general relativity - How do we know the geodesic is a minimum?

The geodesic equation is derived from the Euler-Lagrange equation, which (as I understand it) is a necessary but not sufficient condition to ensure that the geodesic is a minimum.

The introductory GR books I have looked at do not bother with sufficiency.

How do we know for sure that the geodesic is a minimum and not a maximum or inflection?

Answer

In general, the geodesics are "stationary points" which means that they satisfy the appropriate Euler-Lagrange equations. This condition is more fundamental than being a minimum or maximum or a saddle point or some marginal situation involving inflections. Being a minimum etc. is a combination of the fundamental condition specified by the Euler-Lagrange equations; and some kind of an inequality for the matrix of second functional derivatives.

Geodesics in spaces with the positively definite Euclidean signature – not spacetimes! – will tend to be minima of the proper length.

In the Minkowski spacetime, almost all timelike geodesics are actually maxima rather than minima of the proper time measured along the geodesics. That's easy to see: any other timelike curve connecting the two end points is closer to being null (whose proper time is zero), so its proper time is shorter than the time along the straight path (the twin who stays home in the twin paradox will be older). This gets generalized to the curved Minkowski-signature spacetime, too: one can see that if he divides the long curved path to many pieces because each piece de facto follows the same rules as the paths in a flat space.

However, spacelike geodesics are saddle points. Perturbations of the geodesic in the transverse spacelike directions make the proper length longer while perturbations of the geodesic in the timelike transverse direction are shorter.

Null geodesics can't really be uniquely defined as maxima or minima at all: any null curve connecting the same two end points (and there are infinitely many) has the same length, namely zero, while both timelike and spacelike curves are "longer" although the unit of length differs by a factor of $i$. Relatively to some variations, null geodesics will probably be inflections. In that case, the appropriate Euler-Lagrange equations are the only simple way to describe what's special about a geodesic relatively to nearby paths.

The paragraphs above were about "minima vs maxima vs saddle points". There's one more question, whether the stationary points may be global minima or maxima. Saddle points can't be "global" so this possibility is only relevant for timelike geodesics. In many cases "close" to the straight lines in a flat spaces and comparable situations in curved space, the geodesic is a global maximum of the proper time along the curve. In general enough spacetimes, nothing like that can be said.