NR_ADM

The aim of this exercise is to simulate the time evolution of the Schwarzschild metric solving the field equations of General Relativity in ADM formalism. The equations are:

where $A(r,t)$ and $B(r,t)$ are defined such that for a spherically-symmetric system (such as the Schwarzschild spacetime), the spatial part of the metric can be written as

$$dl^2 = A(r,t) dr^2 + r^2 B(r,t) d\Omega^2$$

The fields $D_A$ and $D_B$ are auxiliary variables such that $D_i = \partial_r i$ for $i = A, B$. Lastly $K_i$ are the non vanishing components of the extrinsic curvature, in particular $K_A = K^r_r$ and $K_B = K^\theta_\theta = K^\phi_\phi$

To avoid eccessive gradients at the origin the following transformation has been applied to those variables:

• $$A \rightarrow \tilde{A} = A/\psi^4,\quad B \rightarrow \tilde{B} = B/\psi^4 $$
• $$D_A \rightarrow \tilde{D}_A = D_A - 4\partial_r \log \psi, \quad D_B \rightarrow \tilde{D}_B = D_B - 4\partial_r \log \psi$$

for $$\psi = 1 + \frac{M}{2r}$$

The chosen intial conditions are:

• $$\tilde{A} = \tilde{B} = 1$$
• $$\tilde{D}_A = \tilde{D}_B = 0$$
• $$K_A = K_B = 0$$
• $$\alpha = 1$$

The field $\alpha$ is the lapse function. It follows one of the following evoultion equations: $\partial_t \alpha = 0$ or $\partial_t \alpha = -2\alpha K$, with $K$ being the mean curvature. The first case correspond to the gauge choice called "Geodesic Slicing", the second to the "1+Log Slicing".

Lastly the apparent horizon has been estimated solving the following equation for $r = r_h$: $$\frac{1}{\sqrt{A}} \left(\frac{2}{r} + \frac{\partial_r B}{B}\right) - 2K_B = 0$$