The many ways of the characteristic Cauchy problem

Consider two smooth hypersurfaces N_a, a = 1, 2, in an (n + 1)-dimensional manifold ${\mathscr M}$, with transverse intersection along a smooth submanifold S. As before, we choose adapted null coordinates (x¹, x², x^A) so that N₁ coincides with the set {x¹ = 0}, while N₂ is given by {x² = 0}. We use a 'generalized wave-map gauge' as in [27], with target metric $\hat{g}$ of the form

$$\begin{eqnarray*} &&\hat{g} = 2\,\mathrm{d}x^1\,\mathrm{d}x^2 + \hat{g}_{AB}(x^1,x^2,x^C)\, \mathrm{d}x^A\,\mathrm{d}x^B . \end{eqnarray*}$$

Here $\hat{g}_{AB}$ is any family of Riemannian metrics on S parameterized by x¹ and x², smooth in all variables. The metric $\hat{g}$ is only introduced so that the harmonicity vector H^μ, defined in equation (18), is a vector field, and plays no significant role in what follows.

As gravitational initial data on the initial hypersurfaces, we prescribe all metric components $\overline{g}_{\mu \nu }$ in the coordinates above, as well as a connection coefficient κ; this needs to be supplemented by the initial data $\overline{\phi }$ for ϕ. For instance, in the Einstein–Vlasov case, the supplementary data will be a function $\overline{f}$ defined on the mass-shell $\lbrace \overline{g}_{\mu \nu }p^\mu p^\nu =-m^2\rbrace$, viewed as a subset of the pull-back of $T{\mathscr M}$ to the N_a's.

The hypersurfaces N_a are supposed to be characteristic, which is equivalent to the requirement that, in the coordinates above, on N₁ the metric takes the form

$$\begin{eqnarray} &&{g}|_{N_1}=\overline{g}_{11}(\mathrm{d}x^{1})^{2}+2\overline{g}_{12}\,\mathrm{d}x^{1}\,\mathrm{d}x^{2} +2\overline{g}_{1A}\,\mathrm{d}x^{1}\,\mathrm{d}x^{A}+ \underbrace{\overline{g}_{AB}\,\mathrm{d}x^{A}\,\mathrm{d}x^{B}}_{=:\tilde{g}} , \end{eqnarray} \tag{ 3.2 }$$

similarly on N₂. Here, and elsewhere, the terminology and notation of [27] are used; in particular, an overbar denotes restriction to the initial-data surface N₁∪N₂. (Some obvious renamings need to be applied to the equations in [27], for instance, the variable u there is x¹ on N₁, and x² on N₂; the variable r there is x² on N₁ and x¹ on N₂.)

We want to apply Rendall's existence theorem [14] for an appropriately reduced system of equations. For this, the trace, $\overline{g}_{\mu \nu }$, of the metric on the initial surface N₁∪N₂ needs to be the restriction of a smooth Lorentzian spacetime metric. This will be the case if $\overline{g}_{12}$ is nowhere vanishing, if $\overline{g}_{AB}|_{N_a}$ is a family of Riemannian metrics and if $\overline{g}_{\mu \nu }$ is smooth on N₁ and N₂ and continuous across S ≡ N₁∩N₂. We therefore need to impose the following continuity conditions on S:

$$\begin{eqnarray} &&\lim _{x^2\rightarrow 0} g_{AB}|_{N_1}= \lim _{x^1\rightarrow 0}g_{AB}|_{N_2} , \end{eqnarray} \tag{ 3.3a }$$

$$\begin{eqnarray} &&\lim _{x^2\rightarrow 0}g_{12}|{N_1} = \lim _{x^1\rightarrow 0}g_{12}|{N_2} , \end{eqnarray} \tag{ 3.3b }$$

$$\begin{eqnarray} &&\lim _{x^2\rightarrow 0}g_{1A}|{N_1}= 0 , \quad \lim _{x^1\rightarrow 0} g_{2A}|{N_2} = 0 , \end{eqnarray} \tag{ 3.3c }$$

$$\begin{eqnarray} &&\lim _{x^2\rightarrow 0} g_{11}|{N_1} = 0 , \quad \lim _{x^1\rightarrow 0}g_{22}|{N_2}= 0 . \end{eqnarray} \tag{ 3.3d }$$

Let H^μ be the harmonicity vector, defined as

$$\begin{eqnarray} &&H^{\lambda } := g^{\alpha \beta } \Gamma _{\alpha \beta }^{\lambda }-W^{\lambda } , \hspace{5.0pt}\mathrm{with}\hspace{5.0pt}W^{\lambda }:= \underbrace{g^{\alpha \beta } \hat{\Gamma }_{\alpha \beta }^{\lambda }}_{=:{\hat{W}}^\lambda } + {\mathring{W}}^{\lambda } , \end{eqnarray} \tag{ 3.4 }$$

where ${\mathring{W}}^\lambda$ will be a vector depending only upon the coordinates, and determined by the initial data in a way to be described below. (In principle, ${\mathring{W}}^\lambda$ can be allowed to depend on the metric as well, but not on derivatives of the metric.) To obtain a well-posed system of evolution equations, we will impose the generalized wave-map gauge condition

$$\begin{eqnarray*} &&H^{\lambda }=0 . \end{eqnarray*}$$

More precisely, we view the wave-map gauge constraints [27] as equations for the restriction $\overline{{\mathring{W}}}{}^\mu$ of ${\mathring{W}}{}^\mu$ to N₁ and N₂. We will solve those equations hierarchically. We emphasize that, assuming (15), all components of the energy–momentum tensor restricted to N₁ and N₂ are explicitly known since $\overline{g}_{\mu \nu }$ and $\overline{\phi }$ are.

We present the calculations on N₁; the equations on N₂ are obtained by interchanging index 1 with index 2 in all the formulae.

Let S_μν denote the Einstein tensor. In the notation and terminology of [27], the first constraint, arising from the equation $\overline{S}_{22}\equiv \overline{R}_{22} =\overline{T}_{22}$ evaluated on N₁, reads (see [27, equation (6.11)])

$$\begin{eqnarray} &&{-}\partial _2\tau + { \kappa }\tau - |\sigma |^2 - \frac{\tau ^2}{n-1} = \overline{T}_{22} , \end{eqnarray} \tag{ 3.5 }$$

where τ and σ are defined as in (4) and (5), respectively. Indeed, using the formulae in [27, appendix A], one finds

$$\begin{eqnarray} \overline{\partial _1\Gamma ^1_{22}} &=&\partial _2\overline{\Gamma }{}^1_{12} + \big(\overline{\Gamma }{}^1_{12}\big)^2- \overline{\Gamma }{}^1_{12}\overline{\Gamma }{}^2_{22} \qquad \quad \Longrightarrow \quad \nonumber \\ \overline{S}_{22} &=& \overline{\partial _1\Gamma ^1_{22}} - \partial _2\big(\overline{\Gamma }{}^1_{12} -\overline{\Gamma }{}^A_{2A}\big) + \big(\overline{\Gamma }{}^1_{12}+\overline{\Gamma }{}^A_{2A}\big)\overline{\Gamma }{}^2_{22} - \big(\overline{\Gamma }{}^1_{12}\big)^2 -\overline{\Gamma }{}^A_{2B}\overline{\Gamma }{}^B_{2A} \nonumber \\ &=&- \partial _2\overline{\Gamma }{}^A_{2A} + \overline{\Gamma }{}^A_{2A}\overline{\Gamma }{}^2_{22} -\overline{\Gamma }{}^A_{2B}\overline{\Gamma }{}^B_{2A} \nonumber \\ &=& -\partial _2 \tau + \tau \overline{\Gamma }{}^2_{22} - \chi _A{}^B\chi _B{}^A , \end{eqnarray} \tag{ 3.6 }$$

where

$$\begin{eqnarray*} &&\chi _A{}^B := \case{1}{2}\overline{g}^{BC}\partial _2\overline{g}_{AC} . \end{eqnarray*}$$

Here we adapt the point of view that the function κ is the value on N₁ of the Christoffel coefficient Γ²₂₂, and is part of the initial data. Hence, we view (19) as a constraint equation linking $\overline{g}_{AB}$, κ, and the matter sources (if any).

In the region where τ has no zeros, equation (19) can be trivially solved for κ to give

$$\begin{eqnarray} \kappa &=&\frac{\partial _2\tau + \frac{1}{n-1}\tau ^2 + |\sigma |^2 + \overline{T}_{22}}{\tau } . \end{eqnarray} \tag{ 3.7 }$$

It follows that κ does not need to be included as part of initial data when τ has no zeros, and then the constraint equation (19) can be replaced by the last equation, determining κ.

Equation (21) can still be used, by continuity, to determine κ on the closure of the set where τ has no zeros, keeping in mind that the requirement of smoothness of the function so determined imposes non-trivial constraints on the right-hand side. In any case, equation (21) does not make sense if there are open regions where τ vanishes. It seems therefore best to assume that κ is any smooth function on N₁ such that (19) holds, and view that last equation as a constraint equation relating κ, $\overline{g} _{AB}$ and its derivatives, and the matter fields; similarly on N₂.

It should be kept in mind that once a candidate solution of the Einstein equations has been constructed, one needs to verify that κ is indeed the value of Γ²₂₂ on N₁. We will return to this in (39).

We choose $\overline{{\mathring{W}}}{}^ 1$ to be

$$\begin{eqnarray} \overline{{\mathring{W}}}{}^1 &:=& -\overline{{\hat{W}}}{}^1 -\overline{g}^{12}(2\kappa +\tau ) - 2 \partial _2\overline{g}^{12} . \end{eqnarray} \tag{ 3.8 }$$

Note that the right-hand side is known, so this defines $\overline{{\mathring{W}}}{}^1$. By definition, this is the ∂₁ component of ${\mathring{W}}^\mu$ in the coordinate system (x¹, x², x^A). We will define the remaining components of ${\mathring{W}}^\mu$ shortly, the resulting collection of functions transforming by definition as a vector when changing coordinates.

From [27, appendix A], one then finds

$$\begin{eqnarray} &&\overline{\Gamma }{}^2 _ {22} = \kappa - \case{1}{2} \overline{g_{12}}\overline{H}{} ^1 , \end{eqnarray} \tag{ 3.9 }$$

and (20) together with (19) give

$$\begin{eqnarray} &&\overline{S}_{22} -\overline{T}_{22} = - \case{1}{2} \overline{g_{12}}\overline{H}{} ^1 \tau . \end{eqnarray} \tag{ 3.10 }$$

The corresponding constraint equation on N₂ determines ${\mathring{W}}{}^2|_{N_2}$. We shall return to the question of continuity at S of ${\mathring{W}}{}^1|_{N_1\cup N_2}$ and of ${\mathring{W}}{}^2 |_{N_1\cup N_2}$ shortly.

