In my readings in GR I often come across geodesics characterized as "straightest possible curves." This characterization confuses me. I'd like some clarification as to whether I'm understanding the math concepts correctly.
To fix ideas, my understanding of the basic math of GR is that we can start with a connection, which lets us map a vector in a tangent space to a vector in a nearby tangent space and stipulate that these two vectors count as "the same." This in turn allows us to calculate how a vector field changes in a way that corrects for using different coordinate systems, i.e. the covariant derivative. This in turn lets us define the parallel transport of a vector along a curve. This in turn lets us characterize geodesics and investigate the curvature of the space. (I recognize there are other ways to develop these concepts depending what you start with, but this is most intuitive to me. Hopefully it's correct.)
Now, my understanding of geodesic curves is that they are those whose tangent vectors do not change as we move in the direction of the tangent vector itself. I.e. they parallel transport their tangent vectors.
My question is how this 'parallel transporting its tangent vector' is equivalent to a curve which does so being 'straightest possible.' My understanding is that straightness depends on parallel-ness, which depends on the connection, whicn is not given but something i can choose. So this meant that by choosing a certain connection, i decide what is 'straight'.
But if that is so, what are the constraints that make one kind of curve -e.g. a great circle on a sphere- the 'straightest possible'?
What's to prevent me from defining literally any curve on a sphere and then choosing a connection such the tangent to that curve is thereby defined as parallel to the tangent at the previous point and thereby defining that curve as straight by fiat?
And if i can do that, on what grounds can it be said that a great circle is "straighter" than the curve i have thus defined as straight by construction?