form). On the other hand, if (Σ,ρ,κ ) is a hypersurface with induced metric and second fundamental form in a Lorentz manifold solving Equation (5View Equation), then ρ and κ have to satisfy the constraint equations:

ij 2 r − κijκ + (trκ) = 0, (8 ) Dj κji − Di(trκ) = 0. (9 )
Here r is the scalar curvature and D is the Levi–Civita connection associated with ρ, and indices are raised and lowered with ρ. Equation (8View Equation), the Hamiltonian constraint, is equivalent to the equation G (N, N ) = 0, where G is the Einstein tensor and N is a normal vector to the hypersurface. Equation (9View Equation), the momentum constraint, is equivalent to the equation G (N, X ) = 0, where X is any tangent vector to the hypersurface.

Definition 1 Initial data for Einstein’s vacuum equations consist of a three-dimensional manifold Σ, a Riemannian metric ρ and a covariant symmetric two-tensor κ on Σ, both assumed to be smooth and to satisfy Equations (8View Equation)–(9View Equation). Given initial data, the initial value problem is that of finding a four-dimensional manifold M with a Lorentz metric g such that Equation (5View Equation) is satisfied, and an embedding i : Σ → M such that i∗g = ρ and that if k is the second fundamental form of i(Σ), then i∗k = κ. Such a Lorentz manifold (M, g) is called a development of the data. Furthermore, if i(Σ) is a Cauchy hypersurface in (M, g), then (M, g ) is referred to as a globally-hyperbolic development of the initial data. In both cases, the existence of an embedding i is tacit.

Since the concepts Cauchy hypersurface and globally hyperbolic are referred to above, and will be of some importance below, let us recall how they are defined.

Definition 2 Let (M, g) be a Lorentz manifold. A subset Σ of M is said to be a Cauchy hypersurface if it is intersected exactly once by every inextendible timelike curve. A Lorentz manifold that admits a Cauchy hypersurface is said to be globally hyperbolic.

Remark. Two basic examples of Cauchy hypersurfaces are the t = const. hypersurfaces in Minkowski space and the hypersurfaces of spatial homogeneity in Robertson–Walker spacetimes. The reader interested in the basic properties of globally-hyperbolic Lorentz manifolds and Cauchy hypersurfaces is referred to [60Jump To The Next Citation Point], see also [82Jump To The Next Citation Point] and references cited therein. Cauchy hypersurfaces need neither be smooth nor spacelike, but we shall tacitly assume them to be both. The reason the concept of a Cauchy hypersurface is of such central importance is that it is the natural type of surface on which to specify initial data.

4.1.4 Existence of a development

We are now in a position to ask: given initial data to Einstein’s vacuum equations, is there a development? The answer to this question is yes, due to the seminal work of Choquet-Bruhat [34] (a presentation in book form is also available in, e.g., [82Jump To The Next Citation Point]):

Theorem 1 Given initial data (Σ,ρ,κ ) to Einstein’s vacuum equations, there is a globally-hyperbolic development.

Clearly, this is a fundamental result. In particular, this result is what justifies the terminology “initial data to Einstein’s vacuum equations” as specified in Definition 1. On the other hand, the issue of uniqueness is not addressed. Given initial data, there are infinitely many distinct globally-hyperbolic developments. In order to obtain uniqueness, it is consequently necessary to require some form of maximality.

4.1.5 Existence of a maximal globally-hyperbolic development

The central concept in the study of uniqueness is that of an MGHD:

Definition 3 Given initial data to Einstein’s vacuum equations (5View Equation), a MGHD of the data is a globally hyperbolic development (M, g), with embedding i : Σ → M, such that if (M ′,g ′) is any other globally hyperbolic development of the same data, with embedding i′ : Σ → M ′, then there is a map ′ ψ : M → M, which is a diffeomorphism onto its image, such that ∗ ′ ψ g = g and ′ ψ ∘ i = i.

Note that this definition differs from the standard notion of maximality used in set theory. The standard notion would lead to the definition of a MGHD as a globally hyperbolic development, which cannot be extended (note that this notion of maximality would not a priori rule out the possibility of two maximal elements, neither of which can be embedded into the other, as opposed to Definition 3).

Theorem 2 Given initial data to Equation (5View Equation), there is an MGHD of the data, which is unique up to isometry.

Remark. Uniqueness of a development (M, g) up to isometry is defined as follows: if ′ ′ (M ,g ) is another MGHD, then there is a diffeomorphism ′ ψ : M → M such that ∗ ′ ψ g = g and ′ ψ ∘ i = i, where i and i′ are the embeddings of Σ into M and M ′ respectively.

Theorem 2 is due to the work of Choquet-Bruhat and Geroch; see [13] for the original paper and [82Jump To The Next Citation Point] for a recent presentation. The proof relies, in part, on the local theory and on an argument using what is often referred to as Zorn’s lemma. This leads to the existence of an MGHD in the set theory sense of the word. However, it does not lead to the existence of an MGHD in the sense of Definition 3. In fact, the important part of the result is the uniqueness of the MGHD (in the set theory sense of the word). This requires an additional argument.

Due to Theorem 2, the initial value formulation of Einstein’s equations is meaningful. However, the MGHD might be extendible. In fact, it turns out that there are initial data such that this is the case. If the extensions were unique in their turn, this would not be a serious problem, but it turns out that there are MGHDs with inequivalent maximal extensions [20Jump To The Next Citation Point] (see also [82Jump To The Next Citation Point]). The reason these examples are unfortunate is that they demonstrate that Einstein’s general theory of relativity is not deterministic; given initial data, there is not necessarily a unique corresponding universe. Nevertheless, the examples of this pathological behavior are very special, and there is thus reason to hope that for generic initial data, the MGHD is inextendible. These speculations naturally lead us to the strong cosmic-censorship conjecture.


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