Einstein's theory of General Relativity, put forward in 1915, predicts that space and time do not form a fixed background, but instead are malleable and dynamic quantities themselves. Their union forms something called spacetime, which when curved causes gravitational effects. This framework has led to models of the universe which match observations that the entire universe is expanding. Running these models backwards in time leads to a 'big bang', which is a single point from which the entire known universe came from. This single point is a singularity, a place where the theory breaks down, rendering questions like 'what happened before the big bang' meaningless. However, we can use General Relativity to study what happens near these singularities, which can have profound implications for whatever theory will succeed General Relativity, which will need to explain singularities, and will presumably be a quantum theory of gravity.
In 1970, Belinsky, Khalatnikov, and Lifshitz made a conjecture about the nature of spacetime near any singularity. They proposed that as one asymptotically approaches a singularity, each spatial point decouples from the points around it, and therefore acts like an independent homogeneous universe. An important homogeneous universe is the 'Mixmaster Universe', and in many cases, numerical simulations show that, on approaches to singularities, each point begins to act like its own Mixmaster Universe. The Mixmaster universe features chaotic, oscillatory behavior known as 'Mixmaster Dynamics'.
Mixmaster dynamics are fairly well understood, but in this thesis I will study them in a new way, utilizing an alternative language for understand the curvature of spacetime called Gravito-Electromagnetism. In electromagnetism, the electric and magnetic fields are decomposed from a single quantity which contains all the information of the electromagnetic field. A similar decomposition can be done to the gravitational analogue of the full field quantity, giving rise to the Gravito-Electric and Gravito-Magnetic fields, which have relatively simple physical interpretations, making them ideal for the visualization of spacetimes. Additionally, I will explore the Mixmaster Universe using a related algebraic classification commonly used in General Relativity called the Petrov Classification. While the Mixmaster Universe is known to not be algebraically special according to this classification, we use a recently developed measurement of the 'nearness' of a spacetime to algebraic speciality to gain more insight into Mixmaster Dynamics.