Spherically Symmetric N-body Simulations with General Relativistic
  Dynamics
  Within a cosmological context, we study the behaviour of collisionless
particles in the weak field approximation to General Relativity, allowing for
large gradients of the fields and relativistic velocities for the particles. We
consider a spherically symmetric setup such that high resolution simulations
are possible with minimal computational resources. We test our formalism by
comparing it to two exact solutions: the Schwarzschild solution and the
Lema\^itre-Tolman-Bondi model. In order to make the comparison we consider
redshifts and lensing angles of photons passing through the simulation. These
are both observable quantities and hence are gauge independent. We demonstrate
that our scheme is more accurate than a Newtonian scheme, correctly reproducing
the leading-order post-Newtonian correction. In addition, our setup is able to
handle shell-crossings, which is not possible within a fluid model.
Furthermore, by introducing angular momentum, we find configurations
corresponding to bound objects which may prove useful for numerical studies of
the effects of modified gravity, dynamical dark energy models or even compact
bound objects within General Relativity.
