# Brill Waves

## Brill Waves

This is not actually a single analytic solution to the field equations in itself, but rather a method for numerically solving for the behavior of the metric given initial conditions on an azymuthally symmetric metric. You start out with a metric which has a constant time slice of

$ds^2 |_{t=constant}=e^{2f\left(z,\rho \right)}\left(dz^2 +d\rho ^2 \right)+r^2 d\theta ^2$

where the behavior is restricted to

$f|_{\rho=0}=0$
$\frac{\partial f}{\partial \rho }|_{\rho=0}=0$
$f$ goes to zero faster than $\frac{1}{r}$ where
$r=\sqrt{z^2 + \rho ^2 }$

You start out with your choice of metric initially satisfying these equations, then by computer numerically time step the metric elements by $t_{n+1}=t_{n}+\delta t$ so that the variation in the elements accord with Einstein's field equations. You then look at how a computer simulation describes the time dependent metric. Such simulations indicate that colliding gravitational waves can collapse to form a black hole.

## References

[1]*Relativity - equations 9.3.18-9.3.19