General relativity breaks down at the space-time singularities associated with the Big Bang at the birth of the universe, and at the black-hole singularities which arise from the total gravitational collapse of massive stars which have exhausted their thermonuclear fuel. Global and topological techniques developed by Hawking and Penrose, demonstrated that singularities are totally unavoidable within pure general relativity provided specific conditions hold: namely that matter satisfies the strong energy condition (SEC) such that $\mathbf{T}_{\mu\nu}\xi^{\mu}\xi^{\nu}\ge 0$ or $\rho>0$ and $\sum_{i}p_{i}\ge 0$, for density and pressures, and that the space-time manifold $\mathbf{\mathcal{M}}$ must satisfy appropriate casuality conditions such as global hyperbolicity.

Gravitational collapse and singularity formation are unavoidable once a trapped surface $\mathbf{\mathcal{T}}$ is formed around a sufficiently heavy collapsing star which is well above the Oppenheimer-Volkoff limit, and which has exhausted its fuel. The star can then no longer provide the thermal and radiative transport/pressure required to maintain the equilibrium condition required by the Tolman-Oppenheimer-Volkoff equation –or its Newtonian limit–and which kept it stable for billions of years. Most of the stars in the night sky are within the Oppenheimer-Volkoff limit and will ultimately collapse to super-dense white dwarfs or neutron stars following a red giant phase of nuclear helium burning. Since the (thermo) nuclear reaction rates $R$  for fusion of hydrogen (and heavier elements) in its core varies  as $R \sim M^{3}$, heavier stars burn through their fuel much more rapidly–a star 10 time heavier than the sun will burn through its fuel about a 1000 times faster. In reality, the heaviest stars will supernova but a superheavy central core can still completely collapse to a point.

It is remarkable that the Tolman-Oppenhimer-Volkoff equation follows directly from the energy conservation constraint imposed on the Einstein equations applied to a self-gravitating and isotropic spherically symmetric fluid or gas. The proper density, pressure and gravitational fields within a spherically symmetric star are described by the Einstein equations coupled to a perfect fluid-source tensor:

$\mathbf{G}_{\mu\nu}=\mathbf{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-8\pi G \mathbf{T}_{\mu\nu}$

or

$\mathbf{R}_{\mu\nu}=-8\pi G \mathbf{T}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathbf{T}_{\gamma}{\gamma}$

where $\mathbf{T}_{\mu\nu}=pg_{\mu\nu}+(p+\rho)U_{\mu}U_{\nu}$ and $U_{\mu}$ is the velocity 4-vector defined so that $g^{\mu\nu}U_{\mu}U_{\nu}$. Since the fluid is at rest $U_{r}=U_{\theta}=U_{\phi}=0$ and $U_{t}=-(h(r))^{1/2}$. It is a standard (albeit tedious) calculation to derive the Riemann tensor components and Einstein field equations for a spherical perfect-fluid star (refs) with an exterior metric

$ds^{2}=-f(r)dr^{2}+h(r)dr^{2}+r^{2}(sin^{2}\theta d\theta^{2}+d\varphi^{2})$

so that

$\mathbf{G}_{tt}=-8\pi G \mathbf{T}_{tt}$

$\mathbf {G}_{rr}=8\pi G \mathbf{T}_{rr}$

$\mathbf{G}_{\theta\theta}=\mathbf{G}_{\varphi\varphi}= 8\pi G \mathbf{T}_{\theta\theta}$

The solution is the interior Schwarzchild metric:

$ds^{2}=-\bigg(1-\frac{2GM(r)}{r}\bigg)dt^{2}+\bigg(1-\frac{2GM(r)}{r}\bigg)^{-1}dr^{2}+r^{2}d\Omega^{2}$

where the interior mass up to radius r within the star is

$M(r)=4\pi {\displaystyle \int_{0}^{r}}\rho(r')r'^{2}dr'$.

