**Why the rank-2 stress-energy tensor of general relativity does not imply a spin-2 graviton**

“… reality must take precedence over public relations, for nature cannot be fooled.”

– Richard P. Feynman’s Appendix F to Rogers’ Commission Report into the Challenger space shuttle explosion of 1986.

“If it exists, the graviton must be massless (because the gravitational force has unlimited range) and must have a spin of 2 (because the source of gravity is the stress-energy tensor, which is a second-rank tensor, compared to electromagnetism, the source of which is the four-current, which is a first-rank tensor). To prove the existence of the graviton, physicists must be able to link the particle to the curvature of the space-time continuum and calculate the gravitational force exerted.” – False claim, Wikipedia.

**Above:** spin-1 quantum gravity illustration from the old 2009 version of quantumfieldtheory.org (a PDF linked here, containing useful Feynman quotations about this). To hear to a very brief Feynman tongue-in-cheek talk on spin-1 graviton mechanism problems, please click here.

Previous posts explaining why general relativity requires spin-1 gravitons, and rejects spin-2 gravitons, are linked here, here, here, and here. But let’s take the false claim that gravitons must be spin-2 because the stress-energy tensor is rank-2. A rank 1 tensor is a first-order (once differentiated, e.g. da/db) differential summation, such as the divergence operator (sum of field gradients) or curl operator (the sum of all of the differences in gradients between field gradients for each pair of mutually orthagonal directions in space). A rank 2 tensor is some defined summation over second-order (twice differentiated, e.g. d^{2}a/db^{2}) differential equations. The field equation of general relativity has a different structure from Maxwell’s field equations for electromagnetism: as the Wikipedia quotation above states, Maxwell’s equations of classical electromagnetism are vector calculus (rank-1 tensors or first-order differential equations), while the tensors of general relativity are second order differential equations, rank-2 tensors.

The lie, however, is that this is physically deep. It’s not. It’s purely a choice of how to express the fields conveniently. For simple electromagnetic fields, where there is no contraction of mass-energy by the field itself, you can do it easily with first-order equations, gradients. These equations calculate fields with a first-order (rank-1) gradient, e.g. electric field strength, which is the gradient of potential/distance, measured in volts/metre. Maxwell’s equations don’t directly represent accelerations (second-order, rank-2 equations would be needed for that). For gravitational fields, you have to work with accelerations because the gravitational field contracts the source of the gravitational field itself, so gravitation is more complicated than electromagnetism.

The people who promote the lie that because rank-1 tensors apply to spin-1 field quanta in electromagnetism, rank-2 tensors must imply spin-2 gravitons, offer no evidence of this assertion. It’s arm-waving lying. It’s true that you need rank-2 tensors in general relativity, but it is not necessary in principle to use rank-1 tensors in electromagnetism: it’s merely easiest to use the simplest mathematical method available. You could in principle use rank-2 tensors to rebuild electromagnetism, by modelling the equations to observable accelerations instead of unobservable rank-1 electric fields and magnetic fields. Nobody has ever seen an electric field: only accelerations and forces caused by charges. (Likewise for magnetic fields.)

There is no physical correlation between the rank of the tensor and the spin of the gauge boson. It’s a purely historical accident that rank-1 tensors (vector calculus, first-order differential equations) are used to model fictitious electric and magnetic “fields”. We don’t directly observe electric field lines or electric charges (nobody has seen the charged core of an electron, what we see are effects of forces and accelerations which can merely be described in terms of field lines and charges). We observe accelerations and forces. The field lines and charges are not directly observed. The mathematical framework for a description of the relationship between the source of a field and the end result depends on the definition of the end result. In Maxwell’s equations, the end result of a electric charge which is not moving relative to the observer is a first-order field, defined in volts/metre. If you convert this first-order differential field into an observable effect, like force or acceleration, you get a second-order differential equation, acceleration a = d^{2}x/dt^{2}. General relativity doesn’t describe gravity in terms of a first-order field like Maxwell’s equations do, but instead describes gravitation in terms of a second-order observable, i.e. space curvature produced acceleration, a = d^{2}x/dt^{2}.

So the distinction between rank-1 and rank-2 tensors in electromagnetism and general relativity is *not physically deep:* it’s a matter of human decisions on how to *represent *electromagnetism and gravitation.

We choose in Maxwell’s equations to represent not second-order accelerations but using Michael Faraday’s imaginary concept of a pictorial field, radiating and curving “field lines” which are represented by first-order field gradients and curls. In Einstein’s general relativity, by contrast, we don’t represent gravity by such half-baked unobservable field concept, but in terms of directly-observable accelerations.

Like first-quantization (undergraduate quantum mechanics) lies, the “spin-2” graviton deception is a brilliant example of historical physically-ignorant mathematical obfuscation in action, leading to groupthink delusions in theoretical physics. (Anyone who criticises the lie is treated with a similar degree of delusional, paranoid hostility directed to dissenters of evil dictatorships. Instead of examining the evidence and seeking to correct the problem – which in the case of an evil dictatorship is obviously a big challenge – the messenger is inevitably shot or the “message” is “peacefully” deleted from the arXiv, reminiscent of the scene from *Planet of the Apes* where Dr Zaius – serving a dual role as Minister of Science and Chief Defender of the Faith, has to erase the words written in the sand which would undermine his religion and social tea-party of lying beliefs. In this analogy, the censors of the arXiv or journals like *Classical and Quantum Gravity* are not defending objsctive science, but are instead defending subjective pseudo-science – the groupthink orthodoxy which masquerades as science – from being exposed as a fraud.)

