Test particles are super-useful. Synge in his textbook [1] discusses a five-point curvature detector, which is the formal treatment for a small cloud of non-in

Test particles are super-useful. Synge in his textbook [1] discusses a five-point curvature detector, which is the formal treatment for a small cloud of non-interacting dust that reveals the Riemann curvature tensor. Baez does this succinctly and less formally at [2], using a decomposition of the Riemann curvature tensor into the Weyl tensor and the Ricci tensor. One could substitute ticking clocks for coffee grounds: in e.g. exterior Schwarzschild, when the flat-spacetime round-ball is instead a Weyl-curved spacetime American-football, the long axis of the football is radial to M, so the clock closest to M ticks slower than the others (cf. the Riemann tensor and especially the last sentence at [3]). One may compare with [4] (and the focusing theorem on the same page). - -- [1] Synge, J.L., "Gravitation: The general theory". (1964) Ch. XI, §8 [2] http://www.math.ucr.edu/home/baez/gr/ricci.weyl.html http://www.math.ucr.edu/home/baez/gr/ricci.weyl.html [3] https://en.wikipedia.org/wiki/Schwarzschild_metric#Curvatures https://en.wikipedia.org/wiki/Schwarzschild_metric#Curvature... [4] https://en.wikipedia.org/wiki/Raychaudhuri_equation#Intuitive_significance https://en.wikipedia.org/wiki/Raychaudhuri_equation#Intuitiv...