Kepler’s law (following on from previous post)

The previous post, https://nige.wordpress.com/2006/09/22/gravity-equation-discredits-lubos-motl, has led to an interesting development.

Dr Thomas R. Love of California State University, Dominguez Hills, writes in an email to me: ‘Consider a planet of mass m, orbiting a star of mass M with an average radius of r. The theorem of equipartion of energy requires that the average kinetic energy is equal to the average potential energy [this is because the energy for escape velocity v = (2GM/r) ^1/2 of an orbiting body is exactly equal in magnitude to its existing kinetic energy, so the gravitational potential energy (which is the energy you need to throw an object up to an infinite height and by energy conservation this is equal to the energy an object gets by falling from an infinite height) of an object in orbit is equal to its orbital kinetic energy, E = (1/2) mv ² = (1/2) m((2GM/r) ^1/2 )² = mMG/r ]:

(1/2) mv ² = mMG/r

cancelling the m

(1/2) v ² = MG/r

Since the orbit is close to being a circle, we can take the average velocity to be:

v = 2p r/T

where T is the period.  Substitute

(1/2)(2p r/T)² = MG/r

and simplify to obtain:

r ³ = MGT ² /(2p ² )

which is Kepler’s law.’

This is a nice extension of the idea in the previous post in this weblog.  I’ve sent Dr Love an email stating that if you next consider a photon orbiting the mass M, by simply setting v = c, and using Einstein’s equivalence for mass m = E/c ² , then (1/2) mv ² = mMG/r immediately gives you the correct black hole event horizon radius that general relativity predicts, namely: r = 2GM/c ² . 

This implies that the effective kinetic energy of a photon is E = (1/2) mc ² = (1/2) pc (because the photon has no rest mass, whatever mass is – Higgs field or whatever – momentum p = mc is less objectionable).  This is half the amount in the usual formula relating the energy of a photon to its momentum, which is E = pc.

The factor of two discrepancy here is due to the fact that the photon is a transverse wave of electromagnetic field energy, so it oscillates ar right angles to its propagation direction, and the transverse oscillation carries half of the kinetic energy.  In fact, it has equal energy in its electric and magnetic fields, which oscillate at right angles to one another.  Therefore, the kinetic energy of the electromagnetic vibrations of the photon in the direction of the gravitational field vector (as the photon orbits around the mass) is half its total energy E = pc.

Update (3 October 2006):

The physical dynamics for Dr Love’s (1/2) mv ² = mMG/r is clearly that gravity is trapping the oribiting mass into a closed orbit.  So if the kinetic energy (1/2) mv ² of mass m was bigger than its gravitational potential energy with respect to the bigger mass (M) that it is orbiting, mMG/r, then it would spiral outwards instead of being in a closed orbit.

But if the kinetic energy of the mass m was smaller than its gravitational potential energy with respect to M, then it would obviously spiral inward (until the energies balanced).

See comments on this and the previous post for some more information.  One thing I’d like to add is that in the Yang-Mills gravity dynamics where gauge boson exchanges between masses cause gravity in an orbital situation such as Dr Love considers, the centripetal force (gravity) is often said to be cancelled by a fictitious outward force, called the centrifugal force.  The key equation a = v ² /r leads to F = ma = mv ² /r  for this force, see http://en.wikipedia.org/wiki/Centripetal_force for a couple of derivations of a = v ² /r (sadly, both of the Wikipedia derivations are relatively inelegant and ugly, compared to a really nice derivation which they don’t give; sometime I’ll try to add it).

It is then usually explained that the centrifugal (outward) force is an illusion and the real physics is down to the inertia of the mass (and is thus explained by Newton’s 1st law of motion).  However, when you consider the dynamics of gauge boson exchanges causing gravitational mass, you realise by Einstein’s equivalence principle (the equivalence between inertial and gravitational mass) that quantum gravity is must explain inertial mass as well as gravitational mass, and must therefore explain Newton’s 1st law of motion.

As we know, at least in the part of the universe we inhabit, any gauge boson radiation exchange causing gravitation and inertia normally occurs with isotropic symmetry in all directions with all the other masses in the universe.   Hence, earth’s radius is simply compressed uniformly by the amount general relativity predicts, (1/3)GM/c² = 1.5 mm.  Therefore you only usually feel forces from this Yang-Mills quantum gravity mechanism due to asymmetries such as the presence of nearby, non-receding masses.  The earth is an asymmetry, and you get pushed towards it, because because the earth isn’t receding from us siginficantly like the distant masses in the universe.  Because the earth isn’t receding in spacetime with a velocity that increases with its apparent time past from us (and this a force directed away from us equal to its mass multiplied by the rate of change of velocity as a function of observable time past, F = ma), it doesn’t have a force directed away from us, so the gauge bosons it transmits to us don’t carry a recoil force towards us by Newton’s 3rd law.  Hence, it acts as a shield because it isn’t receding.

