The Large Scale Structure of Space-Time
Einstein's General Theory of Relativity leads to two remarkable predictions: first, that the ultimate destiny of many massive stars is to undergo gravitational collapse and to disappear from view, leaving behind a 'black hole' in space; and secondly, that there will exist singularities in space-time itself. These singularities are places where space-time begins or ends, and the presently known laws of physics break down. They will occur inside black holes, and in the past are what might be construed as the beginning of the universe. To show how these predictions arise, the authors discuss the General Theory of Relativity in the large. Starting with a precise formulation of the theory and an account of the necessary background of differential geometry, the significance of space-time curvature is discussed and the global properties of a number of exact solutions of Einstein's field equations are examined. The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions. A discussion of the Cauchy problem for General Relativity is also included in this 1973 book.
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asymptotically basis body boundary Cauchy surface causality closed collapse compact complete components condition cone connection consider constant contained continuous converge coordinates corresponding covering curvature defined definition density derivatives determined differential direction distance Einstein empty endpoint energy equal equations event horizon example existence expressed extended fact figure finite follows function future give given holds implies incomplete infinity initial integral intersect lemma length light limit lines Lorentz metric manifold mass matter measure metric g negative neighbourhood non-spacelike curve normal null geodesic observer obtain orthogonal parameter particle past positive predictable properties proposition regard region relation represents respect result satisfied Schwarzschild similar singularity solution space space-time spacelike star Suppose surface symmetric takes tangent vector tensor theorem timelike curve topology unique universe values vanishes variation vector field zero