The metric of genera relativity determines the gravitational field. As it is a direct outgrowth of the Minkowski metric for electromagnetic phenomena that propagate at the speed of light, it was naturally assumed (first by Poincare' and later by Einstein) that gravitational perturbations propagate at the speed of light. Thus, it was concluded that gravitational and electrodynamic phenomena have similar propagation properties. The question is when and where does the transformation between electrodynamic and gravitational field propagation occur? Why isn't it as if electromagnetic perturbations propagate through a medium of a differing index of refraction, with respect to the vacuum, when a static gravitational field appears?
As a preliminary, consider the projection of curves on a hyperboloid onto a plane. The hamilonian is
H= (1/2)(v^2-u^2)+1/2
to which we have added a constant term. If u=q^{1/2} and we define a new time ds/dt=1/q, which plays the role of a potential, then, apart from immaterial constants
v=du/ds=q^{-1/2}*dq/dt*dt/ds=q^{1/2}p,
where p=dq/dt. Introducing this into the hamiltonian results in
H=(1/2)q(p^2-1)+1/2.
We now define a new hamiltonian, which is related conformally to the old one:
H'=(ds/dt)^2*H=(1/2)(p^2-1+1/q).
The characteristic surfaces for the propagation of perturbations are determined by the vanishing of the hamiltonians. Setting h=0, we get
q=1/(1-p^2)
and requiring H'=0, gives an identity. The expression for q corresponds to a point on the hyperbola v^2-u^2=-1.
If we had considered a sphere instead of a hyperbola, in three dimensions, we would have obtained stereographic projection onto the lower half of the sphere, where q=1/(1+p^2). This would have united the harmonic oscillator hamiltonian, (1/2)(v^2+u^2), with the Keplerian hamiltonian, (1/2)p^2-1/q. The time s is referred to as "fictitious" time, which measures time on the arc lengths of curves on the sphere, just as time t, measures arc lengths of on the plane of the ellipse. This is referred to as "regularization" between the dual laws of Newton and Hooke, where the singularity of the former (collision orbits) are replaced by non-singular orbits of the latter.
It is important to bear in mind that fictitious time has absolutely nothing whatsoever to do with proper time. Its relation to laboratory time t, reduces the metric to a lagrangian which we will see is related to one of the known non-Euclidean metrics when the mass in the Schwarzschild coefficient is replaced by the mass density. This conforms to Schwarzschild's inner solution.
A similar phenomenon of conformality occurs in the static Schwarzschild solution. There the conformal factor is B=1-2m/r. The radial hamiltonian is
H=(1/2)[(dr/dt)^2-(1-B)B^2].
Its vanishing determines the bicharacteristics along which electromagnetic (gravitational?) perturbations propagate. Setting H=0, and differentiation with respect to time t, gives the hamiltonian equation
-d^2r/dt^2=dH/dr=g(B-3(dr/dt)^2/B),
where g is the gravitational acceleration.
This equation shows "gravitational repulsion" according to Hilbert, Droste, Tredder, etc. The remarkable thing, first noticed by Drumaux, is that by introducing a fictitious time s, according to ds/dt, "gravitational repulsion" disappears completely. That is, defining a new velocity
r'=dr/ds=dr/dt*dt/ds=dr/dt/B,
the hamiltonian becomes
H=(1/2)[r'{2}-(1-B)].
Its vanishing, r'^2=2m/r, determines the null geodesics, and differentiating with respect to time s, gives simply Newton's law:
-d^2r/ds^2=dH/dr=m/r^2.
All vestiges of gravitational repulsion have disappeared, which occurred in the quadratic term in the original equation of motion. This term, involving the Christoffel symbols, is commonly believed to be the result of the gravitational field (see Landau & Lifshitz, Classical Theory of Fields). Yet, nothing could be more removed from the truth. The last equation, obtained by a conformal transformation, describes the gravitational field very well according to Newton. It does not need the nonlinear term in the geodesic equation. And to boot, there is no long the Schwarzschild radius that would signal the presence of a black hole! The gravitational field is completely static, so the propagation of perturbations along the geodesics, determined by the vanishing of the hamiltonian, must necessarily be electromagnetic in origin.
As a final point, the lagrangians are
L=(1/2)[(dr/dt)^2+(1-B)B^2]
and
L'=(1/2)[(dr/ds)^2-(1-B)].
They are related conformally by
L'=L/B^2.
If we had taken into consideration motion on a two sphere, L' would be the Beltrami metric! For greater detail on the Beltrami metric's relation to uniform acceleration of a rotating disc see A New Perspective on Relativity: An Odyssey in Non-Euclidean Geometries.
According to Loinger & Marsico (arXiv:0904.1578), the fictitious time s is the proper time. After a discussion of Hilbert's use of laboratory time t, they say "One could object that if we employ the proper time s for the geodesics of particles, things stand otherwise." That is with respect to gravitational repulsion. Although they find Newton's law, and the disappearance of the "barrier" at r=2m, they conclude that the spatially synchronized clocks are more relevant than the rate of proper time of a "test" particle, "which suffers the influence of the motion."
Yet, they criticize (and correctly) the Kruskal-Szekeres form of ds^2 which "gives a time dependent solution to a static problem." Then what is the proper time of an imaginary "test" particle doing in such a static solution?
We can write the above lagrangian as
2L*dt^2-dr^2=(1-B)ds^2,
which is not at all the metric relation between proper and laboratory times. Yet, the lagrangian equation,
(d/ds)(dL/dr')-dL/dr=0,
does give the correct equation of motion, and coincides with hamilton's equation of motion derived from a hamiltonian that vanishes. It would be interesting to show that the two times are related by geodesics paths on surfaces related by stereographic projection, just like the dual laws are.
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