6.4 Radial=20 Paths in a Spherically Symmetrical Field

It is no = longer clear=20 which way is up even if one wants to rise.

&n= bsp; &nb= sp; &nbs= p; = ; = &= nbsp; &n= bsp; =20 David Riesman, 1950

In this = section we=20 consider the simple spacetime trajectory of an object moving = radially with=20 respect to a spherical mass. As we=92ve seen, according to general = relativity the metric of spacetime in the region surrounding an = isolated=20 spherical mass m is given by

where t is = time=20 coordinate, r is the radial coordinate, and the angles = q and f are the usual angles for polar = coordinates. Since=20 we're interested in purely radial motions the differentials of the = angles=20 dq and df are zero, and we're left with a = 2-dimensional=20 surface with the coordinates t and r, with the = metric

This formula = tells us how=20 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=20 space is given by the diagonal matrix

which has = determinant g =3D=20 -1. The inverse of the covariant tensor=20 g_uv is the contravariant tensor

In order to = make use of=20 index notation, we define x¹ =3D t and x² = =3D r. Then=20 the equations for the geodesic paths on any surface can be = expressed=20 as

where = summation is=20 implied over any indices that are repeated in a given product, and = Gⁱ_jk=20 denotes the Christoffel symbols. Note that the index i can be = either 1 or=20 2, so the above expression actually represents two differential = equations=20 involving the 1st and 2nd derivatives of our coordinates = x¹ and=20 x² (which, remember, are just t and r) with respect to = the=20 "affine parameter" l. This parameter just represents the = normalized=20 "distance" along the path, so it's proportional to the proper time = t for timelike = paths.

The = Christoffel symbol is=20 defined in terms of the partial derivatives of the components of = the=20 metric tensor as follows

Taking the = partials of=20 the components of our g_uv with respect to t and r we = find that=20 they are all zero, with the exception of

Combining = this with the=20 fact that the only non-zero components of the inverse metric = tensor=20 g^uv are g¹¹ and g²², we find that = the=20 only non-zero Christoffel symbols are

So, = substituting these=20 expressions into the geodesic formula (2), and reverting back to = the=20 symbols t and r for our coordinates, we have the two ordinary = differential=20 equations for the geodesic paths on the = surface

These = equations can be=20 integrated in closed form, although the result is somewhat messy. = They can=20 also be directly integrated numerically using small incremental = steps of=20 "dl" for any initial position and = trajectory. This=20 allows us to easily generate geodesic paths in terms of r as a = function of=20 t. If we do this, we will notice that the paths invariably go to = infinite=20 t as r approaches 2m. Is our 2-dimensional surface actually = singular at r=20 =3D 2m, or are the coordinates simply ill-behaved (like longitude = at the=20 North pole)?

As we saw = above, the=20 surface has an invariant Gaussian curvature at each point. Let's = determine=20 the curvature to see if anything strange occurs at r =3D 2m. The = curvature=20 can be computed in terms of the components of the metric tensor = and their=20 first and second partial derivatives. The non-zero first = derivatives for=20 our surface (and the determinant g =3D -1) were noted above. The only non-zero = second=20 derivatives are

So we can = compute the=20 intrinsic curvature of our surface using Gauss's formula for the = curvature=20 invariant K of a two-dimensional surface given in the section on=20 Curvature. Inserting the metric components and derivatives for our = surface=20 into that equation gives the intrinsic = curvature

Therefore, at = r =3D 2m the=20 curvature of this surface is -1/(4m²), which is certainly = finite (and=20 in fact can be made arbitrarily small for sufficiently large m). = The only=20 singularity in the intrinsic curvature of the surface occurs at r = =3D=20 0.

In order to = plot r as a=20 function of the proper time t we would like to eliminate t from the = two=20 equations. To do this, notice that if we define T =3D = dt/dl the first equation can be written in = the=20 form

which is just = an ordinary=20 first-order differential equation in T with variable coefficients. = Recall=20 that the solution of any equation of the form

is given=20 by

where k is a = constant of=20 integration and w =3D . Thus the solution of (4) = is

The integral = in the=20 exponential is just ln(r) - ln(r - 2m) so the result = is

Let's suppose = our test=20 particle is initially stationary at r =3D R and then allowed to = fall freely.=20 Thus the point r =3D R is the "apogee" of the radial orbit. Our = affine=20 parameter l is proportional to the proper time = t along a path, and the value we assign = to "k"=20 determines the scale factor between l and t. From the original metric equation (1) = we know=20 that at the apogee (where dr/dt =3D 0) we have

Multiplying = this with the=20 previous derivative at r =3D R gives

Thus in order = to scale=20 our affine parameter to the proper time t for this radial orbit we need to set k = =3D=20 , and so

(Notice that = this implies=20 the initial value of dt/dl at the apogee is , and of course dr/dl at that point is 0.) Substituting = this into=20 the 2nd geodesic equation (3) gives a single equation relating the = radial=20 parameter r and the affine parameter l, which we have made equivalent to the = proper time=20 t, so we have

At the apogee = r =3D R where=20 dr/dt =3D 0 this reduces to

This is a = measure of the=20 acceleration of a static test particle at the radial parameter r. = More=20 generally, we can use equation (5) to numerically integrate the = geodesic=20 path from any given initial trajectory, and it confirms that the = radial=20 coordinate passes smoothly through r =3D 2m as a function of the = proper time=20 t. This may seem surprising at first, = because the=20 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 =3D 2m. However, = remarkably, the=20 square of dr/dt is invariably forced to 1 = - 2m/R precisely at r =3D 2m, so = the quantity=20 in the square brackets goes to zero, canceling the zero in the=20 denominator.

Interestingly, equation=20 (5) has the same closed-form solution as does radial free-fall in=20 Newtonian mechanics (if t is identified with Newton's=20 absolute time). The solution can be expressed in terms of the = parameter=20 a by the "cycloid = relations"

The = coordinate time t can=20 also be given explicitly in terms of a by the formula

where Q =3D = . A typical timelike radial orbit is illustrated=20 below.

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