Oppenheimer–Snyder Collapse: the Interior Matching

Contents

The interior: a contracting closed dust universe
The boundary, and the matching conditions
The global picture

This appendix proves Theorem 49.23 (Chapter 49): the collapse of a uniform, pressureless, momentarily static ball of dust is an exact solution of the field equations, consisting of a contracting closed Friedmann–Lemaître–Robertson–Walker (FLRW) interior matched smoothly — induced metric and extrinsic curvature both continuous — to a Schwarzschild exterior across the falling surface [Oppenheimer:1939b]. The exterior half of the argument is already in the chapter: Birkhoff's theorem (Theorem 49.4) forces the outside geometry to be Equation (49.6), and the radial-plunge cycloid (Proposition 49.21) is the surface's worldline seen from outside. What remains — the interior solution, the identification of the boundary, and the junction — follows, in the treatment of [Misner:1973].

The interior: a contracting closed dust universe

Inside the ball the matter is homogeneous and isotropic about every comoving point, so the interior metric is of FLRW form; the momentarily static initial state will force the closed (positively curved) case, and we write it, in comoving coordinates \((\tau,\chi,\theta,\varphi)\) with \(x^{0}:=c\tau\),

\begin{equation}\tag{A2.1} \dd s^{2}=-c^{2}\dd\tau^{2} +a^{2}(\tau)\left[\dd\chi^{2} +\sin^{2}\chi\left(\dd\theta^{2} +\sin^{2}\theta\,\dd\varphi^{2}\right)\right]\ec \end{equation}

which is Equation (49.34); \(\tau\) is proper time on every comoving worldline. Write \(g_{ij}=a^{2}\gamma_{ij}\) with \(\gamma_{ij}\) the metric of the unit 3-sphere and primes for \(\dd/\dd x^{0}\). The Christoffel symbols of Equation (A2.1) are, by Equation (17.299),

\begin{equation}\tag{A2.2} \Gamma^{0}{}_{ij}=a\,a'\,\gamma_{ij}\ec\qquad \Gamma^{i}{}_{0j}=\frac{a'}{a}\,\delta^{i}{}_{j}\ec\qquad \Gamma^{i}{}_{jk}=\tilde{\Gamma}^{i}{}_{jk}\ec \end{equation}

with \(\tilde{\Gamma}\) the Christoffel symbols of \(\gamma_{ij}\) alone. Contracting the Riemann tensor Equation (17.306) and using the fact that the unit 3-sphere is maximally symmetric with Ricci tensor \(\tilde{R}_{ij}=2\gamma_{ij}\),

\begin{equation}\tag{A2.3} R_{00}=-3\,\frac{a''}{a}\ec\qquad R_{ij}=\left(a\,a''+2a'^{2}+2\right)\gamma_{ij}\ep \end{equation}

Pressureless dust comoving with these coordinates has \(T_{\mu\nu}=\rho\,u_{\mu}u_{\nu}\) with \(u_{\mu}=(-c,0,0,0)\), hence \(T_{00}=\rho c^{2}\), \(T_{ij}=0\) and trace \(T=-\rho c^{2}\). Two facts follow before any field equation is solved. First, since \(\Gamma^{\mu}{}_{00}=0\) in Equation (A2.2), the comoving worldlines are geodesics — as they must be: \(\nabla_{\mu}T^{\mu\nu}=0\) for dust is exactly the statement that each grain free-falls, and it is what makes the boundary of the truncated ball a geodesic of both geometries below. Second, in the Ricci form \(R_{\mu\nu}=(8\pi G/c^{4})\left(T_{\mu\nu} -\tfrac{1}{2}g_{\mu\nu}T\right)\) (The Einstein Field Equations) the two components of Equation (A2.3) give, restoring \(\tau\) via \(\dot{a}=c\,a'\),

\begin{align} \ddot{a}&=-\frac{4\pi G}{3}\,\rho\,a\ec \tag{A2.4}\\ \dot{a}^{2}+c^{2}&=\frac{8\pi G}{3}\,\rho\,a^{2}\ec \tag{A2.5} \end{align}

the second obtained by eliminating \(a''\) between the \(00\) and \(ij\) equations. (These are the Friedmann equations of Evidence-Based Cosmology for dust and \(k=+1\); the derivation is kept here so that this proof is self-contained.) Differentiating Equation (A2.5) and inserting Equation (A2.4) yields \(\dot{\rho}/\rho=-3\dot{a}/a\), i.e.

\begin{equation}\tag{A2.6} \rho\,a^{3}=\rho_{\mathrm{m}}\,a_{\mathrm{m}}^{3} =\text{const}\ec \end{equation}

mass conservation in a contracting volume.

