In classical General Relativity (meaning not modified) one can think of geodesics in two ways.
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One way is to say that a geodesic is the curve which is the straightest (in analogy with the flat case) among all curves. More or less the story goes like this (correct me if I'm wrong): in the flat case geodesics are of the form $x^\mu(s)=st^\mu +b^\mu$ where $t$ and $b$ are constant vectors and $s$ is the curve parameter. The curve tangent vector is $\frac{dx^\mu}{ds}=t=const$ so $$\frac{dt^\mu}{ds}=t^\nu\partial_\nu t^\mu=0.$$ This is a tensor equation and by general covariance it holds true in a general spacetime, mutatis mutandis: $$t^\nu\nabla_\nu t^\mu=0.$$
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Another way is to obtain the equation by finding the minimum of the length functional $$L=\int \sqrt{g(t,t)} \,\, d\tau,$$ we get $$\frac{dt^\mu}{d\tau}+\Gamma^\mu_{\alpha\beta}t^\alpha t^\beta=0.$$
The above equation turns out to be just $t^\nu\nabla_\nu t^\mu=0$, with a certain parametrization choice.
The two formulations are thus equivalent. My question is when and why is it so? Is there a deep reason?
Looking up Wald's book I found the following, unclear to me, argument:
"On a manifold with a Riemannian metric, one can always find curves of arbitrarily long length connecting any two points. However, the length will be bounded from below, and the curve of shortest length connecting two points (assuming the lower bound in length is attained) is necessarily an extremum of length and thus a geodesic. Thus, the shortest path between two points is always a straightest possible path."
Later in the book he also says something about conjugate points.
I always thought it had to do with torsion: If we relax the condition $\Gamma^\mu_{\alpha\beta}-\Gamma^\mu_{\beta\alpha}=0$ then the connection is not Levi-Civita, hence $\delta_g L=0$ remains unchanged but $t^\nu\nabla_\nu t^\mu=0$ does and gives a different geodesic equation.. so Straightest $\neq$ Shortest anymore.
Can anyone clarify things to me?
Best Answer
This is false.
In the case of spacelike geodesics, your definition of $L$ gives an imaginary number. The complex numbers are not an ordered field, so there is no such thing as "shortest."
There is a similar problem for null geodesics. A null geodesic has $L=0$, and perturbations of a null geodesic may make $L$ either real or imaginary.
Even in the timelike case, it can happen that a geodesic is not a maximal-time curve. There is a discussion of this in Misner, Thorne, and Wheeler, p. 318.
The only general definition of a geodesic that works is that it parallel-transports its own tangent vector, i.e., it's the straightest path.