Epicycles in Einstein's General Theory

The calculation of the advance of the perihelion of Mercury stands as one of the fundamental supports of Einstein's general theory of relativity. It was obtained by him using a modified form of the equation of a trajectory of an elliptical orbit resulting from the Schwarzschild solution of his condition of "emptiness". The additional term, corresponding of an inverse fourth power of the radius in the force, implied that the advance of the perihelion was a relativistic effect since it contained the Schwarzschild radius. The shift in the spectral frequency was found to be 3 times the ratio of the Schwarzschild radius to the semi-latus rectum of the the unperturbed ellipse, and was believed to be the lowest power in a series expansion in terms of the the Schwarzschild radius.

Rather, we will show that the expression for the shift in the perihelion is exact, and is a consequence of the fact that central force problems have conformal duals. The path of a particle moving according to a central force, proportional to a power of the radial coordinate, is a sinus spiral. Recall that a central force is the force joining the object and its source along a straight line. It was conventional wisdom that pointed to Bertrand's theorem that there were only two types of central forces: the harmonic oscillator and Newton's inverse square law. The two were related by the fact that when the source is moved from the center of the ellipse to one of its foci, the harmonic oscillator force is transformed into Newton's inverse square law. These have commonly been referred to as "dual" laws because on squaring the complex expression for an ellipse, one, again, obtains an ellipse except that its centered has been displaced by a distance equal to the ellipse's eccentricity. However, the new expression of the ellipse is no that original ellipse whose center has been moved to one of the foci, but, rather, its square. Hence the nature of the dual laws is less than perfect since the new ellipse is not that of the original ellipse whose center has been displaced by a distance equal to that of the eccentricity.

Rather, there is another form of dualism in the inverses of sinus spirals, representing pairs of conjugate central forces. Examples are the straight line and circle, the parabola and the cardioid, and the orthogonal hyperbola and the lemniscate. All these have integer values of the power of the radius, n. But, that is no restriction since non-integer values of n exist.

The cardioid is an example of an epicycle, and its conjugate is the parabola. The corresponding central force law for the cardioid has an inverse power n=4, while that of the parabola is n=2; that is Newton's inverse square law. The inverse duals of sinus spirals, which is actually a misnomer since the curves are not spirals, is expressed as a Zhukowski transform, coming used to design aircraft wings. The corresponding expression for the force is exactly Einstein's expression for the advance of the perihelion of Mercury: the sum of a parabolic trajectory and an epicycle.

The characteristic length in Newton's law is the semi-latus rectum, or the square of the specific angular momentum and the gravitational parameter. This corresponds to the Bohr radius in quantum theory. When this is multiplied by the gravitational fine structure constant, one obtain the Compton wavelength, and multiplication again by the fine structure constant results in Schwarzschild's radius. This is the characteristic linear distance in the cardioid central force (actually, using the pedal equation to obtain the expression for the central force it is 3 times this quantity). The ratio of the two gives exactly Einstein's expression for the advance of the perihelion, divided by 2 pi. So one can concluded that the advance of the perihelion is due to a relativistic epicycle, which is not a relativistic modification of an elliptical orbit, but, rather, a true epicycle.

The predicted 43 arcseconds per century is the largest of all other dual pairs where n is an integer. Others central force pairs involve the harmonic oscillator and the Bernoulli lemniscate, which should describe binary stars, a constant force and the inverse fifth order force, which was known to Newton, and describes a circular trajectory through the origin. And that is not all. If we perturb the inverse seventh force of the Bernoulli lemniscate we obtain an inverse eleventh order force, which is analogous to the Lennard-Jones 6-12 potential. Recall that there is no theoretical or physical justification for the inverse twelfth order force; it should be replaced by an inverse eleventh order force which implicates a Cassinian oval.

In the hope of obtaining an observable shift, we must turn to non-integer values of n. The exponents -5/2 and -7/2 implicate a limacon and Tschirnhaus curve, in the form of one loop of a bow-tie. The limacon is a true epicycle with respect to a rotating circular orbit, just as Ptolemy envisioned, with a loop that looks like attraction has suddenly been converted into repulsion. The shift in relative frequency is the square root of Einstein's expression for the advance of the perihelion, and, therefore, should be observable.

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