Updated: May 25, 2020
Almost since its inception General Relativity (GR) has been plagued by the arguable phenomenon of repulsive gravity. This was discovered by David Hilbert in 1917 by analyzing the Schwarzschild solution to the Einstein equation R=0, where R is the Ricci tensor. We will show that what should be a "fictitious" time indicating a slowing down of the trajectory at it nears the singularity of the origin is incorrectly interpreted as a "local" time. One may rightly as: A local time of what? That is, the external Schwarzschild solution follows from the Einstein condition of emptiness so it would only appear local that there are no masses. But, masses there are: A central mass M and , because of Newton's second law, another mass whose mass cancels out of the equation. This is commonly referred to as a "test" mass. But neither central nor test masses should exist.
The "external" Schwarzschild metric is
where dS^2 is an element of a 2-sphere, and B=1-2M/r, in "natural" units where G=c=1. Henceforth, we will consider only radial motion. Supposedly, there are two times: Coordinate time, t, and "local" time s. Dividing through by dt^2, we can write the lagrangian as
since the lagrangian L is constant. The conservation of energy can, supposedly, be written as
and introducing it into the lagrangian gives
This was first derived by Droste, and independently by Hilbert, with confirmation given by Treder and Fritze. In those expression, E^2, was set equal to A.
Differentiating with respect to time yields
where g=M/r^2, Newtonian acceleration. The constant A has been eliminated using the above conservation of energy. It is quite clear that for
what was "attractive" gravitation becomes "repulsive" gravitation since d^2r/dt^2 has become positive.
Returning to the conservation of energy, which is usually written as
(dr/dt)^2=B^2+AB^3. and letting r->infinity, reduces it to
We should expect that asymptotically far from the central mass M, v should vanish since it refers to "free fall" at infinity. This is possible if A=-1. But, A was set equal to E^2, and this means an imaginary conservation of energy! It is clear that we are playing with symbols that have no physical meaning.
If we prefer to divide the metric through by ds^2, an entirely different story emerges. We now have
1 =-B(dt/ds)^2+(dr/ds)^2/B. (1)
Now differentiating with respect to local time,
and cancelling the common factor dr/ds. we fin that gravity is always attractive! Moreover, the Schwarzschild barrier at r=2M has also miraculously disappeared!
The conservation of energy,
where V=dr/ds, can give elliptic, E^2<1, as well as hyperbolic, E^2>1, orbits.
Equation (1) was first derived by Droste, and later confirmed by Hilberty.
Taking this at face value, it is argued that the global time, or kosmische zeit,is the important one. Loinger argues that the rates of kosmische zeit "is physically more significant that the rate of proper time of a test-particle, which suffers the influence of motion." But how can there be motion in a static universe where nothing moves? This is trying to justify the absurd, and with it gravitational "repulsion".
What is significant is Newton's second law,
It is well-known that Kepler's problem becomes more translucent in velocity space. In order to achieve this we must express the right hand side in terms of V. This is possible through the conservation of energy
But we have an inverse square instead of an inverse radial distance. If we introduce Levi-Civita's fictitious time dT=ds(2M/r), we get
Squaring gives the Riemann metric
It is well-known that stereographic projection is related to inversion in a circle. We therefore define the inverse velocity wV=1, and write the metric in the more familiar form
where C is an arbitrary scale factor and K=-2E is Gaussian, constant curvature. For E>0, the metric is hyperbolic and corresponds to the Poincare' disc model. For E<0, the geodesics are associated with Kepler orbits.
But, there is a problem here. The Poincare disc metric has been derived on numerous occasion, most notably by Milnor in his article "On the Geometry of the Kepler Problem." However, it requires vanishing angular momentum since dV/dt=d^r/dt^2-h^2/r^3 so that the introduction of a fictitious time will not give |dV/ds|=V^2/2-E because there is a term h^2/r^2.
So the derivation of the Poincare disc requires h=0 , open orbits.
The crucial point is the introduction of fictitious time, dT/ds=1/r so that as dV/ds->infinity, dT/ds->infinity such that dV/dT=r*dV/ds should remain finite. This is if we want to describe Kepler ellipses in velocity space. What was previously interpreted as a conservation of energy ds/dt=B/E, does not fulfill this condition, since ds/dt->0 as r->2M, and not to infinity as it should.
We are therefore stuck with two times, since the "local" time s, can be eliminated to give
having set the arbitrary constant, E, equal to one. It has no significance of an energy at all. If one wants to give physical credence to ds/dt=B, one must do so as the introduction of another "fictitious" time. However, it does not behave as one.
The importance of a velocity space interpretation lies in the stereograph projection of geodesics on a 3-sphere onto a plane involving Kepler circles. Since the Schwarzschild metric must make contact with such a description, the coefficient B must be modified in the metric, and the interpretation of s is not local time, but Levi-Civita's fictitious time.
The upshot is that a conformal transformation can eliminate all vestiges of negative, or repulsive gravity. It is well-known, but hardly understood, that a time change involving Kepler's II law, equal areas in equal times, reduces the inverse square law to linear Hooke law. But, we must remember that it requires the squaring of the original variable which has the effect of shifting the origin of the ellipse from the center to one of the foci. In fact, squaring has the effect of giving the arithmetic of the Kepler equations, one for each foci. The square reducing Kepler's second to Levi-Civita' linear relation between the two times, one real and the other fictitious.
It is quite remarkable that all this was known back in 1909 through the work of Edward Kasner. He correctly understood the relation of dual laws to conformal transformations of the metric. So even if a "local" time does not exist in general relativity because that would require a moving mass with respect to another frame, a fictitious time can be introduced to remove any and all vestiges of repulsive gravity. Kasner deserves much more credit than he has been given.
For the Kepler problem, the general transformation is
where A is a constant. For m=1, we have the eccentric anomaly, while for m=2, there results the true anomaly. In the Schwarzschild problem, the transformation is
which if we take our cue from Kepler can be generalized to any power of r. In comparison with Kepler, Schwarzschild appears as the complementary transform. The introduction of fictitious time is used to smooth out collision orbits by removing the singularity at r=0. Likewise, the Schwarzschild transformation is used to eliminate repulsive gravity, and, remarkably enough, the Schwarzschild radius, as can be clearly seen from the equation of motion which has a sole singularity at the origin.
As a final remark, we observe that we are not limited to radial motion. The full Schwarzschild metric is
where where \phi is the angle in the plane. Rewriting this as
and differentiating with respect to s, give
where g=GM/r^2, and v_s=r^2(d\phi/ds)^2 is the Schwarzschild transverse velocity in the orbital plane. There is absolutely no repulsion, as we would naturally expect!
It's all a matter of conformal equivalence. If the phenomenon does not exist in one metric, it will not exist in its conformally equivalent metric.