For the exterior region $r>R$ for a star of radius $R$, we have $p=\rho=0$ so the metric (-) smoothly joins into the vacuum Schwarzchild metric with total mass $M=M(R)=4\pi \int_{0}^{R}\rho(r)r^{2}dr$. Computing the total mass M of the star of radius R by integrating the density over its volume gives

$M={\displaystyle \int_{o}^{R}}4\pi \rho(r)r^{2} (1-2GM(r)/r)^{-1/2}$

The difference between this mass and the non-relativistic counterpart is

$\delta M={\displaystyle \int_{o}^{R}}4\pi \rho(r)r^{2}\bigg( (1-2GM(r)/r)^{-1/2}-1\bigg)$

which is due to gravitational binding energy. The interior mass term $M(r)$, giving the mass up to given radius within the star, now “runs” within the interior region $0\le r\le R$ then $M(r)\rightarrow 0$ fast enough to remove the central singularity as $r\rightarrow 0$ and $M(r)/r$ remains finite as $r\rightarrow 0$.

An important additional equation is the equation for hydrostatic equilibrium
which is a direct consequence of the energy conservation condition imposed on the matter source tensor $\nabla^{\mu}\mathbf{T}_{\mu\nu}=0$ v ia theEinstein equations. A standard computation yields the Tolman-Oppenheimer-Volkov equilibrium equation:

$-r^{2}\frac{dp(r)}{dr}=GM(r)\rho(r)\left(1+\frac{p(r)}{\rho{r}}\right) \left( 1+\frac{4\pi r^{3}p(r)}{M(r)}\right)\left(1-\frac{2GM(r)}{r}\right)^{-1}$

For lower mass (Newtonian) stars–most of the stars in the night sky–the relativistic terms can be dropped giving the basic Newtonian equilibrium equation.

$-r^{2}\frac{dp(r)}{dr}=GM(r)\rho(r)$

This is one of the fundamental equations of stellar structure theory with general relativist corrections provided by the last three factors in the TOV  equation(ref). It is truly remarkable that this equation can be seen to arise from the purely geometrical Bianchi identify condition imposed on the Einstein tensor $\nabla^{\nu}\mathbf{G}_{\mu\nu}=\nabla\mathbf{T}_{\mu\nu}=0$. The relativistic factors in the Tolman-Oppenheimer-Volkoff equations are all positive definite so that for any value of r, Newtonian gravity seems to become amplified–relativistic corrections essentially strengthen the pull of gravitation.

When supplemented with an equation of state relating pressure to density,  this correctly describes a self=gravitiating spherically symmetric mass of isotropic fluid/gas in ‘gravi-hydrostatic’ equilibrium. They are integrated with respect to the boundary conditions $M(0)=0, P(R)=\rho(R)=0$. When its fuel is exhausted, the star can longer provide the pressure gradient required in the TOV equation so that

$-r^{2}\frac{dp(r)}{dr} \ne GM(r)\rho(r)\left(1+\frac{p(r)}{\rho{r}}\right) \left( 1+\frac{4\pi r^{3}p(r)}{M(r)}\right)\left(1-\frac{2GM(r)}{r}\right)^{-1}$

and if above the Oppenheimer-Volkoff limit, the star collapses under its own weight to a point.  In more precise terms, a trapped surface forms around the star and null geodisic flows from the surface  are incomplete.

It is not necessary to solve the Einstein equations to prove the singularity theorems: they enter as a constraint condition imposed on matter via the strong energy conditions. Central to the proof of singularity theorems are the concepts of geodesic incompleteness, conjugate points, trapped surfaces, and the nonlinear Raychaudhuri equation for the expansion parameter of a congruence of geodesic flows. Additional singularity theorems remove some unwanted assumptions such as global hyperbolicity but the essential feature of each theorem is the concept of geodesic incompleteness in terms of the volume expansion $\theta$ on spacetimes where the Einstein equations hold along with the strong or weak energy conditions for matter. The accepted standard definition of a singularity free spaceime is as follows:

