Optics in curved spacetime

  • Peter Schneider
  • Jürgen Ehlers
  • Emilio E. Falco
Part of the Astronomy and Astrophysics Library book series (AAL)


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 \( {A_{\left( {\alpha \beta } \right)}} = \frac{1}{2}\left( {{A_{\alpha \beta }} + {A_{\left[ {\alpha \beta } \right]}}} \right),{\kern 1pt} {B_{\left[ {\alpha \beta } \right]}} = \frac{1}{2}\left( {{B_{\alpha \beta }} - {B_{\beta \alpha }}} \right) \) . Throughout, we use the Einstein summation convention. The spacetime metric g αβ is taken to have signature -2, or (+ - --); the special-relativistic Minkowski metric in orthonormal coordinates is written ŋαβ = diag(1, -1, -1, -1) . The determinant of g αβ 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 Γ βγ α of g αβ are indicated by semicolons as in gαβ;γ = 0, partial derivatives by commas. The Riemann-, Ricci-, scalar-, and Einstein curvature tensors are defined by \( 2{A_{\alpha ;\left[ {\beta \gamma } \right]}} = {A_\delta }R_{\alpha \beta \gamma }^\delta ,{\kern 1pt} {R_{\alpha \beta }} = R_{\alpha \gamma \beta }^\gamma ,{\kern 1pt} R = R_\alpha ^\alpha ,{\kern 1pt} {G_{\alpha \beta }} = {R_{\alpha \beta }} - \frac{1}{1}R{g_{\alpha \beta }} \) . 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.


Conjugate Point Null Geodesic World Line Eikonal Equation Luminosity Distance 
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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Peter Schneider
    • 1
  • Jürgen Ehlers
    • 2
  • Emilio E. Falco
    • 3
  1. 1.Max-Planck-Institut für AstrophysikGarchingGermany
  2. 2.Max-Planck-Institut für GravitationsphysikAlbert-Einstein-InstitutPotsdamGermany
  3. 3.Harvard-Smithsonian Center for AstrophysicsCambridgeUSA

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