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P: 90
Riemann curvature tensor derivation

 Quote by weio Hey So far this is how I understand it, though I know I could be very wrong. If you have two geodesics parallel to each other, with tangents $$V$$ and $$V'$$ , in which the coordinate $$x^\alpha$$ point along both geodesics. There is some connecting vector $$w^\alpha$$ between them. Let the affine parameter on the geodesics be $$\lambda$$ Riemman tensor calculates the acceleration between these two geodesics. so you calculate the acceleration at some point A, A' on each geodesic , and subtract them. this gives you an expression telling how the components of $$w^\alpha$$ change. $$\frac{d^2w^\alpha} {d\lambda^2} = \frac{d^2x^\alpha} {d\lambda^2} | A' - \frac {d^2x^\alpha} {d\lambda^2} | A = - \Gamma^\alpha_0_0_\beta w^\beta$$ After that you calculate the full 2nd covariant derivative along V, ie , you get something like $$\bigtriangledown v \bigtriangledown v w^\alpha = (\Gamma^\alpha_\beta_0_0 - \Gamma^\alpha_0_0,\beta) w^\beta$$ $$= R^a_0_0\beta w^\beta$$ $$= R^a_u_v_\beta V^u V^v w^\beta$$ That's where the tensor arises. so basically it's a difference in acceleration as geodesics don't maintain their seperation in curved space.
Yes it arises there and in many other places, including the one you asked about and that I told you about.