A spacetime $(\mathcal{M}, g)$ is singularity free if it is geodesically complete. A trapped surface $\mathcal{T}$ is defined as follows: Let $(\mathbf{\mathcal{M}},g)$ be a globally hyperbolic spacetime with Cauchy foliation $\lbrace\mathbf{\Sigma}\rbrace$. A compact 2-dimensional smooth space-like submanifold $\mathbf{\mathscr{T}}\subset \Sigma$, bounding a compact domain $\mathbf{\mathcal{K}}\subset\mathbf{\Sigma}$ is a trapped surface if$\theta\equiv tr(\chi)~<~0$ on $\mathbf{\mathcal{T}}$. And $\mathbf{\mathcal{T}}=\partial\mathbf{\mathcal{K}}$ with expansion $\theta=Tr(\chi)$ and $\chi_{\alpha}^{\beta}=\nabla_{\alpha}u^{\beta}$ for tangent vectors $u^{\alpha}$

At a trapped surface light cones will essentially “tip inward”. All trapped surfaces are entirely contained within black-hole regions of spacetime so that $\mathbf{\mathcal{T}}\in\mathbf{\mathbf{B}}$, where the black-hole region $\mathbf{B}$ is the boundary of the casual past of future-null infinity; that is $\mathbf{B}=\mathbf{\mathcal{M}}-J^{-}(\mathscr{J}^{+})$. The first singularity theorem was given by Penrose in 1965 in relation to trapped surfaces and future-directed null geodesics from the surface
$\bf{Theorem}$
Let $(\mathcal{M},\mathbf{g})$ be a connected globally hyperbolic spacetime with Cauchy hypersurfaces $\mathbf{\Sigma}$ and $\mathbf{\mathcal{M}},\mathbf{g})$ is a development of initial Cauchy data. let $\mathbf{T}_{\mu\nu}$ describe matter on $(\mathcal{M},\mathbf{g})$ such that the Einstein equations hold. so that $\mathbf{R}_{\mu\nu}=8\pi G \mathbf{T}_{\mu\nu}$. For all null vectors $n_{\mu}$, the strong energy condition (SEC) $q=\mathbf{T}_{\alpha\beta}u^{\alpha}u^{\beta}\ge 0$ implies $\mathbf{R}_{\alpha\beta}n^{\alpha}n^{\beta}\ge 0$. Let $\mathbf{\mathcal{T}}$ be a trapped surface, the boundary of a closed compact domain $\mathbf{\mathcal{K}}\subset\mathbf{\Sigma}$ with $\chi_{o}=\theta_{o}<0$. Then at least one inextendible future-directed orthogonal geodesic from $\mathbf{\mathscr{T}}$ has affine length no greater than $|\theta_{o}^{-1}\alpha|$ where $\alpha=\frac{1}{2}$.

There is also an initial-value Cauchy statement of the theorem in that the Cauchy development of initial data for an Einstein-matter system will always be incomplete. An important development of the theorems was the Cosmic Censure Conjecture which states thats Cauchy developments of initial data for an Einstein-matter system always leads to a singularity hidden behind a horizon or outermost trapped surface.

Let $(\mathcal{M},\mathbf{g})$ be a spacetime and let $\mathcal{O}\subset\mathcal{M}$ be an open subset of $\mathcal{M}$. A congruence $\mathcal{C}$ in $\mathcal{O}$ is a set of curves or geodesics $\lbrace \mathscr{G}\rbrace$ such that for each $p\in\mathcal{O}$, one and only one geodesic $\mathscr{G}\subset\mathcal{C}$ passes through $p$. The associated tangent vectors for the congruence are $u^{\alpha}$ so that $u_{\alpha}u^{\alpha}=-1$ for timelike geodesics and $u_{\alpha}u^{\alpha}=0$ for null geodesics. Defining $\chi_{\alpha}^{\beta}=\nabla_{\alpha}u^{\beta}$ then $\chi=\chi_{\alpha}^{\beta}=\theta$.

