the elliptical orbit's semiaxis A and eccentricity e related by the formula to give the precession angle Mathematical derivations of the orbital equation Christoffel symbols The non-vanishing Christoffel symbols for the Schwarzschild-metric are: Geodesic equation According to Einstein's theory of general relativity, particles of negligible mass travel along geodesics in the space-time. In flat space-time, far from a source of gravity, these geodesics correspond to straight lines; however, they may deviate from straight lines when the space-time is curved. The equation for the geodesic lines is where Γ represents the Christoffel symbol and the variable parametrizes the particle's path through space-time, its so-called world line. The Christoffel symbol depends only on the metric tensor , or rather on how it changes with position. The variable is a constant multiple of the proper time for timelike orbits (which are traveled by massive particles), and is usually taken to be equal to it. For lightlike (or null) orbits (which are traveled by massless particles such as the photon), the proper time is zero and, strictly speaking, cannot be used as the variable . Nevertheless, lightlike orbits can be derived as the ultrarelativistic limit of timelike orbits, that is, the limit as the particle mass m goes to zero while holding its total energy fixed. Therefore, to solve for the motion of a particle, the most straightforward way is to solve the geodesic equation, an approach adopted by Einstein and others. The Schwarzschild metric may be written as where the two functions and its reciprocal are defined for brevity. From this metric, the Christoffel symbols may be calculated, and the results substituted into the geodesic equations It may be verified that is a valid solution by substitution into the first of these four equations. By symmetry, the orbit must be planar, and we are free to arrange the coordinate frame so that the equatorial plane is the plane of the orbit. This solution simplifies the second and fourth equations. To solve the second and third equations, it suffices to divide them by and , respectively. which yields two constants of motion. Lagrangian approach Because test particles follow geodesics in a fixed metric, the orbits of those particles may be determined using the calculus of variations, also called the Lagrangian approach. Geodesics in space-time are defined as curves for which small local variations in their coordinates (while holding their endpoints events fixed) make no