This Dissertation covers three aspects of General Relativity: inequivalent Einstein metrics on Lie Group Manifolds, proving the Hoop Conjecture for Black Rings, and investigating ergoregions in magnetised black hole spacetimes. A number of analytical and numerical techniques are employed to that end. It is known that every compact simple Lie Group admits a bi-invariant homogeneous Einstein metric. We use two ansatze to probe the existence of additional inequivalent Einstein metrics on the Lie Group SU (n). We provide an explicit construction of 2k + 1 and 2k inequivalent Einstein metrics on SU (2k) and SU (2k + 1) respectively. We prove the Hoop Conjecture for neutral and charged, singly and doubly rotating black rings. This allows one to determine whether a rotating mass distribution has an event horizon, that it is in fact a black ring. We investigate ergoregions in magnetised black hole spacetimes. We show that, in general, rotating charged black holes (Kerr-Newman) immersed in an external magnetic field have ergoregions that extend to infinity near the central axis unless we restrict the charge to q = amB and keep B below a maximal value. Additionally, we show that as B is increased from zero the ergoregion adjacent to the event horizon shrinks, vanishing altogether at a critical value, before reappearing and growing until it is no longer bounded as B becomes greater than the maximal value.