Paths In Discrete Spacetime: A Numerical Investigation Of The Myrheim Length Conjecture

General relativity and quantum mechanics are two of the most successful theories in physics. However, the problem of getting the two theories to agree with one another in a theory of quantum gravity remains unsolved. String theory and loop quantum gravity receive the most attention as possible quantum gravity theories. The causal set hypothesis provides an alternative to these theories. This hypothesis assumes that spacetime is discrete rather than continuous and that its underlying structure is a partially ordered set of points called a causal set. Before a causal set can be said to represent a spacetime, it must be shown that certain properties of a causal set correspond to certain properties of spacetime. One such property is the length of a path in a causal set. In particular, the Myrheim length conjecture states that the longest path in a causal set corresponds to the geodesic in continuous spacetime in the limit as the number of causal set elements N → ∞. In this thesis, I show that this conjecture can be verified numerically for causal set intervals of relatively small sizes in Minkowski spacetime