General Relativity - geodesic - affine parameter

[binbagsss]

Homework Statement

Question attached:

Homework Equations

see below

The Attempt at a Solution

[/B]
my main question really is

1) what is meant by 'abstract tensors' as I have this for my definition:
to part a)
$V^u\nabla_uV^a=0$

but you do say that $V^u=/dot{x^u}$ ; x^u is a coordinate so, probably a stupid question, how is it really an 'abstract tensor'? I would say $\nabla_u$ is but not $V^u$.

and

b) how this definition specifies/relies on being affinely parameterised. I have the definition as the tangent vector is parallel transported along itself. I would have thought that any tangent vector of a geodesic is parallel transported along itself- does it need to be affinely parameterised and what does this mean?

(I have seen the terminology 'affine connection' used elsewhere and that the Levi-Civita connection is one such connection, which is assumed in GR via the fundamental theorem of Riemann geometry - I posted a thread the other week on showing that definitions of geodesics agree due to this (the above and definition based on calculus of variations), and but in our lecture notes we have only defined a affine parameter as one such that dL/ds =0 , where L is the Lagrangian of a free-ly falling particle, we have not talked about affine connections and so I don't understand the definition of affine parameters in the context of parallel-transported tangent vector definition of geodesic as given above)- I guess the terms affine parameter and affine connection are related and this may cover it?

part b)

simply expand this out to get
$\ddot{x^u}+\Gamma_{ab}^{u}\dot{x^a}\dot{x^b}$

many thanks

 

[andrewkirk]

I would have thought that any tangent vector of a geodesic is parallel transported along itself- does it need to be affinely parameterised and what does this mean?

The direction of the tangent (aka velocity) vector may be constant along the curve, but the magnitude will not, unless the geodesic is affinely parameterised. A curve whose velocity vector is always tangent but which varies in magnitude is not a geodesic. Changing the parameterisation changes the magnitude of the velocity vector. If the change is linear, a velocity vector that was parallel-transported will remain parallel-transported because it will be multiplied by the same constant everywhere - being the gradient of the linear transformation. But if the parameter transformation is nonlinear that will not be the case.