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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?

Dubious
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1 Answers1

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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.

user210915
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