Geodesic defined for a non affine parameter

[victorvmotti]

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?

 

[Orodruin]

We say, therefore, that the tangent vector components are not changing along the geodesic curve and it is parallel transported.

No we don't. Whether the components change or not depends on the coordinate system. What we say is that the tangent vector is parallel along the geodesic.

Is this also seen or defined as parallel transporting the vector?

No, but the change in the vector relative to the parallel transported one is proportional to the tangent vector itself.

How this can be done?

You write a new parameter as a monotonic function of $\alpha$ and solve the ODE resulting from requiring that it is an affine parameter.

Is the general geodesic condition defined above involving

ηη​

\eta only for null curves or is applied to timelike curves for which we are not using an affine parameter like the proper time?

You can change to a non-affine parameter for any curve.

 

[victorvmotti]

No we don't. Whether the components change or not depends on the coordinate system. What we say is that the tangent vector is parallel along the geodesic.No, but the change in the vector relative to the parallel transported one is proportional to the tangent vector itself.

Well, may I ask to refer to Carroll's book p105 where it is said : "We then define parallel transport of the tensor $T$ along the path $x^{\mu}(\lambda)$ to be the requirement that the covariant derivative of the $T$ along the curve vanishes."

Are you suggesting to say that the tangent vector as the tensor in the above definition is not changing along the geodesic instead of saying that components are not changing? What I had in mind was indeed coordinates independent, that the tangent vector is not changing along the curve.

My main question here is that if the directional covariant derivative of the tangent vector along a curve is proportional to the tangent vector then that curve is a geodesic?

And this is the general condition for both timelike and null geodesics using a general parameter?

And only when the directional covariant derivative of the tangent vector vanishes we say it is parallel transported and the curve is defined based on an affine parameter?

In other words is there a difference between parallel transport and geodesic? In the special case of an affine parameter they are the same and no difference?

The geodesic being the more general definition with parallel transport a somewhat limited case?

 

[Orodruin]

Are you suggesting to say that the tangent vector as the tensor in the above definition is not changing along the geodesic instead of saying that components are not changing? What I had in mind was indeed coordinates independent, that the tangent vector is not changing along the curve

This statement does not make sense. Vectors at different points in a manifold cannot be directly compared, they belong to different vector spaces.

Parallel vectors along a curve are defined through the affine connection and is the curved space analogue of "not changing".

And this is the general condition for both timelike and null geodesics using a general parameter?

Yes.

In other words is there a difference between parallel transport and geodesic?

An affinely parametrised geodesic is a special case of parallel transport. Parallel transport can be done of any tensor along any curve. They are definitely not the same thing.

 

[victorvmotti]

An affinely parametrised geodesic is a special case of parallel transport. Parallel transport can be done of any tensor along any curve. They are definitely not the same thing.

So you say that the definition above by Carroll' book is or should be for "an affinely parameterized geodesic" and not the more general parallel transport?

Could you define a parallel transport and a geodesic using tangent vector, (metric compatible torsion free) covariant derivative, and the curve?

I sense that you are giving a better distinction between the two, compared to Carroll.

 

[Orodruin]

Parallel transport along a curve of any tensor just means that the covariant derivative of that tensor in the direction of the curve tangent is equal to zero.

For an affinely parametrised geodesic, the tangent vector itself is parallel transported along the geodesic.

 

[victorvmotti]

For an affinely parametrised geodesic, the tangent vector itself is parallel transported along the geodesic.

And for NON affinely parameterized geodesic, the tangent vector is NOT parallel transported along the geodesic because the covariant derivative is not equal to zero but proportional to the tangent vector.

Right?

 

[Orodruin]

Yes.

 

[victorvmotti]

Thanks now all clear about parallel transport and geodesic in their most general definition.

Back to the point above of distinction between "not changing" and vanishing of directional covariant derivative.

When taking the derivative of a tensor at point $p$ with respect to the parameter of (an integral) curve, what we mean is to take the infinitesimal difference between the tensor at point $q$ pulled back to point $p$ and tensor at point $p$ divided by the infinitesimal difference between the parameter values?

And this is the appropriate and rigorous idea of change or not changing of a tensor along any curve?

 

[Orodruin]

The moving of a vector to a different point is generally not unique. It depends on the connection as well as on the curve. The first issue is resolved by the statement that the connection relevant to GR is the Levi-Civita connection.

In short, you require a way of relating vectors at one point to other points that are infinitesimally close in order to define the derivative. The prescription on how to do this is exactly what the connection tells you.

 

[victorvmotti]

And when using the proper time functional to find its extremal by variation we pick an affine parameter or curve which also defines the Levi-Civita connection. Right?