The next constraint equation follows from $\overline{S}_{2A} \equiv \overline{R}_{2A} =\overline{T}_{2A}$. From the formulae in [27, appendix A], we find

$$\begin{eqnarray*} \overline{\partial _1\Gamma ^1_{2A}} &=& \partial _A\overline{\Gamma }{}^1_{12} +\overline{\Gamma }{}^1_{12}\big(\overline{\Gamma }{}^1_{1A}-\overline{\Gamma }{}^2_{2A}\big) + \overline{\Gamma }{}^B_{12}\overline{\Gamma }{}^1_{AB} -\overline{\Gamma }{}^B_{2A}\overline{\Gamma }{}^1_{1B} , \end{eqnarray*}$$

which gives

$$\begin{eqnarray} \overline{S}_{2A} &=&\overline{\partial _1\Gamma ^1_{2A}} + \partial _2\overline{\Gamma }{}^2_{2A} + \partial _B\overline{\Gamma }{}^B_{2A} - \partial _A\overline{\Gamma }{}^1_{12} - \partial _A\overline{\Gamma }{}^2_{22} - \partial _A\overline{\Gamma }{}^B_{2B} + \overline{\Gamma }{}^1_{1B}\overline{\Gamma }{}^B_{2A} \nonumber \\ &&+ \overline{\Gamma }{}^1_{12}\big(\overline{\Gamma }{}^2_{2A} - \overline{\Gamma }{}^1_{1A}\big) + \overline{\Gamma }{}^B_{2B}\overline{\Gamma }{}^2_{2A} + \overline{\Gamma }{}^B_{BC}\overline{\Gamma }{}^C_{2A} - \overline{\Gamma }{}^1_{AB}\overline{\Gamma }{}^B_{12} - \overline{\Gamma }{}^B_{AC}\overline{\Gamma }{}^C_{1B} \nonumber \\ &=&\partial _2 \overline{\Gamma }{}^2_{2A} + \partial _B\overline{\Gamma }{}^B_{2A} - \partial _A\overline{\Gamma }{}^2_{22} - \partial _A\overline{\Gamma }{}^B_{2B} + \overline{\Gamma }{}^B_{2B}\overline{\Gamma }{}^2_{2A} + \overline{\Gamma }{}^B_{BC}\overline{\Gamma }{}^C_{2A} - \overline{\Gamma }{}^B_{AC}\overline{\Gamma }{}^C_{2B} \nonumber \\ &=& (\partial _2 + \tau )\overline{\Gamma }{}^2_{2A} + \tilde{\nabla }_B\chi _A{}^B - \partial _A\overline{\Gamma }{}^2_{22} - \partial _A\tau , \end{eqnarray} \tag{ 3.11 }$$

where $\tilde{\nabla }$ is the covariant derivative associated with the Riemannian metric g_AB. That leads us to the equation

$$\begin{eqnarray} &&{-}\frac{1}{2}(\partial _2 + \tau )\xi _A + \tilde{\nabla }_B \sigma _A^{\phantom{A}B} - \frac{n-2}{n-1}\partial _A\tau -\partial _A \kappa = \overline{T}_{2A} , \end{eqnarray} \tag{ 3.12 }$$

where the field ξ_A denotes the restriction of the (rescaled) Christoffel coefficient −2Γ²_2A to N₁. We determine ξ_A by integrating (26), with the freedom to prescribe

$$\begin{eqnarray*} &&\xi _A^{N_1}:= \xi _A(x^2 =0) \end{eqnarray*}$$

on S. (One should keep in mind that ξ_A here is unrelated to the corresponding field ξ_A on N₂, determined by an analogous equation where all quantities τ, σ, etc, are calculated using the fields on N₂.) We then define $\overline{{\mathring{W}}}{}^A$ through the formula

$$\begin{eqnarray} &&\fl \overline{{\mathring{W}}}{}^A :=\overline{g}^{AB}[\xi _B + 2\overline{g}^{12} (\partial _2\overline{g}_{1B} - 2 \overline{g}_{1C}\sigma _B{}^C -\overline{g}_{1B} \tau ) -\overline{g}_{1B}(\overline{{\mathring{W}}}{}^1 + \overline{{\hat{W}}}{}^1 ) ] +\overline{g}^{CD}\tilde{\Gamma }^A_{CD} - \overline{\hat{W}}{}^A; \end{eqnarray} \tag{ 3.13 }$$

equivalently

$$\begin{eqnarray} \xi _A &=& -2\overline{g}^{12}\partial _2\overline{g}_{1A} + 4\overline{g}^{12}\overline{g}_{1B}\sigma _A{}^B + 2\overline{g}^{12}\overline{g}_{1A}\tau + \overline{g}_{1A}(\overline{{\mathring{W}}}{}^1+\overline{{\hat{W}}}{}^1) \nonumber \\ &&+\, \overline{g}_{AB}(\overline{{\mathring{W}}}{}^B + \overline{{\hat{W}}}{}^B) - \overline{g}_{AB} \overline{g}^{CD} \tilde{\Gamma }{}^B_{CD} . \end{eqnarray} \tag{ 3.14 }$$

This has been chosen so that, using the formulae in [27, appendix A and section 9],

$$\begin{eqnarray} &&\overline{S}_{2A}-\overline{T}_{2A} = -\case{1}{2}(\partial _{2}+\tau )(\overline{g}_{AB}\overline{H}{}^{B}+\overline{g}_{1A}\overline{H}{}^{1}) +\case{1}{2}\partial _{A}(\overline{g}_{12}\overline{H}{}^{1}). \end{eqnarray} \tag{ 3.15 }$$

Moreover, one finds (cf equation (10.35) in [27])

$$\begin{eqnarray} &&\xi _A = -2\overline{\Gamma }{}^2_{2A} - \overline{g}_{AB} \overline{H}{}^B - \overline{g}_{1A} \overline{H}{}^1 . \end{eqnarray} \tag{ 3.16 }$$

On S, equation (27) takes the form

$$\begin{eqnarray*} &&\overline{{\mathring{W}}}{}^A |_S=\overline{g}^{AB}\big[\xi _B^{N_1} + 2\overline{g}^{12} \partial _2\overline{g}_{1B}\big] +\overline{g}^{BC}(\tilde{\Gamma }^A_{BC} - \hat{\Gamma }^A_{BC}) . \end{eqnarray*}$$

Keeping in mind the corresponding equation on N₂,

$$\begin{eqnarray*} &&\overline{{\mathring{W}}}{}^A |_S=\overline{g}^{AB}\big[\xi _B^{N_2} + 2\overline{g}^{12} \partial _1\overline{g}_{2B} \big] +\overline{g}^{BC}(\tilde{\Gamma }^A_{BC} - \hat{\Gamma }^A_{BC}) , \end{eqnarray*}$$

the requirement of continuity of ${\mathring{W}}{}^A|_{N_1 \cup N_2}$ leads to

$$\begin{eqnarray} &&\xi _A^{N_1} - \xi _A^{N_2} = 2 g^{12} ( \partial _1 g_{2A} - \partial _2 g_{1A} )|_S \equiv 4 \zeta _A . \end{eqnarray} \tag{ 3.17 }$$

Recall that the torsion one-form ζ_A has been defined in (14).

We continue with the equation $\overline{S}_{12} = \overline{T}_{12}$, or, equivalently,

$$\begin{eqnarray*} &&\overline{g}^{AB}\overline{R}_{AB} = -(2\overline{g}^{12} \overline{T}_{12}+\overline{g}^{22}\overline{T}_{22} + 2\overline{g}^{2A}\overline{T}_{2A})= \overline{g}^{AB} \overline{T}_{AB} -\overline{T} , \end{eqnarray*}$$

which we handle in a manner similar to the previous equations. Using identities (10.33) and (a corrected version of²) (10.36) in [27], we find that on N₁, we have

$$\begin{eqnarray*} &&\fl \overline{g}^{AB} \overline{R}_{AB} \equiv \big(\partial _2 + \overline{\Gamma }{}^2_{22} + \tau \big)\big(2\overline{g}^{AB} \overline{\Gamma }{}^2_{AB} +\tau \overline{g}^{22}\big) - 2\overline{g}^{AB} \overline{\Gamma }{}^2_{2A} \overline{\Gamma }{}^2_{2B} - 2 \overline{g}^{AB} \tilde{\nabla }_A \overline{\Gamma }{}^2_{2B} + \tilde{R} , \end{eqnarray*}$$

and we are led to the equation

$$\begin{eqnarray} &&(\partial _2 + \kappa + \tau ) \zeta + \big ( \tilde{\nabla }_A - \case{1}{2}\xi _A\big ) \xi ^A + \tilde{R} = \overline{g}^{AB} \overline{T}_{AB} - \overline{T} , \end{eqnarray} \tag{ 3.18 }$$

with $\xi ^A:=\overline{g}^{AB} \xi _B$, and where the quantity ζ denotes the restriction of

$$\begin{eqnarray*} &&2 \big (\overline{g}^{AB} \overline{\Gamma }{}^2_{AB}+ \tau \overline{g}^{22} \big ) \end{eqnarray*}$$

to N₁. We integrate (32), viewed as a first-order ODE for ζ. The initial data on S are determined by the requirement of continuity of $\overline{{\mathring{W}}}{}^2$ at S, which we choose to be

$$\begin{eqnarray} &&\overline{{\mathring{W}}}{}^2 := \case{1}{2} \zeta - \big(\partial _{2}+\kappa + \case{1}{2} \tau \big)\bar{g}^{22}-\overline{{\hat{W}}}{}^2. \end{eqnarray} \tag{ 3.19 }$$

Indeed, recall that $\overline{{\mathring{W}}}{}^2|_S$ has already been calculated algebraically when analysing the first constraint equation on N₂, in exactly the same way as we calculated $\overline{{\mathring{W}}}{}^1$ in the first step of the analysis above.

Similarly the initial data for the integration of the constraint which determines ${\mathring{W}}{}^1|_{N_2}$ are determined by the requirement of continuity of ${\mathring{W}}{}^1|_{N_1\cup N_2}$.

