Ivancevic_Applied-Diff-Geom

142 Applied Differential Geometry: A Modern IntroductionClearly, this is only true if we know how the fields transform under changeof coordinates. Here we assume that the transformation laws are known.Given this, the Einstein equation (3.1) is equivalent to the fact thatR 00 = T 00 − 1 2 g 00T γ γ (3.8)in every local inertial coordinate system about every point. In such coordinateswe have⎛ ⎞−1 0 0 0g = ⎜ 0 1 0 0⎟⎝ 0 0 1 0 ⎠ (3.9)0 0 0 1so g 00 = −1, as well asT γ γ = −T 00 + T 11 + T 22 + T 33 .Equation (3.8) thus says thatR 00 = 1 2 (T 00 + T 11 + T 22 + T 33 ).By equation (3.5), this is equivalent to the required¨Vlim ∣ = − 1V →0 V t=0 2 (T 00 + T 11 + T 22 + T 33 ).3.1.2 Einstein’s Geometrodynamics in BriefAs a final introductory motivation, we give an ‘express–flight bird–view’on derivation of the Einstein equation from the Hilbert action principle,starting from the Einstein space–time manifold M. For all technical details,see [Misner et al. (1973)], which is still, after 33 years, the core textbook