Induced Data and Matching Maps

Joining two spacetimes begins with a choice that is easy to hide in coordinates: one must decide which event on one boundary is the same event on the other. An abstract interface and two embeddings make that choice explicit. They also separate the first matching condition—the continuity of intrinsic geometry—from the behavior of transverse derivatives, which determines whether the join is regular or carries a thin layer.

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 explains why the same induced metric and extrinsic curvature also control the Dirichlet boundary problem.

Abstract interface and two embeddings

Let $M_-$ and $M_+$ have boundary hypersurfaces $\Sigma_-$ and $\Sigma_+$. Instead of identifying their coordinate labels directly, introduce one abstract three-manifold $\Sigma$ and embeddings

$$X_-:\Sigma\longrightarrow M_-, \qquad X_+:\Sigma\longrightarrow M_+.$$

The corresponding boundary identification is $\varphi=X_+\circ X_-^{-1}:\Sigma_-\to\Sigma_+$, understood on the embedded images. In intrinsic coordinates $y^a$ on $\Sigma$, the tangent vectors are $e^{\mu}_{\pm a}=\partial X_\pm^\mu/\partial y^a$. Pulling the two spacetime metrics back to the same abstract interface gives the first fundamental forms

$$h^{\pm}_{ab}=g^{\pm}_{\mu\nu}e^\mu_{\pm a}e^\nu_{\pm b}.$$

For every quantity that has been pulled back to this common copy of $\Sigma$, this chapter uses

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

Thus the intrinsic matching condition is $[h_{ab}]=0$. It says that lengths, causal character, and angles measured entirely within the interface do not depend on the side from which the interface is approached. This geometric formulation is the coordinate-free content of the first junction condition; Israel develops precisely this embedding-based viewpoint in Israel 1966, §§ 2–4, pp. 3–7.

A normal must be chosen as part of the comparison. Write $n^\mu n_\mu=\epsilon$, with $\epsilon=+1$ for a timelike interface and $\epsilon=-1$ for a spacelike interface, and choose one continuous coorientation: $n^\mu$ points from $M_-$ toward $M_+$. The site convention for the second fundamental form is

$$K^\pm_{ab} =-e^\mu_{\pm a}e^\nu_{\pm b}\nabla^\pm_\mu n^\pm_\nu.$$

This is not the convention used in Israel’s original displayed formulas: his $K_{ab}$ has the opposite sign. Translating his equations therefore reverses every occurrence of his extrinsic curvature before a jump is taken. In particular, a regular nonnull join requires both $[h_{ab}]=0$ and $[K_{ab}]=0$ in the convention above.

Why the matching map is physical data

Suppose the intrinsic coordinates are relabeled by one diffeomorphism $\chi:\widetilde\Sigma\to\Sigma$ on both sides. The new embeddings are $\widetilde X_\pm=X_\pm\circ\chi$, so

$$\widetilde h_\pm=\chi^*h_\pm, \qquad \widetilde K_\pm=\chi^*K_\pm.$$

Consequently $h_+=h_-$ and $K_+=K_-$ are invariant statements. By contrast, independently relabeling only one embedding changes $\varphi$. Unless that relabeling is an isometry of the interface data, it describes a different gluing rather than a harmless coordinate change. This distinction matters, for example, when an angular shift, time translation, or reflection is inserted into a quotient construction.

The first and second fundamental forms answer different questions. Equality of $h_{ab}$ makes the glued metric continuous in Gaussian normal coordinates. Near a nonnull interface one can write

$$ds^2=\epsilon\,d\ell^2+h_{ab}(\ell,y)dy^a dy^b, \qquad K_{ab}=-\frac12\partial_\ell h_{ab},$$

where $\ell$ increases from the minus side to the plus side. Equality of $K_{ab}$ then makes the first normal derivative continuous. If $h_{ab}$ is continuous but $K_{ab}$ jumps, curvature contains a Dirac delta with a well-defined thin-shell interpretation. If $h_{ab}$ itself jumps, the ordinary distributional construction generally fails: the connection already contains a delta, and quadratic connection terms are not canonically defined.

Orientation check and failure modes

The continuous normal is not the pair of outward normals of the two retained regions. Those outward normals point oppositely at a glued boundary. One may formulate an action using two outward normals, but one must first convert both extrinsic curvatures to the single continuous coorientation before forming $[K_{ab}]$. Mixing the two conventions creates a fictitious jump even when the geometry is smooth.

A second failure is to compare component arrays before pulling them back with the proposed embeddings. Equal-looking components can refer to different points or tangent bases, while unequal-looking components can describe the same tensor after a common reparameterization. A third is to regard $[h]=0$ as sufficient for a matter-free join. It controls only the metric itself; the jump of its transverse derivative is encoded in $[K]$ and is tested by the Einstein equation.

Exercises

Two copies of Minkowski spacetime are joined across the timelike cylinder $r=R$. With intrinsic coordinates $(\tau,\theta,\phi)$, take $X_\pm(\tau,\theta,\phi)=(t=\tau,r=R,\theta,\phi)$. Show that a common reparameterization $\tau=f(s)$ preserves matching, while applying it only to $X_+$ generally does not. Check the extrinsic-curvature sign using the continuous normal $n=+\partial_r$.

Solution

Both pullbacks initially give

$$h_+=h_-=-d\tau^2+R^2(d\theta^2+\sin^2\theta\,d\phi^2).$$

After the common change $\tau=f(s)$, both become $-f'(s)^2ds^2+R^2d\Omega^2$, so equality is unchanged. If only the plus embedding is changed, then $h^+_{ss}=-f'(s)^2$ whereas $h^-_{ss}=-1$; the proposed identification fails unless $f'(s)^2=1$.

For $n=+\partial_r$, the site convention gives $K_{\theta\theta}=-R$ and $K_{\phi\phi}=-R\sin^2\theta$ on both sides, hence $[K_{ab}]=0$. The outward normal of the exterior region $r\ge R$ is instead $-\partial_r$. Using that vector only on the plus side reverses $K^+_{ab}$ and manufactures a nonzero jump. It is an orientation error, not a shell.

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.