The choice (33) has been made so that

$$\begin{eqnarray} && \overline{g}^{AB} \overline{R}_{AB} \!-\! \overline{g}^{AB} \overline{T}_{AB} +\overline{T} \!=\! \big(\partial _2 +\kappa + \tau -\case{1}{2}\overline{g}_{12} \overline{H}{}^1 \big) (2\overline{H}{}^2 - \overline{g}_{12}\overline{g}^{22}\overline{H}{}^1 ) -\case{1}{2}\zeta \overline{g}_{12} \overline{H}{}^1 \nonumber \\ &&\quad+\,\big(\tilde{\nabla }_A- \xi _A - \case{1}{2}\overline{g}_{AB}\overline{H}{}^B - \case{1}{2}\overline{g}_{1A}\overline{H}{}^1\big)( \overline{H}{}^A + \overline{g}_{1C} \overline{g}^{AC}\overline{H}{}^1). \end{eqnarray} \tag{ 3.20 }$$

We note that our choice of $\overline{{\mathring{W}}}{}^2$ is equivalent to

$$\begin{eqnarray} &&\zeta = 2\overline{g}^{AB}\overline{\Gamma }{}^2_{AB} + \tau \overline{g}^{22} + \overline{g}_{12}\overline{g}^{22}\overline{H}{}^1 - 2 \overline{H}{}^2 . \end{eqnarray} \tag{ 3.21 }$$

Summarizing, given the fields κ, $\overline{g}_{\mu \nu }$ and $\overline{\phi }$ on N₁∪N₂, satisfying (31), and the sum $\xi ^{N_1}_A+\xi ^{N_2}_A$ on S, we have found a unique continuous vector field $\overline{{\mathring{W}}}$ on N₁∪N₂, smooth up-to-boundary on N₁ and N₂, so that (24), (29) and (34) hold on N₁∪N₂. Letting ${\mathring{W}}$ be any smooth vector field on ${\mathscr M}$ which coincides with $\overline{{\mathring{W}}}$ on N₁∪N₂, and assuming that the reduced Einstein equations (see (43) below) can be complemented by well-posed evolution equations for the matter fields, we obtain a metric, solution of the Cauchy problem for the reduced Einstein equations, in a future neighbourhood of S.

However, the metric so obtained will solve the full Einstein equations if and only if [27] H^μ vanishes on N₁∪N₂, so we need to ensure that this condition holds. Note that at this stage, a smooth metric g, satisfying the reduced Einstein equations, and a smooth vector field W^μ are known to the future of N₁∪N₂ in some neighbourhood of S, and thus H^μ is a known smooth vector field there.

By [27, section 7.6], $\overline{H} {}^1$ will vanish on N₁ if and only if $\overline{H} {}^1$ vanishes on S. Using (22) and (23) together with the equations in [27, appendix A], we find

$$\begin{eqnarray} &&H^1|_S = (g^{12})^2 \partial _1 g_{22} + 2 g^{12} \kappa + 2 \partial _2 g^{12} . \end{eqnarray} \tag{ 3.22 }$$

We conclude that $H{}^1|_{N_1}$ will vanish if and only if the initial data $\overline{g}_{22}$ on N₂ have the property that the derivative ∂₁g₂₂ on S satisfies

$$\begin{eqnarray} &&\partial _1 g_{22}|_S = 2( \partial _2 g_{12} -g_{12} \kappa _{N_1}) \quad \Longleftrightarrow \quad \Gamma ^2_{22}|_S = \kappa _{N_1}, \end{eqnarray} \tag{ 3.23 }$$

where, to avoid ambiguities, we denote by $\kappa _{N_a}$ the function κ associated with the hypersurface N_a, etc. Similarly, $H{}^2|_{N_2}$ will vanish if and only if we choose g₁₁ on N₁ so that

$$\begin{eqnarray} &&\partial _2 g_{11}|_S = 2 ( \partial _1 g_{12} -g_{12} \kappa _{N_2}) \quad \Longleftrightarrow \quad \Gamma ^1_{11}|_S = \kappa _{N_2} . \end{eqnarray} \tag{ 3.24 }$$

With those choices, we have H¹|_S = H²|_S = 0, and the arguments in [27] show that $H^1|_{N_2}= H^2|_{N_1}=0$ as well.

We continue with $\overline{H}^A$. Then by equation (30), we have

$$\begin{eqnarray*} &&\xi _A^{N_1} = -\big(2\Gamma ^{2}_{2A} + g_{AB} H^B + g_{1A}H^1\big)\big|_S \\ &&\hphantom{\xi _A^{N_1} }= -\big(\overline{g}^{12}\big(\partial _A \overline{g}_{12}-\overline{\partial _1g_{2A}}+ \partial _2\overline{g}_{1A}\big) + g_{AB} H^B + g_{1A}H^1\big)\big|_S , \\ &&\xi _A^{N_2} = -\big(2\Gamma ^{1}_{1A} + g_{AB} H^B + g_{1A}H^1\big)\big|_S \\ &&\hphantom{\xi _A^{N_2} }=-\big(\overline{g}^{12}\big(\partial _A \overline{g}_{12}-\overline{\partial _2g_{1A}}+ \partial _1\overline{g}_{2A}\big) + g_{AB} H^B + g_{1A}H^1\big)\big|_S , \end{eqnarray*}$$

and the conditions HA|S = 0 and H1|S = 0 determine ξNaA in terms of the remaining data. Note that (31) is then automatically satisfied, and that we loose the freedom to prescribe ξN1A + ξN2A.

It remains to show that our choice of the parameterization of the null rays is consistent, i.e. we have to make sure that the relations $\Gamma ^2_{22}|_{N_1}=\kappa _{N_1}$ and $\Gamma ^1_{11}|_{N_2}=\kappa _{N_2}$ hold. This follows trivially from the vanishing of the wave-gauge vector H due to the identities

$$\begin{eqnarray} &&H^1|_{N_1} \equiv 2g^{12} \big(\kappa - \Gamma ^2_{22}\big) \quad \mathrm{and} \quad H^2|_{N_2} \equiv 2g^{12} \big(\kappa - \Gamma ^1_{11}\big). \end{eqnarray} \tag{ 3.25 }$$

Similarly, the vanishing of $\overline{H}{}^1$ and $\overline{H}{}^A$ shows via (30) that the identification of ξ_A with the rescaled Christoffel coefficient $-2\overline{\Gamma }{}^2_{2A}$ on N₁ and $-2\overline{\Gamma }{}^1_{1A}$ on N₂ is consistent. The vanishing of $\overline{H}{}^1$ and $\overline{H}{}^2$ together with the identity (35) implies that on N₁ the field ζ indeed represents the value of $2\overline{g}^{AB} \overline{\Gamma }{}^2_{AB} + \tau \overline{g}^{22}$, and the corresponding field on N₂ represents the value of $2\overline{g}^{AB} \overline{\Gamma }{}^1_{AB} + \tau \overline{g}^{11}$ there.

In particular, the above provides a new and simple integration scheme for the vacuum Einstein equations, where all the metric functions are freely prescribable on N₁∪N₂:

Theorem 3.1. Given any continuous functions $(\kappa ,\overline{g}_{\mu \nu })$ on N₁∪N₂ such that

$$\begin{eqnarray} &&g|_{N_1}=\overline{g}_{22}(\mathrm{d}x^{2})^{2}+2\overline{g}_{12}\,\mathrm{d}x^{1}\,\mathrm{d}x^{2} +2\overline{g}_{2A}\,\mathrm{d}x^{2}\,\mathrm{d}x^{A}+ \overline{g}_{AB}\,\mathrm{d}x^{A}\,\mathrm{d}x^{B} , \end{eqnarray} \tag{ 3.26a }$$

$$\begin{eqnarray} &&g|_{N_2}=\overline{g}_{11}(\mathrm{d}x^{1})^{2}+2\overline{g}_{12}\,\mathrm{d}x^{1}\,\mathrm{d}x^{2} +2\overline{g}_{1A}\,\mathrm{d}x^{1}\,\mathrm{d}x^{A}+ \overline{g}_{AB}\,\mathrm{d}x^{A}\,\mathrm{d}x^{B} , \end{eqnarray} \tag{ 3.26b }$$

smooth up-to-boundary on N₁ and N₂, and satisfying (37)–(38) together with the vacuum constraint equations (here $\kappa _{N_1}:= \kappa |_{N_1}$, etc)

$$\begin{eqnarray} &&{-}\partial _2\tau _{N_1} + { \kappa _{N_1}} \tau _{N_1} - |\sigma _{N_1}|^2 - \frac{\tau ^2 _{N_1}}{n-1} = 0 \ \mbox{on $N_1$} , \end{eqnarray} \tag{ 3.27a }$$

$$\begin{eqnarray} &&{-}\partial _1\tau _{N_2} + { \kappa _{N_2}} \tau _{N_2} - |\sigma _{N_2}|^2 - \frac{\tau ^2 _{N_2}}{n-1} =0 \ \mbox{on $N_2$} , \end{eqnarray} \tag{ 3.27b }$$

there exists a smooth metric defined on some neighbourhood of S, solution of the vacuum Einstein equations to the future of N₁∪N₂.

Note that all the conditions are necessary. To see this, let g be any metric solving the Einstein equations to the future of N₁∪N₂, with N_a characteristic. We can introduce adapted coordinates near N₁∪N₂ so that (40a) and (40b) hold. Constraints (41a) and (41b) follow then from the Einstein equations [27], while (37) and (38) follow from our calculations above.

Proof. While the main elements of the proof have already been given, to avoid ambiguities, we summarize the argument: let $(\kappa ,\overline{g})$ be given as above. Set

$$\begin{eqnarray*} &&{\mathscr M}:=[0,\infty )\times [0,\infty )\times S , \end{eqnarray*}$$

where the first [0, ∞) factor refers to the variable x¹, and the second to x². On ${\mathscr M}$ let $\hat{g}$ be the metric $\hat{g} =2 {\rm d}x^1 {\rm d}x^2 + \phi _{AB}{\rm d}x^A {\rm d}x^B$, where ϕ_ABdx^Adx^B is a Riemannian metric on S. Let $\overline{\mathring{W}}{}^\mu$ be constructed as above. Let W^μ be any smooth extension of $\overline{ W}{}^\mu$ to ${\mathscr M}$, and let g be the solution of the wave-map-reduced Einstein equations R^(H)_αβ = 0, with initial data $\overline{g}$, where

$$\begin{eqnarray} &&R_{\alpha \beta }^{(H)}:=R_{\alpha \beta } -{\case{1}{2}}(g_{\alpha \lambda }\hat{D}_{\beta }H^{\lambda }+g_{\beta \lambda }\hat{D}_{\alpha }H^{\lambda }) , \end{eqnarray} \tag{ 3.28 }$$

with H^μ defined by (18), and where $\hat{D}$ is the Lévi-Cività covariant derivative in the metric $\hat{g}$. (It follows from [33, page 163] that R^(H)_αβ is a quasi-linear, quasi-diagonal operator on g, tensor-valued, depending on $\hat{g}$, of the form

$$\begin{eqnarray} &&R_{\alpha \beta }^{(H)}\equiv -{\case{1}{2}}g^{\lambda \mu } {\hat{D}}_{\lambda } {\hat{D}}_{\mu }g_{\alpha \beta }+ \hat{f}[g,{\hat{D}}g]_{\alpha \beta }, \end{eqnarray} \tag{ 3.29 }$$

where $\hat{f}[g,{\hat{D}}g]_{\alpha \beta }$ is a tensor quadratic in $\hat{D}g$ with coefficients depending upon g, $\hat{g}$, ${\mathring{W}}$, $\hat{D} {\hat{W}}$ and $\hat{D} {\mathring{W}}$; the existence of solutions of this problem follows from [14].) As $\overline{H}^\mu =0$ by construction, a standard argument shows that H^μ ≡ 0, and so g is a solution of the vacuum Einstein equations in a suitable neighbourhood of S in ${\mathscr M}$.

