Star Radius in the Oppenheimer-Snyder metric using ADM formalism

Question

I'm working with gravitational collapse models, in particular with the Oppenheimer-Snyder model.

Short list of the assumptions for those unfamiliar with the model, you have a spherical symmetric pressureless dust cloud collapsing.

I'm working also in the Hamiltonian formalism.

The 3-metric inside the ball cloud is then given by a standard FLRW metric:

$$\mathrm{d}\sigma^2 = a(\tau)^2\left[\frac{\mathrm{d}R^2}{1-\epsilon R^2} + R^2(\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2 )\right]$$

With such a metric, the volume of the ball is given by:

$$V(\tau) = a(\tau)^3 V_0$$

Given $$V_0 = 4\pi \int_0 ^{R_s}\frac{r^2\mathrm{d}r}{\sqrt{1-\epsilon r^2}}$$

Here $\tau$ and $R$ are coordinates comoving with the dust particles.

The radius of the star is given by $R_B(\tau)$

For $R>R_B$ we get the Schwarzschild geometry, with 3-metric:

$$d\sigma ^2 = \Lambda(r,t) ^2 dr^2 +\rho(r,t) ^2 ( \mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2 )$$

Here $r$ and $t$ are the standard coordinates used in Schwarzschild geometries.

My question is:

In the Hamiltonian formulation I find that the only dynamical degrees of freedom for the system are $a(\tau)$ and the momentum conjugated. Solved all the dynamical equations and obtained the function $a(\tau)$, how can I relate such a function to the star radius? Both to obtain the radius equation and to give to the system an adequate initial value condition, like $R_B(\tau=0)=R_0$?

Is there a way to express the radius trajectory in Schwarzschild coordinates?

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Extra reference for those interested: Hamiltonian Formulation of Oppenheimer-Snyder collapse

Answer 1

To find what is the surface radius, you need to match the exterior Schwarzschild geometry to the interior metric. The matching criteria will involve:

1. Mass conservation $(\rho u^\mu)_{;\mu} = 0$. You will see that this implies that the total star mass is constant. This will give you a relationship between the star mass, the homogenenous (but time-dependent) density $\rho$ inside the contracting star and the star radius $R(\tau)$. The density $\rho$ is in fact related to your canonical momentum through Einstein equations, so this essentially answers your question.
2. The induced metric on the stellar surface and its extrinsic curvature must match on both sides. This reduces to a matching of the external gravitating mass and the internal one obtained by integrating over the star. The extrinsic curvature conditions amount to geodesic equations for the surface radius on both sides of the matching, at least when assuming dust. (This is only relevant if you are looking to determine the Schwarzschild-coordinate curve the surface traces out.)

For more, you can read the Junction condition chapter in the Advanced GR notes by Eric Poisson. (These also came out in print as "A Relativist's Toolkit" in 2004.)