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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achronal affine parameter basis black hole boundary bundle Cauchy development Cauchy surface closed timelike curves closed trapped surface collapse compact set components conjugate constant contained converge coordinate neighbourhood covariant derivatives curvature defined density diffeomorphism differential Einstein equations energy energy-momentum tensor ergosphere event horizon existence field equations figure finite function future future-directed geodesic curve geodesically complete implies incomplete inextendible infinity initial data integral curves intersect isometry J+(p Kerr solution Killing vector lemma Lie derivative light cone linear Lorentz metric manifold metric g Minkowski space non-spacelike curve non-zero normal null cone null geodesic obtain open set partial Cauchy surface particle past-directed Penrose proposition radiation region regular predictable represents respect Riemann tensor satisfied Schwarzschild solution singularity space-time spacelike surface star static stationary strong causality tangent vector tensor field theorem timelike geodesics topology two-sphere two-surface unique universe values vanishes variation vector field world-line zero