Now let us consider the same problem on a light-cone C_O with vertex O. We prescribe the metric functions $\overline{g}_{\mu \nu }$ on the cone in adapted null coordinates (cf [27]) as well as $\overline{\phi }$ and, if τ has zeros, κ (note that $\tau \equiv \frac{1}{2} \overline{g}^{AB} \partial _1 \overline{g}_{AB}$ has no zeros in a sufficiently small neighbourhood around the vertex). In this section, the notations and conventions from [27] are used again; in particular, the x¹-coordinate will be frequently denoted by r, and the light-cone is given as the surface {x⁰ ≡ u = 0}.

For C_O to be a characteristic cone, we need to have $\overline{g}_{11}=0=\overline{g}_{1A}$ in our adapted coordinates. To end up with a Lorentzian metric, the component $\nu _0\equiv \overline{g}_{01}$ has to be nowhere vanishing, while $\overline{g}_{AB}$ has to be a family of Riemannian metrics on S^{n − 1}. We consider initial data which satisfy

$$\begin{eqnarray} \overline{g}_{00} \,=\, -1 + O(r^2) , \quad &\partial _r\overline{g}_{00} \,=\, O(r) , \end{eqnarray} \tag{ 3.30a }$$

$$\begin{eqnarray} \nu _0 \,=\, 1 + O(r^2) , \quad &\partial _r\nu _0\,=\, O(r) , \end{eqnarray} \tag{ 3.30b }$$

$$\begin{eqnarray} \nu _A \,=\, O(r^3) , \quad &\partial _r \nu _A \,=\, O(r^2) , \end{eqnarray} \tag{ 3.30c }$$

$$\begin{eqnarray} \overline{g}_{AB} \,=\, r^2s_{AB} + O_2(r^4) , \quad &\partial _r\partial _C \overline{g}_{AB} \,=\, 2r\partial _C s_{AB} + O_1(r^3) , \end{eqnarray} \tag{ 3.30d }$$

$$\begin{eqnarray} \partial _r^2\partial _C\partial _D \overline{g}_{AB} \,=\, 2\partial _C\partial _Ds_{AB} & \hspace{-3.00003pt}+ O(r^2), \end{eqnarray} \tag{ 3.30e }$$

for small r, where f = O_n(r^α) means that ∂ⁱ_r∂_A^βf = O(r^{α − i}) for i + |β| ⩽ n. The tensor s_AB denotes the round sphere metric.

These conditions ensure that the metric is of the same form near the vertex as in [27]. The assumptions concerning the derivatives, which are compatible with relations (4.41)–(4.51) in [27], are made to compute the behaviour of $\overline{\mathring{W}}$ near the vertex³: although we do not attempt to tackle the regularity problem at the vertex here, as a necessary condition we want to make sure that $\overline{\mathring{W}}$ remains bounded near the vertex, which in our adapted coordinates means

$$\begin{equation*} \overline{\mathring{W}}{}^0 = O(1), \quad \overline{\mathring{W}}{}^1 = O(1), \quad \overline{\mathring{W}}{}^A = O(r^{-1}) . \end{equation*}$$

In fact it turns out that with (44a)–(44e) and the subsequent assumptions on the target metric and the energy–momentum tensor, the vector $\overline{\mathring{W}}$ goes to zero.

We present the scheme for an arbitrary target metric $\hat{g}$ that satisfies the relations

$$\begin{eqnarray} \hat{\nu }_0 &=& 1+ O_1(r^2), \quad \hat{\nu }_A \,= \,O_1(r^3) , \quad \overline{\hat{g}}_{00} \,= \, -1 + O(r^2), \end{eqnarray} \tag{ 3.31a }$$

$$\begin{eqnarray} \partial _r\overline{ \hat{g}}_{00} &= & O(r), \quad \overline{\hat{g}}_{AB}\, =\, r^2 s_{AB} + O_1(r^4), \end{eqnarray} \tag{ 3.31b }$$

$$\begin{eqnarray} \overline{\partial _0 \hat{g}_{11}} &=& O(r) , \quad \overline{\partial _0 \hat{g}_{1A}} \,= \,O(r^2) , \quad \overline{g}^{AB} \overline{\partial _0 \hat{g}_{AB}} \,= \, O(r) . \end{eqnarray} \tag{ 3.31c }$$

Again, these assumptions are to ensure that the behaviour of $\overline{\mathring{W}}$ can be determined at the vertex.

Additionally, we take a look at two particular target metrics: a Minkowski target $\hat{g} = \eta$ as in [27] and a target metric $\hat{g} = C$ which satisfies $\overline{C}= \overline{g}$ and which simplifies the expressions for the components of $\overline{\mathring{W}}$.

Let us now solve the constraint equations. The first constraint yields (supposing that τ has no zeros, the case where it does have zeros can be treated as the case of two transversally intersecting hypersurfaces)

$$\begin{eqnarray} \kappa &=&\frac{\partial _1\tau + \frac{1}{n-1}\tau ^2 + |\sigma |^2 + \overline{T}_{11}}{\tau } \end{eqnarray} \tag{ 3.32 }$$

and

$$\begin{eqnarray*} \overline{{\mathring{W}}}{}^0 &=& -\overline{{\hat{W}}}{}^0- \nu ^0(2\kappa +\tau ) - 2 \partial _1\nu ^0 . \end{eqnarray*}$$

If we assume⁴

$$\begin{eqnarray*} &&\overline{T}_{11} = O(1), \quad \partial _A\overline{T}_{11} = O(1), \quad \partial _A\partial _B\overline{T}_{11} = O(1), \end{eqnarray*}$$

we obtain with our assumptions (44a)–(44e) and with assumptions (45a) and (45b) concerning the target metric

$$\begin{eqnarray*} &&\kappa = O(r), \quad \partial _A\kappa = O(r) , \quad \partial _A\partial _B\kappa = O(r), \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&\overline{{\mathring{W}}}{}^0 = O(r) . \end{eqnarray*}$$

Let us write $\overset{\eta }{=}$ for an equality which holds when $\hat{g}$ is the Minkowski metric, with an obvious similar meaning for $\overset{C}{=}$. Then

$$\begin{eqnarray*} &&\overline{{\hat{W}}}{}^0 \overset{\eta }{=} -r \overline{g}^{AB} s_{AB}, \end{eqnarray*}$$

and also

$$\begin{eqnarray*} &&\overline{{\mathring{W}}}{}^0 \overset{C}{=} 2\nu ^0\big(\hat{\Gamma }^1_{11}- \kappa \big) . \end{eqnarray*}$$

From the second constraint equation, one first determines ξ_A. Recall that this is a first-order ODE. The integration constant which arises is determined by the requirement of finiteness of ξ_A at the vertex (cf [27, section 9.2]),

$$\begin{eqnarray} &&\fl \xi _A = \displaystyle 2\frac{{\rm e}^{-\int _1^r(\tau -\frac{n-1}{\tilde{r}})\,\mathrm{d}\tilde{r}}}{r^{n-1}}\int _0^r \tilde{r}^{n-1} {\rm e}^{\int _1^{\tilde{r}}(\tau -\frac{n-1}{\tilde{\tilde{r}}})\,\mathrm{d}\tilde{ \tilde{r}}} \left( \tilde{\nabla }_B\sigma _A{}^B - \frac{n-2}{n-1}\partial _A\tau - \partial _A \kappa - \overline{T}_{1A} \right)\,\mathrm{d}\tilde{r} . \end{eqnarray} \tag{ 3.33 }$$

If we assume

$$\begin{eqnarray*} &&\overline{T}_{1A} = O(r), \qquad \partial _B \overline{T}_{1A} = O(r)\; \end{eqnarray*}$$

and employ (44a)–(45c), we find

$$\begin{eqnarray*} &&\xi _A = O_1(r^2) . \end{eqnarray*}$$

The function $\overline{{\mathring{W}}}{}^A$ can then be computed algebraically,

$$\begin{eqnarray*} \overline{\mathring{W}}{}^A &=& \overline{g}^{AB} \xi _B + 2\nu ^0\overline{g}^{AB}(\partial _1\nu _B - 2\nu _C\chi _B{}^C) -\nu _B\overline{g}^{AB}(\overline{\mathring{W}}{}^0 + \overline{\hat{W}}{}^0) + \overline{g}^{BC} \tilde{\Gamma }^A_{BC} - \overline{\hat{W}}{}^A \\ &=& O(1) , \end{eqnarray*}$$

where

$$\begin{eqnarray*} &&\chi _A{}^B \equiv \case{1}{2}\overline{g}^{BC} \partial _1 \overline{g}_{AC} . \end{eqnarray*}$$

In particular,

$$\begin{eqnarray*} \overline{\hat{W}}{}^A&\overset{\eta }{=}& -\frac{2}{r}\nu ^0\overline{g}^{AB}\overline{g}_{0B} + \overline{g}^{BC} S^A_{BC} , \end{eqnarray*}$$

and

$$\begin{eqnarray*} \overline{\mathring{W}}{}^A&\overset{C}{=}& 2\overline{g}^{1A}\big(\hat{\Gamma }^1_{11}-\kappa \big) + \overline{g}^{AB}\big(2 \hat{\Gamma }^1_{1B} + \xi _B \big) . \end{eqnarray*}$$

The functions S^A_BC denote the Christoffel coefficients associated with the round sphere metric.

