According to Mach's principle, inertial systems have coordinate axes that are, on the average, at rest relative to the fixed stars in the heavens, or, have uniform motion with respect to it.
Classical mechanics cannot satisfy it because, whereas the potential energy depends on on the relative coordinates of the particles, the kinetic energy is absolute. It was Schrodinger's idea, in his paper on the "Fullment of Relativity in Classical Mechanics", that the kinetic energy, too, depends on the relative positions of the particles, but, in addition, upon their relative motion. But Einstein's calculation of the advance of the perihelion of Mercury didn't depend upon the motion at all, since it was contained in the modification of the potential energy of interaction. In any event it was not clear how General Relativity could satisfy Mach's principle since the boundary conditions required the vanishing of the potentials at infinity. So Schrodinger's idea was to treat the kinetic energy as an interaction between two masses so that it would become relative, and not absolute.
If he had only looked at Weber's rebuttal to Helmholtz when the latter claimed that Weber's formula violated the conservation of energy for, even though two electrified particles started out with finite velocities, they could upon the completion of a circuit acquire infinite kinetic energy and perform an infinite amount of work. This would be the result of introducing negative terms into the potential energy that dependent on the square of the velocity
V= m/r{ 1-(1/2c^2)(dr/dt)^2},
where m=GM, the gravitational parameter, and c is the velocity of light. Helmholtz pointed to the fact that when dr/dt >c, the potential becomes negative, which we now know is no problem at all.
So Schrodinger could have saved himself a publication by referring to Weber's rebuttal. What he did do beyond Weber was to introduced an unknown parameter in the second term which he set equal to 3 to make it correspond to the known result. He could have also found all this in the discussion in Sec 854 of the second volume of Maxwell's A Treatise on Electricity and Magnetism.
The idea that the precession of Mercury is the result of a motional force, that is, one that depends on the velocity, goes back to Gerber who showed that it resulted from the expansion of a retarded Lienard-Wiechert potential of the form
V= m/r (1-dr/dt/c)^2.
The novelty, or should I say, the sticking point, is the square in the denominator. For it is the square that gives correct factor of 6 in the inertial term which translates into the Binet equation
(1+6m/c^2 u)u" + u = m/h^2 - 3(mc^2)u'^2 (1)
where u=1/r and the prime denotes differentiation with respect to the angle variable and h is the conserved angular momentum per unit mass. This equation also happens to be the Weber force [cf. The Physics of Gravitation, p. 117]
When the last term is neglected on account of its smallness with respect to the other terms, the coefficient of the inertial term is equivalent to multiplying the angle variable, phi, by
phi -> (1-6 alpha^2) phi
where alpha= m/h c, the gravitational "fine structure" constant, and u has been set equal to m/h^2, the inverse of the gravitational "Bohr radius". The factor produces a new orbit with a different angular speed while maintaining the same radial motion. The new orbit will precess since alpha is not the ratio of two integers. Newton developed this theorem of precessing orbits to explain the apsidal precession of the moon. We can think of two particles one with angular momentum h, and the other with alpha h. Since alpha<<1, the second order should have a much smaller radius. But, to keep the radii the same, the orbit will fail to close, which will be counterclockwise in this case since alpha is less than unity.
Why Einstein was so very lucky to derive the anomalous precession of Mercury was due to an expansion of the pedal equation, p, in odd powers of u, i.e.,
1/p^2 = u'^2 +u^2 =2{ u/r_B + r_C u^3 + (r_C r_B)^(3/2) u^5+....} (2)
where r_B=h^2/m, the Bohr radius, and r_C= m/c^2, the "classical" of the electron, only now m replaces e, the charge. The two lengths are related by the square of the fine structure constant, i.e. r_C= alpha^2 r_B. In between the two is the Compton wavelength which is independent of the gravitation parameter.
Actually, Einstein did not use the pedal equation at all, but Schwarzschild's solution to his vacuum equations. Upon expansion in powers of r_C/r, there appears the coupling term
r_C/r h^2/r^3. Neglecting the last term in the pedal equation, this term appears when we differentiate the pedal equation with respect to phi, thereby obtaining
u" + u = m/h^2 +3 r_C u^2.
Linearizing the last term and identifying u_0 as the inverse of the Bohr radius, the Binet equation becomes
u" + (1-6 alpha^2)u = m/h^2. (3)
Dividing through by the coefficient of u in (3), this becomes equation (1) when 1+6 alpha^2 is replaced by 1/1-6 alpha^2, and the last term is neglected---except for the fact that Einstein's equation predicts a diminution of the Bohr radius by the factor
(1-6 alpha^2) r_B
as we have pointed out in another occasion. The static, or Einstein explanation, requires a diminution in the Bohr radius, while the inertial one does not.
Moreover, the expansion of the pedal equation in even powers of u leads to spiral trajectories. In order to get rotating orbits, it would be necessary to mix even and odd powers, e.g.,
1/p^2 = 2 u/r_B + alpha^2 u^2,
and this would give us a Binet equation as the same form as (3) without linearizing, but with a coefficient that would be off by a factor of 6. Again, the effect is entirely static, and it would also predict a decrease in the Bohr radius by an amount (1- alpha^2).
The inertial interpretation requires no modification in the Bohr orbit, but it does lead to the introduction of velocity dependent terms in the gravitational potential, like in the Weber potential. That being the case, it would imply that the speed of light is the upper limit to the propagation of gravity, in contraction with the Poincare' conjecture that it propagates at the speed of light. Such considerations were undertaken by Tisserand and others in the 1870's modelling gravitation after the Weber electromagnetic force. Without such terms, gravity does not propagate; it remains completely static as Newton imagined.
To say that it propagates at the speed of light would necessitate the introduction of a second field--the magnetic field--just as in Maxwell's equations. But then we would get just another theory of light, and it behooves all those that make such a claim as the existence of a gravito-magnetic field to show that gravity, like light, manifests the properties of diffraction. And whereas solid bodies diffract light, gravity just passes right through them, i.e., gravity knows no shielding.
Consequently, the Achilles' heel of general relativity is to show that the amazing results can be obtained by other means, and that such means lead to a contradiction in phenomena like the propagation of gravity at the speed of light.
Comments