Internal Momentum Changes Manifesting as Clock Rate Changes in GPS Clocks
| Scientific Paper | |
|---|---|
| Title | Internal Momentum Changes Manifesting as Clock Rate Changes in GPS Clocks |
| Read in full | Link to paper |
| Author(s) | Viraj Fernando |
| Keywords | GPS, clocks |
| Published | 2009 |
| No. of pages | 13 |
Read the full paper here
Abstract
Hitherto all physical theories have considered bodies as mass-points without internal structure. These theories therefore have been impotent and incapable of considering how internal energy (mc2) / internal momentum (mc) of a body interacts with applied momentum. Or how the internal momentum functions in the gravitational process in relation to other bodies. Since these theories have not been able to capture and visualise the changes of internal momentum and the consequent phenomena that arise in these interactions, these have appeared bizarre to them and they have therefore been labeled as "relativistic phenomena". The present paper has discerned for the first time, how the internal momentum of a body interacts with external constraints. This enables the explanation of all "relativistic phenomena" as arising due to changes of internal momentum of a body (or a particle). As a sampling of how the theory works, the author has thought it opportune to demonstrate how it can be applied to a problem that cuts across both gravitation and inertial motion (invading the exclusive territories of GRT and SRT) at one and the same time. This is done by showing how the changes of internal momentum, cause clock rates changes in GPS clocks and obtaining results of extreme accuracy. Along with it, the delay in the decay of a fast moving muon is also explained by the same principle that the so-called "relativistic phenomena" occur as a result of changes of internal momentum.
Overview
Viraj Fernando's paper argues that the whole family of effects physics calls "relativistic" are in fact bookkeeping consequences of something classical mechanics never worked out: how a body's internal momentum interacts with momentum applied from outside. His starting complaint is textual and goes back to the Principia. Newton's Definition III recognises vis insita, the innate force of matter, which Fernando reads as internal momentum and therefore as vector-like; Definition II makes momentum mass and velocity conjointly, as two aspects of a single entity. Yet when physics comes to treat a body's interaction with an applied force, "the reference to the velocity aspect of this conjoint entity has been ignored as if it did not exist, and the mass aspect has been considered as if it were the only aspect that matters."
Fernando's remedy is to treat a body not as a point mass m but as a quantity of internal momentum mc, where c is the root-mean-square speed of its atomic vibrations. He then constructs two geometric constructions — an "algorithm of gravitation" and an "algorithm of motion" — which partition that total internal momentum into sectors as the body's position in a gravitational field and its state of motion change. The fraction remaining available to drive atomic vibrations is the "clock momentum", and it fixes the atomic frequency and hence the clock rate. As a demonstration, he computes both the altitude gain and the orbital loss of a GPS satellite clock, and the decay delay of a fast muon. Where general and special relativity are two separate theories applied to the same satellite, Fernando's claim is that one construction handles both at once.
The argument
Omni-directional vectors
The obstacle Fernando identifies is that internal momentum is the sum of the momenta of randomly vibrating atoms, and so has no ascertainable resultant magnitude and direction. Physics, he says, dealt with this by sweeping the velocity aspect under the carpet and treating a body as a point mass. His alternative is to change the classification. He draws the analogy with the randomly moving molecules of a gas, whose resultant momentum is likewise unascertainable but which is handled statistically by the root-mean-square speed. Speed is a scalar; root-mean-square speed, he argues, is an omni-directional vector — and can even acquire the status of a universal constant.
He quotes Maxwell twice in support of the method: that conservation of energy remains "a guide to our researches" even where phenomena have not been explained dynamically, and that molecular science must proceed "by the method of hypothesis, and comparison of the results of the hypothesis with the observed facts."
The hypothesis itself has two parts: (a) the root-mean-square velocity of a body's atomic vibrations stays at the constant magnitude c whatever the body's state of motion or position, and (b) that velocity is potentially omni-directional and "responds to external constraints as if it had acquired the direction perpendicular to the direction of the external constraint acting on the body."
Fernando argues the hypothesis is already confirmed by an accepted relation. Writing the energy-momentum relation as E2/c2 = E02/c2 + p2, he reads it as a right triangle with AC = E/c, AD = E0/c and DC = p. The equation is tenable, he says, only because E0/c has the same dimensional quality as p — and its structure shows that E0/c "responds to [p] by acquiring a direction perpendicular to that of p", whatever p's empirical direction. Dividing the relation by c2 shows energy/c has the dimensions of momentum; by c4, that momentum/c has the dimensions of mass. So the same construction may be read in energy, momentum or mass units interchangeably, and Fernando uses whichever suits the context.