Finally, we have a first-order equation for

$$\begin{eqnarray} &&\zeta = (2\partial _1 + 2\kappa + \tau )\overline{g}^{11} + 2\overline{\mathring{W}}{}^1 + 2 \overline{\hat{W}}{}^1 . \end{eqnarray} \tag{ 3.34 }$$

It reads

$$\begin{eqnarray} &&\fl (\partial _1 + \kappa + \tau )\zeta + \tilde{R} + \overline{g}^{AB}\big(\tilde{\nabla }_A \xi _B - \case{1}{2}\xi _A\xi _B) + \overline{g}^{11} \overline{T}_{11} + 2\overline{g}^{1A}\overline{T}_{1A} + 2\nu ^0\overline{T}_{01} =0 . \end{eqnarray} \tag{ 3.35 }$$

This can be integrated,

$$\begin{eqnarray*} &&\fl \zeta = \frac{ {\rm e}^{-\int _1^r (\kappa +\tau -\frac{n-1}{\tilde{r}})\,\mathrm{d}\tilde{r}}}{r^{n-1}}\bigg[ c - \int _{0}^r \tilde{r}^{n-1}\, {\rm e}^{\int _1^{\tilde{r}} (\kappa +\tau -\frac{n-1}{\tilde{\tilde{r}}})\,\mathrm{d}\tilde{\tilde{r}}} \bigg( \tilde{R} + \overline{g}^{AB}\tilde{\nabla }_A \xi _B \\ &&-\, \frac{1}{2}\overline{g}^{AB}\xi _A\xi _B + \overline{g}^{11} \overline{T}_{11} + 2\overline{g}^{1A}\overline{T}_{1A} + 2 \nu ^0\overline{T}_{01} \bigg)\, \mathrm{d}\tilde{r}\bigg] , \end{eqnarray*}$$

where c is an integration constant.

Using again relations (44a)–(44e) and our assumptions on the target metric, we deduce that

$$\begin{eqnarray*} &&\overline{g}^{11} = 1 + O(r^2) , \qquad \partial _1 \overline{g}^{11} = O(r) , \\ &&\tilde{R} = (n-1)(n-2) r^{-2} + O(1) . \end{eqnarray*}$$

Assuming that $\overline{T}_{01} = O(1),$ we find that a general solution ζ has a term of order r^{−(n − 1)} due to which $\overline{\mathring{W}}{}^1$ would not converge at the vertex. We thus set c = 0. That yields

$$\begin{eqnarray*} &&\zeta = -(n-1) r^{-1} + O(1) . \end{eqnarray*}$$

Inserting this result into (48), we end up with

$$\begin{eqnarray*} &&\overline{{\mathring{W}}}{}^1 = O(r) . \end{eqnarray*}$$

For that we employed

$$\begin{eqnarray*} &&\overline{{\hat{W}}}{}^1 = -(n-1) r^{-1} + O(r) . \end{eqnarray*}$$

In the special case of a Minkowski target, we have

$$\begin{eqnarray*} &&\overline{{\hat{W}}}{}^1 \overset{\eta }{\equiv } \overline{{\hat{W}}}{}^0 \overset{\eta }{\equiv } -r\overline{g}^{AB} s_{AB}. \end{eqnarray*}$$

Moreover, we find

$$\begin{eqnarray*} &&\overline{\mathring{W} }{}^1 \overset{C}{=} \case{1}{2}\zeta - \overline{g}^{AB} \hat{\Gamma }^1_{AB} -\case{1}{2}\tau \overline{g}^{11}+ \overline{g}^{11}\big(\hat{\Gamma }^1_{11} - \kappa \big) . \end{eqnarray*}$$

Let us assume that the vector field $\overline{{\mathring{W}}}{}^{\lambda }$ can be extended to a smooth spacetime vector field ${\mathring{W}}{}^{\lambda }$ on the spacetime manifold ${\mathscr M}$. If we further assume, as in the case of two transversally intersecting null hypersurfaces, that the reduced Einstein equations can be complemented by well-posed evolution equations for the matter fields, for sufficiently well-behaved initial data, we obtain [34] a solution of the Cauchy problem in a future neighbourhood of the tip of the cone. The metric obtained this way solves the full Einstein equations if and only if H vanishes on C_O, as shown in sections 7.6, 9.3 and 11.3 of [27].

Let us assume now that the initial data $(\kappa ,\overline{g}_{\mu \nu })$ and a target metric $\hat{g}$ have been specified. In order to prove that the wave-map gauge vector H^λ vanishes on the cone, one first establishes that it is bounded near the vertex. In our adapted coordinates, that means

$$\begin{eqnarray} &&\overline{H}{}^0=O(1), \quad \overline{H}{}^1 = O(1) , \quad \overline{H}{}^A=O(r^{-1}). \end{eqnarray} \tag{ 3.36 }$$

If we assume that those transverse derivatives which appear in the generalized wave-map gauge condition $\overline{H}=0$ satisfy (compare [27] for a justification under the conditions there)⁵

$$\begin{eqnarray*} &&\overline{\partial _0 g_{11}}= O(r), \quad \overline{\partial _0 g_{1A}} = O(r^2), \quad \overline{g}^{AB} \overline{\partial _0 g_{AB}} = O(r), \end{eqnarray*}$$

and the initial data fulfil, additional to (44a)–(44e), the relations

$$\begin{eqnarray*} &&\partial _A \nu _0 = O(r^2), \quad \partial _B \nu _A = O(r^3) , \end{eqnarray*}$$

then one finds (using (44a)–(45c)), say in vacuum,

$$\begin{eqnarray*} &&\overline{H}^0=O(r), \quad \overline{H}^1 = O(r) , \quad \overline{H}^A=O(1), \end{eqnarray*}$$

which more than suffices for (50).

Let us present a more geometric description of initial data on a characteristic surface.

A triple $({\mathscr N},\tilde{g},\kappa )$ will be called a characteristic initial-data set if ${\mathscr N}$ is a smooth n-dimensional manifold, n ⩾ 3, equipped with a degenerate quadratic form $\tilde{g}$ of signature (0, +, ..., +), as well as a connection form κ on the one-dimensional degeneracy bundle $\mathrm{Ker }\, \tilde{g}$, understood as a bundle above its own integral curves. The data are moreover required to satisfy a constraint equation, as follows.

We can always locally introduce an adapted coordinate system where $\mathrm{Ker }\, \tilde{g}$ is Span ∂₁.⁶ (There only remains the freedom of coordinate transformations of the form $(x^1,x^A) \mapsto (\bar{x}^1 (x^1, x^A), \bar{x}^B (x^A))$.) Then the connection form κ reduces to one connection coefficient:

$$\begin{eqnarray} &&\nabla _{\partial _1} \partial _1 = \kappa \partial _1 . \end{eqnarray} \tag{ 4.1 }$$

In this coordinate system, we have $\tilde{g} = \overline{g}_{AB}\,\mathrm{d}x^A\,\mathrm{d}x^B$. Denoting by ${\overline{g} }^{AB}$ the matrix inverse to $\overline{g} _{AB}$, set

$$\begin{eqnarray} &&\chi _B{}^A:= \case{1}{2} {\overline{g} }^{AC}\partial _{1} {\overline{g} }_{CB} , \qquad \tau := \chi _A{}^A . \end{eqnarray} \tag{ 4.2 }$$

Under redefinitions of the adapted coordinates, the field τ transforms as a covector, which leads to the natural covariant derivative operator

$$\begin{eqnarray} &&\nabla _1 \tau := (\partial _{1} -\kappa )\tau . \end{eqnarray} \tag{ 4.3 }$$

With those definitions, the characteristic constraint equation reads

$$\begin{eqnarray} &&\nabla _1 \tau = - \chi _B{}^A\chi _A{}^B - \rho \hspace{5.0pt}\Longleftrightarrow \hspace{5.0pt}\nabla _1 \tau + \frac{1}{n-1}\tau ^2= - \sigma _B{}^A\sigma _A{}^B - \rho , \end{eqnarray} \tag{ 4.4 }$$

where ρ represents the component $T_{11}|_{{\mathscr N}}$ of the energy–momentum tensor of the associated spacetime $({{\mathscr M}},g)$ and, as before, σ is the trace-free part of χ.

A triple $({\mathscr N}, \tilde{g}, \kappa )$ satisfying (52) with ρ = 0 will be called vacuum characteristic initial data.

An initial-data set on a light-cone is a characteristic data set where ${\mathscr N}$ is a star-shaped neighbourhood of the origin in $\mathbb {R}^n$ from which the origin has been removed, with the tangents to the half-rays from the origin lying in the kernel of $\tilde{g}$, and with $(\tilde{g},\kappa )$ having specific behaviour at the origin as described e.g. in [27].

The reader is referred to [35] for a clear discussion of the geometry of null hypersurfaces, and to [36–38] for a further analysis of the objects involved.

An alternative geometric point of view, closely related to that in [37], is a slight variation of the above, as follows: instead of considering a connection on the degeneracy bundle ${\mathrm{Ker}}\,{\widetilde{g}}$, viewed as a bundle over the integral curves of ${\mathrm{Ker}}\,{\widetilde{g}}$, one considers a connection on this bundle viewed as a bundle over ${\mathscr N}$. For this, one needs the connection coefficients κ and ξ_A, defined by the equations

$$\begin{eqnarray} &&\nabla _{\partial _1} \partial _1 = \kappa \partial _1, \quad \nabla _{\partial _A} \partial _1 = -\case{1}{2}\xi _A \partial _1 + \chi _A{}^B \partial _B . \end{eqnarray} \tag{ 4.5 }$$

The coefficient κ satisfies the same constraint equation as before. The remaining coefficients ξ_A are obtained from (26), in notation adapted to the current setting:

$$\begin{eqnarray} &&{-}\frac{1}{2}(\partial _1 + \tau )\xi _A + \tilde{\nabla }_B \sigma _A^{\phantom{A}B} - \frac{n-2}{n-1}\partial _A\tau -\partial _A \kappa = \overline{T}_{1A} . \end{eqnarray} \tag{ 4.6 }$$

The fact that this system of ODEs can be solved in a rather straightforward way (compare (47)) given κ and g_AB should not prevent one to view this equation as a constraint on the initial data.

A useful observation here is that in [37, appendix C], where it is shown that (54) and (56) can be obtained from the usual vector constraint equation on a spacelike hypersurface by a limiting process, when considering a family of spacelike hypersurfaces which become null in the limit.

