Geometry and Data on Null Hypersurfaces

A null hypersurface carries less intrinsic metric data than a spacelike slice because its normal direction lies inside its tangent space. The radiative information is encoded in the evolving conformal geometry of two-dimensional screen sections. Expansion, nonaffinity, and several connection components are then determined by transport constraints once corner values, matter data, and a generator gauge have been fixed.

Required background. Timelike Congruences and Spatial Projection supplies the deformation, connecting-field, and caustic logic that is adapted here to null generators.

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.

Null basis and screen geometry

Let $\mathcal N$ be a future-directed null hypersurface generated by $\ell^\mu$. Choose a parameter $\lambda$ and labels $y^A$ constant along each generator, so $\ell=\partial_\lambda$. Complete the basis with an auxiliary future-directed null vector $n^\mu$ and impose

$$\ell^\mu\ell_\mu=n^\mu n_\mu=0, \qquad \ell^\mu n_\mu=-1.$$

The screen metric and projector are

$$q_{\mu\nu} =g_{\mu\nu}+\ell_\mu n_\nu+n_\mu\ell_\nu, \qquad q^\mu{}_\nu\ell^\nu=q^\mu{}_\nu n^\nu=0.$$

Its pullback $q_{AB}$ is positive definite before a caustic. The orientation choice is explicit: $\ell^\mu$ is future directed along $\mathcal N$, while $n^\mu$ points along the other future null direction at each screen. Neither vector is an outward unit normal.

The generator may be nonaffine:

$$\ell^\nu\nabla_\nu\ell^\mu=\kappa\ell^\mu.$$

Its screen deformation is

$$B_{AB} =q_A{}^\mu q_B{}^\nu\nabla_\nu\ell_\mu =\frac12\mathcal L_\ell q_{AB} =\frac12\theta q_{AB}+\sigma_{AB},$$

where $\theta=q^{AB}B_{AB}$ and $q^{AB}\sigma_{AB}=0$. Hypersurface orthogonality makes the screen twist vanish. The area element evolves by

$$\ell(\ln\sqrt q)=\theta.$$

This basis and the distinction between intrinsic screen geometry and generator gauge underlie the worldtube–nullcone construction reviewed in Winicour 2012, § 2.1.

Free conformal data and transport constraints

The two local gravitational polarizations are represented by the conformal class of $q_{AB}$ along $\mathcal N$. Prescribing a representative with a determinant condition fixes an area gauge; its tracefree derivative fixes $\sigma_{AB}$. The area scale and expansion are not an additional independent pair. In four dimensions the null Raychaudhuri equation is

$$\ell(\theta) =\kappa\theta-\frac12\theta^2 -\sigma_{AB}\sigma^{AB} -8\pi G\,T_{\ell\ell}.$$

Once $\kappa$, the conformal screen history, and matter flux are specified, Raychaudhuri transports the expansion from its corner value. The area equation then needs its own corner area. In a fixed generator parameter, both values are needed; they are not interchangeable.

To see the count directly, write $q_{AB}=a^2\widehat q_{AB}$ with a fixed determinant for $\widehat q_{AB}$. Then $\theta=2\ell(a)/a$, and Raychaudhuri becomes

$$\ell\bigl(\ell(a)\bigr)-\kappa\ell(a) =-\frac a2\left(\sigma_{AB}\sigma^{AB}+8\pi G T_{\ell\ell}\right).$$

This second-order equation requires $a$ and $\ell(a)$ at the corner, equivalently area and expansion. In an areal coordinate gauge, $a$ itself is used as the parameter, and the same constraint instead determines a lapse-like coefficient. The data count has been reorganized by gauge choice, not removed. Other components of Einstein’s equation transport connection and shift variables. Thus null data consist of free conformal data and compatible corner values that seed a constraint hierarchy.

Data on a single null hypersurface generally do not determine a spacetime on both sides. A second intersecting null hypersurface, a timelike worldtube, or another transverse prescription supplies the missing information. Winicour separates initial null data from the radial hierarchy in Winicour 2012, § 6.6.

Generator scaling and invariant content

For any smooth function $\alpha$ on $\mathcal N$, the boost

$$\ell^\mu\longmapsto\ell'^\mu=e^\alpha\ell^\mu, \qquad n^\mu\longmapsto n'^\mu=e^{-\alpha}n^\mu$$

preserves $\ell\cdot n=-1$ and leaves $q_{\mu\nu}$ unchanged. Direct substitution gives

$$\theta'=e^\alpha\theta, \qquad \sigma'_{AB}=e^\alpha\sigma_{AB}, \qquad \kappa'=e^\alpha\bigl(\kappa+\ell(\alpha)\bigr).$$

Consequently the numerical values of expansion and shear depend on generator normalization, while the conformal screen geometry and the statement $\sigma_{AB}=0$ do not. Affine parameterization, $\kappa=0$, still permits a rescaling constant along each generator and a shift of its affine origin. Choosing the initial screen fixes the origin; a corner normalization fixes the scale.

The construction fails when neighboring generators cross or focus. Then the label map $(\lambda,y^A)\mapsto x^\mu$ is no longer one-to-one and $\det q_{AB}$ can vanish. This is a failure of the null chart and data hierarchy, not automatically a curvature singularity.

Exercises

On an outgoing Minkowski light cone, use $(\lambda,y^A)=(r,\theta,\phi)$, $\ell=\partial_r$, and $q_{AB}=r^2\Omega_{AB}$. Compute $\theta$, $\sigma_{AB}$, and $\kappa$. Then apply a constant rescaling $\ell'=a\ell$, $n'=a^{-1}n$.

Solution

Because $\mathcal L_\ell q_{AB}=2r\Omega_{AB}$,

$$B_{AB}=r\Omega_{AB}=\frac1r q_{AB}.$$

Its trace is $\theta=2/r$, and its tracefree part vanishes: $\sigma_{AB}=0$. The coordinate $r$ is affine along each radial null generator, so $\kappa=0$.

For constant $a\gt 0$,

$$\theta'=\frac{2a}{r}, \qquad \sigma'_{AB}=0, \qquad \kappa'=0.$$

The expansion’s value changes because the parameter speed changes, but the round conformal screen data and the absence of shear are invariant. Also $\theta'\,d\lambda'=\theta\,d\lambda$ when $\lambda'=\lambda/a$, so the integrated area change is unchanged.

References

• Winicour, Jeffrey. “Characteristic Evolution and Matching.” Living Reviews in Relativity 15 (2012): article 2. arXiv:0810.1903. doi:10.12942/lrr-2012-2.