Optics in curved spacetime

Abstract

In this and the next chapter, we use standard tensor notation. Greek indices refer to spacetime, as in x α = (x 0,x i), whereas Roman indices label spatial coordinates and components; x 0= ct. Round and square brackets indicate symmetrization and alternation, respectively, as in \( {\text{ }}{{A}_{{(\alpha \beta )}}} = \frac{1}{2}({\text{ }}{{A}_{{\alpha \beta }}} + {\text{ }}{{A}_{{\beta \alpha }}}),{\text{ }}{{B}_{{\left[ {\alpha \beta } \right]}}} = \frac{1}{2}({\text{ }}{{B}_{{\alpha \beta }}} - {\text{ }}{{B}_{{\beta \alpha }}}). \)Throughout, we use the Einstein summation convention.The spacetime metric gα β is taken to have signature -2, or (+ - - -); the special-relativistic Minkowski metric in orthonormal coordinates is written ηα β= diag(l,-1,-1,-1). The determinant of gα β is denoted by g. Spacetime is always assumed to be time-oriented so that we may speak, e.g, of future-directed vectors, future (half-) lightcones. Covariant derivatives with respect to the Levi-Civita connection\( \Gamma _{{\beta \gamma }}^{\alpha } \) g αβ are indicated by semicolons as in g α β;γ = 0, partial derivatives by commas. The Riemann-, Ricci-, scalar-, and Einstein curvature tensors are defined by \( 2{{A}_{{\alpha ;[\beta \gamma ]}}} = {\text{ }}{{A}_{\delta }}{\text{ }}R_{{\alpha \gamma \beta }}^{\delta },{\text{ }}{{R}_{{\alpha \beta }}} = {\text{ }}R_{{\alpha \gamma \beta }}^{\gamma },R = {\text{ }}R_{\alpha }^{\alpha },{\text{ }}{{G}_{{\alpha \beta }}} = {\text{ }}{{R}_{{\alpha \beta }}} - \frac{1}{2}R{{g}_{{\alpha \beta }}} \).The Weyl conformal curvature tensor is defined in (3.62). More special notations will be explained where they are used for the first time. We employ electromagnetic units and dimensions according to the system of Heaviside-Lorentz, specialized in Chap. 3 to c = 1; see, e.g., [JA75.1], appendix.KeywordsConjugate PointNull GeodesicWorld LineEikonal EquationLightlike HypersurfaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.