Dissimilarities in tensor ranks used to describe two different fields originate from dissimilarities in the field definitions for those two different fields, not to the spin of the field quanta. Any gauge field whose field is written in a second order differential equation, e.g., acceleration, can be classically approximated by rank-2 tensor equation. Comparing Maxwell’s equations in which fields are expressed in terms of first-order gradients like electric fields (volts/metre) with general relativity in which fields are accelerations or curvatures, is comparing chalk and cheese. They are not just different units, but have different purposes. For a summary of textbook treatments of curvature tensors, see Dr Kevin Aylward’s General Relativity for Teletubbys: “*the* fundamental point of the Riemann tensor [the Ricci curvature tensor in the field equation general relativity is simply a cut-down, rank-2 version Riemann tensor: the Ricci curvature tensor, **R**_{ab} = **R**^{x}_{axb}, where **R**^{x}_{axb} is the Riemann tensor], as far as general relativity is concerned, is that it describes the *acceleration* of geodesics with respect to one another. … I am led to believe that many people don’t have a … clue what’s going on, although they can apply the formulas in a sleepwalking sense. … The Riemann curvature tensor is what tells one what that acceleration between the [particles] will be. This is expressed by

[Beware of some errors in the physical understanding of some of these general relativity internet sites, however. E.g., some suggest – following a popular 1950s book on relativity – that the inverse-square law is discredited by general relativity, because the relativistic motion of Mercury around the sun can be *approximated within Newton’s framework* by increasing the inverse-square law power slightly from its value of 1/R^{2} to 1/R^{2 + X} where X is a small fraction, so that the force appears to get stronger nearer the sun. This is fictitious and is just an approximation to *roughly* accommodate relativistic effects that Newton ignored, e.g. the small increase in planetary mass due to its higher velocity when the planet is nearer the sun on part of its elliptical orbit, than it has when it is moving slower far from sun. This isn’t a physically correct model; it’s just a back-of-the-envelope fudge. A physically correct version of planetary motion in the Newtonian framework would keep the geometric inverse square law and would then correctly modify the force by making the right changes for the relativistic mass variation with velocity. Ptolemy’s epicycles demonstrated the danger of constructing approximate mathematical model which have no physical validity, which then become fashion.]”

Maxwell’s theory based on Faraday’s field lines concept employs only rank-1 equations, for example the divergence of the electric field strength, E, is directly proportional to the charge density, q (charge density is here defined as the charge per unit surface area, not the charge per unit volume): div.E ~ q. The reason this is a rank-1 equation is simply because the divergence operator is the sum of gradients in all three perpendicular directions of space for the operand. All it says is that a unit charge contributes a fixed number of diverging radial lines of electric field, so the total field is directly proportional to the total charge.

But this is *just Faraday’s way of visualizing the way the electric force operates!* Remember that nobody has yet seen or reported detecting an “electric field line” of force! With our electric meters, iron filings, and compasses we only see the results of forces and accelerations, so the number and locations of electric or magnetic field lines depicted in textbook diagrams is due to *purely arbitrary conventions*. It’s *merely* an abstract aetherial legacy from the Faraday-Maxwell era, not a physical reality that has any experimental evidence behind it. If you are going to confuse Faraday’s and Maxwell’s imaginary concept of field “lines” with experimentally defensible reality, you might as well write down an equation in which the invisible intermediary between charge and force is an angel, a UFO, a fairy or an elephant in an imaginary extra dimension. Quantum field theory tells us that there are no physical lines. Instead of Maxwell’s “physical lines of force”, we have known since QED was verified that there are *field quanta being exchanged between charges.*

So if we get rid of our *ad hoc* prejudices, getting rid of “electric field strength, E” in volts/metre and just expressing the *result* of the electric force in terms of what we can actually measure, namely accelerations and forces, we’d have a rank-2 tensor, basically the same field equation as is used in general relativity for gravity. The only differences will be the factor of ~10^{40} difference between field strengths of electromagnetism and gravity, the differences in the *signs* for the curvatures (like charges repel in electromagnetism, but attract in gravity) and the absence of the contraction term that makes the gravitational field contract the source of the field, but supposedly does not exist in electromagnetism. *The tensor rank will be 2 for both cases, thus disproving the arm-waving yet popular idea that the rank number may be correlated to the field quanta spin.* In other words, the electric field could be modelled by a rank-2 equation if we simply make the electric field consistent with the gravitational field by expressing both field in terms of accelerations, instead of using the gradient of the Faraday-legacy volts/metre “field strength” for the electric field. This is however beyond the understanding of the mainstream, who are deluded by fashion and historical *ad hoc* conventions. Most of the problems in understanding quantum field theory and unifying Standard Model fields with gravitational fields result from the legacy of field definitions used in Maxwellian and Yang-Mills fields, which for purely *ad hoc* historical reasons are different from the field definition in general relativity. * If all fields are expressed in the same way as accelerative curvatures, all field equations become rank-2 and all rank-1 divergencies automatically disappear, since are merely an historical legacy of the Faraday-Maxwell volts/metre field “line” concept, which isn’t consistent with the concept of acceleration due to curvature in general relativity!*

However, we’re not advocating the use of any particular differential equations for any quantum fields, because discontinuous quantized fields can’t in principle be correctly modelled by differential equations, which is why you can’t properly represent the source of gravity in general relativity as being a set of discontinuities (particles) in space to predict curvature, but must instead use a physically false averaged distribution such as a “perfect fluid” to represent the source of the field. The rank-2 framework of general relativity has relatively few easily obtainable solutions compared to the simpler rank-1 (vector calculus) framework of electrodynamics. But both classical fields are false in ignoring the random field quanta responsible for quantum chaos (see, for instance, the discussion of first-quantization versus second-quantization in the previous post here, here and here).

