Constant along a geodesic vs covariantly constant

[binbagsss]

some questions I have seen tend to word as show that some quantity/tensor/scalar (e.g let this be $K$) is constant along an affinely parameterised geodesic, others ask show covariantly constant.

the definition of covariantly constant/ parallel transport is:

$V^a\nabla_u K = 0$for the quantity $K$ where $V^a$ is the tangent vector to the geodesicsimply constant is wr.t the affine paramter

$\frac{d}{ds} K =0$

but, it is often the case, to show the latter case, we use the chain rule , i.e. that $\frac{d}{ds} = V^a \nabla_a$ when showing covariantly constant

e.g for the proof that given a KVF $K^u$, we make use of the chain rule (connections not needed since we are acting on a scalar) to show that along a geodesic $V^uK^u$ is conserved.

But are, simply being constant w.r.t the affine parameter, and being covariantly constant/parallel transported not different things physically?

thanks.

 

[PeterDonis]

are, simply being constant w.r.t the affine parameter, and being covariantly constant/parallel transported not different things physically?

Why do you think they would be?

 

[martinbn]

What exactly is the confusion here? If $X$ is a vector field and $f$ a function, then the derivative of $f$ in the direction of $X$ is $\nabla_Xf=X(f)$ and doesn't depend on the connection.

 

[stevendaryl]

Are you asking about the intuitive difference between the Lie derivative and the covariant derivative?