“Momentarily static” means \(\dot{a}=0\) at \(\tau=0\); call the initial values \(a_{\mathrm{m}}\), \(\rho_{\mathrm{m}}\). Then Equation (A2.5) at \(\tau=0\) reads

\begin{equation}\tag{A2.7} \frac{8\pi G}{3}\,\rho_{\mathrm{m}} =\frac{c^{2}}{a_{\mathrm{m}}^{2}}\ec \end{equation}

which is why the curvature had to be positive: a momentarily static homogeneous dust ball has nowhere to go but down its own cycloid. With Equations (A2.6) and (A2.7), Equation (A2.5) becomes \(\dot{a}^{2}=c^{2}\left(a_{\mathrm{m}}/a-1\right)\), which the cycloid solves: substituting

\begin{equation}\tag{A2.8} a=\frac{a_{\mathrm{m}}}{2}\left(1+\cos\eta\right)\ec\qquad c\tau=\frac{a_{\mathrm{m}}}{2}\left(\eta+\sin\eta\right) \end{equation}

gives, exactly as in the proof of Proposition 49.21, \(\dot{a}^{2}=c^{2}(1-\cos\eta)/(1+\cos\eta) =c^{2}\left(a_{\mathrm{m}}/a-1\right)\). The scale factor reaches \(a=0\) at \(\eta=\pi\), i.e. at proper time

\begin{equation}\tag{A2.9} \tau_{\mathrm{collapse}}=\frac{\pi a_{\mathrm{m}}}{2c}\ec \end{equation}

the same on every comoving worldline: by homogeneity, all the dust — centre and surface alike — arrives at the singularity simultaneously in comoving time. This is part (iii) of Theorem 49.23.

The boundary, and the matching conditions

Truncate the interior at \(\chi=\chi_{0}\) (with \(\chi_{0}<\pi/2\) for a ball less than half the 3-sphere). The boundary consists of comoving dust worldlines, which are geodesics of the interior geometry by the first fact above; its areal radius — read off the angular part of Equation (A2.1) — is

\begin{equation}\tag{A2.10} R(\tau)=a(\tau)\sin\chi_{0}\ec\qquad R_{0}:=R(0)=a_{\mathrm{m}}\sin\chi_{0}\ep \end{equation}

Outside, spherical symmetry and Theorem 49.4 force the Schwarzschild geometry Equation (49.6) with some mass parameter \(M\); the boundary, being made of freely falling dust with no pressure acting on it from either side, must follow a radial timelike geodesic of that exterior, released from rest at \(R_{0}\). By Proposition 49.21 its areal radius is the cycloid Equation (49.31),

\begin{equation}\tag{A2.11} R=\frac{R_{0}}{2}\left(1+\cos\eta\right)\ec\qquad \tau=\sqrt{\frac{R_{0}^{3}}{8GM}}\, \left(\eta+\sin\eta\right)\ep \end{equation}

The interior assigns the same boundary the areal radius Equation (A2.10) with \(a(\tau)\) the cycloid Equation (A2.8): the same shape \(R=(R_{0}/2)(1+\cos\eta)\), with proper time \(\tau=\left(a_{\mathrm{m}}/2c\right)(\eta+\sin\eta)\). The two descriptions of one worldline agree for all \(\eta\) iff

\begin{equation}\tag{A2.12} \sqrt{\frac{R_{0}^{3}}{8GM}}=\frac{a_{\mathrm{m}}}{2c} \qquad\Longleftrightarrow\qquad r_{\mathrm{s}}=\frac{2GM}{c^{2}} =\frac{R_{0}^{3}}{a_{\mathrm{m}}^{2}} =a_{\mathrm{m}}\sin^{3}\chi_{0}\ec \end{equation}

which is Equation (49.35). The mass identity follows at once: by Equation (A2.7),

\begin{equation}\tag{A2.13} \frac{4\pi}{3}\,\rho_{\mathrm{m}}R_{0}^{3} =\frac{c^{2}}{2G\,a_{\mathrm{m}}^{2}}\, a_{\mathrm{m}}^{3}\sin^{3}\chi_{0} =\frac{c^{2}}{2G}\,a_{\mathrm{m}}\sin^{3}\chi_{0} =\frac{c^{2}r_{\mathrm{s}}}{2G}=M\ep \end{equation}

The Schwarzschild mass is the initial density times \(\tfrac{4}{3}\pi R_{0}^{3}\) — the Euclidean volume of areal radius \(R_{0}\), not the (larger) proper volume of the curved interior. The deficit is not an error: \(Mc^{2}\) is the total energy including the (negative) gravitational binding energy, and the curved-volume excess of rest mass is exactly what binding has removed.

It remains to show that agreement of the boundary worldline is the whole of the junction condition — that the two geometries fit with no surface layer. Two conditions must hold on the matching hypersurface: the induced metrics must agree, and the extrinsic curvatures must agree.

Induced metric. Coordinatize the boundary by \((\tau,\theta,\varphi)\). From inside, Equation (A2.1) restricted to \(\chi=\chi_{0}\) gives \(-c^{2}\dd\tau^{2}+a^{2}\sin^{2}\chi_{0}\,\dd\Omega^{2} =-c^{2}\dd\tau^{2}+R(\tau)^{2}\dd\Omega^{2}\). From outside, along a radial worldline parametrized by its proper time at areal radius \(R(\tau)\), the induced metric is likewise \(-c^{2}\dd\tau^{2}+R(\tau)^{2}\dd\Omega^{2}\). Given Equation (A2.12) the two functions \(R(\tau)\) coincide, so the induced metrics agree identically.

