15 - Neutron and X-ray Optics in General Relativity and Cosmology

Abstract

Surveyed in this chapter are general relativity and cosmology, including such topics as the equivalence principle, manifolds, covariant derivatives, metric coefficients, Gauss’s remarkable theorem, Gaussian curvature, curvilinear coordinate systems, metric tensor, distance invariance, Jacobian determinant, scalar and vector fields, contravariant vectors and tensors, basis vectors, covariant vectors and tensors, dual basis vectors. Next, we discuss differentials, parallel transport, absolute and covariant derivatives of vectors in contravariant and covariant representation, and geodesic curves and connection coefficients. In addition, we present a description of the covariant and contravariant metric and connection coefficients for cylindrical and spherical coordinates and general relativistic versions of Newton’s laws and Maxwell’s equations. Also covered are the curvature, Ricci, and Einstein tensors and Ricci scalar, stress tensor, and universe fluid model. We derive the Einstein field equations, with alternative derivation by the principle of least action. Gravity waves are derived from linearized, Einstein field equations for weak gravity conditions, followed by the Schwarzschild solution to Einstein field equations, experimental predictions of general relativity, Friedmann equation, cosmological models, Orr space, and equidynamics.