Aspects of Axisymmetric Spacetimes

<p><strong>In recent times, interest in axisymmetric spacetimes has been intense, in part due to observations of Sagittarius A* and the supermassive black hole at the centre of galaxy M87 by the event horizon telescope (EHT). Knowledge of axisymmetric spacetimes (both Kerr and its generalisations) will prove useful when analysing present and future data from the EHT. We begin this text with a brief introduction to general relativity. We then turn to the work undertaken as part of the PhD, starting with a new and novel derivation of the Kerr solution. We then analyse a 3-free function generalisation of the Kerr spacetime, which we find to be the most general axisymmetric spacetime which still models real astrophysical black holes in our universe. We derive this model and prove the existence of Killing horizons, calculate the surface gravity and show that the zeroth law of black hole mechanics is satisfied. We then turn to considering more general axisymmetric spacetimes, where we calculate the surface gravity and show that the horizons are Killing and the zeroth law is satisfied if a certain set of conditions are satisfied. We then turn to calculating photon escape cones for various static spherically symmetric spacetimes and both a low rotation limit of Kerr and extremal Kerr. The photon escape cones are key to analysing data from the EHT since black hole silhouettes/shadows are just the time reversed compliment of the escape cone.</strong></p>