The geodesic equation is useful in establishing one of the necessary theoretical foundations of relativity, which is the uniqueness of geodesics for a given set of initial conditions. We will seldom have occasion to resort to this technique, an exception being example 19. The solution to this chicken-and-egg conundrum is to write down the differential equations and try to find a solution, without trying to specify either the affine parameter or the geodesic in advance. Likewise, we can’t do the geodesic first and then the affine parameter, because if we already had a geodesic in hand, we wouldn’t need the differential equation in order to find a geodesic. We can’t start by defining an affine parameter and then use it to find geodesics using this equation, because we can’t define an affine parameter without first specifying a geodesic. There is a factor of two that is a common gotcha when applying this equation. Recall that affine parameters are only defined along geodesics, not along arbitrary curves. Recognizing T d x d as a total non-covariant derivative, we find. If this differential equation is satisfied for one affine parameter \(\lambda\), then it is also satisfied for any other affine parameter \(\lambda' = a \lambda + b\), where a and b are constants (problem 5). \) implies that when \(\kappa\) and \(\nu\) are distinct, the same term will appear twice in the summation.
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