**Summary:**

1. The electric field is *defined* by Michael Faraday as simply the gradient of volts/metre, which Maxwell correctly models with a first-order differential equation, which leads to a rank-1 tensor equation (vector calculus). Hence, electromagnetism with spin-1 field quanta has a rank-1 tensor purely because of the way it is formulated. Nobody has ever seen Faraday’s electric field, only accelerations/forces. There is no physical basis for electromagnetism being intrinsically rank-1; it’s just one way to mathematically model it, by describing it in terms of Faraday rank-1 fields rather than the directly observable rank-2 accelerations and forces which we see/feel.

2. The gravitational field has historically never been expressed in terms of a Faraday-type rank-1 field gradient. Due to Newton, who was less pictorial than Faraday, gravity has always been described and modelled *directly* in terms of the end result, i.e. accelerations/forces we see/feel.

This difference between the human formulations of the electromagnetic and gravitational “fields” is the *sole* reason for the fact that the former is currently expressed with a rank-1 tensor and the latter is expressed with a rank-2 tensor. If Newton had worked on electromagnetism instead of aether crackpots like Maxwell, we would undoubtedly have a rank-2 mathematical model of electromagnetism, in which electric fields are expressed not in volts/metre, but directly in terms of rank-2 acceleration (curvatures), just like general relativity.

Both electromagnetism and gravitation should define fields the same way, with rank-2 curvatures. The discrepancy that electromagnetism uses instead rank-1 tensors is due to the inconsistency that in electromagnetism fields are not *defined* in terms of curvatures (accelerations) but in terms of a Faraday’s imaginary abstraction of field lines. This has nothing whatsoever to do with particle spin. Rank-1 tensors are used in Maxwell’s equations because the electromagnetic fields are defined (inconsistently with gravity) in terms of rank-1 unobservable field gradients, whereas rank-2 tensors are used in general relativity purely because the definition of a field in general relativity is acceleration, which requires a rank-2 tensor to describe it. The difference is purely down to the way the field is described, not the spin of the field.

**The physical basis for rank-2 tensors in general relativity**

I’m going to rewrite the paper linked here when time permits.

**Groupthink delusions**

The real reason why gravitons supposedly “must” be spin-2 is due to the mainstream investment of energy and time in worthless string theory, which is designed to permit the existence of spin-2 gravitons. We know this because whenever the errors in spin-2 gravitons are pointed out, they are ignored. These stringy people aren’t interested in physics, just grandiose fashionable speculations, which is the story of Ptolemy’s epicycles, Maxwell’s aether, Kelvin’s vortex atom, Piltdown Man, S-matrices, UFOs, Marxism, fascism, etc. All were very fashionable with bigots in their day, but:

“… reality must take precedence over public relations, for nature cannot be fooled.” – Feynman’s Appendix F to Rogers’ Commission Report into the Challenger space shuttle explosion of 1986.

**Above:** the mainstream groupthink on the spin of the graviton goes back to Pauli and Fierz’s paper of 1939, which insists that gravity is attractive (that we’re not being pushed down), which leads to a requirement for the spin to be an even number, not an odd number:

‘In the particular case of spin 2, rest-mass zero, the equations agree in the force-free case with Einstein’s equations for gravitational waves in general relativity in first approximation …’

– Conclusion of the paper by M. Fierz and W. Pauli, ‘On relativistic wave equations for particles of arbitrary spin in an electromagnetic field’, *Proc. Roy. Soc. London.,* v. A173, pp. 211-232 (1939).

There is evidence for a spin-1 graviton. For example, the following is from a *New Scientist* page:

‘Some physicists speculate that dark energy could be a repulsive gravitational force that only acts over large scales. “There is precedent for such behaviour in a fundamental force,” Wesson says. “The strong nuclear force is attractive at some distances and repulsive at others.”’

This possibility was ignored by Pauli and Fierz when first proposing that the quanta of gravitation has spin-2.

(1) gives cosmological repulsion of large masses, and

Above: Perlmutter’s discovery of the acceleration of the universe, based on the redshifts of fixed energy supernovae, which are triggered as a critical mass effect when sufficient matter falls into a white dwarf. A type Ia supernova explosion, always yielding 4 x 10^{28} megatons of TNT equivalent, results from the critical mass effect of the collapse of a white dwarf as soon as its mass exceeds 1.4 solar masses due to matter falling in from a companion star. The degenerate electron gas in the white dwarf is then no longer able to support the pressure from the weight of gas, which collapses, thereby releasing enough gravitational potential energy as heat and pressure to cause the fusion of carbon and oxygen into heavy elements, creating massive amounts of radioactive nuclides, particularly intensely radioactive nickel-56, but half of all other nuclides (including uranium and heavier) are also produced by the ‘R’ (rapid) process of successive neutron captures by fusion products in supernovae explosions. *Because we can model how much energy is released using modified computer models of nuclear fusion explosions developed originally by weaponeer Sterling Colgate at Lawrence Livermore National Laboratory to design the early H-bombs, the brightness of the supernova flash tells us how far away the Type Ia supernova is, while the redshift of the flash tells us how fast it is receding from us. That’s how the acceleration of the universe was discovered.* Note that “tired light” fantasies about redshift are disproved by Professor Edward Wright on the page linked here.

