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I'm learning general relativity from the book " Einstein's General Theory of Relativity - Øyvind Grøn and Sigbjorn Hervik". The field equations are derived by the Hilbert - Einstein action and are written in the form

$$ R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}+\Lambda\,g_{\mu\nu}=kT_{\mu\nu}$$
where $k$ is acostant that should be determined. The authors find this $k=\frac{8\pi G}{c^4}$ by imposing the well known conditions of "Newtonian Limit". The first requirement is the following

The particles in free fall moving along geodesics induced by $g$.

This is a postulate? By the Einstein equivalence principle we know that locally we can eliminate the effects of gravity and so living in a (flat) Minkowskian space-time but there is not mention of the above statement. I agree on the fact that once we have a curved space-time, in presence of gravitational fields, it is natural to require that the free falling particles move along the geodesics, but this can be proved or it is postulated?


marked as duplicate by joshphysics, Alfred Centauri, Emilio Pisanty, Brandon Enright, Qmechanic Nov 12 '13 at 23:39

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The important point is that the effects of gravity can only be eliminated locally. We actually live in a more complicated metric, and any observer will only feel Minkowski if she is free-falling. Mathematically that means that you can make $g_{ab}(x) = (-1,1,1,1)$, but you'll still have non-zero second derivatives $\partial_a\partial_b g_{cd}\neq 0$ (for this reason, you can't make the curvature $R_{ab},R$ vanish by choosing appropriate coordinates).

You can actually prove the geodesic equation by considering the theory of a point particle (with an action given by minus the potential energy $S= -\int m d\tau $, where $d\tau^2 ={g_{ab}d x^a d x^b}$ is given by the metric in terms of the trajectory $x^a(\lambda)$, and $\lambda$ is a parameter), deriving the energy-momentum tensor $T^{ab}$ for that theory (by deriving with respect to the metric) and applying energy momentum conservation $\nabla_a T^{ab} = 0$. Energy momentum conservation is a consequence of Einstein Equations (due to the Bianchi identities, which are a property of $G_{ab}$), so you can consider that Einstein equation implies the geodesic equation.


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