The first and still the most important rigorous solution of the Einstein field equations was found by Schwarzschild in 1916. Although it's quite difficult to find exact analytical solutions of the complete field equations for general situations, the task is immensely simplified if we restrict our attention to highly symmetrical physical configurations. For example, it's obvious that the flat Minkowski metric trivially satisfies the field equations. The simplest non-trivial configuration in which gravity plays a role is a static mass point, for which we can assume the metric has perfect spherical symmetry and is independent of time. Let r denote the radial spatial coordinate, so that every point on a surface of constant r has the same intrinsic geometry and the same relation to the mass point, which we fix at r = 0. Also, let t denote our temporal coordinate. Any surface of constant r and t must possess the two-dimensional intrinsic geometry of a 2-sphere, and we can scale the radial parameter r such that the area of this surface is 4pr2. (Notice that since the space may not be Euclidean, we don't claim that r is "the radial distance" from the mass point. Rather, at this stage r is simply an arbitrary radial coordinate scaled to give the familiar Euclidean surface area.) With this scaling, we can parameterize the two-dimensional surface at any given r (and t) by means of the ordinary "longitude and latitude" spherical metric
(dS)2 = r2 (dq )2 + r2 sin(q )2 (df )2
where dS is the incremental distance on the surface of an ordinary sphere of radius r corresponding to the incremental coordinate displacements dq and df. The coordinate q represents "latitude", with q = 0 at the north pole and q = p/2 at the equator. The coordinate f represents the longitude relative to some arbitrary meridian.
On this basis, we can say that the complete spacetime metric near a spherically symmetrical mass m must be of the form
(dt)2 = gtt (dt)2 + grr (dr)2 + gqq (dq)2 + gff (df)2
where gqq = -r2, gff = -r2 sin(q)2, and gtt and grr are (as yet) unknown functions of r and the central mass m. Of course, if we set m = 0 the functions gtt and -grr must both equal 1 in order to give the flat Minkowski metric (in polar form), and we also expect that as r increases to infinity these functions both approach 1, regardless of m, since we expect the metric to approach flatness sufficiently far from the gravitating mass.
This metric is diagonal, so the non-zero components of the contravariant metric tensor are gaa = 1/gaa. In addition, the diagonality of the metric allows us to simplify the definition of the Christoffel symbols to
(no implied summations)
Now, the only non-zero partial derivatives of the metric coefficients are
along with gtt/dr and grr/dr, which are yet to be determined. Inserting these values into the preceding equation, we find that the only non-zero Christoffel symbols are
These are the coefficients of the four geodesic equations near a spherically symmetrical mass. We assume that, in the absence of non-gravitational forces, all natural motions (including light rays and massive particles) follow geodesic paths, so these equations provide a complete description of inertial/gravitational motions of test particles in a spherically symmetrical field. All that remains is to determine the metric coefficients gtt and grr.
We expect that one possible solution should be circular Keplerian orbits, i.e., if we regard r as corresponding (at least approximately) to the Newtonian radial distance from the center of the mass, then there should be a circular geodesic path at constant r that revolves around the central mass m with an angular velocity of w, and these quantities must be related (at least approximately) in accord with Kepler's third law
m = r3 w 2
(It's interesting that the original deductions of an inverse-square law of gravitation by Hooke, Wren, Newton, and others were based on this same empirical law. See Section 8.1 for a discussion of the origin of Kepler's law.) If we consider purely circular motion on the equatorial plane (q = p/2) at constant r, the metric reduces to
(dt)2 = gtt (dt)2 - r2 (df )2
and since dr/dt = 0 the geodesic equations are simply
Multiplying through by (dt/dt)2 and identifying the angular speed w with the derivative of f with respect to the coordinate time t, the right hand equation becomes
For consistency with Kepler's Third Law we must have w2 equal (or very nearly equal) to m/r3, so we make this substitution to give
Integrating this equation, we find that the metric coefficient gtt must be of the form k - (2m/r) where k is a constant of integration. Since gtt must equal 1 when m = 0 and/or as r approaches infinity, it's clear that k = 1, so we have
Also, for a photon moving away from the gravitating mass in the purely radial direction we have dt = 0, and so our basic metric for a purely radial ray of light gives
gtt (dt)2 = -grr (dr)2
Invoking the symmetry v 1/v, we select the factorization gtt = dr/dt and grr = -dt/dr, which implies grr = -1/gtt. This gives the complete Schwarzschild metric
from which nearly all of the experimentally accessible consequences of general relativity follow.
In matrix form the Schwarzschild metric is written as
Now that we've determined gtt and grr, we have the partials
so the Christoffel symbols that we previously left undetermined are
Therefore, the complete set of geodesic equations for the Schwarzschild metric are
There are all parametric equations, where l denotes a parameter that monotonically varies along the path. When dealing with massive particles, which travel at sub-light speeds, we must choose l proportional to t, the integrated lapse of proper time along the path. On the other hand, the lapse of proper time along the path of a massless particle (such as a photon) is zero by definition, so this raises an interesting question: How is it possible to extremize the "length" of a path whose length is identically zero? Even though the path of a photon has singular proper time, the path is not singular in all respects, so we can still parameterize the path by simply assigning monotonic values of l to the points on the path. (Notice that, since geodesics are directionally symmetrical, it doesnt matter whether l is increasing or decreasing in the direction of travel.) An alternative approach to solving for light-like geodesics, based on Fermats principle of least time, will be discussed in Section 8.4.
We applied Kepler's Third Law as a heuristic guide to these equations of motion, but there is a certain ambiguity in the derivation, due to the distinction between coordinate time t and the orbiting object's proper time t. Recall that we defined the angular speed w of the orbit as df/dt rather than df/dt. This illustrates the unavoidable ambiguity in carrying over Newtonian laws of mechanics to the relativistic framework. Newtonian physics didn't distinguish between the proper time along a particular path and coordinate time - not surprisingly - since the two are practically indistinguishable for objects moving at much less than the speed of light. Nevertheless, the slight deviation between these two time parameters has observable consequences, and provides important tests for distinguishing between the space geodesic approach and the Newtonian force-at-a-distance approach to gravitation. We've assumed that Kepler's Third law is exactly satisfied with respect to coordinate time t, but only approximately with respect to the orbiting object's proper time t. It's interesting that the Newtonian free-fall formulas for purely radial paths are also applicable exactly in relativity, but only if time is interpreted as the proper time of the falling particle. Thus we can claim an exact correspondence between Newtonian and relativistic laws in each of these two fundamental cases by a suitable correspondence of the time coordinates, but no single correspondence works for both of them.
To show that the equations of motion derived above are fully equivalent to those of Newtonian gravity in the weak slow limit, we need only note that the scale factor between r and t is so great that we can neglect any terms that have a factor of dr/dt unless that term is also divided by r, in which case the scale factor cancels out. Also we can assume that dt/dt is essentially equal to 1, and it's easy to see that if the motion of a test particle is initially in the plane q = p/2 then it remains always in that plane, and by spherical symmetry this applies to all planes. So we can assume q = p/2 and with the stated approximations the equations of motion reduce to the familiar Newtonian equations
where w is the angular velocity.