You can go to an internet page and see the correct predictions on the linked page here or the about page. This isn’t based on speculations, cosmological acceleration has been observed since 1998 when CCD telescopes plugged live into computers with supernova signature recognition software detected extremely distant supernova and recorded their redshifts (see the article by the discoverer of cosmological acceleration, Dr Saul Perlmutter, on pages 53-60 of the April 2003 issue of *Physics Today,* linked here). The outward cosmological acceleration of the 3 × 10^{52} kg mass of the 9 × 10^{21} observable stars in galaxies observable by the Hubble Space Telescope (page 5 of a NASA report linked here), is approximately *a = Hc* = 6.9 x 10^{-10} ms^{-2} (L. Smolin, *The Trouble With Physics,* Houghton Mifflin, N.Y., 2006, p. 209), giving an immense outward force under Newton’s 2nd law of *F = ma* = 1.8 × 10^{43} Newtons. Newton’s 3rd law gives an equal inward (implosive type) reaction force, which predicts gravitation quantitatively. What part of this is speculative? Maybe you have some vague notion that scientific laws should not for some reason be applied to new situations, or should not be trusted if they make useful predictions which are confirmed experimentally, so maybe you vaguely don’t believe in applying Newton’s second and third law to masses accelerating at 6.9 x 10^{-10} ms^{-2}! But why not? What part of “fact-based theory” do you have difficulty understanding?

It is usually by applying facts and laws to new situations that progress is made in science. If you stick to applying known laws to situations they have already been applied to, you’ll be less likely to observe something new than if you try applying them to a situation which nobody has ever applied them to before. We should apply Newton’s laws to the accelerating cosmos and then focus on the immense forces and what they tell us about graviton exchange.

*Above:* The mainstream 2-dimensional ‘rubber sheet’ interpretation of general relativity says that mass-energy ‘indents’ spacetime, which responds like placing two heavy large balls on a mattress, which distorts more between the balls (where the distortions add up) than on the opposite sides. Hence the balls are pushed together: ‘Matter tells space how to curve, and space tells matter how to move’ (Professor John A. Wheeler). This illustrates how the mainstream (albeit arm-waving) explanation of general relativity is actually a theory that gravity is produced by space-time distorting to *physically push* objects together, not to pull them! (When this is pointed out to mainstream crackpot physicists, they naturally freak out and become angry, saying it is just a pointless analogy. But when the checkable predictions of the mechanism are explained, they may perform their always-entertaining “hear no evil, see no evil, speak no evil” act.)

Above: LeSage’s own illustration of quantum gravity in 1758. Like Lamarke’s evolution theory of 1809 (the one in which characteristics acquired during life are somehow supposed to be passed on genetically, rather than Darwin’s evolution in which genetic change occurs due to the inability of inferior individuals to pass on genes), LeSage’s theory was full of errors and is still derided today. The basic concept that mass is composed of fundamental particles with gravity due to a quantum field of gravitons exchanged between these fundamental particles of mass, is now a frontier of quantum field theory research. What is interesting is that quantum gravity theorists today don’t use the arguments used to “debunk” LeSage: they don’t argue that quantum gravity is impossible because gravitons in the vacuum would “slow down the planets by causing drag”. They recognise that gravitons are not real particles: they don’t obey the energy-momentum relationship or mass shell that applies to particles of say a gas or other fluid. Gravitons are thus off-shell or “virtual” radiations, which cause accelerative forces but don’t cause continuous gas type drag or the heating that occurs when objects move rapidly in a real fluid. While quantum gravity theorists realize that particle (graviton) mediated gravity is possible, LeSage’s mechanism of quantum gravity is still as derided today as Lamarke’s theory of evolution. Another analogy is the succession from Aristarchus of Samos, who first proposed the solar system in 250 B.C. against the mainstream earth-centred universe, to Copernicus’ inaccurate solar system (circular orbits and epicycles) of 1500 A.D. and to Kepler’s elliptical orbit solar system of 1609 A.D. Is there any point in insisting that Aristarchus was the original discoverer of the theory, when he failed to come up with a detailed, convincing and accurate theory? Similarly, Darwin rather than Lamarke is accredited with the theory of evolution, because he made the theory useful and thus scientific.

If someone fails to come up with a detailed, accurate and successfully convincing theory, and merely gets the basic idea right without being able to prove it against the mainstream fashions and groupthink, then the history of science shows that the person is not credited with a big discovery: science is not merely guesswork. Maxwell based his completion of the theory of classical electrodynamics upon an ethereal displacement current of virtual charges in the vacuum, in order to correct Ampere’s law for the case of open circuits such as capacitors using the permittivity of free space (a vacuum) for the dielectric. Maxwell believed, by analogy to the situation of moving ions in a fluid during electrolysis, that current appears to flow through the vacuum between capacitor plates while the capacitor charges and discharges; although in fact the real current just spreads along the plates, and electromagnetic induction (rather than ethereal vacuum currents) produces the current on the opposite place.

