Matching Interior and Exterior Spacetimes

The surface of a star is a matching problem, not a place where every bulk field must be continuous. A regular fluid–vacuum join requires equality of the first and second fundamental forms. Einstein’s constraint projections then select the physically necessary boundary conditions: for a comoving perfect-fluid surface next to vacuum, the pressure vanishes, while the density may jump.

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 relates the same induced and extrinsic data to a well-posed nonnull boundary variation.

Timelike stellar surface and regular matching

Let $M_-$ be the material interior and $M_+$ the exterior. Their timelike boundaries are identified with one abstract world tube $\Sigma$. Choose the continuous spacelike unit normal $n^\mu$ to point from the interior toward the exterior, so $n^\mu n_\mu=+1$. After pullback to $\Sigma$, use

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

The intrinsic metric $h_{ab}$ fixes proper time and the two-geometry of each stellar cross-section. With the site convention,

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

A join without a surface layer obeys

$$[h_{ab}]=0, \qquad [K_{ab}]=0.$$

The normal is not chosen outward from each retained region. The outward normal of the exterior manifold’s inner boundary points toward the star and is opposite to the continuous $n^\mu$. It must be reversed before the two extrinsic curvatures are compared.

These conditions do not demand one coordinate system across the surface. Interior comoving coordinates and exterior static or radiative coordinates can be used independently; the embeddings identify the same proper time, angular points, and areal radius. Israel emphasizes this coordinate-independent characterization of ordinary boundary surfaces in Israel 1966, §§ 3–4, pp. 6–7.

What the Einstein constraints require

The Gauss and Codazzi equations express $G_{\mu\nu}n^\mu n^\nu$ and $G_{\mu\nu}n^\mu e^\nu_a$ entirely in terms of $h_{ab}$, $K_{ab}$, and derivatives intrinsic to $\Sigma$. Therefore regular matching implies

$$[G_{\mu\nu}n^\mu n^\nu]=0, \qquad [G_{\mu\nu}n^\mu e^\nu_a]=0.$$

If the cosmological constant is the same on both sides, Einstein’s equation turns these into

$$[T_{\mu\nu}n^\mu n^\nu]=0, \qquad [T_{\mu\nu}n^\mu e^\nu_a]=0.$$

Now suppose the interior is a perfect fluid,

$$T^-_{\mu\nu} =(\rho+p)u_\mu u_\nu+p\,g_{\mu\nu},$$

and the surface is comoving, so $u\cdot n=0$. Then

$$T^-_{\mu\nu}n^\mu n^\nu=p.$$

For a vacuum exterior, $T^+_{\mu\nu}=0$, and the first normal constraint gives

$$p\big|_\Sigma=0.$$

The density $\rho$ does not appear in this normal projection and may approach a nonzero interior limit before jumping to zero outside. This is why a sharp density surface need not be a thin shell. The Oppenheimer–Snyder dust boundary is the simplest limiting check: $p=0$ identically, the density jumps, and no surface stress tensor is required.

If the exterior contains electromagnetic fields, radiation, or a different effective vacuum stress, the condition is equality of normal traction rather than $p=0$. If the surface is not comoving, $(u\cdot n)\ne0$ and

$$T^-_{nn}=(\rho+p)(u\cdot n)^2+p;$$

the normal-tangential equation also tracks matter flux through the moving interface. Calling such a surface “the same fluid particles” would be inconsistent.

Spherical mass check and shell alternative

For this mass check set $\Lambda=0$ and impose spherical symmetry. Let $R$ denote areal radius. The angular extrinsic curvature is

$$K^\theta{}_\theta=-\frac{n^\mu\nabla_\mu R}{R}.$$

Continuity of $h_{ab}$ equates $R$ and its derivative along the surface velocity $u^\mu$. Continuity of $K^\theta{}_\theta$ equates the normal derivative of $R$. Since

$$1-\frac{2Gm}{R} =g^{\mu\nu}\nabla_\mu R\nabla_\nu R =-(u^\mu\nabla_\mu R)^2 +(n^\mu\nabla_\mu R)^2$$

on the world tube, the Misner–Sharp mass also matches: $m_-=m_+$. For a Schwarzschild exterior this gives $m_-(\Sigma)=M$. This check uses only invariant surface data, not equality of radial coordinate components.

If $p|_\Sigma\ne0$ next to vacuum, a regular perfect-fluid join is impossible under the stated assumptions. One may change the exterior matter, move the interface relative to the fluid, or introduce a genuine surface layer with $8\pi G S_{ab}=[K_{ab}]-h_{ab}[K]$. It is not consistent simply to keep $[K_{ab}]=0$ and ignore the normal Einstein equation.

Exercises

A comoving perfect-fluid interior is matched across a timelike surface to vacuum with no thin shell. Derive the pressure condition and explain why it imposes no analogous requirement that $\rho$ vanish. Then state what changes if the exterior has normal stress $P_{\mathrm{ext}}=T^+_{\mu\nu}n^\mu n^\nu$.

Solution

Regular matching makes the normal Einstein projection continuous. With the same cosmological constant on both sides,

$$0=[T_{nn}]=T^+_{nn}-T^-_{nn}.$$

For a comoving surface $u\cdot n=0$ and $n^2=+1$, so

$$T^-_{nn} =(\rho+p)(u\cdot n)^2+p\,n^2=p.$$

Vacuum has $T^+_{nn}=0$, hence $p|_\Sigma=0$. The energy density multiplies $(u\cdot n)^2$ in this projection, so the equation does not set $\rho|_\Sigma$ to zero. A finite density jump changes the regular bulk curvature on the two sides but creates no delta function when $h$ and $K$ match.

With exterior normal stress $P_{\mathrm{ext}}$, the same calculation gives $p|_\Sigma=P_{\mathrm{ext}}$. This is traction balance, not a universal zero-pressure rule.

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.