General relativity begins with a generalization of the hyperbolic metric of Special relativity. This, already, introduces two times: Proper time, as measured by a clock on the body in motion, and coordinate time, as measured in the stationary laboratory. Are there other times, and are these times for real?
The gravitational shift of spectral lines begins with the expression:
where the time s measures the rate of a "standard" clock at rest in a gravitational field, and t measures the rate of a coordinate clock, displaced far from the field so that it is not influenced by it. Inverting times for frequencies, inverse of the time t becomes the proper time of light emitted by an atom. The coordinate clock with have a frequency which is affected by the gravitational field, and will be "red" shifted by an amount B^(1/2), where
B=1-2GM/r, in "natural" units where c=1.
In the non-stationary case of the Schwarzschild metric, we can write the square of the proper time as
in the equatorial plane. From the Schwarzschild solution of the exterior problem, it is known that the angular momentum is not conserved, but, rather, given by
r^2d phi/dt=B h,
where h is the angular momentum. Now, if we introduce the "fictitious" time of Levi-Civita, we do get a conservation of angular momentum,
where s and t are related by
ds= B dt.
This differs from our first expression by a square root of B!
Now, the radial equation of motion of the Schwarzschild field is
where the prime stands for the differentiation with respect to r: B'=2g, where g is Newtonian acceleration. The velocity v=dr/dt, all in coordinate time. Passing to inverse velocity w=1/v, we find after some algebraic steps:
which, except for a scale change, is just the form given by Rieman in his inaugural dissertation, for a radius of curvature, R=1/2^(1/2)B. Since B depends on r, the radius of curvature is not constant; it goes to infinity as r approaches the Schwarzschild radius, 2GM.
Introducing the conservation of angular momentum into the distance formula, and integrating it becomes
The argument of the logarithm has the familiar Doppler form so that it represents a distance in velocity space. The left-hand side is the angular average of the radius of curvature. Only if it is considered constant will it reduce to a difference in angles. And, if it is considered constant it obliterates the difference of the two times, s and t! There are no other times: the "local" time is a vistage of the Special theory which has no place in the General theory---and, consequently, neither does the line element.
Thus, the fictitious time of Levi-Civita was introduced in order to save the condition of the conservation of angular momentum. Therefore, the first equation we introduced is incorrect! The concept of a "local" or proper time has no meaning in GR. It is a left-over of the Lorentz factor, which is a second-order Doppler effect in Special Relativity. To obtain a correspondence we would necessary have to set v^2=2GM/r, which is related to the "escape" velocity instead of the velocity of a Kepler orbit.
Furthermore, the condition:
is the energy condition that one normally obtains from the variational expression when the lagrangian is taken to be (ds/dt)^2. If the above energy condition holds, it is apparent that B^(1/2)dt/ds cannot be a constant! Then, in order to obtain complete generality by including electromagnetic waves where ds=0, an affine parameter is introduced to measure the progression along a trajectory of the motion. The affine parameter is another "time", which muddies the water still further.
Since the increment in fictitious time, ds, cannot vanish along any trajectory, be it geodesic or not, there is a fundamental problem in dealing with the assumed generalized hyperbolic metric of Special relativity. We can even ask the more embarrassing question of the "local" time of what? The "external" Schwarzschild metric was solved under the condition that the universe be empty. It was already a leap of faith to introduce a mass, M, as a constant of integration which results when the eigenvalues of the Ricci tensor are set equal to zero and subsequently integrated.
Yet, to introduce yet another mass for which there is a "local" time is pushing things too far. This is usually circumvented by considering a "test" mass. But, where does this "test" mass come from? and how do we know that Newton's law of attraction is satisfied? We don't! It is well-known that for velocities greater than
v^2 > B^2/3-(2/3)v_s^2*B,
the radial acceleration becomes positive, implying gravitational repulsion! v_s is the Schwarzschild transverse velocity (r*d phi/dt). However, gravitational repulsion of what? We only have a central mass in an otherwise empty universe.
As Moller shows in his derivation of the advance of the perihelion of Mercury in his book, The Theory of Relativity, the correct equation (18) is replaced by the approximate one (21) in Chapter XII. This cannot be right because it is tantamount to considering dt=ds. Angular momentum is not conserved via an approximation but rather in time s. The only justification for this is the smallness of the Schwarzschild radius. But, by the same token, the cubic equation in his equation for the orbit should also be dropped, which, if done, obliterates the advance of the perihelion.