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According to the book Introduction to Cosmology by J.V. Narlikar, if we have x^i (λ) as the parametric representation of a curve in spacetime, having the tangent vector

u^i = (dx^i)/(dλ)

Along traversing a curve from λ to λ+dλ, the change in u^i is given by:

Image from Narlikar

but according to https://en.m.wikipedia.org/wiki/Christoffel_symbols under 'Relationship to parallel transport and derivation of Christoffel symbols in Riemannian space', it is mentioned:

enter image description here

there is a minus sign before the Christoffel symbol. I'm not getting this sign anomality. Please help! Thank You.

Tanay Gupta
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1 Answers1

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No, they’re saying the essentially the same thing. Rearrange the second, and relabel the dummy indices ($(m,j)\to(l,k)$) and use the fact that the connection is torsion free (so $\Gamma^i_{lk}=\Gamma^i_{kl}$) equation to get that for each $1\leq i\leq n$, \begin{align} \frac{d\xi^i}{ds}+\Gamma^i_{kl}\xi^k\frac{dx^l}{ds}&=0. \end{align} Just FYI: you can write this equation equivalently as $\nabla_{\dot{x}}\xi=0$.

If you compare with the first equation, then you see that this equation says that the vector $\xi$ is being parallel-transported along the curve $x(s)$. If you had a non-zero term on the right, then it means the vector field $\xi(s)$ along the curve $x(s)$ is not parallel-transported.

peek-a-boo
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