the relation higher up between and , we can also write Since asymptotically the integrand is inversely proportional to , this shows that in the frame of reference if approaches it does so exponentially without ever reaching it. However, as a function of , does reach . The above solutions are valid while the integrand is finite, but a total solution may involve two or an infinity of pieces, each described by the integral but with alternating signs for the square root. When and , we can solve for and explicitly: and for photonic geodesics () with zero angular momentum (Although the proper time is trivial in the photonic case, one can define an affine parameter , and then the solution to the geodesic equation is .) Another solvable case is that in which and and are constant. In the volume where this gives for the proper time This is close to solutions with small and positive. Outside of the solution is tachyonic and the "proper time" is space-like: This is close to other tachyonic solutions with small and negative. The constant tachyonic geodesic outside is not continued by a constant geodesic inside , but rather continues into a "parallel exterior region" (see Kruskal–Szekeres coordinates). Other tachyonic solutions can enter a black hole and re-exit into the parallel exterior region. The constant solution inside the event horizon () is continued by a constant solution in a white hole. When the angular momentum is not zero we can replace the dependence on proper time by a dependence on the angle using the definition of which yields the equation for the orbit where, for brevity, two length-scales, and , have been defined by Note that in the tachyonic case, will be imaginary and real or infinite. The same equation can also be derived using a Lagrangian approach or the Hamilton–Jacobi equation (see below). The solution of the orbit equation is This can be expressed in terms of the Weierstrass elliptic function . Local and delayed velocities Unlike in classical mechanics, in Schwarzschild coordinates and are not the radial and transverse components of the local velocity (relative to a stationary observer), instead they give the components for the celerity which are related to by for the radial and for the transverse component of motion, with . The coordinate bookkeeper far away from the scene observes the shapiro-delayed velocity , which is given by