Matter-Supported and Interior Spacetimes

Vacuum metrics are only one part of the exact-spacetime landscape. Matter-supported solutions make Einstein’s equation a coupled geometric problem: the metric must have the advertised curvature, the stress tensor must have the advertised algebraic form, and both must remain meaningful on the stated domain. This chapter develops a compact set of exact fluid, null-radiation, scalar, anisotropic, and charged geometries, with regularity and matching data treated as part of each solution rather than as afterthoughts.

Helpful background. The Einstein Equation supplies the field equations; The Riemann Curvature Tensor supplies invariant curvature checks; Hilbert Stress–Energy from Metric Variation fixes stress–energy conventions; Exact Solutions of Einstein’s Equations places the families in this volume; GR Formulations develops constraint and evolution formulations; and Continuum Kinematics and Balance Laws provides the matter variables used below.

This chapter uses geometrized units $G=c=1$. Its mass variables $m$ and $M$ are lengths, with $m_{\mathrm{geom}}=G M_{\mathrm{phys}}$; for example, the compactness $2M/R$ means $2G M_{\mathrm{phys}}/R$ when dimensions are restored.

Enter this chapter

For a static spherical perfect fluid, use areal radius $r$ and write

$$ds^2=-e^{2\Phi(r)}dt^2+\frac{dr^2}{1-2m(r)/r}+r^2d\Omega^2, \qquad T^{\hat a}{}_{\hat b}=\operatorname{diag}(-\rho,p,p,p).$$

Einstein’s equation and $\nabla_aT^{ab}=0$ reduce to

$$m'=4\pi r^2\rho, \qquad \Phi'=\frac{m+4\pi r^3p}{r(r-2m)}, \qquad p'=-(\rho+p)\Phi'.$$

These equations expose three different questions. Local geometry asks whether $m(r)$, $\Phi(r)$, and the curvature are regular. Matter interpretation asks whether the orthonormal-frame eigenvalues really are the claimed density and pressures. Global construction asks whether an interior can be joined to an exterior without a distributional surface stress. The static-fluid page derives this reduction and turns regularity at $r=0$ into explicit Taylor conditions.

The equations are not an astrophysical equation-of-state model by themselves. Choosing $\rho(r)$, $p(r)$, or a relation $p(\rho)$ closes the system, but physical acceptability requires more than solving the resulting ODEs. Positivity, monotonicity, causal response, stability, and a plausible material model are logically distinct tests. This chapter concentrates on exact geometry, source identification, domains, and junction data; dynamical stability and stellar inference belong to later volumes.

Six complementary exact-solution problems

The pages are ordered by which assumption is relaxed.

1. Static Spherical Fluid Geometry establishes the mass, redshift, pressure, and central-regularity variables.
2. The Constant-Density Interior Solution integrates them exactly and locates the finite-radius pressure pole. It is a clean analytic model, not a realistic incompressible star.
3. Fluid–Vacuum Matching uses induced metric and extrinsic curvature to decide which jumps are harmless and which create a thin shell.
4. Vaidya and Null-Fluid Spacetimes replaces hydrostatic matter by directed null flux and makes the sign of $dm/dv$ or $dm/du$ physically decisive.
5. Scalar-Supported Exact Spacetimes compares geometries supported by minimally or nonminimally coupled scalar fields, where an apparent horizon in one coordinate form may instead be a curvature singularity.
6. Anisotropic and Charged Interior Families shows how $p_t-p_r$ and electric charge alter the force balance and compactness bounds.

The constant-density solution and Vaidya geometry have especially transparent checks. The former must recover the exterior Schwarzschild mass $M=4\pi\rho R^3/3$ and its pressure must vanish at $r=R$. The latter must become Schwarzschild when its mass function is constant, while its Einstein tensor must have rank-one null-fluid form when the mass varies. These checks distinguish a correct metric from a plausible-looking line element. The classic exact integrations and their coordinate domains are organized in Stephani et al. 2003, chs. 14–16 and Griffiths and Podolský 2009, chs. 10 and 16.

Regularity, matching, and invariant diagnosis

A regular spherical center is not merely the coordinate value $r=0$. The area of the symmetry spheres must approach $4\pi r^2$, the mass must satisfy $m=O(r^3)$, the radial metric coefficient must approach unity, and scalar curvature invariants must remain finite. For an isotropic fluid with smooth central data, $p'(0)=\rho'(0)=0$. A formula that violates these conditions may still solve the field equations on a punctured domain; it is then an exterior or singular solution, not a regular stellar interior.

At a timelike surface $r=R$, absence of a surface layer requires continuity of the first and second fundamental forms. In the static perfect-fluid case this implies $p(R)=0$ when the exterior is vacuum, but it need not imply $\rho(R)=0$. The distinction is essential: a finite density jump changes derivatives of the matter profile without automatically producing a Dirac delta in curvature. Israel’s geometric junction formalism makes this statement precise Israel 1966, pp. 1–14.

Every family in this chapter is therefore read through the same four-part test:

• state the metric, coordinate domain, parameters, and matter action or stress tensor;
• substitute into the field equations and identify the orthonormal-frame source eigenvalues;
• inspect the center, axes, candidate horizons, and curvature scalars rather than trusting coordinates;
• test a vacuum, isotropic, uncharged, static, or weak-field limit and state the junction data.

Review the chapter

For the static ansatz above, suppose $\rho(0)=\rho_c$ and $p(0)=p_c$ are finite. Show that regularity forces $m(r)=(4\pi/3)\rho_c r^3+O(r^5)$ and $p'(0)=0$.

Solution

Smooth spherical symmetry makes $\rho(r)=\rho_c+O(r^2)$. Integrating $m'=4\pi r^2\rho$ gives

$$m(r)=\frac{4\pi}{3}\rho_c r^3+O(r^5).$$

Consequently

$$\Phi'(r)=4\pi\left(\frac{\rho_c}{3}+p_c\right)r+O(r^3).$$

The conservation equation then gives $p'=-(\rho+p)\Phi'=O(r)$, hence $p'(0)=0$. The same expansion shows $1-2m/r=1+O(r^2)$, so the areal-radius chart is locally regular rather than conical.

References

• Jerry B. Griffiths and Jiří Podolský, Exact Space-Times in Einstein’s General Relativity, Cambridge University Press, 2009. DOI.
• Werner Israel, “Singular Hypersurfaces and Thin Shells in General Relativity,” Il Nuovo Cimento B 44, 1–14 (1966); erratum 48, 463 (1967). DOI.
• Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt, Exact Solutions of Einstein’s Field Equations, 2nd ed., Cambridge University Press, 2003. DOI.