Holon Physics Seminar

 

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It is commonly believed that a static gravitational potential causes a shift in frequency that is derived from the temporal part of the indefinite space-time metric of general relativity, relating local to coordinate times. Rather, it will be shown that the shift in wavelength is analogous to the behavior of light passing from one medium to another with a different index of refraction.

 

Time, as well as frequency, is not involved; rather, since the local velocity of light varies, the wavelength changes at constant frequency. In contrast, if time were involved, the frequency would change with the local speed of propagation of light leaving the wavelength invariant. In 1911 Einstein showed that the deflection of light by a massive body is the result of a Doppler shift in which the uniform velocity was replaced by the product of the gravitational acceleration and time. Then expressing time as the ratio of the distance to the speed of light, Einstein associated a static gravitational potential with the red-shift.

 

As Louis Essen has emphasized, there was an illogical train of logic leading from a uniform velocity to uniform acceleration, and finally to a completely static gravitational potential. All these different physical conditions cannot hold simultaneously. Slightly later, Einstein attempted to use a varying velocity of light as a potential in the construction of a theory of gravity.

 

This occurs when light transverses media of different indices of refraction, and like Eddington was later to observe, all perturbative effects described by general relativity can be accounted for by an optical analogy. Shortly after Einstein’s formulation of his equations, Schwarzschild derived a metric from the condition that the individual components of the Ricci tensor vanish. This was Einstein’s condition of "emptiness”. Yet, an arbitrary mass entered as an arbitrary constant of integration that wasn’t there previously.

 

From the Schwarzschild metric, the correct values for the deflection of light by a massive body (the famous doubling of Einstein’s earlier result), and the advance of the perihelion follow. A short time later, Hilbert noticed that the Schwarzschild (gravitational) acceleration could become repulsive at sufficiently high velocities. It was concluded that Newton’s theory was valid for only small particle velocities, where a "test” particle has been introduced into an otherwise "empty” universe.

 

Ohanian has argued that the gravitational force contains the variable particle mass; however, this mass does not enter into Newton’s law of gravitational attraction. Newton’s law for radial motion is upheld by the Schwarzschild metric when the acceleration is expressed in terms of "local” time, and not "coordinate” time, as it should be. On the other hand, Kepler’s third law is found to be valid when the angular velocity is expressed in "coordinate” time, and not "local” time.

 

It is hard to believe that different laws of physics require different times! And when the angular velocity is expressed in local time, and not coordinate time, the corrective term to the angular velocity leads to the deflection of light and the perihelion shift. In fact, the prediction of repulsive gravity by the Schwarzschild metric is the Achilles’ heel of general relativity. Theories based on general principles must show monotonic behavior. For example, consider thermodynamics where a single experiment suffices to show whether the entropy will always increase or decrease Entropy cannot both increase or decrease for a critical value of some parameter without losing its claim to universal validity.

 

Likewise, gravity must be always attractive (or repulsive), and cannot change at a critical value of a particle’s velocity. This is all the more surprising insofar as the perturbations to the conic sections are obtained that give the correct predictions to the angle of deflection by a massive body, and the advance of the perihelion. This indicates that there is something basically wrong with the indefinite space-time metric.

 

The vanishing of the increment in local time gives the local speed of light. General relativity claims this is also the speed at which gravity propagates. Laplace long ago argued that such a low speed would cause havoc with the stability of planetary orbits, and optical effects like aberration would be observed. In particular, the delay in propagation would create a couple leading to observable effects on planetary motion.

 

The reduced metric for the local speed of light is precisely the Beltrami metric of hyperbolic geometry, and the different speeds in the radial and transverse motions are what are responsible for the deflection of light by a massive body and the advance of the perihelion. Following Ives, we derive the Beltrami metric by considering an interferometer placed in a non-uniform gravitational field. Ives did not appreciate the optical analogy with that of a birefringent material; the ordinary and extraordinary rays with respect to the optical axis are analogous to the velocities of light in the radial and transverse directions. When an index of refraction is appended on to the Beltrami metric that describes the path of a light ray as a particle in a gravitational field, the same equation of motion is derived as in general relativity, showing that gravity is always attractive.

 

This is even more surprising in that the derivation of the equation of motion follows from the Schwarzschild metric. However, certain simplifying assumptions, such as the conservation of angular momentum, render gravity always attractive. Moreover, the components of the Ricci tensor do not vanish attesting to the fact that the space is not "empty” because gravity need mass just like electromagnetism needs charge. In conclusion a few remarks are made regarding Mach’s principle.