Fluid-Vacuum Matching

A fluid surface can carry a finite density jump without carrying a thin shell. The deciding data are geometric: a nonsingular timelike matching requires the induced metric and extrinsic curvature to agree across the surface. For a static spherical perfect fluid joined to vacuum, those conditions reduce to continuity of the enclosed mass and lapse together with zero normal pressure. They do not require the energy density itself to vanish.

Required background. Hilbert Stress–Energy from Metric Variation supplies the bulk and distributional stress tensors.

Helpful background. The Einstein Equation fixes the field equations; The Riemann Curvature Tensor identifies curvature singularities; Exact Solutions of Einstein’s Equations supplies the family context; GR Formulations develops hypersurface geometry; and Continuum Kinematics and Balance Laws supplies fluid balance.

This page uses geometrized units $G=c=1$. The mass functions are lengths, $m_{\mathrm{geom}}=G M_{\mathrm{phys}}$; the representative dimensionful shell formula below restores an explicit factor of $G$.

Darmois data at a stellar surface

Let the timelike matching hypersurface $\Sigma$ be $r=R$ in static areal-radius coordinates. On each side write

$$ds^2=-e^{2\Phi_\pm(r)}dt_\pm^2 +\frac{dr^2}{F_\pm(r)}+r^2d\Omega^2,$$

where $-$ denotes the interior and $+$ the exterior. Choose a single normal orientation from the interior toward the exterior,

$$n^a=\sqrt{F}\,(\partial_r)^a.$$

The first fundamental form is the pullback $h_{ij}$ of the spacetime metric. After a constant rescaling of one time coordinate, continuity $[h_{ij}]=0$ gives

$$e^{2\Phi_-(R)}=e^{2\Phi_+(R)}, \qquad r_-(\Sigma)=r_+(\Sigma)=R.$$

Square brackets mean exterior minus interior, evaluated using the same normal orientation. The second fundamental form $K_{ij}=h_i{}^a h_j{}^b\nabla_a n_b$ has nontrivial components

$$K_{tt}=-\sqrt{F}\,e^{2\Phi}\Phi', \qquad K_{\theta\theta}=R\sqrt{F}, \qquad K_{\phi\phi}=\sin^2\theta\,K_{\theta\theta}.$$

Thus $[K_{\theta\theta}]=0$ gives $F_-(R)=F_+(R)$. If the interior has $F_-=1-2m(r)/r$ and the exterior is Schwarzschild, $F_+=1-2M/r$, this is

$$m(R)=M.$$

The remaining condition $[K_{tt}]=0$ says that $\Phi'$ agrees at the surface. The exterior derivative is $\Phi'_+=M/[R(R-2M)]$. The interior radial Einstein equation gives

$$\Phi'_-(R)=\frac{M+4\pi R^3p(R)}{R(R-2M)}.$$

Therefore a shell-free matching requires $p(R)=0$. Conversely, continuity of the lapse and mass together with $p(R)=0$ makes the displayed $K_{ij}$ continuous. This is the spherical static specialization of the Darmois junction conditions.

Density jumps and genuine thin shells

The density enters $m'=4\pi r^2\rho$. If $\rho$ jumps from a finite interior value to zero, then $m'$ jumps but $m$ remains continuous. The metric is piecewise $C^1$ in a suitable chart, its extrinsic curvature has no jump, and its curvature contains a step but no Dirac delta. The constant-density interior provides the canonical example.

If $K_{ij}$ does jump, the singular part of Einstein’s equation defines a surface stress tensor. Restoring $G$ in this representative observable gives

$$S_{ij}=-\frac{1}{8\pi G} \left([K_{ij}]-[K]h_{ij}\right),$$

for the stated convention and a fixed normal orientation. Reversing the normal reverses both extrinsic curvatures and changes the intermediate sign convention; physical conclusions must be translated consistently. Israel’s original derivation and the distinction between boundary surfaces and surface layers are given in Israel 1966, pp. 1–14.

A nonzero surface pressure or an explicitly prescribed material membrane generally produces $S_{ij}\ne0$. Such a construction can be valid, but the shell is then part of the spacetime’s source and must be stated. Hiding it behind discontinuous coordinate derivatives changes the physical solution.

What may and may not jump

For the shell-free perfect-fluid matching just derived:

• $r$, the induced lapse, $m(R)$, and $K_{ij}$ are continuous;
• the normal pressure reaches the exterior value, hence $p(R)=0$ for vacuum;
• $\rho$, $d\rho/dr$, and tangential matter derivatives may have finite jumps;
• second coordinate derivatives of the metric may jump, so nonsingular curvature components may be discontinuous;
• no curvature invariant may diverge merely because a finite density step is present.

For an anisotropic material, it is the radial pressure $p_r=T_{ab}n^an^b$ that must match the exterior normal stress. With an electromagnetic field on both sides, the total normal stress includes the Maxwell contribution, and charge continuity depends on whether a surface current is present. The rule is always applied to the total stress tensor and the full geometry, not to one matter sector in isolation.

The derivation assumes a non-null surface. At a horizon, $F(R)=0$ makes the unit-normal construction degenerate; null junction data must be used instead. A coordinate limit of the timelike formulas is not automatically a valid null matching.

Common pitfalls

Using oppositely directed normals on the two sides. That convention can be used only with a correspondingly modified jump definition. Insert both sides into one declared convention before deciding that a shell exists.

Demanding continuity of every coordinate derivative. Darmois matching is geometric. Coordinate derivatives beyond those encoded in $h_{ij}$ and $K_{ij}$ can jump without a distributional source.

Exercises

For the constant-density interior with $m(r)=Mr^3/R^3$, verify directly that a finite jump in $\rho$ produces no delta function in $m'$.

Solution

Represent the global mass function as

$$m(r)=\frac{Mr^3}{R^3}\Theta(R-r)+M\Theta(r-R).$$

Distributional differentiation gives ordinary derivatives plus a coefficient of $\delta(r-R)$ equal to the jump in $m$:

$$M-\frac{MR^3}{R^3}=0.$$

Hence

$$m'(r)=\frac{3Mr^2}{R^3}\Theta(R-r),$$

with a finite step but no delta. Since $m'=4\pi r^2\rho$, the density is discontinuous and finite; it is not a surface mass density.

References

• Werner Israel, “Singular Hypersurfaces and Thin Shells in General Relativity,” Il Nuovo Cimento B 44, 1–14 (1966); erratum 48, 463 (1967). DOI.