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This thesis is concerned with several problems in general relativity and low energy string theory that are pertinent to the time evolution of the gravitational field. We present a formulation of the Einstein field equations in terms of variational techniques borrowed from geometric analysis. These equations yield the evolution equations for the Cauchy problems of both general relativity and low energy string theory. We then proceed to investigate the evolutionary linear stability of Schwarzschild-like solutions in higher dimensional relativity called black strings. These objects are determined to be linearly unstable. This motivates a further stability analysis of the charged p-brane solutions of low energy string theory. We show that one can eliminate linear instabilities in p-branes for sufficiently large values of charge. We also consider the characteristic problem of general relativistic magnetohydrodynamics (GRMHD). We compute the eigenvalues and eigenvectors of GRMHD and establish degeneracy conditions. Finally, we consider the initial value problem for axisymmetric GRMHD. We formulate the general Einstein and MHD equations under the assumption of a stationary axisymmetric spacetime without assuming the circularity condition.