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## Main Question or Discussion Point

The geodesic general condition, i.e. for a non affine parameter, is that the directional covariant derivative is an operator which scales the tangent vector:

$$\zeta^{\mu}\nabla_{\mu}\zeta_{\nu}=\eta(\alpha)\zeta_{\nu}$$

I have three related questions.

When $$\alpha$$ is an affine parameter the scale factor $$\eta$$ vanishes. This makes intuitive sense because the derivative of the vector along the geodesic curve vanishes. We say, therefore, that the tangent vector components are not changing along the geodesic curve and it is parallel transported.

But what is our sense of the geodesic curve definition when $$\alpha$$ is not an affine parameter, noting that the derivative along the curve doesn't vanish and simply scales the tangent vector by $$\eta$$ Is this also seen or defined as parallel transporting the vector?

This GR text by Carroll on p444 says even if given the general scale $$\eta(\alpha)$$ we can indeed rescale the vector for a null curve, to re-parameterize it and make the right hand of the geodesic equation vanish. How this can be done?

Is the general geodesic condition defined above involving $$\eta$$ only for null curves or is applied to timelike curves for which we are not using an affine parameter like the proper time?

$$\zeta^{\mu}\nabla_{\mu}\zeta_{\nu}=\eta(\alpha)\zeta_{\nu}$$

I have three related questions.

When $$\alpha$$ is an affine parameter the scale factor $$\eta$$ vanishes. This makes intuitive sense because the derivative of the vector along the geodesic curve vanishes. We say, therefore, that the tangent vector components are not changing along the geodesic curve and it is parallel transported.

But what is our sense of the geodesic curve definition when $$\alpha$$ is not an affine parameter, noting that the derivative along the curve doesn't vanish and simply scales the tangent vector by $$\eta$$ Is this also seen or defined as parallel transporting the vector?

This GR text by Carroll on p444 says even if given the general scale $$\eta(\alpha)$$ we can indeed rescale the vector for a null curve, to re-parameterize it and make the right hand of the geodesic equation vanish. How this can be done?

Is the general geodesic condition defined above involving $$\eta$$ only for null curves or is applied to timelike curves for which we are not using an affine parameter like the proper time?