The dynamics of inertia are not very simple: http://thumbsnap.com/v/ZF9FQD7v.jpg shows some dynamics but not the FitzGerald-Lorentz contraction of the atoms at different places in the mass.  The orbital speed of the atoms at different places in the mass is slightly different: those further from the origin of the curvature (eg, the centre of the orbit) move faster than those located closer.  However, the spatial distribution of the atoms in the mass does not vary the overall effect, what counts is the mass and its speed. 

When a mass moves along a straight line, the paths of successive gauge bosons emitted by perpendicular to its trajectory by atoms of the mass (which is spread out spatially) are parallel, but when it moves on a curved trajectory, the paths of successive transmission of gauge bosons emitted on the side facing the origin of the curvature (say the centre of a circular orbit, or a focus in an elliptical orbit) are not parallel but instead converge at the centre or focus of the orbit.  On the other side of the orbital mass, successive gauge bosons emitted perpendicular to its direction of motion diverge from one another.  The difference in the angular distribution of the gauge bosons on the two sides emitted by a mass moving on a curved trajectory causes a real centrifugal force, ie, it is the origin of the inertial force which opposes gravity and keeps the mass orbiting without either falling inward or flying outward.  It is fairly clear that to prove this rigorously will be the next step, following the kind of dynamics described at http://feynman137.tripod.com/#a.

If you consider a gyroscope’s physics, see http://www.mariner.connectfree.co.uk/html/gyro.htm, the angular momentum effects are subtle when you get away from mathematical models and try to use simple physical concepts; for example see http://www.newton.dep.anl.gov/askasci/phy99/phy99191.htm:

‘If you push sideways a speeding car you do not expect the path of the car to suddenly change so as to lie along the direction of the push.  Rather, you expect the car to acquire a little extra velocity in the direction of the push, and the combined action of this new velocity and the car’s original velocity to result in a path mostly along the original direction but deflected slightly towards the direction of the push.  The key insight is that a force changes directly the velocity of an object and not its path, and the path only changes eventually, via the change in velocity.’

Professor Eric Laithwaite turned the gyroscope into a tool for mocking the mainstream of physics in the 1974 Royal Institution Christmas Lectures he delivered, causing uproar.  It is dangerous to go down that road, see the videos of the lectures at http://www.gyroscopes.org/1974lecture.asp:

‘Air powered gyroscope (5000rpm – 8lb). Searching for centrifugal force. Gyroscope hanging over the top of a table. Out of balance by 2kg. … Gyroscope on an arm with a second pivot point. Making a body lighter than it is. … Denis lifts a 18lb gyroscope with a 6lb shaft running at 2000rpm. … The energy contained within a gyroscope. … What’s wrong with the scientific world? … Ohm’s law only applies to DC and not AC.’

Laithwaite showed evidence that Newton’s laws don’t apply in situations where the acceleration of mass is changing (they do apply where the velocity is changing).  Laithwaite may have made a mistake in trying to question empirical laws, after all the equations which Einstein got from special relativity were the already-known FitzGerald-Lorentz contraction and time dilation, and other electromagnetic theory results.  Nobody sensible attacks empirically defensible laws.

Poor old Royal Institution, having such a load of crackpotism transmitted on TV!  Little did they expect a crackpot lecture when they invited the distinguished Professor Laithwaite to explain the gyroscope at the lectures initiated over a century earlier by Michael Faraday.  The problem is that, as you can see in the lectures he gave, he did experiments which were transmitted on TV and demonstrated all of his claims.

(I haven’t replicated Laithwaite’s experiments with gyroscopes, but I can tell you that Ohm’s law only applies to steady state systems: when you send logic pulses, the logic pulses can be shorter than the size of the circuit, so they certainly can’t tell if the circuit is complete or not (or what its complete resistance is) when they start out.  In fact, logic pulses start out the same regardless of whether the circuit is complete.  You can send a logic pulse into an unterminated transmission line, where Ohm’s law would say has infinite resistance because the two conductors are separate by insulators.  What happens in this case was worked out by Heaviside around 1875, when he was experimenting with and mathematically modelling the undersea telegraph cable between Newcastle and Denmark.  Heaviside found that electric signals travel at the speed of light, and they have no way of telling in advance of travelling around the circuit, what the resistance of the complete circuit will turn out to be.  Instead, Heaviside found that there is what he considered – like Maxwell – to be an aetherial effect called impedance which has the same units as resistance (Ohm) but behaves very differently, being dependent only on the geometry of the conductors involved and the insulator used.)