Maxwell nevertheless suggested (in an *Encyclopedia Britannica* article) an experiment to test whether light is carried at an absolute velocity by a mechanical spacetime fabric. After the Michelson-Morley experiment was done in 1887 to test Maxwell’s conjecture, it was clear that no absolute motion was detectable: suggesting (1) that motion appears relative, not absolute, and (2) that light always appears to go at the same velocity in the vacuum. In 1889, FitzGerald published an *explanation* of these “relativity” results in *Science:* he argued that the physical vacuum contracted moving masses like the Michelson-Morley experiment, by analogy to the contraction of anything moving in a fluid due to the force from the head-on fluid pressure (wind drag, or hydrodynamic resistance). This fluid-space based explanation predicted quantitatively the relativistic contraction law, and Lorentz showed that since mass depends inversely on the classical radius of the electron, it predicted a mass increase with velocity. Given the equivalence of space and time via the velocity of light, Lorentz showed that the contraction predicted time-dilation due to motion.

**Above**: In *Science* in 1889, FitzGerald used the Michelson-Morley result to argue that moving objects at velocity *v* must contract in length in the direction of their motion by the factor (1 – *v*^{2}/*c*^{2})^{1/2} so that there is no difference in the travel times of light moving along two perpendicular paths. Groupthink crackpots claim that if the lengths of the arms of the instrument are different, FitzGerald’s argument for absolute motion is destroyed since the travel times are still cancelled out. Actually, the arms of the Michelson-Morley instrument can never be the same length to within the accuracy of the relative times implied by interference fringes! The instrument does not measure the absolute times taken in two different directions: it merely determines if there is a difference in the *relative* times (which are always slightly different, since the arms can’t be machined to perfectly identical length) when the instrument is rotated by 90 degrees. Another groupthink crackpot argument is that although the FitzGerald theory predicts relativity from length contraction in an absolute motion universe, other special relativity results like time dilation, mass increase, and *E = mc*^{2} can only be obtained from Einstein. Actually, all were obtained by Lorentz and Poincare: Lorentz showed that evidence for space-time from electromagnetism implies that apparent time dilates like distance when an clock moves, while he argued that since the classical electromagnetic electron radius is inversely proportional to its mass, its mass should thus increase with velocity by a factor equal to the reciprocal of the FitzGerald contraction factor. Likewise, a force *F = d(mv)/dt* acting on a body moving distance *dx* imparts kinetic energy *dE = F.dx = d(mv).dx/dt = d(mv)v = v.d(mv) = v*^{2}*dm + mvdv.* Comparison of this purely Newtonian result with the derivative of Lorentz’s relativistic mass increase formula *m _{v} = m_{0}*(1 –

*v*

^{2}/

*c*

^{2})

^{-1/2}gives us

*dm = dE/c*

^{2}or

*E = mc*

^{2}. (See for example, Dr Glasstone’s

*Sourcebook on Atomic Energy*, 3rd ed., 1967.)

Carlos Barceló and Gil Jannes, ‘A Real Lorentz-FitzGerald Contraction’, *Foundations of Physics*, Volume 38, Number 2 / February, 2008, pp. 191-199 (PDF file: http://digital.csic.es/bitstream/10261/3425/3/0705.4652v2.pdf):

“Many condensed matter systems are such that their collective excitations at low energies can be described by fields satisfying equations of motion formally indistinguishable from those of relativistic field theory. The finite speed of propagation of the disturbances in the effective fields (in the simplest models, the speed of sound) plays here the role of the speed of light in fundamental physics. However, these apparently relativistic fields are immersed in an external Newtonian world (the condensed matter system itself and the laboratory can be considered Newtonian, since all the velocities involved are much smaller than the velocity of light) which provides a privileged coordinate system and therefore seems to destroy the possibility of having a perfectly defined relativistic emergent world. In this essay we ask ourselves the following question: In a homogeneous condensed matter medium, is there a way for internal observers, dealing exclusively with the low-energy collective phenomena, to detect their state of uniform motion with respect to the medium? By proposing a thought experiment based on the construction of a Michelson-Morley interferometer made of quasi-particles, we show that a real Lorentz-FitzGerald contraction takes place, so that internal observers are unable to find out anything about their ‘absolute’ state of motion. Therefore, we also show that an effective but perfectly defined relativistic world can emerge in a fishbowl world situated inside a Newtonian (laboratory) system. This leads us to reflect on the various levels of description in physics, in particular regarding the quest towards a theory of quantum gravity. …

“… Remarkably, all of relativity (at least, all of special relativity) could be taught as an effective theory by using only Newtonian language. …In a way, the model we are discussing here could be seen as a variant of the old ether model. At the end of the 19th century, the ether assumption was so entrenched in the physical community that, even in the light of the null result of the Michelson-Morley experiment, nobody thought immediately about discarding it. Until the acceptance of special relativity, the best candidate to explain this null result was the Lorentz-FitzGerald contraction hypothesis. … we consider our model of a relativistic world in a fishbowl, itself immersed in a Newtonian external world, as a source of reflection, as a *Gedankenmodel.* By no means are we suggesting that there is a world beyond our relativistic world describable in all its facets in Newtonian terms. Coming back to the contraction hypothesis of Lorentz and FitzGerald, it is generally considered to be *ad hoc.* However, this might have more to do with the caution of the authors, who themselves presented it as a hypothesis, than with the naturalness or not of the assumption. … The ether theory had not been *disproved,* it merely became *superfluous.* Einstein realised that the knowledge of the elementary interactions of matter was not advanced enough to make any claim about the relation between the constitution of matter (the ‘molecular forces’), and a deeper layer of description (the ‘ether’) with certainty. Thus his formulation of special relativity was an advance within the given context, precisely because it avoided making any claim about the fundamental structure of matter, and limited itself to an *effective* macroscopic description.”

