The Positive Mass Theorem
The important positive mass theorem roughly says that—in contrast to Newtonian gravity theory—it is impossible to construct an object out of ordinary matter, i.e., matter with positive local energy density, whose total energy (including gravitational cont
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Einstein’s Field Equations
There is something else that I have learned from the theory of gravitation: No collection of empirical facts, however comprehensive, can ever lead to the setting up of such complicated equations. A theory can be tested by experience, but there is no way from experience to the formulation of a theory. Equations of such complexity as are the equations of the gravitational field can be found only through the discovery of a logically simple mathematical condition which determines the equations completely or (at least) almost completely. Once one has those sufficiently strong formal conditions, one requires only little knowledge of facts for the setting up of a theory; in the case of the equations of gravitation it is the fourdimensionality and the symmetric tensor as expression for the structure of space which, together with the invariance concerning the continuous transformation-group, determine the equation almost completely. —A. Einstein (Autobiographical Notes, 1949)
In the previous chapter we examined the kinematical framework of the general theory of relativity and the effect of gravitational fields on physical systems. The hard core of the theory, however, consists of Einstein’s field equations, which relate the metric field to matter. After a discussion of the physical meaning of the curvature tensor, we shall first give a simple physical motivation for the field equations and will then show that they are determined by only a few natural requirements.1
3.1 Physical Meaning of the Curvature Tensor In a local inertial system the “field strengths” of the gravitational field (the Christoffel symbols) can be transformed away. Due to its tensor character, this is not possible for the curvature. Physically, the curvature describes the “tidal forces” as we shall see in the following. This was indicated already in the discussion of (2.181). 1 For an excellent historical account of Einstein’s struggle which culminated in the final form of his gravitational field equations, presented on November 25 (1915), we refer to A. Pais, [71]. Since the publication of this master piece new documents have been discovered, which clarified what happened during the crucial weeks in November 1915 (see, e.g., [72]).
N. Straumann, General Relativity, Graduate Texts in Physics, DOI 10.1007/978-94-007-5410-2_3, © Springer Science+Business Media Dordrecht 2013
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Einstein’s Field Equations
Fig. 3.1 Geodesic congruence
Consider a family of timelike geodesics, having the property that in a sufficiently small open region of the Lorentz manifold (M, g) precisely one geodesic passes through every point. Such a collection is called a congruence of timelike geodesics. It might represent dust or a swarm of freely falling bodies. The tangent field to this set of curves, with proper time s as curve parameter, is denoted by u. Note that u, u = −1. Let γ (t) be some curve transversal to the congruence. This means that the tangent vector γ˙ is never parallel to u in the region under consideration, as indicated
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