Never has so much confusion, and conflicting results, have been made that a theory can be made compatible with just about anything.
Einstein built into his general theory of relativity that particles should follow geodesic orbits. Somehow, gravity should raise its head once the indefinite metric of Minkowski 4D should have the gravitational potentials depend on space and time. But, these gravitational potentials are never functions of time in a static universe, and why time should enter at all is enigmatic. Not only does simple time not appear, but its two-time form is related to one of the gravitational potential potentials via the geodesic equation--which is not a conservation of energy condition at all!
The indefinite metric,
ds^2 = F^2(r) dt^2- dr^2 - G(r) d(phi)^2, (1)
which is (3.37e) of Besse's Einstein's Manifolds, gives rise to a Lagrangian
L=(1/2){F^2 (dt/dp)^2- (dr/dp)^2 - G(d(phi)/dp)^2}, (2)
where p is an affine parameter, chosen so as not to run into trouble if we consider photon null geodesics, ds=0. The geodesic equations coincide with the Euler-Lagrange equation,
(d/dp)(dL/d(dx/dp)) - dL/dx = 0. (3)
Since t and phi are cyclic coordinates, they lead immedialy for first integrals:
F^2(dt/dp) = constant (4)
and
G^2(d(phi)/dp)= constant. (5)
The first has been confused with some type of "energy" conservation equation. If the affine parameter coincides with a meaure of proper time, s, then (4) is some type of two-timing condition. The second integral is said to be a manifestation of the conservation of angular momentum, which is not conserved in general relativity, in general.
The final geodesic equation reads
d^2 r/dp^2 = FF'(dt/dp)^2 - GG' (d(phi)/dp)^2 (16)
Now, Besse tells us that an orbit is said to be circular if r=a=const. so that (16) reduces
to
(d(phi)/dp)^2= (FF')/(GG') (dt/dp)^2 = (m/G^3) (dt/dp)^2, (17)
where we have introduced F=(1-2m/G)^(1/2) and F=G', the prime denoting differentiation with respect to r. But what about the first integrals, (4) and (5)? Well, they reduce the geodesic equation (16) to nonsense.
Reverting to the case where s is a measure of proper time, as in (1), the metric can be written as
F^2(dt/ds)^2-G^2(d(phi)/ds)^2= 1, (18)
and introducing (17), results in
(dt/ds) =1/{1-3m/G}^(1/2) . (19)
However, we know from the geodesic equation with p=s, that (19) can't be true, since it is not equal to 1/F! The computation of the period of the orbit is even more absurd.
For proper tme, the period of the orbit is
T=|2 pi/d(phi)/ds|. (20)
Introducing (19) into (18) results in
(m/G^3)/(1-3m/G) = (d(phi)/ds)^2. (21)
Hence, from the defintion of the period (20), we have
T^2= G^3/m, (22)
in the limit G(a)-> infinity. Supposedly, (22) is an expression of Kepler's III law, yet (20) together with the geodesic condition (5) would have the proper period increasing as the square of G, where G is interpreted as the distance from the center of the star [cf. Sec. 3.43 of Besse].
If we now introduce the transform R=G(r), the Lagrangian can be written as
L=(1/2){F^2(dt/dp)^2-(dR/dp)^2/F^2 - R^2(d(phi)/dp^2). (23)
Instead of (16) we now get the raidal geodesic equation,
d^2 R/dp^2 - R(d(phi)^2/dt =g{3(dR/dp)^2/F^2 -2V_s^2- (1-2m/G) (24)
which was found by McGrudder (and references therein), Hilbert, Tredder, etc. We appreciate with McGrudder, that the first term on the rhs of (24) gives positive acceleration. But, this has absolutely no physical content as we can make it disappear by a mere coordinate transform, R=G(r).
At the hands of mathematicans therefore, there is a strong dose of Riemann geometry but lacks any physical explanation. Periodic orbits do no exist because the F and G coefficients have c^2 in the denominator, and cannot, therefore lead to Keplerian orbits. Kepler's laws have become mangled with the geodesic conditions being conveniently forgotten. The so-called "energy conservation condition" relates two-times, or two-frequences to the gravitational potential. This is due to the indefinite form of the metric, which applies to electromagnetic phenomena, and not to gravity. Gravity, on the other hand, serves as the inhomogeneous media through which electromagnetic waves pass in full agreement with Snell's law. The reddening of spectral lines is really the elongation of the wavelengths of the electromagnetic waves when they pass from a medium of a given index of refraction to one of higher index in the presence of a static gravitational field.