Whenever there's no acceleration then: (Geodesic equation)(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \nabla _{u} u =0[/tex]

where the covariant derivative includes "Christoffel symbols" so [tex] \Gamma_{kl}^{i} =0 [/tex] for an Euclidean space-time..however when generalizing to a system under an acceleration then Gedesic equation becomes:

[tex] \nabla _{u} u -a^{\mu}=0 [/tex] (foruth dimensional acceleration)

for a test particle [tex] \acute{{R}^{\mu}}_{\alpha \nu \beta} \acute{u}^{\alpha} \acute{x}^{\nu} \acute{u}^{\beta} = - \acute{f}^{\mu} [/tex]

where does this 4-dimensional force come from?? but if you use "Weak field approximation" then the potential of the particle (and force) is: [tex] \Gamma^{i}_{00} [/tex]

and using all that how you derive Einstein field equation so [tex] R_{ab}=0 [/tex]

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# Understanding this equation

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