In this chapter we'll describe how to compute the spacetime trajectory of an object moving radially with respect to a spherical mass. According to general relativity the metric of spacetime in the region surrounding an isolated spherical mass m is given by the Schwarzschild solution
where t is time coordinate, r is the radial coordinate, and the angles q and f are the usual angles for polar coordinates. Since we're interested in purely radial motions the differentials of the angles dq and df are zero, and we're left with a 2-dimensional surface with the coordinates t and r, with the metric
This formula tells us how to compute the absolute lapse of proper time dt along a given path corresponding to the coordinate increments dt and dr. The metric tensor on this 2-dimensional space is given by the diagonal matrix
which has determinant g = - 1. The inverse of the covariant tensor guv is the contravariant tensor
In order to make use of index notation, we define x1 = t and x2 = r. Then the equations for the geodesic paths on any surface can be expressed as
where summation is implied over any indices that are repeated in a given product, and G ijk denotes the Christoffel symbols. Note that the index i can be either 1 or 2, so the above expression actually represents two differential equations involving the 1st and 2nd derivatives of our coordinates x1 and x2 (which, remember, are just t and r) with respect to the "affine parameter" l. This parameter just represents the normalized "distance" along the path, so it's proportional to the proper time t for timelike paths.
The Christoffel symbol is defined in terms of the partial derivatives of the components of the metric tensor as follows
Taking the partials of the components of our guv with respect to t and r we find that they are all zero, with the exception of
Combining this with the fact that the only non-zero components of the inverse metric tensor guv are g11 and g22, we find that the only non-zero Christoffel symbols are
So, substituting these expressions into the geodesic formula (2), and reverting back to the symbols t and r for our coordinates, we have the two ordinary differential equations for the geodesic paths on the surface
These equations can be integrated in closed form, although the result is somewhat messy. They can also be directly integrated numerically using small incremental steps of "dl" for any initial position and trajectory. This allows us to easily generate geodesic paths in terms of r as a function of t. If we do this, we will notice that the paths invariably go to infinite t as r approaches 2m. Is our 2-dimensional surface actually singular at r = 2m, or are the coordinates simply ill-behaved (like longitude at the North pole)?
As we saw above, the surface has an invariant Gaussian curvature at each point. Let's determine the curvature to see if anything strange occurs at r = 2m. The curvature can be computed in terms of the components of the metric tensor and their first and second partial derivatives. The non-zero first derivatives for our surface (and the determinant g = -1) were noted above. The only non-zero second derivatives are
So we can compute the intrinsic curvature of our surface using Gauss's formula for the curvature invariant K of a two-dimensional surface given in the section on Curvature. Inserting the metric components and derivatives for our surface into that equation gives the intrinsic curvature
Therefore, at r = 2m the curvature of this surface is -1/(4m2), which is certainly finite (and in fact can be made arbitrarily small for sufficiently large m). The only singularity in the intrinsic curvature of the surface occurs at r = 0.
In order to plot r as a function of the proper time t we would like to eliminate t from the two equations. To do this, notice that if we define T = dt/dl the first equation can be written in the form
which is just an ordinary first-order differential equation in T with variable coefficients. Recall that the solution of any equation of the form
is given by
where k is a constant of integration and w = 
. Thus the solution of (4) is
The integral in the exponential is just ln(r) - ln(r -2m) so the result is
Let's suppose our test particle is initially stationary at r = R and then allowed to fall freely. Thus the point r = R is the "apogee" of the radial orbit. Our affine parameter l is proportional to the proper time t along a path, and the value we assign to "k" determines the scale factor between l and t . From the original metric equation (1) we know that at the apogee (where dr/dt = 0) we have
Multiplying this with the previous derivative at r = R gives
Thus in order to scale our affine parameter to the proper time t for this radial orbit we need to set k = 
, and so
(Notice that this implies the initial value of dt/dl at the apogee is 
, and of course dr/dl at that point is 0.) Substituting this into the 2nd geodesic equation (3) gives a single equation relating the radial parameter r and the affine parameter l, which we have made equivalent to the proper time t , so we have
At the apogee r = R where dr/dt = 0 this reduces to
This is a measure of the acceleration of a static test particle at the radial parameter r. More generally, we can use equation (5) to numerically integrate the geodesic path from any given initial trajectory, and it confirms that the radial coordinate passes smoothly through r = 2m as a function of the proper time t . This may seem surprising at first, because the denominator of the leading factor contains (r - 2m), so it might appear that the second derivative of r with respect to proper time t "blows up" at r = 2m. However, remarkably, the square of dr/dt is invariably forced to 1 - 2m/R precisely at r = 2m, so the quantity in the square brackets goes to zero, canceling the zero in the denominator.
Interestingly, equation (5) has the same closed-form solution as does radial free-fall in Newtonian mechanics (if t is identified with Newton's absolute time). The solution can be expressed in terms of the parameter a by the "cycloid relations"
The coordinate time t can also be given explicitly in terms of a by the formula
where Q = 
. A typical timelike radial orbit is illustrated below.