In 1905, Einstein took the two implications of the Michelson-Morley research (that motion appears relative not absolute, and that the observed velocity of light in the vacuum is always constant) and used them as postulates to derive the FitzGerald-Lorentz transformation and Poincare mass-energy equivalence. Einstein’s analysis was preferred by Machian philosophers because it was purely mathematical and did not seek to *explain* the principle of relativity and constancy of the velocity of light in the vacuum by invoking a physical contraction of instruments. Einstein *postulated* relativity; FitzGerald *explained* it. Both predicted a similar contraction quantitatively. Similarly, Newton’s theory or gravitation is the combination of Galileo’s principle that dropped masses all accelerate at the same rate due to the constancy of the Earth’s mass, with Kepler’s laws of planetary motion. Newton *postulated* his universal gravitational law based on this evidence plus the guess that the gravitational force is directly proportional to the mass producing it, and he checked it by the Moon’s centripetal acceleration; LeSage tried to *explain* what Newton had postulated and checked.

The previous post links to Peter Woit’s earlier article about string theorist Erik Verlinde’s arXiv preprint *On the Origin of Gravity and the Laws of Newton,* which claims: “Gravity is explained as an entropic force caused by changes in the information associated with the positions of material bodies.” String theorist Verlinde derives Newton’s laws and other results using only “high-school mathematics” (which brings contempt from mathematical physicist Woit, probably one of the areas of agreement he has with string theorist Jacques Distler), i.e. no tensors, and he is derives the Newtonian weak field approximation for gravity, not the relativistic Einsteinian gravity law which also includes contraction. This contraction is physically real but small for weak gravitational fields and non-relativistic velocities: Feynman famously calculated in his published *Lectures on Physics* that the contraction term in Einstein’s field equation contracts the Earth’s radius by *MG*/(3*c*^{2}) = 1.5 mm. Consider two ways to predict contraction using Einstein’s equivalence principle.

**First, Einstein’s way.** Einstein began by expressing Newton’s law of gravity in tensor field calculus which allows gravity to be represented by non-Euclidean geometry, incorporating the equivalence of inertial and gravitational mass: Einstein started with a false hypothesis that the curvature of spacetime (represented with the Ricci tensor) which causes acceleration (“curvature” is literally the curve of a line on a graph of distance versus time, i.e. it implies acceleration) simply equals the source of gravity (the stress-energy tensor, since in Einstein’s earlier special relativity, mass and energy are equivalent, albeit via the well-known very large conversion factor, *c*^{2}). (Non-Euclidean geometry wasn’t Einstein’s innovation; it was studied by Riemann and Minkowski, while Ricci and Levi-Civita pioneered tensors to generalize vector calculus to any number of dimensions.)

Einstein in 1915 found that this this simple equivalence was wrong: the Ricci curvature tensor could not be equivalent to the stress-energy tensor because the divergence (the sum of gradients in all spatial dimensions) of the stress-energy tensor is not zero. Unless this divergence is zero, mass-energy will not be conserved. So Einstein used Bianchi’s identity to alter source of gravity, subtracting from the stress-energy tensor, **T**_{ab}, half the product of the metric tensor **g**_{ab}, and the trace of the stress-energy tensor, T (the trace of a tensor is simply the sum of the top-left to bottom-right diagonal elements of that tensor, i.e. energy density plus pressure, or trace T = **T**_{00} + **T**_{11} + **T**_{22} + **T**_{33}), because this combination: (1) does have zero divergence and thereby satisfies the conservation of mass-energy, and (2) reduces the stress-energy tensor for weak fields, thereby correctly corresponding to Newtonian gravity in the weak field limit. This is how Einstein found that the Ricci tensor **R**_{ab} = **T**_{ab} – (1/2)**g**_{ab}T, which is exactly equivalent to the oft-quoted Einstein equation **R**_{ab} – (1/2)**g**_{ab}R = **T**_{ab}, where R is the trace of the Ricci tensor (R = **R**_{00} + **R**_{11} + **R**_{22} + **R**_{33}).

Secondly, Feynman’s way. A more physically intuitive explanation to the modification of Newton’s gravitational law implied by Einstein’s field equation of general relativity is to examine Feynman’s curvature result: space-time is non-Euclidean in the sense that the gravitational field contracts the Earth’s radius by (1/3)*MG/c*^{2} or about 1.5 mm. This is unaccompanied by a transverse contraction, i.e. the Earth’s circumference is unaffected. To *mathematically* keep “Pi” a constant, therefore, you need to invoke an extra dimension, so that the n-1 = 3 spatial dimensions we experience are in string theory terminology a (mem)brane on a n = 4 dimensional bulk of spacetime. Similarly, if you draw a 2-dimensional circle upon the surface of the interior of a sphere, you will obtain Pi from the circle only by drawing a straight line through the 3-d bulk of the volume (i.e. a line that does *not* follow the 2-dimensional curved surface or “brane” of the sphere upon which the circle is supposed to exist). If you measure the diameter upon the curved surface, it will be different, so Pi will appear to vary.