The alert reader will note that the constraint equations (54) and (56) exhaust all tangential components of T_μνℓ^ν. We are not aware of a geometric interpretation of equation (49), which involves the remaining, transverse, component of T_μνℓ^ν. Pursuing the analogy with the spacelike Cauchy problem, one could be tempted to think of this equation as corresponding to the scalar spacelike constraint equation. However, this analogy is wrong since it is shown in [37, appendix C] that the scalar constraint equation and one of the vector constraint equations degenerate to the same single equation when a family of spacelike hypersurfaces degenerates to a characteristic one.

Given a vacuum characteristic data set on a light-cone, or two vacuum characteristic data sets with a common boundary S (where some further data might have to be prescribed, as made clear in previous sections), one can impose various supplementary conditions to construct an associated spacetime metric. For example, one can redefine x¹ so that κ = 0 and impose wave-coordinate conditions or wave-map coordinate conditions in the light-cone case, to obtain the required spacetime metric, or one can prescribe the remaining metric functions as in section 3, with appropriate conditions at the tip of the light-cone or at the intersection surface. In [27, section 7.1], a scheme is presented where $g_{12}|_{{\mathscr N}}$ is prescribed, together with wave-map conditions. It is obvious that there exist further schemes which are mixtures of the above and which might be more appropriate for some specific physical situations, or for matter fields with exotic coupling to gravity.

Rendall's analysis, or that in [27], makes it clear that every vacuum characteristic data set as defined in section 4.1 leads to a unique, up to isometry, associated spacetime, either near the tip of the light-cone, or near the intersection surface S. Here uniqueness is understood locally, though again it is clear that unique maximal globally hyperbolic developments should exist in the current context.

For any $\tilde{g}$ for which τ has no zeros, equations (53) and (54) can be solved algebraically for κ. This appears to be the most natural choice near the tip of a light-cone, where τ is nowhere vanishing.

Another way of solving (54) is to prescribe $[\overline{g}_{AB}]$ and the mean null extrinsic curvature τ. Here one can simply choose τ to be nowhere vanishing such that (54) is solvable for κ. Regularity conditions on τ in the light-cone case are discussed in an appendix.

Rendall's proposal is to reparameterize the characteristic curves so that κ = 0; equation (54) can then be rewritten as a linear equation for a conformal factor, say Ω, such that g_AB = Ω²γ_AB, where $\gamma _{AB}=[\overline{g}_{AB}]$ is freely prescribed; compare (7).

In some situations, it might be convenient not to assume that κ = 0, but retain a version of the conformal approach of Rendall. This requires only a few minor modifications in section 2: it suffices to replace (6) by (19), (8) by (26), (11) by (33) and (10) by (32) with $\overline{T}_{\mu \nu }$, $\overline{\mathring{W}}{}^{\mu }$ and $\overline{\hat{W}}{}^{\mu }$ set to zero. As a matter of course, the corresponding equations on N₂ have to be adjusted analogously.

By an appropriate choice of κ, i.e. by choosing an adapted parameterization of the null rays, equation (19), which determines τ, can sometimes be simplified; an example will be given in the following section.

An elegant approach is due to Hayward [25] who, in space dimension n = 3, proposes to use a parameterization where κ = τ/(n − 1). Then, in vacuum, (54) becomes a linear equation for τ, in terms of the trace-free part of χ which depends only upon the conformal class of $\tilde{g}$,

$$\begin{eqnarray} &&\partial _1\tau + |\sigma |^2 = 0 . \end{eqnarray} \tag{ 5.1 }$$

The solution τ can then be used to determine a conformal factor Ω² relating $\overline{g}_{AB}$ to a freely prescribed representative γ_AB of the conformal class,

$$\begin{eqnarray*} &&\partial _1\Omega - \frac{\Omega }{n-1}\left(\tau - \frac{1}{2}\gamma ^{AB}\partial _1\gamma _{AB}\right) =0 . \end{eqnarray*}$$

In the case where data are given on a light-cone, one has to face the question of boundary conditions for (57), of the (necessary and/or sufficient) conditions on the data which will guarantee regularity at the vertex and a possible relation between those.

To address those questions, we start by comparing the κ = τ/(n − 1)-gauge with the geometric $\mathring{\kappa }=0$ gauge. Those quantities computed in the latter gauge will be labelled by $\mathring{}$ in what follows. Both gauges are related by an angle-dependent rescaling of the coordinate r: using the transformation law of the Christoffel symbols, we find

$$\begin{eqnarray} &&\tau /(n-1) \,= \, \kappa \,= \, \Gamma ^1_{11}\,= \, \frac{\partial r}{\partial \mathring{r}}\underbrace{\mathring{\Gamma }^1_{11}}_{=\mathring{\kappa }=0} + \frac{\partial r}{\partial \mathring{r}}\frac{\partial ^2 \mathring{r}}{\partial r^2} \end{eqnarray} \tag{ 5.2 }$$

for $r =r(\mathring{r},\mathring{x}^A)$. That yields

$$\begin{eqnarray} &&\mathring{r}(r) = \int ^r {\rm e}^{\frac{1}{n-1}\int ^{r_1} \tau (r_2)\,\mathrm{d}r_2 }\, \mathrm{d}r_1\, =\, \int ^r {\rm e}^{-\frac{1}{n-1}\int ^{r_1} \int ^{r_ 2}|\sigma (r_3)|^2\,\mathrm{d}r_3\,\mathrm{d} r_2}\, \mathrm{d}r_1 , \end{eqnarray} \tag{ 5.3 }$$

where we have suppressed any angle dependence, and left unspecified any potential constants of integration. This defines the desired local diffeomorphism.

As an example (and to obtain some intuition for this gauge scheme), consider the flat case where |σ|² ≡ 0 and for which we can compute everything explicitly. There is no difficulty in determining the transformed data which satisfy $|\mathring{\sigma }|^2\equiv 0$. The general solution of (57) is

$$\begin{eqnarray*} &&\tau (r,x^A) = \tau _0(x^A) . \end{eqnarray*}$$

Now we can explicitly compute (59),

$$\begin{eqnarray} &&\mathring{r}(r) = A^{(1)} + A^{(2)} {\rm e}^{\frac{\tau _0}{n-1}r} \end{eqnarray} \tag{ 5.4 }$$

for some integration functions A⁽ⁱ⁾, with A⁽²⁾ and τ₀ nowhere vanishing since we seek a map $r\mapsto \mathring{r}$ which is a diffeomorphism on each generator. Then

$$\begin{eqnarray} &&\mathring{\tau }(\mathring{r}) = \left(\frac{\partial \mathring{r}}{\partial r}\right)^{-1} \tau (r(\mathring{r})) = \frac{n-1}{\mathring{r} - A^{(1)}} . \end{eqnarray} \tag{ 5.5 }$$

We choose, as usual, the affine parameter $\mathring{r}$ in such a way that $\lbrace \mathring{r}=0\rbrace$ represents the vertex and such that $\mathring{\tau }= \frac{n-1}{\mathring{r}}$. This leads to A⁽¹⁾ = 0. Consequently, we either have to place the vertex at r = −∞ and choose a positive τ₀ or at r = +∞ with a negative τ₀ (w.l.o.g. we shall prefer the first alternative). We conclude that in the κ = τ/(n − 1)-gauge, we need to prescribe initial data for all $r\in \mathbb {R}$.

The regularity condition for $\mathring{\tau }$ translated into the κ = τ/(n − 1)-gauge does not lead to any boundary conditions for τ, except for the requirement of constant sign. It determines instead the position of the vertex, which in the new coordinates is located at infinity.

Let us come back to the general case, which we tackle from the other side, namely by starting in the $\mathring{\kappa }=0$-gauge. We use the identity $\frac{\partial ^2\mathring{r}}{\partial r^2} = -\left(\frac{\partial \mathring{r}}{\partial r}\right)^3 \frac{\partial ^2 r}{\partial \mathring{r}^2}$ to rewrite (58),

$$\begin{eqnarray} &&\frac{\partial \mathring{r}}{\partial r}\frac{\mathring{\tau }}{n-1} = \frac{\tau }{n-1}= - \left(\frac{\partial \mathring{r}}{\partial r}\right)^2 \frac{\partial ^2 r}{\partial \mathring{r}^2} \nonumber \\ && \Longleftrightarrow \quad \frac{\partial ^2 r}{\partial \mathring{r}^2} + \frac{\mathring{\tau }}{n-1} \frac{\partial r}{\partial \mathring{r}} =0 \nonumber \\ && \Longleftrightarrow \quad r(\mathring{r}) = \int ^{\mathring{r}} {\rm e}^{-\frac{1}{n-1}\int ^{\mathring{r}_1} \mathring{\tau }(\mathring{r}_2)\, \mathrm{d}\mathring{r}_2 }\, \mathrm{d}\mathring{r}_1. \end{eqnarray} \tag{ 5.6 }$$

This provides the inverse coordinate transformation.

Now, for a smooth metric, in adapted null coordinates which are constructed starting from normal coordinates, the generators are affinely parameterized and we have

$$\begin{eqnarray} &&\mathring{\tau }= \frac{n-1}{\mathring{r}} + O(\mathring{r}) . \end{eqnarray} \tag{ 5.7 }$$

But this behaviour remains unchanged under all reparameterizations of the generators which preserve the affine parameterization as well as the position of the vertex. It follows that (63) holds for all smooth metrics in the gauge $\mathring{\kappa }=0$.

From (62) and (63), we obtain

$$\begin{eqnarray*} &&r(\mathring{r}) = A^{(1)} + A^{(2)}\log \mathring{r} + O(\mathring{r}^2), \quad A^{(2)}\ne 0 \hspace{5.0pt}\forall x^A . \end{eqnarray*}$$

If we start in the $\mathring{\kappa }=0$-gauge, with the vertex at $\mathring{r}=0$, and transform into the κ = τ/(n − 1)-gauge, then, similarly to Minkowski spacetime, the vertex is shifted to, w.l.o.g., r = −∞. Thus, spacetime regularity forces the vertex to be located at infinity in the κ = τ/(n − 1)-gauge.

