Hypersurface Geometry and Junction Conditions

Spacetime matching asks when two smooth geometries can be joined across a hypersurface and what distribution of matter is required when their transverse derivatives disagree. The answer is geometric: first choose the identification of boundary points, then compare intrinsic metrics, and only then compare extrinsic or transverse data. This chapter develops that hierarchy for regular nonnull interfaces, material thin shells, null layers, corners, and stellar boundaries.

Helpful background. The chapter draws on Hypersurface Normals and Tangential Projectors, The Einstein Equation, Flat-Background Hilbert Stress-Energy from Metric Variation, and The Gibbons-Hawking-York Boundary Term.

Enter the matching problem

Let an abstract interface $\Sigma$ be embedded into regions $M_-$ and $M_+$. All side data must first be pulled back with those embeddings. Only after that comparison is defined do we write

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

For a nonnull interface, choose one unit normal $n^\mu$ directed continuously from the minus side to the plus side. With $h^\mu{}_\nu$ the tangential projector, this volume uses

$$K_{ab} =-e^\mu_a e^\nu_b\nabla_\mu n_\nu.$$

The pair of outward normals used for two separate boundary actions point oppositely at a glued interface. They are not the pair to use in $[K_{ab}]$; convert them to the continuous normal first. This single rule prevents the most common sign error in junction calculations.

The logical levels are then distinct. Equality of the first fundamental forms, $[h_{ab}]=0$, makes intrinsic distances and causal structure agree. Equality of the second fundamental forms, $[K_{ab}]=0$, makes a nonnull join free of a surface layer. If $h$ is continuous but $K$ jumps, the Einstein tensor has a controlled delta function. If $h$ jumps, ordinary distribution theory generally no longer defines the curvature. Israel’s invariant construction and its Gaussian-coordinate realization are developed in Israel 1966, §§ 2–7, pp. 3–10.

Route through the chapter

The pages are ordered so that the comparison itself is fixed before any dynamical equation is used.

1. Induced Data and Matching Maps introduces one abstract interface, two embeddings, and the pullbacks that make tensor equality meaningful. It distinguishes a common reparameterization from a genuinely different gluing map.
2. Darmois-Israel Junction Conditions integrates the singular part of Einstein’s equation through a nonnull interface. In the site’s sign convention the result is $8\pi G S_{ab}=\epsilon([K_{ab}]-h_{ab}[K])$, where $\epsilon=n^2$.
3. Thin-Shell Dynamics reduces the timelike spherical equation to signed square roots. It retains the orientation branches before deriving an effective potential and adds the surface flux law.
4. Null Junction Conditions replaces unit-normal extrinsic curvature by a transverse null vector and transverse curvature. Density, screen current, and pressure arise from different components of its jump.
5. Corners, Joints, and Intersecting Boundaries supplies the oriented boost or angle missing from the intrinsic joint metric. It also derives the localized action term and tests it by subdivision.
6. Matching Interior and Exterior Spacetimes applies the regular conditions to stellar world tubes. Normal Einstein constraints become traction and flux conditions without demanding that the bulk density be continuous.

The nonnull and null formalisms are complementary, not interchangeable limits. At a null surface the induced metric is degenerate and its normal is a tangent generator, so no normalized $n^\mu$ exists. The Barrabès–Israel data instead use $k^\mu$, a screen metric $q_{AB}$, and an auxiliary $N^\mu$ with $N\cdot k=-1$, as organized in Poisson 2002, §§ II–IV, pp. 2–7, Eqs. (2.1)–(4.18), PDF.

Cross-checks that travel between pages

Several tests recur because they detect different kinds of mistakes.

Regular limit. Setting the surface tensor to zero in the nonnull Israel equation must return $[K_{ab}]=0$, not merely $[K]=0$. The induced metric must already have been matched.

Orientation covariance. Reversing the continuous coorientation while consistently exchanging the side description cannot change the physical surface tensor. Reversing only one side is not a convention change; it inserts the wrong normal into the jump.

Branch recovery. A squared spherical-shell potential is only a necessary equation. Each candidate solution must be substituted into the original signed-root equation and checked against $\operatorname{sign}(n\cdot\nabla r)$ on both sides.

Null normalization. Rescaling a generator changes the bare coefficients called density, current, and pressure. The observer-normalized distribution remains invariant because the basis and delta-function coefficient transform together.

Action additivity. For spacelike joints with a common additive prescription, an artificial internal boundary leaves no residual segment or joint contribution. Null-segment normalizations must be matched as part of that cancellation. Timelike joints are excluded from this statement: their usual smoothing prescription can be nonadditive. See Lehner et al. 2016, §II H, pp. 25–27, and p. 38, footnote 23, PDF.

Matter boundary. A regular perfect-fluid–vacuum surface imposes vanishing pressure only when it is comoving and the exterior normal stress is zero. Density continuity is not one of the normal constraints.

Review the chapter

Consider a timelike spherical interface with surface stress $S^a{}_b=\operatorname{diag}(-\sigma,p,p)$. Let $n^\mu$ point from minus to plus and define $\beta_\pm=\varepsilon_\pm\sqrt{\dot R^2+f_\pm(R)}$, where $\varepsilon_\pm=\operatorname{sign}(n\cdot\nabla r)_\pm$.

1. Derive the unsquared angular junction equation in the site’s $K$ convention.
2. State what its $\sigma=0$ limit says for a regular join.
3. If the minus side is a comoving perfect fluid and the plus side is vacuum, use the regular limit to identify the surface pressure condition.
4. Explain why the analogous null calculation needs more than the degenerate metric $q_{AB}$.

Solution

The site convention gives $K^\theta{}_{\theta\,\pm}=-\beta_\pm/R$. For a timelike shell,

$$8\pi G S^a{}_b=[K^a{}_b]-\delta^a_b[K].$$

Its $\tau$ component gives $[K^\theta{}_\theta]=4\pi G\sigma$, hence

$$\beta_- -\beta_+=4\pi G\sigma R.$$

If $\sigma=0$ and there is no other surface stress, the full tensor equation requires $[K_{ab}]=0$; the angular equation alone tests only one component. Together with $[h_{ab}]=0$, this is the regular Darmois join.

The Gauss–Codazzi normal constraint then makes $T_{nn}$ continuous. For a comoving perfect fluid, $u\cdot n=0$ and $T^-_{nn}=p$. Vacuum has $T^+_{nn}=0$, so $p|_\Sigma=0$. No corresponding equation forces $\rho$ to vanish.

On a null interface, $q_{AB}$ contains no derivative transverse to the generators and has no three-dimensional inverse. One must choose $N^\mu$ with $N\cdot k=-1$ and compare the transverse curvatures $C_{ab}=-N_\mu e^\nu_a\nabla_\nu e^\mu_b$. Their jumps determine the null surface density, current, and pressure, while generator rescaling is checked on the complete distribution rather than on a bare coefficient.

References

• Israel, Werner. “Singular Hypersurfaces and Thin Shells in General Relativity.” Il Nuovo Cimento B 44, no. 1 (1966): 1–14; erratum 48, no. 2 (1967): 463. doi:10.1007/BF02710419. Erratum doi:10.1007/BF02712210.
• Lehner, Luis, Robert C. Myers, Eric Poisson, and Rafael D. Sorkin. “Gravitational Action with Null Boundaries.” Physical Review D 94, no. 8 (2016): 084046. arXiv:1609.00207. doi:10.1103/PhysRevD.94.084046.
• 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.