Path Lengths and Coordinates
The derivation of Fermat's Principle of Least Time in the context of Schwarzschild spacetime relies on the interesting proposition that we arrive at the same geodesic paths on a manifold by extremizing any one of the squared differentials - including the path length - in a diagonal metric, provided the metric coefficients are independent of the extremized variable. The metric line element for an n-dimensional manifold with coordinates x1, x2, ..., xn is conventionally written in the form
where summation from 1 to n is implied over repeated indices, but this can also be written in homogeneous form by treating the path length parameter s as a pseudo-independent variable x0, so the line element is
where the summations over repeated indices now extend from 0 to n. The gmn need not be positive, but obviously we have g0n = gm0 = 0 for all non-zero indices m,n, and the gmn are independent of x0. If any other variable xj satisfies the same (respective) conditions, then extremizing xj must give the same geodesic paths on the manifold as given by extremizing x0. This might seem implausible because, after all, x0 is a path length parameter, not a coordinate. However, in terms of the differentials and their first and second derivatives at a given point, the metric does not distinguish between the path length parameter and coordinates that satisfy the "diagonal and independence" conditions. In effect, the path length parameter is a valid ("polar") independent coordinate at any given point, at least as far as the differentials and their derivatives at that point, which is sufficient to ensure unique geodesic paths on the manifold, regardless of which coordinate is extremized (provided, again, that the coordinate satisfies the "diagonal and independence" conditions).
To illustrate, consider the simple Pythagorean metric on the flat plane
The geodesics on this manifold are simply straight lines, i.e., loci such that x(s) and y(s) are both linear functions of the path length parameter s evaluated along the locus. This can be formally deduced by extremizing (ds)2 along any given path. Since the metric coefficients are constants, the Christoffel symbols vanish, and the geodesic equations for the manifold are
which implies that the x and y variables along any geodesic path can be expressed parametrically as functions of the path length s in the form
for constants A, B, C, and D. The nature of the extremum represented by this solution can be seen in the figure below, which show the geodesic extrapolation of a curve with the initial trajectory given by the differentials dx,dy.
Our objective is to extrapolate the path AB through the origin to a point C with coordinates dx,dy at a fixed incremental distance ds from A while traversing the minimum possible distance, i.e., in such a way that the distance ds from B to C is minimized. Conversely, for a fixed distance ds from B to C, we wish to find the point C that maximizes the distance ds from A to C. The metric of the surface gives the relations
Expanding the second relation gives
Solving the first relation for dy and substituting into this last expression gives
Everything on the right hand side is fixed expect for dx, so we can differentiate with respect to dx to find the value that maximizes ds. This gives
Setting this to zero, we find that dx/dy = dx/dy, which confirms that the geodesics on this surface are the linear paths.
Now suppose we write the basic metric relation between the squared differentials in the form
and regard the path length parameter s as an pseudo-independent coordinate variable and y as the quantity to be extremized. In this form the metric is a non-positive definite, but the metric coefficients are still constants, so the Christoffel symbols still vanish. Thus the geodesic equations are
so the variables x and s along the path of any geodesic must be of the form
This is equivalent to the previous parametric equations for the geodesics, as can be seen by setting
Notice that in this case we are actually maximizing dy along the path. (Likewise we can arrive at the same set of geodesics if we maximize dx.) Analogously to the positive-definite case discussed previously, the nature of the extremum represented by this solution can be seen in the figure below, which (again) shows the geodesic extrapolation of a curve with a given initial trajectory, this time represented by the differentials dx,ds.
Here our objective is to extrapolate the path AB through the origin to a point C with coordinates dx,ds at a fixed incremental "distance" dy from A while traversing the maximum possible "distance", i.e., in such a way that the "distance" dy from B to C is maximized. Conversely, for a fixed "distance" dy from B to C, we wish to find the point C that minimizes the "distance" dy from A to C. (Notice that the minima and maxima have been reversed relative to the previous case.) In this case our "metric" gives the relations
Expanding the second relation gives
Solving the first relation for ds and substituting into this last expression gives
Again, everything on the right hand side is fixed expect for dx, so we can differentiate with respect to dx to find the value that maximizes dy. This gives
Setting this to zero, we find that dx/ds = dx/ds, which again confirms that the geodesics on this surface are the linear paths. (Also, we saw previously that linearity in the x,s plane is equivalent to linearity in the x,y plane.)
This has some interesting applications in physics. For example, the Minkowski spacetime metric is
and strictly speaking the geodesics are the paths that extremize dt, but since this is a diagonal metric with constant coefficients, it follows that the geodesics are also given by extremizing any of the coordinate variables (treated as a pseudo-path length). For timelike worldlines the geodesics maximize the lapse of proper time dt, because the signature of the Minkowski metric is negative, but if we write this relations between the differentials in the form
we see that the geodesics are the paths on which this coordinate dt is minimized. This accounts for Fermat's Principle of Least Time, which historically was applied to the propagation of light, but which also applied to time-like geodesics of massive objects. This is also valid in the context of the Schwarzschild metric, which represents the spacetime manifold in a spherically symmetrical gravitational field, as discussed in Accelerating in Place.