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The key notions of the differential and Riemannian geometry necessary for understanding the General Relativity are introduced here. We provide the reader with the necessary tools for study the properties of black holesand their interaction with matter and fields. We tried to omit unnecessary details and to make the presentation simpler. We focus on an explanation of the basic concepts and explicit formulas, which can be used for concrete calculations, and give hints, which might be used to simplify these calculations.

This chapter studies conformal curvature tensors of a pseudo-Riemannian metric g. These are defined in terms of the covariant derivatives of the curvature tensor of an ambient metric in normal form relative to g. Their transformation laws under conformal change are given in terms of the action of a subgroup of the conformal group O(p + 1, q + 1) on tensors. It is assumed throughout this chapter that n ≥ 3.

This chapter fleshes out the strategy of iteratively decomposing any P(g) = unconverted formula 1 for which ∫P(g)dVsubscript g is a global conformal invariant. It makes precise the notions of better and worse complete contractions in P(g) and then spells out (1.17), via Propositions 2.7, 2.8. In particular, using the well-known decomposition of the curvature tensor into its trace-free part (the Weyl tensor) and its trace part (the Schouten tensor), it reexpresses P(g) as a linear combination of complete contractions involving differentiated Weyl tensors and differentiated Schouten tensors, as in (2.47). The chapter also proves (1.17) when the worst terms involve at least one differentiated Schouten tensor.

This chapter develops tensor calculus: integration on manifolds, Cartan calculus for differential forms, connections and covariant derivatives, and the Levi-Civita connection used in general relativity. It then introduces the Riemann curvature tensor in several different ways, including the most directly physical picture of the curvature as a measure of the convergence of neighboring geodesics. The chapter concludes with a discussion of Cartan’s beautiful formulation of the connection and curvature in the language of differential forms.

The intrinsic curvature of a metric space is captured by the Riemann curvature tensor, which can be contracted to the Ricci tensor and the Ricci scalar. Einstein took these curvature quantities and constructed the Einstein field equations that relate the curvature of space-time to energy and mass density. For an isotropic density, a solution to the field equations is the Schwarzschild metric, which contains mass terms that modify both the temporal and the spatial components of the invariant element. Consequences of the Schwarzschild metric include gravitational time dilation, length contraction, and redshifts. Trajectories in curved space-time are expressed as geodesics through the Schwarzschild metric space. Solutions to the geodesic equation lead to the precession of the perihelion of Mercury and to the deflection of light by the Sun.

This chapter is about Riemannian manifolds. It first discusses the metric manifold and the Levi-Civita connection, determining if the metric is Riemannian or Lorentzian. Next, the chapter turns to the properties of the curvature tensor. It states without proof the intrinsic versions of the properties of the Riemann–Christoffel tensor of a covariant derivative already given in Chapter 2. This chapter then performs the same derivation as in Chapter 4 by obtaining the Einstein equations of general relativity by varying the Hilbert action. However, this will be done in the intrinsic manner, using the tools developed in the present and the preceding chapters.