6.4  Radial Paths in a Spherically Symmetrical Field

It is no longer clear which way is up even if one wants to rise.

                                                                                                David Riesman, 1950

In this section we consider the simple spacetime trajectory of an object moving radially with respect to a spherical mass. As we’ve seen, according to general relativity the metric of spacetime in the region surrounding an isolated spherical mass m is given by

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 g_uv 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 Gⁱ_jk 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 g_uv 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 g^uv 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 the quantity in the square brackets is unity and this equation reduces to

This is a measure of the acceleration of a static test particle at the radial parameter r. Even before we solve equation (5), it’s clear that it can be integrated to give the geodesic path from any given initial trajectory, and we can confirm 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 right hand side contains (1 - 2m/r), so it might appear that the second derivative of r with respect to proper time t "blows up" at r = 2m. However, it will be shown below that the square of dr/dt is invariably forced to 1 - 2m/R precisely at r = 2m, so the numerator goes to zero at this point as well, canceling the zero in the denominator.

To solve equation (5) analytically, note that the derivative (with respect to t) of the quantity in the square brackets is

The expression in square brackets on the right hand side vanishes if and only if equation (5) is satisfied, so it follows that the quantity in brackets on the left side is constant (noting again that the singularity at r = 2m is removable), and therefore equal to unity (assuming it is unity at any point on the trajectory, which it is at the apogee for a bound trajectory). Therefore, the inverse square relation (6) is satisfied at all times. In addition, since the quantity in brackets on the left side of the above equation is unity, we have

Taking the square root and re-arranging terms, this gives

We have the integral

To simplify this result, we make a change of variables by defining the argument of the inverse sine to be the cosine of some angle a. Thus we define a such that cos(a) = 2r/R – 1, which implies

Inserting this into the preceding equation gives the elapsed proper time between r₁ = R and r₂ = r as

This shows that 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), namely, the parametric "cycloid relations". A plot of this r versus t corresponds to the position of a point on the rim of a rolling wheel of radius R/2, where a is the angle of the wheel. It follows from the above derivation that this parametric solution also satisfies the basic inverse square equation d²r/dt² = -m/r², which is why it applies in the Newtonian case as well (as shown in Section 4.3).

We can also express the Schwarzschild coordinate time t explicitly in terms of a by multiplying the two relations

to give

Substituting the parametric expression for r into this equation, multiplying through by da, and integrating both sides, we get

The integral can be evaluated explicitly to give

Now, making use of the trigonometric identities

the equation can be written in the form

where Q = . A typical timelike radial orbit is illustrated below, both in terms of proper time and Schwarzschild coordinate time.

A notable feature of this trajectory is its temporal symmetry. Not only is there a continuous geodesic path from the apogee down through the Schwarzschild radius to the singularity at r = 0, there is also a continuous geodesic path from the singularity up through the Schwarzschild radius to the apogee. This was to be expected in view of the temporal symmetry of the field equations in general, and the Schwarzschild metric in particular, but it might seem inconsistent with the well-known fact that once a particle has crossed from outside to inside the Schwarzschild radius it can never re-emerge. However, there is no inconsistency, because the emerging particle has never crossed from outside to inside that radius.

To understand the full set of possible trajectories consistent with the Schwarzschild metric, it’s useful to first note an ambiguity present in all pseudo-Riemannian metrics due to their quadratic character. Consider the Minkowski metric (dt)² = (dt)² – (dx)², which obviously doesn’t constrain the signs of the differentials, because they each appear squared. At constant x the metric requires (dt/dt)² = 1, but the ratio dt/dt itself can be either +1 or -1. Strictly speaking, we are free to choose whether the proper time along a given path increases or decreases as the coordinate time increases. We might fancifully imagine that the Minkowski metric actually entails two separate universes, with proper time increasing with coordinate time in one, and decreasing with coordinate in the other. Alternatively we could imagine a single universe with two families of particles, whose proper times increase in opposite directions of the coordinate time t. (In fact, John Wheeler once speculated that anti-matter particles might be modeled as particles moving backward in time.) However, with a fixed metric like the Minkowski metric, it’s easy to just arbitrarily stipulate the same sign for dt and dt for every path, and then continuity requires that this always remains true (since timelike paths cannot “turn around” in Minkowski space).

The same quadratic ambiguity arises when considering the Schwarzschild metric, but in this case the various possible signs of the differentials are more inter-related, because the coefficients of the metric change signs at r = 2m. For values of r greater than 2m we have a metric of the form (dt)² = (dt)² – (dr)² neglecting scale factors, whereas for values of r less than 2m the metric takes the form (dt)² = (dr)² – (dt)². In the former case, |dt| must always equal or exceed |dt|, but in the latter case |dr| must equal or exceed |dt|. Thus, outside the Schwarzschild radius we must choose the sign of dt/dt, and inside that radius we must choose the sign of dr/dt. The signs of these ratios cannot change along any particle’s path in their respective regions. In effect, the radius r serves as the “time” coordinate inside the Schwarzschild radius.

Now, by analytic continuation, it can be shown that a path crossing the Schwarzschild radius from an outer region with positive dt/dt must enter an inner region with negative dr/dt. This is why a particle falling inward through the Schwarzschild must thereafter continue to reach smaller and smaller values of r. It cannot “turn around”, but must continue down to r = 0. However, conversely, it can be shown that a particle passing outward through the Schwarzschild radius into an outer region with positive dt/dt must have come from an inner region of positive dr/dt. Hence if we observe object falling into the inner region, and other object emerging from the inner region, we seem forced to conclude that there are two physically distinct inner regions, or else that there exist closed spacetime loops if we insist on a single interior region. One or the other of these consequences is unavoidable if we take seriously the analytic continuation of all geodesics consistent with the Schwarzschild metric. The existence of two distinct inner regions is perhaps not surprising if we note that an infalling object requires infinite coordinate time to cross the boundary at r = 2m, and conversely an out-going object requires infinite coordinate time to emerge. Clearly these are two very different classes of objects, one coming from the beginning of coordinate time, and the other departing to the end of coordinate time. The same reasoning leads to the potential existence of a second outer region, with negative dt/dt, so the full extent of the manifold entailed by the Schwarzschild metric, if fully developed, consists of four distinct regions. Thus the consideration of radial trajectories in Schwarzschild spacetime leads unavoidably to cosmological issues, which are described more fully in the discussions of “black holes” in Section 7.

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