The algorithm of gravitation
The total internal momentum of a body is the segment NA, which Fernando calls the "nascent momentum" Mnc — what the body would have with no gravitation and no motion, all of it devoted to atomic vibration. In a gravitational field it splits into two main sectors: ND, the gravitational momentum, and AD, what remains. When the body also moves, AD subdivides further into AG (the clock momentum, driving the atomic vibrations that set the frequency) and GD (latent momentum). Crucially, NA is conserved throughout: "fractions of that momentum pass from one part to the other within the system as enforced by the law of proportions."
The partition is geometric, not arithmetic, and Fernando makes a point of this: were the proportions determined arithmetically the internal relationships would be linear, and it is precisely the trigonometric determination that produces the observed non-linearity.
For a body at rest at distance R from the centre of a gravitational field, the gravitational constraint OD is set perpendicular to NA, with AD = NA cos2α, ND = NA sin2α and OD = NA sinα cosα; O always lies on the semicircle ALN. Since the velocity component of AD still has the value c, AD = (Mncos2α)c, and
- tan α = (GM/Rc2)1/2
so that AD = MRc with MR = Mncos2α. Fernando's interpretive move is that MR is only the apparent mass at position R — and that "the rest mass of a body is its apparent mass when at a given position R in a gravitational field." Rest mass, on this view, is not intrinsic but positional.
The GPS altitude term
Applying this, the clock momentum at radius R is Mnc/(1 + GM/Rc2). The gain in clock momentum on moving a clock from the equator (Re) to the orbital radius (Ro) is the difference of two such terms — an expression Fernando flags as "the accurate formula" — which, because RoRec2/GM greatly exceeds the remaining terms, simplifies to
- Δp ≈ Mnc GM(Ro − Re)/(RoRec2)
He is careful to note this second form is approximate and "there will bound to be long term corrections to be made" when it is used. Because time is counted in cycles of atomic vibration, the fractional gain in time equals the fractional gain in clock momentum, giving a gain per day of 86400 × GM(Ro − Re)/(RoRec2). With Ro = 26,600 km, Re = 6378 km, GM = 3.986×105 km3/s2 and c = 2.99792×105 km/s, he obtains 45,674 ns per day, which he compares with the 45,900 ns/day recorded in Van Flandern's paper.
The algorithm of motion
For the motion term Fernando invokes Eötvös: since gravitational and inertial mass are equal, and the sector AD represents the apparent mass MR at position R, the momentum needed to set a body in motion must be proportional to MR — that is, to the clock momentum divided by c. Every body is always in some gravitational field, so the gravitational algorithm is the necessary starting point and the applied momentum p acts on AD, not on NA.
Because AD is omni-directional, it aligns perpendicular to whatever direction p is applied in; the pair then rotate about D through an angle φ = tan-1(DC′/A′D). Fernando remarks pointedly that "Minkowski and Einstein both have referred to the 'rotation of axes', without explaining how it happens or for what reason this happens." In his account the rotation is a real internal redistribution: it renders part of the applied momentum latent, leaves an effective momentum of motion, and induces a centrifugal component. On the internal side there is a "double rotation back to back", which renders part of the internal momentum latent and induces a component equal and opposite to the effective motion — a dynamic equilibrium which Fernando identifies with Newton's third law, noting that although the third law is usually seen as applying to statics, "how this law operates in the motion of bodies has hitherto not been demonstrated."
The upshot is that the clock momentum available for atomic vibration falls from AD to AG = AD cosφ, so the atomic frequency slows by the fraction 1 − cosφ. From the construction, sinφ = orbital velocity / velocity of atomic vibrations = (GM/Ro)1/2/c. Hence the fractional slowing is 1 − (1 − GM/Roc2)1/2 per second, or 7.202766 × 10-6 s per day.
The combined result and the muon
Subtracting, the net gain for a GPS clock in orbit is 45.67398579 × 10-6 minus 7.202766 × 10-6, giving 38.471219 × 10-6 s per day — about 38,471 ns/day.
The same geometry gives, for a moving clock, t′ = t secφ for the time to complete a given number of cycles. Fernando tests this against muon decay: with a laboratory lifetime t = 2.2 × 10-6 s, v = 0.9c and sinφ = v/c, t′ = t(1 − v2/c2)-1/2 = 5.047 × 10-6 s, which he says matches the displacement of cosmic-ray muons as recorded by Feynman.