Following Christodoulou [28], we let the second fundamental form χ of a null hypersurface ${{\mathscr N}}$ with null tangent ℓ be defined as

$$\begin{eqnarray} &&\chi (X,Y)=g(\nabla _X \ell ,Y) , \end{eqnarray} \tag{ 5.8 }$$

where $X,Y\in T{{\mathscr N}}$. Choosing ℓ to be ∂_r, we then have, using [27, appendix A],

$$\begin{eqnarray} \nonumber \chi _{AB} &=& \overline{g(\nabla _A \partial _r,\partial _B)} = \overline{g}_{\mu B} \overline{\Gamma }^\mu _{Ar} = \overline{g}_{C B} \overline{\Gamma }^C _{Ar} + \overline{g}_{u B} \overline{\Gamma }^u _{Ar} \\ & = & \frac{1}{2} \partial _r \overline{g}_{AB} , \end{eqnarray} \tag{ 5.9a }$$

$$\begin{eqnarray} \chi _{rr} &=& \overline{ g(\nabla _r \partial _r,\partial _r)} = 0 , \end{eqnarray} \tag{ 5.9b }$$

$$\begin{eqnarray} \chi _{Ar} &=& \overline{ g(\nabla _A \partial _r,\partial _r)} = \overline{g}_{\mu r} \overline{\Gamma }^\mu _{Ar} = \overline{g}_{u r} \overline{\Gamma }^u _{rr} = 0 . \end{eqnarray} \tag{ 5.9c }$$

Let σ be the trace-free part of χ on the level sets of r:

$$\begin{eqnarray*} &&\sigma _{AB}:= \chi _{AB} - \frac{1}{n-1} \overline{g}^{CD} \chi _{CD} \overline{g}_{AB} \;; \end{eqnarray*}$$

σ is often called the shear tensor of ${\mathscr N}$. It has been proposed (cf, e.g., [28]) to consider σ as the free gravitational data at ${{\mathscr N}}$. There is an apparent problem with this proposal, because to define a trace-free tensor, one needs a conformal metric; but if a conformal class $[{\widetilde{g}}(r)]$ is given on ${{\mathscr N}}$, there does not seem to be any need to supplement this class with σ. This issue can be taken care of by working in a frame formalism, as follows.

Let ${\mathscr N}$ be an n-dimensional manifold threaded by a family of curves, which we call characteristic curves or generators. We assume moreover that each curve is equipped with a connection: if, in local coordinates, ∂_r is tangent to the curves, then we let κ denote the corresponding connection coefficient, as in (51).

Suppose, for the moment, that ${\mathscr N}$ is a characteristic hypersurface embedded as the submanifold {u = 0} in a spacetime ${\mathscr M}$. Choose some local coordinates so that ∂_r is tangent to the characteristic curves of ${\mathscr N}$. Let e_a denote a basis of $T{\mathscr M}$ along ${\mathscr N}$ such that

$$\begin{eqnarray} &&\nabla _r e_a = 0 . \end{eqnarray} \tag{ 5.10 }$$

Let S denote an (n − 1)-dimensional submanifold of ${\mathscr N}$ (possibly, but not necessarily, its boundary) which intersects the generators transversally. We will further require on S that e₀ is null, and that for a = 2, ..., n, the family of vectors e_a is orthonormal. These properties will then hold along all those generators that meet S.

Let x^A be the local coordinates on S; we propagate those along the generators of ${\mathscr N}$ to a neighbourhood ${\mathscr U}\subset {\mathscr N}$ of S by requiring ${\mathscr L}_{\partial _r}x^A =0$.

We choose e₁ ∼ ∂_r at S; equation (51) implies then that this will hold throughout ${\mathscr U}$:

$$\begin{eqnarray} &&e_1 = e_1{}^r \partial _r \mbox{ on ${\mathscr U}$.} \end{eqnarray} \tag{ 5.11 }$$

We choose the e_as, a = 2, ...n, to be tangent to S; since $T{\mathscr N}$ coincides with e^⊥₀, the e_as, a = 2, ...n, will remain tangent to ${\mathscr N}$:

$$\begin{eqnarray} &&e_a = e_a{}^r \partial _r + e_a{}^B \partial _B \mbox{ on ${\mathscr U}$, $\quad a=2,\ldots ,n$.} \end{eqnarray} \tag{ 5.12 }$$

On S, we choose the vector e₀ to be null, orthogonal to S, with

$$\begin{eqnarray} &&g(e_0,e_1)=1 . \end{eqnarray} \tag{ 5.13 }$$

Let {θ^a}_{a = 0, 1, ..., n}, be a spacetime coframe, defined on ${\mathscr U}\subset {\mathscr N}$, dual to the frame {e_a}_{a = 0, 1, ..., n}. From what has been said, we have

$$\begin{eqnarray} &&g = \underbrace{ \theta ^0 \otimes \theta ^1 + \theta ^1 \otimes \theta ^ 0 + \theta ^2 \otimes \theta ^2 + \cdots + \theta ^n \otimes \theta ^n}_{ =: \eta _{ab}\theta ^a \theta ^b} . \end{eqnarray} \tag{ 5.14 }$$

By construction, the one-forms θ^a are covariantly constant along the generators of ${\mathscr N}$:

$$\begin{eqnarray} &&\nabla _r \theta ^a = 0 . \end{eqnarray} \tag{ 5.15 }$$

Here ∇ is understood as the covariant-derivative operator acting on one-forms.

Again by construction, θ⁰ annihilates $T{\mathscr N}$; thus, θ⁰ ∼ du along ${\mathscr N}$:

$$\begin{eqnarray} &&\theta ^0 =\theta ^0{}_u\, \mathrm{d} u \mbox{\ on ${\mathscr U}$.} \end{eqnarray} \tag{ 5.16 }$$

We further note that

$$\begin{eqnarray} &&\theta ^a(\partial _r) =0 \mbox{\ on ${\mathscr U}$ for $a=2,\ldots ,n$.} \end{eqnarray} \tag{ 5.17 }$$

To see this, recall that ∂_r is orthogonal to ∂_A; hence,

$$\begin{eqnarray*} &&0 =g(\partial _r, \partial _A) = \eta _{ab}\theta ^a(\partial _r) \theta ^b (\partial _B) = \sum _{a=2}^N \theta ^a(\partial _r) \theta ^a (\partial _B) . \end{eqnarray*}$$

The result follows now from the fact that the (n − 1) × (n − 1) matrix (θ^a(∂_B))_{a ⩾ 2} is non-degenerate.

We would like to calculate $\tilde{g}:= \overline{g}_{AB}\,\mathrm{d} x^A\, \mathrm{d} x^B$ using the covectors θ^a. For this, note that (74) and (76) imply

$$\begin{eqnarray} &&\overline{g}_{AB} = \sum _{a=2}^N \theta ^a(\partial _A) \theta ^a (\partial _B) . \end{eqnarray} \tag{ 5.18 }$$

Thus, to determine ${\widetilde{g}}$ it suffices to know the components (θ^a_B := θ^a(∂_B))_{a ⩾ 2}. Now, using (77) together with [27, appendix A], we have for a ⩾ 2

$$\begin{eqnarray} &&0 = \nabla _r \theta ^a{}_B = \partial _r \theta ^a{}_B - \overline{\Gamma }^\mu {}_{rB} \theta ^a{}_ \mu = \partial _r \theta ^a{}_B - \case{1}{2} \overline{g}^{AC}\partial _r g_{CB} \theta ^a{}_ A . \end{eqnarray} \tag{ 5.19 }$$

There holds thus the following evolution equation for (θ^a_B)_{a ⩾ 2}:

$$\begin{eqnarray} &&\partial _r \theta ^a{}_B - \overline{g}^{AC}\chi _{CB} \theta ^a{}_ A =0 , \end{eqnarray} \tag{ 5.20 }$$

where $\overline{g}^{AC}$ denotes the matrix inverse to ∑^N_{a = 2}θ^a_Aθ^a_B.

Let, as before, ℓ = ∂_r and define

$$\begin{eqnarray} &&B_{\mu \nu }:= \nabla _\mu \ell _\nu . \end{eqnarray} \tag{ 5.21 }$$

Let B_ab denote the frame components of B:

$$\begin{eqnarray} &&B_{ab}= e_a{}^\mu e_b{}^\nu B_{\mu \nu } \quad \Longleftrightarrow \quad B_{\mu \nu } = \theta ^a{}(\partial _\mu )\theta ^b{}(\partial _\nu ) B_{ab} . \end{eqnarray} \tag{ 5.22 }$$

It follows from definition (68) that χ encodes the information on components of B in directions tangent to ${\mathscr N}$:

$$\begin{eqnarray} &&\chi _{AB}= B_{AB} , \quad \chi _{rr}= B_{rr} =0 , \quad \chi _{Ar}= B_{Ar} =0 . \end{eqnarray} \tag{ 5.23 }$$

The key distinction between B and χ is that B has components along directions transverse to ${\mathscr N}$, while χ has not. Also note that for a, b ⩾ 1 the frame components B_ab only involve the coordinate components of B_μν tangential to ${\mathscr N}$, so the frame formula

$$\begin{eqnarray*} &&\chi _{ab} = B_{ab} , \ a,b\ge 1, \end{eqnarray*}$$

is geometrically sensible.

The last two equations in (83) give

$$\begin{eqnarray} &&\chi _{AC}=\sum _{a,b=2}^n \theta ^a{}_ A \theta ^b {}_C B_{ab} \equiv \sum _{a,b=2}^n \theta ^a{}_ A \theta ^b {}_C \chi _{ab} , \end{eqnarray} \tag{ 5.24 }$$

which allows us to rewrite (80) as

$$\begin{eqnarray} &&\partial _r \theta ^a{}_B - \sum _{b,c=2}^n \overline{g}^{AC} \theta ^b{}_ B \theta ^c {}_C\chi _{bc} \theta ^a{}_ A =0 . \end{eqnarray} \tag{ 5.25 }$$

Now, for a, c ⩾ 2,

$$\begin{eqnarray} &&\eta ^{ac}= \overline{g}^{\mu \nu } \theta ^a{}_\mu \theta ^c{}_\nu = \overline{g}^{AC} \theta ^a{}_A \theta ^c{}_C , \end{eqnarray} \tag{ 5.26 }$$

which leads to

$$\begin{eqnarray} &&\partial _r \theta ^a{}_B - \sum _{b,c=2}^{n} \eta ^{ac} \theta ^b{}_ B \chi _{bc} =0 . \end{eqnarray} \tag{ 5.27 }$$

This equation leads naturally to the following picture, assuming for simplicity vacuum Einstein equations. Consider, first, two null transversely intersecting hypersurfaces ${\mathscr N}_1$ and ${\mathscr N}_2$, with ${\mathscr N}_1\cap {\mathscr N}_2 =S$. For a, b ⩾ 2, let η^ab be 1 when a and b coincide, and 0 otherwise. In addition to κ, the gravitational data on ${\mathscr N}={\mathscr N}_1\cup {\mathscr N}_2$ can be encoded in a field of symmetric η-trace-free (n − 1) × (n − 1) matrices σ_ab, a, b = 2, ..., n.

