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On p138 of Wald's General Relativity, 4th para, he says:
Secondly, in GR, if one actually computes the connection coefficients for Schwarzschild, to write down the geodesic equations explicitly, one then sees immediately that ##\theta=\pi/2## (say) is a convenient choice that one can make without loss of generality.
But I don't follow Wald's particular argument, i.e., that the symmetry ##\theta\to \pi-\theta## allows one to conclude that "if the initial position and tangent vector of a geodesic lie in the equatorial plane ##\theta=\pi/2##, then the entire geodesic must lie in this plane.''
What am I missing?
Firstly, I already understand (of course) that this sort of simplification is easy and immediate in Newtonian mechanics, since conservation of angular momentum always occurs in central force problems (the proof is just a few lines) and one can re-orient the coordinates so that motion occurs in the plane perpendicular to the angular momentum axis.Wald said:[...] First, we note that because of the parity reflection symmetry, ##\theta \to \pi - \theta##, of the Schwarzschild metric, if the initial position and tangent vector of a geodesic lie in the "equatorial plane" ##\theta = \pi/2##, then the entire geodesic must lie in this "plane." Since every geodesic can be brought to an initially (and hence everywhere) equatorial geodesic by a rotational isometry, this means that without loss of generality we may restrict attention to study of the equatorial geodesics, and we shall do so.
Secondly, in GR, if one actually computes the connection coefficients for Schwarzschild, to write down the geodesic equations explicitly, one then sees immediately that ##\theta=\pi/2## (say) is a convenient choice that one can make without loss of generality.
But I don't follow Wald's particular argument, i.e., that the symmetry ##\theta\to \pi-\theta## allows one to conclude that "if the initial position and tangent vector of a geodesic lie in the equatorial plane ##\theta=\pi/2##, then the entire geodesic must lie in this plane.''
What am I missing?