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In standard GR the equation of motion of a (classical) particle is a geodesic. A geodesic is a curve along which the tangent vector is parallel transported. If the tangent vector is denoted by V this condition is

[tex]\nabla_V V=0[/tex]

In components this is

[tex]\frac{d^2x^\alpha}{ds^2}+{\Gamma^\alpha}_{\mu\nu}\frac{dx^\mu}{ds}\frac{dx^\nu}{ds}=0[/tex]

where s is the parameter of the curve. [itex]{\Gamma^\alpha}_{\mu\nu}[/itex] are the connection coefficients. The connection coefficients are the component representation of a given connection (covariant derivative). Only the symmetric part [itex]{\Gamma^\alpha}_{(\mu\nu)}[/itex] contributes to the equation above.

For a metric connection, the connection coefficents can be expressed as

[itex]{\Gamma^\alpha}_{(\mu\nu)}=\frac{1}{2}g^{\alpha\beta}(g_{\mu\beta,\nu}+g_{\nu\beta,\mu}-g_{\mu\nu,\beta})+\frac{1}{2}( {{T_\mu}^\alpha}_\nu + {{T_\nu}^\alpha}_\mu)[/itex]

[itex]{\Gamma^\alpha}_{[\mu\nu]}=\frac{1}{2}{T^\alpha}_{\mu\nu}[/itex]

where T is the torsion tensor.

If the length of a curve between two fixed points is

[tex]\int{\sqrt{g_{\alpha\beta}\frac{dx^\alpha}{ds}\frac{dx^\beta}{ds}}ds}[/tex]

then it can be shown that the equation of the curve which minimizes (or maximizes maybe?) this length is

[tex]\frac{d^2x^\alpha}{ds^2}+\frac{1}{2}g^{\alpha\beta}(g_{\mu\beta,\nu}+g_{\nu\beta,\mu}-g_{\mu\nu,\beta})\frac{dx^\mu}{ds}\frac{dx^\nu}{ds}=0[/tex]

This equation matches the geodesic condition in the absence of torsion.

so now my question: which one do we use if torsion is non-zero? I would imagine we would use the geodesic equation since, otherwise, the presence of torsion would have no measurable effect. But if so we then find that particle paths do NOT follow extremal curves. Is that acceptable?

Or maybe we need to use another measure of length in the presence of torsion?

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# How would torsion affect the particle equation of motion?

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