selfAdjoint
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I think some of the questions can be answered by some quotes from MTW's Gravitation, at the start of their Chapter10,"Affine Geometry: Geodesics, Parallel Transport, and Covariant Derivative".
> Free fall is the "natural state of motion," so natural, in fact, that the path through spacetime of a freely falling, neutral test body is independent of its structure and composisiton (the "weak equivalence principle"...)
> Ask for the maximum amount of information tied up in each trajectory. Is it merely the sequence of points along which the test body falls? No; there is more. Each test body can carry a clock ([independent] of structure or composition of test body). The clock ticks as the body moves, labelling each event on its trajectory with a number: the time \lambda when the body was there. Result: the free-fall trajectory isnot just a sequence of points; it is a parametrized sequence, a "curve".
> in the curved spacetime of Einstein...these parametrized free-fall trajectories are the straightest of all possible curves. Consequently one gives these trajectories the same name, "geodesics", that mathematicians use for the straight lines of a curved manifold...
Finnally, from Chapter 13,
>Concord between locally straight lines and geodesics of curved spacetimes demands that timelike geodesics have extremal proper length...
> Free fall is the "natural state of motion," so natural, in fact, that the path through spacetime of a freely falling, neutral test body is independent of its structure and composisiton (the "weak equivalence principle"...)
> Ask for the maximum amount of information tied up in each trajectory. Is it merely the sequence of points along which the test body falls? No; there is more. Each test body can carry a clock ([independent] of structure or composition of test body). The clock ticks as the body moves, labelling each event on its trajectory with a number: the time \lambda when the body was there. Result: the free-fall trajectory isnot just a sequence of points; it is a parametrized sequence, a "curve".
> in the curved spacetime of Einstein...these parametrized free-fall trajectories are the straightest of all possible curves. Consequently one gives these trajectories the same name, "geodesics", that mathematicians use for the straight lines of a curved manifold...
Finnally, from Chapter 13,
>Concord between locally straight lines and geodesics of curved spacetimes demands that timelike geodesics have extremal proper length...