A simple physical mechanism of Feynman’s (1/3)*MG/c*^{2} excess radius for symmetric, spherical mass *M* is that the gravitational field quanta compress a mass radially when being exchanged with distant masses in the universe: the exchange of gravitons pushes against masses. By Einstein’s principle of the equivalence of inertial and gravitational mass, the cause of this excess radius is exactly the same as the cause of the FitzGerald-Lorentz contraction of moving bodies in the direction of their motion, first suggested in *Science* in 1889 by FitzGerald. FitzGerald explained the apparent constancy of the velocity of light regardless of the relative motion of the observer (indicated by the null-result of the Michelson-Morley experiment of 1887) as the physical effect of the gravitational field. In the fluid analogy to the gravitational field, if you accelerate an underwater submarine, there is a head-on pressure from the inertial resistance of the water which it is colliding with, which causes it to contract slightly in the direction it is going in. This head-on or “dynamic” pressure is equal to half the product of the density of the water and the square of the velocity of the submarine. In addition to this “dynamic” pressure, there is a “static” water pressure acting in all directions, which compresses the submarine slightly in all directions, even if the submarine is not moving. In this analogy, the FitzGerald-Lorentz contraction is the “dynamic” pressure effect of the graviton field, while the Feynman excess radius or radial contraction of masses is the “static” pressure effect of the graviton field. Einstein’s special relativity *postulates* (1) relativity of motion and (2) constancy of *c*, and derives the FitzGerald-Lorentz transformation and mass-energy equivalence from these postulates; by contrast, FitzGerald and Lorentz sought to physically *explain* the mechanism of relativity by *postulating* contraction. To contrast this difference:

(1) Einstein: **postulated relativity** and **produced contraction**.

(2) Lorentz and FitzGerald: **postulated contraction **to **produce “apparent” observed Michelson-Morley relativity** as just an instrument contraction effect within an absolute motion universe.

These two relativistic contractions, the contraction of relativistically moving inertial masses and the contraction of radial space around a gravitating mass, are simply related under Einstein’s principle of the equivalence of inertial and gravitational masses, since Einstein’s other equivalence (that between mass and energy) then applies to both inertial and gravitational masses. In other words, the equivalence of inertial and gravitational mass implies an effective energy equivalence for each of these masses. The FitzGerald-Lorentz contraction factor [1 – (*v/c*)^{2}]^{1/2} contains velocity *v,* which comes from the kinetic energy of the moving object. By analogy, when we consider a mass *m* at rest in a gravitational field from another much larger mass *M* (like a person standing on the Earth), it has acquired gravitational potential energy *E = mMG/R*, equivalent to a kinetic energy of *E* = (1/2)*mv*^{2}, so by Einstein’s equivalence principle of inertial and gravitational field energy it can be considered to have an “effective” velocity of *v* = (*2GM/R*)^{1/2}. Inserting this velocity into the FitzGerald-Lorentz contraction factor [1 – (*v/c*)^{2}]^{1/2} gives [1 – 2*GM*/(*Rc*^{2})]^{1/2} which, when expanded by the binomial expansion to the first couple of terms as a good approximation, yields 1 – *GM*/(*Rc*^{2}). This result assumes that all of the contraction occurs in one spatial dimension only, which is true for the FitzGerald-Lorentz contraction (where a moving mass is only contracted in the direction of motion, not in the two other spatial dimensions it has), but is not true for radial gravitational contraction around a static spherical, uniform mass, which operates equally in all 3 spatial dimensions. Therefore, the contraction in any one of the three dimensions is by the factor 1 – (1/3)*GM*/(*Rc*^{2}). Hence, when gravitational contraction is included, radius *R* becomes *R*[1 – (1/3)*GM*/(*Rc*^{2})] = *R* – *GM*/(3*c*^{2}), which is the result Feynman produced in his *Lectures on Physics* from Einstein’s field equation.

The point we’re making here is that general relativity isn’t mysterious unless you want to ignore the physical effects due to energy conservation and associated contraction, which produce its departures from Newtonian physics. Physically understanding the mechanism for how general relativity differs from Newtonian physics therefore immediately takes you to the facts of how the quantum gravitational field physically distorts static and moving masses, leading to checkable predictions which you cannot make with general relativity alone. It is therefore helpful if you want to understand physically how quantum gravity must operate in order to be consistent with general relativity within its domain of validity. Obviously general relativity breaks down outside that domain, which is why we need quantum gravity, but within the limits of validity for that classical domain, both theories are consistent. The reason why quantum gravity of the LeSage sort needs to be fully reconciled with general relativity in this way is that one objection to LeSage was by Laplace, who ignored the gravitational and motion contraction mechanisms of quantum gravity for relativity (Laplace was writing long before FitzGerald and Einstein) and tried to use this ignorance to debun LeSage by arguing that orbital aberration would occur in LeSage’s model due to the finite speed of the gravitons. This objection does not apply to general relativity due to the contractions incorporated into the general relativity theory by Einstein: similarly, Laplace’s objection does not apply to quantum gravity which inherently includes the contractions as physical results of quantum gravity upon moving masses.

