Various interpretations of the Riemann Curvature Tensor, Ricci Tensor, and Scalar Curvature are described. Also the physical meanings of the Einstein Tensor and Einstein's Equations are discussed. Finally, a derivation of Newtonian Gravity from Einstein's Equations is given.

We prove by explicit construction that there exists a maximal slicing of the Schwarzschild spacetime such that the lapse has zero gradient at the puncture. This boundary condition has been observed to hold in numerical evolutions, but in the past it was not clear whether the numerically obtained maximal slices exist analytically. We show that our analytical result agrees with numerical simulation. Given the analytical form for the lapse, we can derive that at late times the value of the lapse at the event horizon approaches the value 3 16 ͱ3Ϸ0.3248, justifying the numerical estimate of 0.3 that has been used for black hole excision in numerical simulations. We present our results for the nonextremal Reissner-Nordström metric, generalizing previous constructions of maximal slices.

We study how different types of blow-ups can occur in systems of hyperbolic evolution equations of the type found in general relativity. In particular, we discuss two independent criteria that can be used to determine when such blow-ups can be expected. One criteria is related with the so-called geometric blow-up leading to gradient catastrophes, while the other is based upon the ODE-mechanism leading to blow-ups within finite time. We show how both mechanisms work in the case of a simple one-dimensional wave equation with a dynamic wave speed and sources, and later explore how those blow-ups can appear in one-dimensional numerical relativity. In the latter case we recover the well known "gauge shocks" associated with Bona-Masso type slicing conditions. However, a crucial result of this study has been the identification of a second family of blow-ups associated with the way in which the constraints have been used to construct a hyperbolic formulation. We call these blow-ups "constraint shocks" and show that they are formulation specific, and that choices can be made to eliminate them or at least make them less severe.