Appendix: extensive and intensive components
An appendix supplies the general rationale, quoting Newton's Query 31 on deriving "two or three general Principles of Motion from Phaenomena" rather than assigning occult specific qualities, and Maxwell on the superiority of hypotheses framed at the most general level. Fernando's generalisation is that every quantity of energy of motion is the product of an extensive component (mass, or charge in electrical motions) and an intensive component (velocity), which vary conjugately — explicitly on the model of thermodynamics, where volume and pressure, or entropy and temperature, play those roles. Dalton's law of partial pressures, VP1 + VP2 = V(P1 + P2), is the pattern for the classical addition of velocities at constant mass. What has not been recognised, he argues, is the converse theorem, the division of a fixed total internal momentum into gravitational, latent and clock sectors — the three summing back to Mnc because Mnsin2α + Mncos2α = Mn. This, he says, is what expresses "the principle of constancy of the velocity of atomic vibrations under all states of motion and position of a body."
Assessment
There is something genuinely appealing in the paper's organising picture. Fernando takes a fixed budget — the body's total internal momentum — and makes every clock effect a redistribution within that budget, with nothing created or destroyed. That gives a single mechanism where the standard treatment uses two theories, and it supplies an intuitive answer to a question students really do ask: why does a clock deep in a potential well or moving fast run slow? "Because less of its momentum is available to drive the atomic vibrations" is at least a picture, and Fernando is right that the geometry of the energy-momentum relation invites exactly this reading. His close attention to Newton's Definitions II and III is also more than ornamental; the observation that mechanics kept the mass aspect of vis insita and quietly dropped the velocity aspect is a fair historical point. And he is admirably explicit about which of his formulas are exact and which approximate.
But the paper's central claim — that it derives relativistic results from a new principle — does not survive inspection of what is actually being done. The two GPS terms are the standard ones. The altitude term GM(Ro − Re)/(RoRec2) is the weak-field gravitational potential difference divided by c2, which is precisely general relativity's first-order prediction. The velocity term 1 − (1 − GM/Roc2)1/2 is 1 − (1 − v2/c2)1/2 written with the circular-orbit relation v2 = GM/Ro substituted in — that is, the Lorentz factor. And t′ = t secφ with sinφ = v/c is identically t′ = t(1 − v2/c2)-1/2, as Fernando himself writes out in the muon calculation. The geometry has not produced new formulas; it has re-derived the familiar ones by defining an angle whose sine is v/c. The agreement with the GPS and muon numbers therefore confirms relativity exactly as much as it confirms the internal-momentum picture, and cannot discriminate between them.
The claim to greater accuracy is also not borne out by the paper's own figures. Fernando's altitude result is 45,674 ns/day against the 45,900 ns/day he cites from Van Flandern — a discrepancy of about 0.5%, which is enormous by the standards of the actual GPS correction, where the pre-launch frequency offset is set to a fractional precision far finer than that. The difference is fully accounted for by his choices of Re (a nominal equatorial radius, ignoring the observatory's altitude, the Earth's oblateness and the rotational velocity of the ground station, all of which enter the standard budget) and by his use of the approximate rather than the exact formula. That is not a criticism of the physics so much as of the presentation: a result 0.5% from the reference value is described as "of extreme accuracy" when the standard calculation reproduces the same number to several more digits.
The foundational apparatus is where the difficulties are deepest. The "omni-directional vector" is introduced to license treating E0/c as a vector perpendicular to p, but nothing establishes that a root-mean-square speed is a vector of any kind; the perpendicularity is read off the Pythagorean form of the energy-momentum relation and then offered as an independent premise explaining it, which is circular. The identification of c with "the velocity of atomic vibrations" is asserted throughout and never justified: atoms in a solid vibrate at speeds many orders of magnitude below c, and the c appearing in E = mc2 is not a mechanical vibration speed. Nor does the theory address what makes it a theory rather than a re-parameterisation — it produces no prediction that differs from relativity anywhere, so there is no measurement that could distinguish them. Finally, the claim that "the rest mass of a body is its apparent mass when at a given position R in a gravitational field" conflicts with a well-tested fact: rest masses inferred from atomic and nuclear spectra, and from particle physics, do not vary with location in the solar potential, and the null results of Eötvös-type and gravitational-redshift experiments constrain any such positional dependence of m0 far below what this construction would require. Fernando invokes Eötvös in his own support, but the experiment establishes the universality of free fall, not the positional variability of rest mass.
Taken as a heuristic — a way of visualising why clocks run at different rates as a conserved internal quantity being reapportioned — the paper is inventive and internally consistent. Taken as the claim in its abstract, that "all 'relativistic phenomena'" are hereby explained for the first time, it re-labels rather than replaces.