Consider a force-free particle moving on a geodesic with four-velocity [tex]v^\nu[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

The formula for the four-acceleration in any coordinate system is

[tex]

\frac{dx^\mu}{d\tau} = - \Gamma^\mu_{\nu\lambda} v^\nu v^\lambda[/tex]

Since the four-acceleration on the left side is orthogonal to the four-velocity, this implies

[tex]\Gamma^\mu_{\nu\lambda} v^\nu v^\lambda v_\mu=0[/tex]

Is this correct? I've never seen this equation anywhere. (Perhaps because it is trivial?)

It seems to imply that for any coordinate system with 0 as a time-like coordinate

[tex]\Gamma^0_{00}=0[/tex]

because I can always find a particle which is at rest in this system (i.e. with a four-velocity of (1,0,0,0) ). Is that a valid conclusion?

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# I Is the Christoffel symbol orthogonal to the four-velocity?

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