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?

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

• You can take any radius, for example the initial radius of your dust ball, it scales with a(t) anyway. The real question is what function of t the a is, that should depend on your initial radius. The a(t) for a closed matter dominated ΩK=1-ΩM universe is not exactly elegant though, at least not the solution I got Commented Aug 25, 2023 at 16:24
• I already solved my constraint to get $a(\tau)$, my problem is that i have 2 separate spacetimes. $a(\tau)$ works only for the dust, it's for a comoving coordinate system. My question is how $a(\tau)$ is described if seen in the external Schwartzschild (stationary) coordinates Commented Aug 28, 2023 at 8:27

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.