Extrinsic curvature. By spherical symmetry the extrinsic curvature has two independent components, \(K^{\tau}{}_{\tau}\) and \(K^{\theta}{}_{\theta}=K^{\varphi}{}_{\varphi}\). The \(\tau\tau\) component measures the normal component of the boundary worldlines' acceleration, and vanishes on both sides because the boundary is geodesic in both geometries — comoving dust inside, radial free fall outside. For the angular component, from inside the outward unit normal is \(n=a^{-1}\pp_{\chi}\); for a hypersurface of constant \(\chi\) in a metric with no \(\chi\)-cross terms, \(K_{ij}=\tfrac{1}{2}\,n^{\chi}\pp_{\chi}g_{ij}\), and with \(g_{\theta\theta}=a^{2}\sin^{2}\chi\),

\begin{equation}\tag{A2.14} K^{\theta}{}_{\theta} =\frac{K_{\theta\theta}}{g_{\theta\theta}} =\frac{a\sin\chi_{0}\cos\chi_{0}}{a^{2}\sin^{2}\chi_{0}} =\frac{\cos\chi_{0}}{a\sin\chi_{0}} =\frac{\cos\chi_{0}}{R}\ep \end{equation}

From outside, write \(f:=1-r_{\mathrm{s}}/R\) and let \(u^{\mu}=(u^{t},u^{r})\) be the falling boundary's four-velocity, with \(u^{t}=\mathcal{E}/(fc^{2})\) and \(c^{2}(u^{r})^{2}=\mathcal{E}^{2}-fc^{4}\) (Proposition 49.21). The outward unit normal \(n^{\mu}=(n^{t},n^{r})\) is fixed by \(n_{\mu}u^{\mu}=0\) and \(n_{\mu}n^{\mu}=1\): orthogonality gives \(n^{t}=u^{r}n^{r}/(f^{2}c^{2}u^{t})\), and substituting into the normalization \(-fc^{2}(n^{t})^{2}+f^{-1}(n^{r})^{2}=1\),

\[ \left(n^{r}\right)^{2} \left[\frac{1}{f} -\frac{\left(u^{r}\right)^{2}}{f^{3}c^{2}\left(u^{t}\right)^{2}} \right] =\frac{\left(n^{r}\right)^{2}}{f}\, \frac{\mathcal{E}^{2}-c^{2}\left(u^{r}\right)^{2}} {\mathcal{E}^{2}} =\left(n^{r}\right)^{2}\frac{c^{4}}{\mathcal{E}^{2}}=1\ec \]

using \(f^{3}c^{2}(u^{t})^{2}=f\,\mathcal{E}^{2}/c^{2}\) in the first step and \(\mathcal{E}^{2}-c^{2}(u^{r})^{2}=fc^{4}\) in the second; so \(n^{r}=\mathcal{E}/c^{2}\). Since the only angular Christoffel symbol involved is \(\Gamma^{\theta}{}_{r\theta}=1/R\) (Equation (49.2)),

\begin{equation}\tag{A2.15} K^{\theta}{}_{\theta}=n^{r}\,\Gamma^{\theta}{}_{r\theta} =\frac{\mathcal{E}}{c^{2}R} =\frac{\sqrt{1-r_{\mathrm{s}}/R_{0}}}{R}\ep \end{equation}

The two expressions Equations (A2.14) and (A2.15) agree iff \(\cos\chi_{0}=\sqrt{1-r_{\mathrm{s}}/R_{0}}\), i.e. iff \(r_{\mathrm{s}}=R_{0}\sin^{2}\chi_{0} =a_{\mathrm{m}}\sin^{3}\chi_{0}\) — precisely the condition Equation (A2.12) already imposed by the worldline. The junction is therefore smooth, with no surface shell, exactly when Equation (49.35) holds; parts (i) and (ii) of Theorem 49.23 are proved.

The global picture

The matched spacetime settles the questions the vacuum extension of the chapter left open. The surface crosses \(r=r_{\mathrm{s}}\) at the finite comoving time given by \(\eta_{\mathrm{h}}\) with \(\cos\eta_{\mathrm{h}}=2r_{\mathrm{s}}/R_{0}-1\); from that moment the exterior region contains trapped surfaces (Remark 49.26) and the vacuum part of the spacetime is the regions I and II of the Kruskal diagram (Remark 49.20) — the white-hole region III and the second universe IV are excised, their place taken by the matter-filled interior, which is why they are artefacts of vacuum eternity rather than predictions about collapse. A distant observer, meanwhile, receives the exponentially redshifted last light of Proposition 49.22: the “frozen star” outside, the finite-time singularity inside, with no contradiction between them [Oppenheimer:1939b] [Misner:1973].