Null Congruences and Screen Geometry

A null congruence is a smooth family of null curves with tangent $k^\mu$. Because $k^\mu$ is orthogonal to itself, its orthogonal complement contains the propagation direction and does not inherit a nondegenerate metric. The physically transverse geometry is therefore a quotient space, represented concretely by a two-dimensional screen in four spacetime dimensions after an auxiliary null direction is chosen. The optical and hypersurface-orthogonality statements below concern geodesic curves, with affine or nonaffine tangents; the pointwise screen construction itself does not require geodesicity.

This construction isolates the geometry seen by an infinitesimal pencil of rays while discarding displacement along the rays, which merely relabels points on a generator.

Required background. The Riemann Curvature Tensor fixes the curvature convention needed when this screen construction is used for null deviation on the following pages; Affine and Variational Geodesics supplies null geodesics, affine parameters, and the curve-variation setup. Helpful background. Calculus and Linear Algebra for Relativity supplies quotient spaces and projectors; Frames, Coframes, and Anholonomy supplies null-frame components.

Two related choices should be separated from the outset. Selecting the curves fixes a null line at every event, but not a preferred tangent vector on that line: multiplying the tangent by a positive function preserves the unparametrized curves. Selecting a screen additionally chooses how transverse equivalence classes are represented inside the tangent space. Observable comparisons are therefore built from quantities whose normalization has been fixed by stated boundary data, or from combinations invariant under the corresponding changes. This bookkeeping is essential near horizons and null boundaries, but it is already visible in the elementary flat-space example below.

Null directions and their quotient

At an event, let

$$k^\mu k_\mu=0, \qquad k^\mu\neq0.$$

The subspace $k^\perp$ has dimension $n-1$, but its restricted metric is degenerate because $k\in k^\perp$. The intrinsic transverse space is

$$\mathcal S_p=k^\perp/\operatorname{span}\{k\},$$

which has dimension $n-2$ and a positive-definite metric. Two vectors in $k^\perp$ represent the same transverse displacement if they differ by a multiple of $k^\mu$. This quotient is the invariant object; a screen is a useful representative of it.

Choose an auxiliary null vector $\ell^\mu$ such that

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

Then

$$q_{\mu\nu} =g_{\mu\nu}+k_\mu\ell_\nu+\ell_\mu k_\nu$$

annihilates both $k^\mu$ and $\ell^\mu$, has trace $n-2$, and satisfies $q^\mu{}_\alpha q^\alpha{}_\nu=q^\mu{}_\nu$. In four dimensions its image is the two-dimensional screen. Changing $\ell^\mu$ while preserving the normalization changes the embedded screen but not the quotient geometry represented by transverse equivalence classes.

Generator normalization and inaffinity

Null tangents have no unit normalization. If their curves are geodesic, a general parameter gives

$$k^\nu\nabla_\nu k^\mu=\kappa k^\mu,$$

where $\kappa=0$ only for an affine parameter. Under a positive rescaling $k'^\mu=e^f k^\mu$,

$$\kappa'=e^f\left(\kappa+k^\mu\nabla_\mu f\right).$$

Thus, if $k^\mu$ is affine, the rescaled tangent $k'^\mu$ is affine precisely when $k(f)=k^\mu\nabla_\mu f=0$. The factor $e^f$ may vary from one generator to another, but it must be constant along each generator to preserve affine parametrization. Taking $\ell'^\mu=e^{-f}\ell^\mu$ keeps $k'\cdot\ell'=-1$ and leaves $q_{\mu\nu}$ unchanged. The freedom is often called a null boost. It is not an error or an extra physical polarization; it records the normalization and parametrization of the generators.

For a chosen generator normalization, define the screen deformation tensor

$$\mathcal B_{\mu\nu} =q_\mu{}^\alpha q_\nu{}^\beta\nabla_\beta k_\alpha.$$

Algebraically, under the paired boost above,

$$\mathcal B'_{\mu\nu}=e^f\mathcal B_{\mu\nu}, \qquad \theta'=e^f\theta.$$

Thus whether the expansion vanishes is invariant, but its numerical value is tied to the generator scale. When both representatives are required to remain affine, the comparison is restricted to $k(f)=0$. Statements involving affine distance or optical area must specify that scale.

