The formation of cosmic large scale structure is a highly non-linear process and is therefore studied with numerical simulations. For this purpose, N-body simulations using Newton's law of gravity have been very successful. However, they are ultimately limited by the fact that Newtonian gravity is only an approximation to General Relativity and does not match its full dynamical content. Relativistic effects can become important for the interpretation of current and future high precision observations, in particular since there are some exotic sources of stress-energy in cosmology, like e.g. Dark Energy, which are poorly understood and can be modelled in various ways. Furthermore, it is hoped that one can gain a deeper understanding of these "dark" phenomena by studying even the most subtle effects they can have on structure formation.
To achieve this goal, it is a great advantage if one can overcome the limitations imposed by the use of the Newtonian limit. We present in our work a major technological advancement in this direction, making it possible for the first time to simulate structure formation within a truly dynamical spacetime. In order to avoid the overwhelming complexity of full General Relativity, we employ some controlled approximations which are suitable for cosmology, where one is interested in a description of the Universe at large. On the relevant scales, gravitational fields are generally weak, but they typically show variations on small scales. Our approximation scheme therefore gives spatial gradients a larger weight in the perturbative expansion of Einstein's equations. We present algorithms for the evolution of all metric components and test our numerical approach in a plane-symmetric setting, which has the advantage of requiring only very little computational resources. We plan to implement our algorithms in a 3D N-body framework in the future.