Updated: Feb 13, 2020
Instead of dealing with an ellipse in configuration space, Hamilton found it simpler to deal with a circle in velocity space, which is true for any object moving in an inverse square law.
According to Hamilton, Newton's law could be characterized as being the law of the circular hodograph. A hodograph depicts how the velocity vectors all tied to a common point change with time. Since the speed of the planet varies as it travels along its orbit, the common origin of the velocity vectors is not at the center of the circle, but displaced by an amount that depends on its eccentricity. Before Kepler, the ancients believed all heavenly bodies traveled in circles. It is rather ironic that the circles that were banished by Kepler in 1609 made their comeback 235 years later in Hamilton's hodographic interpretation of elliptic orbits.
It is likewise surprising that no one ever thought that the exterior Schwarzschild metric could be simplified to a metric of (negative) constant curvature in velocity space.
Let us being with the classical Hamilton-Jacobi (HJ) equation for an inverse potential
where H is the total energy per unit mass. The equation of motion is given by Hamilton's equation
The Hamiltonian possess a collision at r=0. To "slow" down the motion, we introduce Levi-Civita's fictitious time, s, according to
into the equation of motion to obtain
Inversion in a circle can be considered as the two dimensional analogue of three-dimensional stereographic projection. Introducing the inverted velocity, w=c^2/v, the time increment becomes
where the curvature, K=2H/c^4, can be either positive, negative, or zero for zero energy. For H>0, we get the Lobachevsky "straight" line
while for H<0,
and finally for H=0, the metric becomes flat, and geodesics get ironed out becoming straight lines in Euclidean space. The unconstrained hyperbolic velocity for H>0, becomes a constrained angular variable for H<0 to angles less than 90 degrees.
In the relativistic limit, Landau and Lifshitz, in their Theory of Classical Fields, give the HJ radial equation as:
where B=1-R_s/r, and R_s=2GM/c^2 is the Schwarzschild radius. They have been criticized for this since it does not rely on the indefinite metric, and gives results in contradiction to General Relativity (GR). In particular, the Hamilton equation of motion is
In the limit of small velocities |H|->c^2 so that the equation reduces to Newton's law of gravitation.
In comparison, GR predicts a radial acceleration of:
This was first derived by Droste in his doctoral thesis and later by Hilbert (1917,1924).
From this, it is concluded that dv/dt>0 for v^2>B^2/3, resulting in "repulsive" gravitation.
No such thing can happen in the "classical" HJ formulation.
Parenthetically, we recall that the fictitious time
eliminates the repulsive gravitational contribution and restores Newtonian gravitational attraction. Since the two expressions differ by a "mere" conformal transformation no physical explanation can be given to repulsive gravity. It appears as a mere "fluke" of a contorted formulation. For if we begin with the indefinite metric,
for radial motion, and divide through by ds^2, we get, using the above fictitious time
Taking the derivative of both sides with respect to s, gives Newton's law of gravitational attraction unconditionally!
Rather, if we begin with the Droste-Hilbert equation of motion above and divide through by B,
where V=v/B. Now, introducing the fictitious time, ds/dt=g, where s has the nature of a velocity, we obtain
Inverting in a circle of radius c, namely V*W=c^2, gives the line element
There is not an inkling of gravitational repulsion, or the Schwarzschild radius. This is none other than the Riemann metric for a hyperbolic system with a radius of curvature, c! The geodesics are segments of arcs that intersect the limit circle of radius c orthogonally or straight lines through the center.
Although this is a bona fide Riemann metric, it must be considered as a hybrid between configuration and velocity space since it depends on the ratio v/B. We will now introduce a velocity space metric via the HJ equation.
We now introduce a fictitious time given by
where g is Newtonian gravitational acceleration, into the equation of motion to obtain:
Eliminating B with the help of the conservation of energy, i.e.,
the equation of motion becomes:
where the curvature, K=(2H/c^2)^2, is always positive. But, the denominator is the square root of the denominator in the Kepler case so that changes things.
Integrating the last equation and inverting give:
v = c^2 K^(-1/2) sinh(K^(1/2)s),
which is the semiperimeter of a geodesic circle of radius s, in the hyperbolic velocity space of constant curvature. This expression was first derived by Gauss, and later used extensively by Beltrami in his groundbreaking paper on hyperbolic geometry of constant curvature. According to Beltrami ["Essay on the Interpretation of Noneuclidean Geometry", Giornale di Matematica 6 (1868) 284-312], the geodesics on the surface are presented in their totality by chords of the limit circle of radius, (Mc)^2c/2|H|, "while their prolongation past the circle are devoid of a real interpretation...To avoid circumlocution, we call the surface of constant negative curvature pseudospherical, and we retain the term radius for the constant, K^(-1/2), on which the curvature depends."
The line element is
where the velocity can run through all values from 0 to infinity. The ratio of the circumference of the velocity circle to its radius is greater than 2 pi, telling us that the non-Euclidean geometry is hyperbolic. Had the denominator in the above equation been squared, we would have come to the opposite conclusion that the geometry would be elliptic. It would then coincide with the case H<0 in the Kepler problem for the inverse velocity above.
In summary, conformal transformations have to handled with care. The additional parameter is not a local time, for such a parameter does not exist. Rather, it is a fictitious time which slows down the collision orbits that was first introduced by Levi-Civita in 1920. The above shows that Hamilton (1846) was correct to introduce hodograph transformations since the picture in velocity space becomes more translucent than the distortions in configuration space, e.g., ellipses in configuration space transform into circles in velocity space.