Screen expansion, shear, and twist

Let $s=n-2$, with $n\ge3$. Decompose

$$\mathcal B_{\mu\nu} =\frac{\theta}{s}q_{\mu\nu} +\sigma_{\mu\nu} +\omega_{\mu\nu}.$$

Here $\theta=q^{\mu\nu}\mathcal B_{\mu\nu}$, the shear is symmetric and screen-trace-free, and the twist is antisymmetric. In four dimensions the screen map has four components: one expansion, two independent shear components, and one twist component.

For a hypersurface-orthogonal family of null generators, the twist vanishes. Conversely, for a smooth nonzero geodesic null field, vanishing screen twist implies local hypersurface orthogonality. The geodesic assumption removes the transverse acceleration components that a screen–screen projection alone cannot test. This is the null version of the Frobenius criterion, but the null hypersurface’s induced metric remains degenerate. It must not be obtained by taking an $\epsilon\to0$ limit of a unit-normal non-null projector.

Witten develops affine normalization and transverse null geometry in Witten 2020, §5.4, pp. 49–53, PDF. The screen construction and its optical interpretation for ray bundles are developed in Perlick 2004, §2.3, “Sachs basis” and “Jacobi matrix”.

Radial Minkowski check

In spherical Minkowski coordinates,

$$ds^2=-dt^2+dr^2+r^2d\Omega^2,$$

take outgoing affine generators $k=\partial_t+\partial_r$ and $\ell=(\partial_t-\partial_r)/2$. They satisfy $k\cdot\ell=-1$. The screen metric is the round two-sphere metric $q_{AB}=r^2\Omega_{AB}$. Along $k$, $dr/d\lambda=1$, so

$$\mathcal B_{AB} =\frac12\mathcal L_k q_{AB} =r\Omega_{AB} =\frac1r q_{AB}.$$

Consequently

$$\theta=\frac2r, \qquad \sigma_{AB}=0, \qquad \omega_{AB}=0.$$

The nonzero expansion is just the logarithmic derivative of the area $4\pi r^2$. Spacetime curvature is zero, so this example cleanly separates transverse beam growth from curvature.

Common pitfalls

The orthogonal complement is not the screen. It still contains $k^\mu$. Either pass to the quotient or choose an auxiliary $\ell^\mu$ and project with $q_{\mu\nu}$.

Expansion has a boost weight. Among affine representatives, a rescaling constant along each generator changes its numerical value. Comparisons require a common normalization, usually fixed by boundary data or an observer.

Null and non-null hypersurfaces are different cases. A null normal is also tangent, and its induced metric is degenerate. Setting a unit-normal sign $\epsilon$ to zero destroys, rather than limits, the non-null projector formula.

Exercises

Verify that the paired rescaling $k'=e^f k$, $\ell'=e^{-f}\ell$ leaves the screen projector unchanged, determine the transformed expansion, and find the condition under which an affine $k$ remains affine after the rescaling.

Solution

The two mixed products are invariant:

$$k'_\mu\ell'_\nu=k_\mu\ell_\nu, \qquad \ell'_\mu k'_\nu=\ell_\mu k_\nu.$$

Hence $q'_{\mu\nu}=q_{\mu\nu}$. In

$$\nabla_\beta k'_\alpha =e^f\left(\nabla_\beta k_\alpha+k_\alpha\nabla_\beta f\right),$$

the second term is killed by $q_\mu{}^\alpha k_\alpha=0$. Therefore $\mathcal B'_{\mu\nu}=e^f\mathcal B_{\mu\nu}$ and $\theta'=e^f\theta$. Finally, the inaffinity law gives $\kappa'=e^f k(f)$ when $\kappa=0$, so the rescaled tangent remains affine exactly when $k(f)=0$.

References

• Perlick, Volker. “Gravitational Lensing from a Spacetime Perspective.” Living Reviews in Relativity 7, 9 (2004). doi:10.12942/lrr-2004-9.
• Witten, Edward. “Light Rays, Singularities, and All That.” Reviews of Modern Physics 92, 045004 (2020). DOI; Open PDF.