### Regge Calculus as a Numerical Approach to General Relativity

### Parandis Khavari

Doctor of Philosophy 2009

Graduate Department of Astronomy and Astrophysics, University of Toronto
A (3+1)-evolutionary method in the framework of Regge Calculus, known
as "Parallelisable Implicit Evolutionary Scheme", is analysed and
revised so that it accounts for causality. Furthermore, the
ambiguities associated with the notion of time in this evolutionary
scheme are addressed and a solution to resolving such ambiguities is
presented. The revised algorithm is then numerically tested and shown
to produce the desirable results and indeed to resolve a problem
previously faced upon implementing this scheme.

An important issue that has been overlooked in "Parallelisable
Implicit Evolutionary Scheme" was the restrictions on the choice of
edge lengths used to build the space-time lattice as it evolves in
time. It is essential to know what inequalities must hold between the
edges of a 4-dimensional simplex, used to construct a space-time, so
that the geometry inside the simplex is Minkowskian. The only known
inequality on the Minkowski plane is the "Reverse Triangle Inequality"
which holds between the edges of a triangle constructed only from
space-like edges. However, a triangle, on the Minkowski plane, can be
built from a combination of time-like, space-like or null edges. Part
of this thesis is concerned with deriving a number of inequalities
that must hold between the edges of mixed triangles.

Finally, the Raychaudhuri equation is considered from the point of
view of Regge Calculus. The Raychaudhuri equation plays an important
role in many areas of relativistic Physics and Astrophysics, most
importantly in the proof of singularity theorems. An analogue to the
Raychaudhuri equation in the framework of Regge Calculus is derived.
Both (2+1)-dimensional and (3+1)-dimensional cases are considered and
analogues for average expansion and shear scalar are found.