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The normal and normal-tangential projections of the field equation are the constraint equations

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), let $T$ be a smooth symmetric covariant $2$ -tensor field on $M$ (Smooth covariant 2-tensor fields on a smooth manifold), let $κ, Λ \in R$ and let $G$ be the Einstein tensor of the metric-selected connection (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). Suppose that the field equation $G + Λ g = κ T$ holds at every point of $M$ . Let $Σ \subseteq M$ be a spacelike embedded hypersurface with future unit normal field $n$ , induced metric $h$ and second fundamental form $K$ , and let $ρ : = T (n, n)$ and $j (V) : = - T (n, V)$ for $V \in T_{p} Σ$ , $p \in Σ$ , be the matter data of Spacelike initial data, the constraint equations, developments and maximal developments clause (b). Then:

(a) The tuple $(Σ, h, K, ρ, j)$ satisfies the constraint equations for $(κ, Λ)$ in the sense of Spacelike initial data, the constraint equations, developments and maximal developments clause (b), $G (n, n) - Λ = κ ρ, G (n, V) = κ T (n, V) for every V \in T_{p} Σ, p \in Σ;$ equivalently $G (n, V) = - κ j (V)$ on $Σ$ .

(b) The fields $ρ$ and $j$ are smooth on $Σ$ : in every slice chart their expressions in the slice coordinates are $C^{\infty}$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold with metric-selected connection, a smooth symmetric field $T$ , constants $κ, Λ$ , a spacelike embedded hypersurface $Σ$ with data $(h, K, ρ, j)$ of the definition, and a smooth vector field $V$ tangent to $Σ$ at the point under consideration.

[L1]

The Einstein tensor is $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ , a smooth covariant $2$ -tensor field on $M$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[L2]

The future unit normal $n$ of $Σ$ is smooth along $Σ$ and satisfies $g (n, n) = - 1$ with $g_{p} (n_{p}, N_{p}) < 0$ ; the induced metric $h$ and the second fundamental form $K$ of Spacelike initial data, the constraint equations, developments and maximal developments clause (a) are as constructed there (The future unit normal, induced metric and second fundamental form of a spacelike hypersurface clauses (a)-(c), Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (a)).

[L3]

A covariant $2$ -tensor field is a family of bilinear maps varying smoothly in charts; evaluating it on smooth vector fields (or on one smooth field twice) therefore gives a smooth function, and $T$ is symmetric (Smooth covariant 2-tensor fields on a smooth manifold, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L4]

For every $p \in Σ$ the equation $G + Λ g = κ T$ holds at $p$ ; both sides are evaluated on pairs of vectors of $T_{p} M$ , so equality is preserved under evaluation on any pair (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, given).

[L5]

In a slice chart $(U, y)$ with $Σ \cap U = {y^{4} = 0}$ , the restriction of $n$ to $Σ \cap U$ is expressible in the slice coordinates $y^{1}, y^{2}, y^{3}$ with $C^{\infty}$ coefficients, because the local extension $\tilde{n}$ of [L2] has $C^{\infty}$ components in $y$ and is evaluated at $y^{4} = 0$ (Embedded submanifolds and slice charts).

Proof

technique · direct

1.1givenL2L3L4

(Normal projection.) Let $p \in Σ$ and evaluate the field equation at $p$ on the pair $(n_{p}, n_{p})$ , an evaluation of the smooth covariant $2$ -tensor field $G$ of [L1]; by [L4] and [L3], $G (n_{p}, n_{p}) + Λ g_{p} (n_{p}, n_{p}) = κ T (n_{p}, n_{p})$ . By [L2], $g_{p} (n_{p}, n_{p}) = - 1$ , and $T (n_{p}, n_{p}) = ρ (p)$ by the definition of $ρ$ ; hence $G (n_{p}, n_{p}) - Λ = κ ρ (p)$ .

1.2givenL2L3L4

(Normal-tangential projection.) Let $p \in Σ$ and $V \in T_{p} Σ$ , and evaluate the field equation at $p$ on the pair $(n_{p}, V)$ ; by [L4] and [L3], $G (n_{p}, V) + Λ g_{p} (n_{p}, V) = κ T (n_{p}, V)$ . By [L2], $V$ is tangent to $Σ$ and $n_{p}$ is normal, so $g_{p} (n_{p}, V) = 0$ ; hence $G (n_{p}, V) = κ T (n_{p}, V) = - κ j (V)$ by the definition of $j$ .

2.1step 1.1step 1.2given

(Conclusion for the constraints.) Steps 1.1 and 1.2 give the two displays of clause (a) for arbitrary $p \in Σ$ and $V \in T_{p} Σ$ , which is exactly the condition defining that $(Σ, h, K, ρ, j)$ satisfies the constraint equations in Spacelike initial data, the constraint equations, developments and maximal developments clause (b).

3.1step 1.1step 1.2givenL3L5∎

(Smoothness of the matter data.) In a slice chart $(U, y)$ as in [L5], the components $n^{a}$ of the normal, the components $g_{a b}$ of the metric and the components $T_{a b}$ of the stress-energy field are $C^{\infty}$ functions of $(y^{1}, y^{2}, y^{3})$ on $Σ \cap U$ ; hence $ρ = T_{a b} n^{a} n^{b}$ and $j (V) = -$ the corresponding contraction of $T$ with $n$ and $V$ have $C^{\infty}$ coordinate expressions in the slice coordinates, for every smooth tangent vector field $V$ on $Σ$ . Since the slice charts cover $Σ$ , $ρ$ is a smooth function on $Σ$ and $j$ a smooth field on $Σ$ (a covariant $1$ -tensor field in the sense of the restriction to $Σ$ of the formulas above). This is (b) and completes the proof.

Remarks

What is and is not claimed. The lemma is a two-line evaluation identity: it says that a solution of the field equation on the ambient manifold has data satisfying the constraint equations. It does not prove that arbitrary data satisfying the constraints admit a development, and it does not derive the equivalent form of the constraints in terms of the induced scalar curvature and the divergence of $K$ ; that form requires the Gauss-Codazzi relations and is deferred with the PDE treatment.
Smoothness. Clause (b) records that the matter data are genuine smooth fields on $Σ$ , which is what the definition asserts; the proof reads the components in slice charts and uses smoothness of $n$ from the normal lemma.

The normal and normal-tangential projections of the field equation are the constraint equations

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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