Null Junction Conditions

A null interface has no unit normal transverse to itself: its normal is also a tangent generator, and its induced metric is degenerate. Consequently the ordinary extrinsic curvature cannot be inserted into the nonnull Israel equation. The Barrabès–Israel construction replaces it with an auxiliary transverse null direction and a transverse curvature whose jump separates null matter from an independent impulsive gravitational wave.

Required background. We use Hypersurface Normals and Tangential Projectors, The Einstein Equation, and Flat-Background Hilbert Stress-Energy from Metric Variation.

Helpful background. The Gibbons-Hawking-York Boundary Term clarifies why the unit-normal construction must be replaced, rather than continued formally, at a null segment.

Generator, screen metric, and transverse direction

Choose intrinsic coordinates $y^a=(\lambda,\theta^A)$ on the null hypersurface $\Sigma$. The vector

$$k^\mu=e^\mu_\lambda$$

is chosen future directed and tangent to its generators, while $e^\mu_A$ span a spacelike two-dimensional screen. The pullback metric has only

$$q_{AB}=g_{\mu\nu}e^\mu_Ae^\nu_B$$

as nonzero independent components. Match the same $(\lambda,\theta^A)$ and require $[q_{AB}]=0$. Here and throughout this page, after pulling both sides to one abstract $\Sigma$,

$$[Q]\equiv Q_+-Q_-.$$

The side labels are fixed by a transverse coordinate that increases from $M_-$ to $M_+$; they are not reassigned when the generator is rescaled. Complete the basis with an auxiliary transverse null vector $N^\mu$ satisfying

$$N^\mu N_\mu=0, \qquad N^\mu k_\mu=-1, \qquad N^\mu e_{\mu A}=0.$$

The null analogue of extrinsic data is the symmetric transverse curvature

$$C_{ab} =-N_\mu e^\nu_a\nabla_\nu e^\mu_b.$$

These definitions and the matching algorithm are set out in Poisson 2002, § II, pp. 2–5, Eqs. (2.1)–(2.19), PDF. For comparison, on a nonnull hypersurface this site uses $K_{ab}=-e^\mu_a e^\nu_b\nabla_\mu n_\nu$. There is no normalized $n^\mu$ at $\Sigma$, so $C_{ab}$ is a replacement for $K_{ab}$, not its value at $n^2=0$.

Surface density, current, and pressure

In the Poisson reformulation of the Barrabès–Israel formalism, the jump in $C_{ab}$ gives

$$\mu=-\frac{1}{8\pi G}q^{AB}[C_{AB}], \qquad j^A=\frac{1}{8\pi G}q^{AB}[C_{\lambda B}], \qquad p=-\frac{1}{8\pi G}[C_{\lambda\lambda}].$$

The corresponding intrinsic spacetime tensor is

$$S^{\mu\nu} =\mu k^\mu k^\nu +j^A\!\left(k^\mu e^\nu_A+e^\mu_Ak^\nu\right) +p\,q^{AB}e^\mu_Ae^\nu_B.$$

The trace $q^{AB}[C_{AB}]$ supplies the surface mass density, the mixed components supply a screen current, and the generator component supplies an isotropic screen pressure. These are the Ricci, or material, data. The part of the singular Weyl tensor not fixed by them can describe an impulsive gravitational wave; a vanishing $S^{\mu\nu}$ therefore does not by itself prove that the full distributional curvature vanishes. The original lightlike-shell construction and its relation to the nonnull limit are developed in Barrabès and Israel 1991, pp. 1129–1142.

To state the spacetime distribution without hiding its normalization, let freely falling observers cross $\Sigma$ with four-velocity $u^\mu$ and proper time $\tau=0$ at the crossing. Then

$$T_\Sigma^{\mu\nu} =(-k\cdot u)^{-1}S^{\mu\nu}\delta(\tau).$$

The prefactor is part of the physical distribution. Omitting it makes the apparent surface density depend spuriously on the parameter chosen along the generators.

Generator rescaling as a consistency check

Take a constant reparameterization $\bar\lambda=a\lambda$ with $a>0$. Keeping the screen labels fixed requires

$$\bar k^\mu=a^{-1}k^\mu, \qquad \bar N^\mu=aN^\mu, \qquad \bar e^\mu_A=e^\mu_A.$$

It follows directly from the definition of $C_{ab}$ that

$$\bar C_{AB}=aC_{AB}, \qquad \bar C_{\lambda A}=C_{\lambda A}, \qquad \bar C_{\lambda\lambda}=a^{-1}C_{\lambda\lambda}.$$

Therefore $\bar\mu=a\mu$, $\bar j^A=j^A$, and $\bar p=a^{-1}p$, while $\bar S^{\mu\nu}=a^{-1}S^{\mu\nu}$. At the same time $(-\bar k\cdot u)^{-1}=a(-k\cdot u)^{-1}$, so $T_\Sigma^{\mu\nu}$ is invariant. General generator-dependent reparameterizations also mix the screen basis and current; the complete transformation is given in Poisson 2002, § IV, pp. 6–7, Eqs. (4.6)–(4.18), PDF.

Two failures are especially common. First, choosing $N^\mu$ independently on the two sides breaks the comparison: the same normalization and screen identification must be used before taking a jump. Second, demanding that $\lambda$ be affine on both sides can exclude a physical pressure layer. Since $C_{\lambda\lambda}$ is the generator nonaffinity, its jump is proportional to $-p$.

Exercises

An ingoing spherical null shell separates a Minkowski interior from a Schwarzschild exterior with $f_+=1-2GM/r$. Choose $\lambda=-r$ as an affine generator parameter on both sides. Suppose

$$C^-_{AB}=\frac{1}{2r}q_{AB}, \qquad C^+_{AB}=\frac{f_+}{2r}q_{AB}, \qquad C^\pm_{\lambda A}=C^\pm_{\lambda\lambda}=0.$$

Find $\mu$, $j^A$, and $p$.

Solution

The angular jump is

$$[C_{AB}] =\frac{f_+-1}{2r}q_{AB} =-\frac{GM}{r^2}q_{AB}.$$

Because the screen is two-dimensional, $q^{AB}q_{AB}=2$. Hence

$$\mu =-\frac{1}{8\pi G} \left(-\frac{2GM}{r^2}\right) =\frac{M}{4\pi r^2}.$$

The other jumps vanish, so $j^A=0$ and $p=0$. Thus the shell’s mass is distributed uniformly over area $4\pi r^2$. This reproduces the explicit calculation in Poisson 2002, § V, pp. 8–9, Eqs. (5.1)–(5.6), PDF.

References

• Barrabès, Claude, and Werner Israel. “Thin Shells in General Relativity and Cosmology: The Lightlike Limit.” Physical Review D 43, no. 4 (1991): 1129–1142. doi:10.1103/PhysRevD.43.1129.
• Poisson, Eric. “A Reformulation of the Barrabès–Israel Null-Shell Formalism.” Physical Review D 66, no. 2 (2002): 024012. arXiv:gr-qc/0207101. doi:10.1103/PhysRevD.66.024012.