In ordinary hydrodynamics, coordinates moving the flow or streamline can be chosen, then any observer moving with the stream observes rates of shear, separation, vorticity and volume variations relative to a chosen nearby flowline. By analogy an observer moving with a geodesic congruence uses a proper time $s\in\mathbb{R}^{+}$ and has 4-velocity $u^{\alpha}$ can also observe rates of shear, vorticity and volume variations with respect to neighboring geodesic flows.
Let $\mathcal{V}\subset\Sigma$ be volume elements on spacelike Cauchy surfaces/slices $\Sigma$ which are transverse to timelike geodesic congruences. Let $\mathcal{A}\subset\mathcal{N}$ be area elements on null slices transverse to a null geodesic congruence. (Latin indices can be on these hypersurfaces):
[1] The expansion is $\theta_{ab}=h_{a}^{c}$ $h_{b}^{d} u_{cd}$
[2] The volume expansion is $\chi\equiv\theta=\theta_{ab}h^{ab}=\nabla_{a}u^{a}$
[3] The shear is $\sigma_{ab}=\theta_{ab}-\frac{1}{3}h_{ab}\theta$
[4] The vorticity is $\omega_{ab}=h_{a}^{c}$ $h_{bd}u_{cd}$

where $h_{ab}$ is the spacelike positive-definite3-metric or 2-metric on $\latex \Sigma$ or $\mathcal{N}$. The trace of the extrinsic curvature of $\Sigma$ is $\chi=\chi_{\alpha}^{\alpha}=\nabla_{\alpha}u^{\alpha}\equiv\theta(s)$. The geodesic deviation equation or Jacobi equation gives the Raychaudhuri equation so that in terms of $\chi$

$\frac{d\chi(s)}{ds}=-\frac{1}{3}(\chi(s))^{2}-\mathbf{R}_{\alpha\beta}u^{\alpha}u^{\beta}-\sigma_{\alpha\beta}\sigma^{\alpha\beta}+\omega_{\alpha\beta}\omega^{\alpha\beta}$

For null geodesics, with $u_{\alpha}=\ell_{\alpha}$, the Raychauhuri equation essentially describes geometric optics in curved space and has a similar form so that

$\frac{d\chi(s)}{ds}=-\frac{1}{2}(\chi(s))^{2}-\mathbf{R}_{\alpha\beta}\ell^{\alpha}\ell^{\beta}-\sigma_{\alpha\beta}\sigma^{\alpha\beta}+\omega_{\alpha\beta}\omega^{\alpha\beta}$

The energy conditions on matter require that via the Einstein equations

$\mathbf{R}_{\alpha\beta}u^{\alpha}u^{\beta}=8\pi\mathbf{T}_{\alpha\beta}u^{\alpha}u^{\beta}\ge 0$

The Raychuadhuri equation has very useful properties because it is a relatively simple nonlinear differential equation. It plays a central role in establishing the singularity theorems, but the Einstein-matter system itself is not solved explicitly–its only function is to impose a constraint or inequality via the energy conditions. The vorticity terms and the shear terms are usually dropped for hypersurface orthogonality so that the general basic form of the Raychaudhuri equation is the Riccati-type nonlinear differential equation

$\frac{d\chi(s)}{ds}=-\alpha(\chi(s))^{2}=q\equiv \alpha[(\chi(s))^{2}+q/\alpha]\equiv H(\chi(s))$

where $\alpha\in\lbrace\frac{1}{2},\frac{1}{3}\rbrace$. Since $q\ge 0$ then

$\frac{d\chi(s)}{ds}\le -\alpha(\chi(s))^{2}$

so that for initial data $\chi(0)=\chi_{o}<0$, for the trace $\chi_{o}$ of the extrinsic curvature of a trapped surface $\mathcal{T}$ the solution blows up at $s=\chi_{o}^{-1}/\alpha|$ or $|\chi(\chi_{o}^{-1}/\alpha)|=\infty$. It can be proved that this is more than a blowup in the caustic but that the spacetime is itself geodesically incomplete (refs.). For any points $(p,q)$ a conjugate point is inflicted at q relative to p if $\chi(q)=\infty$.

Let $\mathscr{G}\in\mathcal{C}$ be a geodesic of the congruence $\mathcal{C}$. If $(p,q) \in\mathscr{G}$ with $q\gg p$ then q is a conjugate point relative to p if $\chi(q)=-\infty$

Advertisements