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I am studying General Relativity from Schutz's book. In chapter 7 he starts with the geodesics equation with lowered indices: $$ p^{\alpha}p_{\beta;\alpha} = 0 \qquad(7.25) $$

and goes on to derive the equation : $$ m\frac{dp_\beta}{dt} = \frac12 g_{\nu\alpha,\beta} p^\nu p^\alpha \qquad(7.29)$$

I understand his derivation but i do not understand how we get equation 7.25 from the geodesics equation: $$ \nabla_p p =0 \qquad (7.10)$$ can someone help please?

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    $\begingroup$ They are pretty much the same by definition. $\endgroup$ Feb 16, 2022 at 12:29
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    $\begingroup$ You are referring to the geodesic equation and not to the geodesic deviation. $\endgroup$ Feb 16, 2022 at 13:11

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@CameronGibson is right: $\nabla_p:=p^\alpha\nabla_\alpha$, so $\nabla_p q_\beta=p^\alpha\nabla_\alpha q_\beta=p^\alpha q_{\beta;\,\alpha}$.

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  • $\begingroup$ Thank you. I think i had misunderstood the nabla notation. $\endgroup$ Feb 17, 2022 at 8:55

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