$$\begin{eqnarray*} &&\sigma _{ab} = \sigma _{ba}, \quad \eta ^{ab} \sigma _{ab} = 0 . \end{eqnarray*}$$

Let τ be a solution of the equation

$$\begin{eqnarray} &&(\partial _r -\kappa ) \tau + \frac{\tau ^2}{n-1} +|\sigma |_\eta ^2=0,\quad \mbox{where\quad $ |\sigma |^2_\eta := \eta ^{ac} \eta ^{bd} \sigma _{ab} \sigma _{cd}$} . \end{eqnarray} \tag{ 5.28 }$$

There remains the freedom to prescribe τ ≡ g^ABχ_AB = ∑ⁿ_{a, b = 2}η^abχ_ab on S (one such function for each surface N₁ and N₂). Define

$$\begin{eqnarray} &&\chi _{ab} = \sigma _{ab} + \frac{\tau }{n-1} \eta _{ab} . \end{eqnarray} \tag{ 5.29 }$$

Solving (87) for θ^a_B along the generators of ${\mathscr N}_1$ and ${\mathscr N}_2$, we can calculate $\overline{g}_{AB}$ on ${\mathscr N}$ from (78), as long as the determinant of the matrix (θ^a_B)_{a ⩾ 2} does not vanish (which will be the case in a neighbourhood of S). Here one has the freedom of prescribing θ^a_B on S for a ⩾ 2. The characteristic constraint equation R_μνℓ^μℓ^ν = 0 holds by construction. Indeed, the relation σ_ab = e_a^Ae_b^Bσ_AB, a, b ⩾ 2, can be justified in an analogous manner as equation (84). That gives, using (86), with a, b, c, d ⩾ 2,

$$\begin{eqnarray*} |\sigma |_{\eta }^2 \equiv \eta ^{ac}\eta ^{bd}\sigma _{ab}\sigma _{cd} &=& \overline{g}^{AC} \theta ^a{}_A\theta ^c{}_C \overline{g}^{BD}\theta ^b{}_B \theta ^d{}_D e_a{}^Ee_b{}^F \sigma _{EF}e_c{}^G e_d{}^H\sigma _{GH} \\ &=& \overline{g}^{AC} \overline{g}^{BD} \sigma _{AB}\sigma _{CD} \equiv |\sigma |^2 , \end{eqnarray*}$$

and the assertion follows immediately.

We can now apply any of the methods described previously (e.g., Rendall's original method if κ = 0) to obtain a solution of the characteristic Cauchy problem to the future of ${\mathscr N}$.⁷

One should keep in mind the following: prescribing the data e_a^A|_S, or equivalently θ^a_B|_S, determines the metric g_AB|_S on S. There is a supplementary freedom of making an O(n − 1)-rotation of the frame:

$$\begin{eqnarray*} &&e_a{}^A|_S (x^A) \mapsto \omega ^b{}_a (x^A) e_b{}^A|_S (x^A) , \end{eqnarray*}$$

where the ω^b_a(x^A)s are O(n − 1)-matrices. Any such rotation needs to be reflected in the σ_abs:

$$\begin{eqnarray*} &&\sigma _{ab}(r,x^A) \mapsto \omega ^c{}_a (x^A)\omega ^d{}_b (x^A)\sigma _{cd} (r,x^A) . \end{eqnarray*}$$

So in this construction, σ_ab undergoes gauge transformations which are constant along the generators, and are thus non-local in this sense.

In the case of a light-cone, the above construction can be implemented by first choosing an orthonormal coframe $\mathring{\phi }^a \equiv \mathring{\phi }^a{}_A\, {\rm d}x^A$, a ⩾ 2, for the unit round metric s_AB dx^A dx^B on S^{n − 1}. The solutions θ^a := rϕ^a ≡ rϕ^a_A dx^A, a ⩾ 2, of (87) are then chosen as the unique solutions asymptotic to $r \mathring{\phi }_a$. It would be of interest to settle the question, ignored here, of sufficient and necessary conditions on σ_ab so that the resulting initial data on the light-cone can be realized by restricting a smooth spacetime metric to the light-cone.

In [19, 39], Friedrich proposes alternative initial data on ${{\mathscr N}}$, based on the identity⁸

$$\begin{eqnarray} &&\partial _r^2 \overline{g}_{AB} -\kappa \partial _r \overline{g}_{AB} - \case{1}{2} \overline{g}^{CD} \partial _r \overline{g}_{AC} \partial _r \overline{g}_{BD} = - 2 \overline{R}_{A r B r } ; \end{eqnarray} \tag{ 5.30 }$$

equivalently

$$\begin{eqnarray} &&\partial _rB_{AB} -\kappa B_{AB} - \overline{g}^{CD}B_{AC}B_{BD} = - \overline{R}_{A r B r } . \end{eqnarray} \tag{ 5.31 }$$

Equation (90) shows that the component $\overline{R}_{A r B r }$ of the Riemann tensor can be calculated in terms of κ and the field $\overline{g}_{AB}$.

Alternatively, given the fields $\overline{C}_{A r B r }$, $\rho \equiv \overline{T}_{rr}$ and κ, together with suitable boundary conditions, one can solve (90) to determine $\overline{g}_{AB}$. So in spacetime dimension 4, Friedrich [19] proposes to use frame components of $\overline{C}_{A r B r }$ as the free data on ${{\mathscr N}}$. There is, however, a problem, in that $\overline{C}_{A r B r }$ is traceless,

$$\begin{eqnarray*} &&0= \overline{g}^{\mu \nu } \overline{C}_{\mu r \nu r} =\overline{g}^{AC} \overline{C}_{A r C r } . \end{eqnarray*}$$

So this condition has to be built-in into the formalism. But, as in the previous section, the tracelessness condition does not seemingly make sense unless the inverse metric g^AB, or at least its conformal class, is known.

This issue can again be taken care of by a frame formalism, whatever the dimension, as follows: let the orthonormal frame e_a, a = 0, ..., n, and its dual coframe θ^a be as in the last section. The property that the frame is parallel along the generators implies

$$\begin{eqnarray} &&\partial _r e_a {}^B= - \Gamma ^B_{rC}e_a{}^C = -\overline{g}^{BA}B _{AC} e_a{}^C \ \mbox{for $a\ge 2$} . \end{eqnarray} \tag{ 5.32 }$$

We then have, for a, b ⩾ 2,

$$\begin{eqnarray} \nonumber \partial _r B_{ab} & = & \partial _r\big( e_a {}^\mu e_ b {} ^\nu B_{\mu \nu }\big) \nonumber \\ & = & \partial _r\big( e_a {}^A\big) e_ b {} ^C B_{AC} + e_a {}^A \partial _r\big(e_ b {} ^C\big) B_{AC} + e_a {}^A e_ b {} ^C \partial _r B_{AC} \nonumber \\ & = & -\overline{g}^{AD}B_{DE}e_a {}^E e_ b {} ^C B_{AC} - e_a {}^A \overline{g}^{CD}B_{DE} e_ b {} ^E B_{AC} \nonumber \\ && + e_a {}^A e_ b {} ^C ( \kappa B_{AC} + \overline{g}^{ED}B_{AE}B_{CD} - \overline{R}_{A r C r } ) \nonumber \\ & = & - e_a {}^A e_ b {} ^E \overline{g}^{CD} B_{AC}B_{DE} + \kappa B_{ab} - e_a {}^A e_ b {} ^B \overline{R}_{A rB r } . \end{eqnarray} \tag{ 5.33 }$$

Using

$$\begin{eqnarray*} &&\overline{g}^{CD}= \eta ^{cd} e_c{} ^C e_d {}^D = \sum _{c,d=2}^n\eta ^{cd}e_c{} ^C e_d {}^D, \end{eqnarray*}$$

we conclude that

$$\begin{eqnarray*} \nonumber \partial _r B_{ab} & = & - \sum _{c,d=2}^n\eta ^{cd} B_{ac}B_{db} + \kappa B_{ab} - e_a {}^A e_ b {} ^B \overline{R}_{A rB r } . \end{eqnarray*}$$

From the definition of the Weyl tensor in dimension n + 1,

$$\begin{eqnarray} &&\fl \nonumber C_{\mu \nu \sigma \rho } := R_{\mu \nu \sigma \rho }-\frac{1}{n-1}\left(g_{\mu \sigma }R_{\nu \rho }- g_{\mu \rho }R_{\nu \sigma }-g_{\nu \sigma }R_{\mu \rho }+g_{\nu \rho }R_{\mu \sigma }\right) \\ &&+ \frac{1}{ n (n-1)} R (g_{\mu \sigma }g_{\nu \rho }-g_{\mu \rho }g_{\nu \sigma }) , \end{eqnarray} \tag{ 5.34 }$$

we find

$$\begin{eqnarray*} \nonumber \overline{C}_{ArBr} &=& \overline{R}_{ArBr} -\frac{1}{n-1}\overline{g}_{AB}\overline{R}_{rr} . \end{eqnarray*}$$

For a, b ⩾ 2, let

$$\begin{eqnarray*} &&\psi _{ab} := e_a{}^A e_b {}^B \overline{C}_{A r B r} \end{eqnarray*}$$

represent the components of $\overline{C}_{A r B r }$ in the current frame. Then ψ_ab is symmetric, with vanishing η-trace. We finally obtain the following equation for B_ab ≡ χ_ab, a, b ⩾ 2:

$$\begin{eqnarray} (\partial _r -\kappa )\chi _{ab} & = & - \sum _{c,d=2}^n\eta ^{cd} \chi _{ac}\chi _{db} - \psi _{ab} -\frac{1}{n-1}\eta _{ab}\overline{T}_{rr} . \end{eqnarray} \tag{ 5.35 }$$

This equation shows that (κ, ψ_ab) can be used as the free data for the gravitational field: indeed, given κ, ψ_ab and the component $\overline{T}_{rr}$ of the energy–momentum tensor, we can integrate (95) to obtain χ_ab. Note that by taking the η-trace of (95), one recovers constraint (88) (here with $\overline{T}_{rr}$ possibly non-vanishing).

In the case of two transverse hypersurfaces, the integration leaves the freedom of prescribing two tensors χ_ab on S, one corresponding to N₁ and another to N₂. Then one proceeds as in the previous section to construct the remaining data on the initial surfaces, keeping in mind the further freedom to choose θ^a_B|_S, a ⩾ 2, on S.

On a light-cone, equation (95) should be integrated with vanishing data at the vertex.