Double-Null Formulations

A double-null formulation uses two transverse families of null hypersurfaces. One family transports outgoing geometry and the other transports incoming geometry, so the Einstein equations become coupled equations along the two characteristic directions. The price is a corner problem: the data induced by both families must agree where they intersect, and their relative generator normalization must be fixed there.

Required background. Timelike Congruences and Spatial Projection supplies the congruence deformation and focal-point concepts generalized to each null family.

Helpful background. We use The Einstein Equation, Hypersurface Normals and Tangential Projectors, Formulations of General Relativity, Causality and Global Structure, and PDEs and Green Functions for Gravitation.

Two null foliations and their screen sections

Let $u=\mathrm{constant}$ and $v=\mathrm{constant}$ define transverse null hypersurfaces. A useful local metric form is

$$ds^2=-2\Omega^2\,du\,dv +q_{AB}(dx^A+b^Adu)(dx^B+b^Bdu).$$

Here $\Omega\gt0$ is a null lapse, $q_{AB}$ is the positive screen metric, and $b^A$ is a tangential shift. This $\Omega$ is not a compactifying conformal factor and is not taken to vanish at a boundary. The notation is local to this double-null metric. Choose both null directions future directed:

$$\ell=\Omega^{-1}\partial_v, \qquad n=\Omega^{-1}(\partial_u-b^A\partial_A), \qquad \ell\cdot n=-1.$$

The intersection $S_{u,v}$ is spacelike and carries $q_{AB}$. The two null second fundamental forms are

$$\chi_{AB}=\frac12\mathcal L_\ell q_{AB}, \qquad \underline\chi_{AB} =\frac12\mathcal L_n q_{AB}.$$

Each decomposes into an expansion and tracefree shear. A normal-bundle connection,

$$\zeta_A=-q_A{}^\mu n_\nu\nabla_\mu\ell^\nu,$$

records how the null frame varies across a screen. Under a boost $\ell\mapsto e^\alpha\ell$, $n\mapsto e^{-\alpha}n$,

$$\zeta_A\mapsto\zeta_A+D_A\alpha, \qquad \operatorname{tr}\chi\mapsto e^\alpha\operatorname{tr}\chi, \qquad \operatorname{tr}\underline\chi \mapsto e^{-\alpha}\operatorname{tr}\underline\chi.$$

Thus $\zeta_A$ is a connection rather than an invariant one-form, while the boost weight of each expansion is explicit.

The two transport hierarchies

Einstein’s equation splits by contraction with $\ell$, $n$, and the screen basis. The $\ell\ell$ and $nn$ equations are the two Raychaudhuri constraints. The $\ell A$ and $nA$ equations transport normal-connection and shear-divergence data. Mixed and tracefree screen equations couple the two families and propagate the geometry away from the initial null sheets.

On two initial hypersurfaces $\mathcal N_u$ and $\mathcal N_v$ meeting at $S$, one may prescribe conformal screen data along each family. Their restrictions to $S$ must define the same conformal two-metric. Corner values for the area scale, the two expansions, $\zeta_A$, and the boost normalization seed the transport equations. They cannot be chosen twice independently. Rendall’s construction makes this hierarchy and its corner functions explicit in Rendall 1990, § 5, pp. 230–234.

The advantage over a one-cone formulation is structural. Incoming and outgoing radiation are treated on equal geometric footing, trapped or marginal screens are visible through the signs of the two expansions, and constraints are integrated along generators rather than solved as spacelike elliptic equations. The formulation remains local: either null family can form a caustic and destroy its coordinates.

Minkowski reconstruction and orientation check

Introduce flat double-null coordinates

$$U=\frac{t-r}{\sqrt2}, \qquad V=\frac{t+r}{\sqrt2}.$$

Then

$$ds^2=-2\,dU\,dV+r^2d\Omega_2^2, \qquad r=\frac{V-U}{\sqrt2}.$$

With $\ell=\partial_V$ future outgoing and $n=\partial_U$ future ingoing, one has $\ell\cdot n=-1$. The screen is round, both shears vanish, and $\zeta_A=0$. The expansions have opposite signs:

$$\theta_{(\ell)}=\frac{\sqrt2}{r}, \qquad \theta_{(n)}=-\frac{\sqrt2}{r}.$$

This is the flat reference for corner orientation. Reversing one generator would change its time orientation and invalidate the declared future–future normalization. A legitimate boost rescales the two expansions inversely and leaves their product unchanged.

The spherically symmetric double-null hierarchy and its relation to characteristic evolution are reviewed in Winicour 2012, § 3.1. Outside spherical symmetry, vanishing shear is no longer automatic; it is free radiative information along the initial null sheets.

Exercises

Verify the outgoing Raychaudhuri equation for the Minkowski data above, and show that the product $\theta_{(\ell)}\theta_{(n)}$ is invariant under a constant corner boost.

Solution

The outgoing generator is affine, $\kappa_{(\ell)}=0$, and the shear and curvature term vanish. Since $\partial_Vr=1/\sqrt2$,

$$\ell(\theta_{(\ell)}) =\partial_V\left(\frac{\sqrt2}{r}\right) =-\frac1{r^2}.$$

The right-hand side of Raychaudhuri is

$$-\frac12\theta_{(\ell)}^2 =-\frac12\frac{2}{r^2} =-\frac1{r^2},$$

so the transport equation holds.

For $\ell'=c\ell$ and $n'=c^{-1}n$ with $c\gt 0$,

$$\theta_{(\ell')}=c\,\theta_{(\ell)}, \qquad \theta_{(n')}=c^{-1}\theta_{(n)}.$$

Their product remains $-2/r^2$, while the screen metric and its corner area remain unchanged.

References

• Rendall, Alan D. “Reduction of the Characteristic Initial Value Problem to the Cauchy Problem and Its Applications to the Einstein Equations.” Proceedings of the Royal Society of London A 427, no. 1872 (1990): 221–239. doi:10.1098/rspa.1990.0009.
• Winicour, Jeffrey. “Characteristic Evolution and Matching.” Living Reviews in Relativity 15 (2012): article 2. arXiv:0810.1903. doi:10.12942/lrr-2012-2.