 ...An Exact Classical Mechanics leads toward Quantum Gravitation... Contents
2.6 Vibration of a hypothetical matter clock
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Fig. 2 Magnetic force on current elements |
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Fig. 3 Hypothetical matter clock |
For the revised Newtonian, it is necessary to define two kinds
of clock: "light clocks" and "matter clocks". Light clocks will depend on photons,
generally involving laser beams, whilst matter clocks will depend on mechanical
vibrations, such as those of quartz crystals. Vibration of crystals is made
complex by an induced motion, perpendicular to the direction of main excitation
caused by the so-called "Poisson's ratio". A hypothetical matter clock, which
is mathematically less complex, will therefore be considered in detail.
A special clock is to be imagined built from three identical spheres arranged in a straight line, as shown in Fig. 3. The outer spheres are fixed to a frame and the centre one is allowed one degree of freedom so that it can vibrate only along the line of centres. Its rest position is centrally located at distance A from each end sphere. All spheres carry identical electrostatic charge Q and, at absolute rest, a restoring force is produced proportional to the net electrostatic force. Additional magnetic forces, caused by the linear motion of all charges, will come into play at absolute speed v.
When the central sphere vibrates it will generate electromagnetic waves, so the resulting motion will be a damped oscillation. Forced at resonance, however, this energy loss can be offset so that the effect of wave generation can be completely ignored. The magnetic force for each of two short lengths of wire dl and dl1 carrying currents i and i1 is illustrated in Fig.2. Each reacts to the magnetic field produced by the other so that forces are perpendicular to motion. It is interesting to note that Newton's third law, action and reaction are equal and opposite, is apparently violated when the line joining these elements have angles θ and θ1 relative to directions of current flow. It is restored when reaction against the background medium is considered. For complete circuits there is not even apparent violation. The motion of the charges relative to the i-ther is equivalent to a current such that current i = vQ/δl. Furthermore since the magnetic μ0 and electrostatic ε0 constants are linked to the speed of light c such that:
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[22] |
It is convenient to convert entirely to electrostatic units so that the magnetic force Fm on the upper moving charge can be expressed as:
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[23] |
The arrangement of the clock is illustrated in Fig. 3 together with its absolute motion at speed v, at an angle θ with respect to the in-line direction and in which the centre sphere has displacement x. The net force F on this sphere, after binomial expansion with second and higher order terms ignored, becomes:
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[24] |
If the mass of the centre sphere is m then, together with equations [8] & [24] a simple harmonic motion is specified which yields the angular velocity ω:
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[25] |
By putting v = 0 the value ω0 for the vibration of the clock at rest is obtained and this can be divided into equation [25] to yield ratios for finding how frequency changes with linear speed v, i.e.:
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[26] |
This expression can be used to investigate the way the clock will change frequency as it is accelerated to a higher speed. It has been derived from a Newtonian basis but is equally applicable to special relativity since, in the latter case, the observer can always be considered at rest. It is convenient to define β = v2/c2 . The two cases are compared below for v<<c: Special relativity for θ = 90° then A/A0 = 1 and Q/Q0 = 1 has to be assumed. For θ = 0 the Lorentz contraction has to be applied so that A/A0 = 1 - 1/2β . The two results yield Δ ω /ω 0 of -3/4β and 1/2β respectively so that the arithmetic mean is -1/4β. Revised Newtonian. In both cases A/A0 = 1 = Q/Q0 is assumed giving: Δ ω/ω0 = -3/4β and Δ ω/ω0 = -1/4β respectively. In this case, for v<<c an integration yields an exact average for a smoothly rotating clock which becomes -1/2β. Only this value is consistent with experimental observation. For the case of special relativity an inconsistency is evident because, from a time dilation prediction, the result has to be Δ ω/ω0 = -1/2β. No practical clock could be expected to show the full anisotropy suggested by the revised Newtonian case. For caesium-beam clocks the atoms would be rotating and so only the average would appear. The nearest approximation to the hypothetical clock analysed would be a quartz crystal oscillator. It could not be expected to give the complete anisotropy, however, owing to Poisson's ratio. When materials are compressed they expand laterally by between 1/4 and 1/3 of the direct strain and this will tend to reduce the anisotropy. It will not be eliminated, however, and so a new experimental check has emerged! It could only be measured for an experiment in orbit because terrestrial speeds would not give adequate resolution. A pair of identical quartz crystals with axes arranged mutually perpendicular would be required with output signals added to produce a beat frequency. They could be mounted on a rotating table and then the beats should cycle when the clock is in Earth orbit. Calculations show that at an absolute speed of 7.79 km/s and using 460 MHz oscillators, a beat frequency of .155 Hz should be returned. It is known that such clocks can achieve stabilities better than 10-10 s/s when in temperature controlled environments but this is about equal to the signal. At least a tenfold improvement is required but this only applies to the difference signal and so should not pose an insoluble problem. Unfortunately a problem arises for both relativity and the Newtonian in that the value of A will vary in direct proportion to the size of atoms and this size cannot be assumed fixed simply by assuming electric charge invariant. Indeed the Bohr radius can be investigated. Fitzgerald made a similar study on atoms. He found that a lateral expansion Δ x/x equal to 1/2β would occur taking the magnetic force into account. He suggested this could be the reason for the null result given by the Michelson Morley experiment. Michelson and Morley had used an interferometer to measure the absolute speed of the Earth, expecting a value of at least 30 km/s to be shown. This depended on the speed of light appearing different for two mutually perpendicular directions and the theoretical difference would be -1/2β. Hence the Fitzgerald expansion would just cancel the effect. (Lorentz had suggested a contraction in the direction of motion to do the same and this idea, despite the lack of any theoretical basis, was adopted: presumably because it fitted in better with the predictions of Special Relativity) When this expansion is allowed Δ ω/ω0 becomes -3/2β for the transverse direction. If this were accompanied by a Lorentz contraction then the correct average value would arise but the anisotropy is still further increased. Such a contraction is not disallowed by the revised Newtonian but will apply only to objects built of atoms: it will not apply to empty space, in addition, as it does in relativity. To keep A constant Q has to increase as speed increases. If e is the electronic charge then Δ e/e0 = 3/4β. Some basic quantum analysis along the lines of the Schrödinger wave model seems to be required backed by experiment to discover exactly what really happens. No such modifications can be accommodated by the relativistic approach because this has to maintain A constant in the transverse direction and ignore the inconsistency with charge also remaining invariant. The Lorentz contraction must also be accepted in the direction of motion. This inflexibility is due to the absence of preferred frames of reference, with the consequent need for the observer to see the same mechanics whether travelling with a moving object or not. Fortunately for the revised Newtonian, not needing this constraint, some flexibility is allowable and any changes needed could be accommodated in this case. For adequate sensitivity only experiments in orbit could resolve the issue, using a practical version of the electromagnetic clock previously analysed. If transverse charge does in fact vary it could have serious implications for high energy physics and so its investigation would be worthwhile. To achieve a high enough clock vibrational frequency, however, the charged spheres (Fig.3) would need to be about the size of c60 buckyballs. The next problem to be resolved, however, is the manner in which rest-energy varies with potential. This will take us into a minefield of misconception. | 
