How is the length functional L defined for a geodesic?

Minimizing the length functional $L$ is described as tricky. When $\lambda$ is an affine parameter, minimizing the simpler functional $F = \int_A^B g_{ij}\dot{x}^i\dot{x}^j\,d\lambda$ is equivalent to minimizing $L$ and is much easier.

Conditions: $\lambda$ must be an affine parameter.; $F$ is used as an easier surrogate for $L$ under that condition.

The geodesic equation is obtained by applying the Euler-Lagrange equation to the Lagrangian $\mathscr{L} = g_{ij}\dot{x}^i\dot{x}^j$. The resulting differential equation for the curve components is $\frac{d^2x^k}{d\lambda^2} + \Gamma^k_{ij}\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda} = 0$, where $\Gamma^k_{ij}$ are the connection coefficients.

Conditions: $\lambda$ is an affine parameter.; $\Gamma^k_{ij}$ are connection coefficients.; The equation is presented as the general solution of the Euler-Lagrange step for F.

A geodesic is treated as a curve in space-time and is written componentwise as functions of a parameter $\lambda$. Every space-time component of the curve can be expressed as $x^\mu = x^\mu(\lambda)$, which converts the geometric object into coordinate functions suitable for differentiation.

Conditions: $\lambda$ is the chosen curve parameter.; Each space-time component $x^\mu$ is expressed as a function of $\lambda$.

A null geodesic is characterized by the vanishing of the tangent-vector norm, expressed as $\tilde{T}\cdot\tilde{T} \equiv g_{ij}T^iT^j = 0$. A non-null geodesic is distinguished by having a nonzero tangent-vector norm, $g_{ij}T^iT^j \neq 0$.

Conditions: $\tilde{T}$ is the tangent vector to the geodesic.; $g_{ij}$ are metric components.; The equality to zero defines the null case, and nonzero defines the non-null case.

An affine parameter is the special choice of the curve parameter $\lambda$ for which the scalar proportionality function $f(\lambda)$ in the tangent evolution equation vanishes identically. Specifically, the tangent vector $\tilde{T}$ satisfies $\frac{d\tilde{T}}{d\lambda} = f(\lambda)\tilde{T}$, and the parameter is affine if $f(\lambda)=0$ for all $\lambda$.

Conditions: $\tilde{T}$ is the tangent to the geodesic.; $f(\lambda)$ is a scalar function of the parameter.; The affine case is the subcase $f(\lambda)=0$ for all $\lambda$.