The Royal Institution refused to publish the text of Laithwaite’s lectures (although the lectures were transmitted live on TV by the BBC and video recorded on tape).  Wikipedia states that Laithwaite responded by quoting a cynical comment by quantum field theorist, Professor Freeman Dyson:

“The scientific establishment, in the form of the Royal Institution, rejected his theory and his lecture was not published by the RI. His feelings on this can be seen in one of the 1974–1975 Royal Institution Christmas Lectures which he presented. In an apparent defence of his position he quoted Freeman Dyson: ‘Most of the crackpot papers that are submitted to the Physical Review are rejected, not because it is impossible to understand them, but because it is possible. Those that are impossible to understand are usually published.’ (Freeman Dyson, Innovations in Physics, Scientific American, September 1958).”

(That 1958 Dyson article in Sci. Am. Vol. 199, No. 3, pp. 74-82, is very important historically.  It quotes Niels Bohr’s statement to Wolfgang Pauli: ‘We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that is not crazy enough.’  Dyson also states in the article: ‘I have observed in teaching quantum mechanics (and also in learning it) that students go through the following experience: The student begins by learning how to make calculations in quantum mechanics and get the right answers; it takes about six months. This is the first stage in learning quantum mechanics, and it is comparatively easy and painless. The second stage comes when the student begins to worry because he does not understand what he has been doing. He worries because he has no clear physical picture in his head. He gets confused in trying to arrive at a physical explanation for each of the mathematical tricks he has been taught. He works very hard and gets discouraged because he does not seem able to think clearly. This second stage often lasts six months or longer, and it is strenuous and unpleasant. Then, quite unexpectedly, the third stage begins. The student suddenly says to himself, “I understand quantum mechanics”, or rather he says, “I understand now that there isn’t anything to be understood”. The difficulties which seemed so formidable have mysteriously vanished. What has happened is that he has learned to think directly and unconsciously in quantum mechanical language, and he is no longer trying to explain everything in terms of pre-quantum conceptions.’  This is a gutless surrender to the Copenhagen Interpretation.)

It is significant that Laithwaite was a Professor at Imperial College of London University, which was a hotbed of dissent in theoretical physics: Professor Herbert Dingle was there at the same time (note that the Wikipedia article on him is prejudiced by a disgraceful error that I have pointed out on the discussion page of the article) and also Theo Theocharis who graduated there in the early 1980s and stayed on to do- as I understand it – do a PhD on the errors of stringy stuff in mainstream physics (naturally that had to be stopped).  Theocharis and M. Psimopoulos did succeed in getting an attack on the Copenhagen Interpretation etc into a peer-reviewed journal: ‘Where Science Has Gone Wrong’, Nature, v329, p595, 1987.  However, that just caused more uproar:

‘Teachers of history, philosophy, and sociology of science … are up in arms over an attack by two Imperial College physicists … who charge that the plight of … science stems from wrong-headed theories of knowledge. … Scholars who hold that facts are theory-laden, and that experiments do not give a clear fix on reality, are denounced. … Staff on Nature, which published a cut-down version of the paper after the authors’ lengthy attempts to find an outlet for their views, say they cannot recall such a response from readers. ‘It really touched a nerve,’ said one. There was unhappiness that Nature lent its reputation to the piece.’ – Jon Thurney, Times Higher Education Supplement, 8 Jan 88, p2. [This refers to the paper by T. Theocharis and M. Psimopoulos, ‘Where Science Has Gone Wrong’, Nature, v329, p595, 1987.]

The dangers of pointing out errors in orthodoxy without correcting them at the same time are potentially massive.  Hans Christian Anderson and George Orwell effectively explain problems in modern physics between the research and the teaching orthodoxy:

‘The Emperor realized that the people were right but could not admit to that. He though it better to continue the procession under the illusion that anyone who couldn’t see his clothes was either stupid or incompetent. And he stood stiffly on his carriage, while behind him a page held his imaginary mantle.’ – Hans Christian Anderson, 1837.

‘Crimestop means the faculty of stopping short, as though by instinct, at the threshold of any dangerous thought. It includes the power of not grasping analogies, of failing to perceive logical errors, of misunderstanding the simplest arguments if they are inimical to (an authority) and of being bored or repelled by any train of thought which is capable of leading in a heretical direction. Crimestop, in short, means protective stupidity.’ – George Orwell, 1949.

String theory being ‘not even wrong’ demonstrates this very nicely.