In the past, however, FitzGerald’s physical contraction of moving masses as miring by fluid pressure has been controversial in physics, and Einstein tried to dispose of the fluid. The problem with the fluid was investigated by citics of Fatio and LeSage, who promoted a shadowing theory of gravity, whereby masses get pushed together by mutually shielding one another from the pressure of the fluid in space. These critics included some of the greatest classical physicists the world has ever known: Newton (Fatio’s friend), Maxwell and Kelvin. Feynman also reviewed the major objection, drag, to the fluid in his broadcast lectures on the *Character of Physical Law*. The criticisms of the fluid is that it the force it needs to exert to produce gravity would classically be expected to cause fast moving objects in the vacuum

(1) to heat up until they glow red hot or ablate at immense temperature,

(2) to slow down and (in the case of planets) thus spiral into the sun,

(3) while the fluid would diffuse in all directions and on large distance scales fill in the “shadows” like a gas, preventing the shadowing mechanism from working (this doesn’t apply to gravitons exchanged between masses, for although they will take all possible paths in a path integral, the resultant, effective graviton motion for force delivery will along the path of least action, due to the cancellation of the amplitudes of paths which interfere off the path of least action: see Feynman’s 1985 book *QED*),

(4) the mechanism would not give a force proportional to mass if the fundamental particles have a large gravitational interaction cross-sectional area, which would mean that in a big mass some of the shadows would “overlap” one another, so the net force of gravity from a big mass would be less than directly proportional to the mass, i.e. it would increase not in simple proportion to *M* but instead statistically in proportion to: 1 – e^{–bM}, where *b* is a gravity cross-section and geometry-dependent coefficient, which allows for the probability of overlap. This 1 – e^{–bM} formula has two asymptotic limits:

(a) for small masses and small cross-sections, *bM* is much smaller than 1, so: e^{–bM} ~ 1 – *bM*, so 1 – e^{–bM} ~ *bM*. I.e., for small masses and small cross-sections, the theory agrees with observations (there is no significant overlap).

(b) for larger masses and large cross-sections, *bM* might be much larger than 1, so e^{–bM} ~ 0, giving 1 – e^{–bM} ~ 1. I.e., for large masses and large cross-sections, the overlap of shadows would prevent any increase in the mass of a body from increasing the resultant gravitational force: once gravitons are stopped, they can’t be stopped again by another mass.

This overlap problem is not applicable for the solar system or many other situations because *b* is insignificant owing to the small graviton scattering cross-section of a fundamental particle of mass, since the total inward force is trillions upon trillions of times higher than the objectors believed possible: the force is simply determined by Newton’s 2nd and 3rd laws to be the product of the cosmological acceleration and the mass of the accelerating universe, 1.8 × 10^{43} Newtons, and the cross-section for shielding is the black hole event horizon area, which is so small that overlap is insignificant in the solar system or other tests of Newton’s weak field limit.

(5) the LeSage mechanism suggested that the gravitons which cause gravity would be slowed down by the energy loss when imparting a push to a mass, so that they would not be travelling at the velocity of light, contrary to what is known about the velocity of gravitational fields. However this is false and is due to the real (rather than virtual “off-shell”) radiation that LeSage assumed. The radiation goes at light velocity and merely shifts in frequency due to energy loss. For static situations, where no acceleration is produced (e.g. an apple stationary hanging on a tree) the graviton exchange results in no energy change; it’s a perfectly elastic scattering interaction. No energy is lost from the gravitons, and no kinetic energy is gained by the apple. Where the apple is accelerated, the kinetic energy it gains is that lost due to a shift to lower energy (longer wavelength) of the “reflected” or scattered gravitons. Notice that Dr Lubos Motl has objected to me by falsely claiming that virtual particles don’t appear to have wavelengths; on the contrary, the empirically confirmed Casimir effect is due to inability of virtual photons of wavelength longer than the distance between two metal plates, to exist and produce pressure between the plates (so the plates are pushed together from the complete spectrum of virtual photon wavelengths in the vacuum surrounding the places, which is stronger than the cut-off spectrum between the plates). Like the reflection of light by a mirror, the process is consists of the absorption of a particle followed by the emission of a new particle.

However, quantum field theory, which has been very precisely tested for electrodynamics, resurrects a quantum fluid or field in space which consists of gauge boson radiation, i.e. virtual (off-shell) radiation which carries “borrowed” or off-mass shell energy, not real energy. It doesn’t obey the relationship between energy and momentum that applies to real radiation. *This is why the radiation can exert pressure without causing objects to heat up or to slow down: they merely accelerate or distort instead.*

The virtual radiation is not like a regular fluid. It carries potential energy that can be used to accelerate and contract objects, but it cannot directly heat them or cause continuous drag to non-accelerating objects by carrying away their momentum in a series of impacts in the way that gas or water molecules cause continuous drag on non-accelerating objects. There is only resistance to accelerations (i.e., inertia and momentum) because of these limitations on the energy exchanges possible with the virtual (off-shell) radiations in the vacuum.

In a new blog post, Dr Woit quotes a New Scientist article about Erik Verlinde’s “entropic gravity”:

Like Woit, I don’t see much hope in Verlinde’s entropic gravity since it doesn’t make falsifiable predictions, just *ad hoc* ones, but the idea that gravity is an “emergent property of the way objects are organised, much as fluidity arises as a property of water” is correct: gravity predicted accurately from the shadowing of the implosive pressure from gravitons exchanged with other masses around us. At best, mainstream quantum gravity theories such as string theory and loop quantum gravity are merely compatible with a massless spin-2 excitation and thus are wrong, *ad hoc* theories of quantum gravity, founded on error and which fail to make any quantitative, falsifiable predictions.

So Dr Woit has finally flipped, giving up on precise mathematical expressions and coming round to the “much better” vague and mysterious ideas of the mainstream string theorists. Well, I think that’s sad, but I suppose it can’t be helped. Newton in 1692 scribbled in his own printed copy of his Principia that Fatio’s 1690 gravity mechanism was “the unique hypothesis by which gravity can be explained”, although Newton did not publish any statement of his interest in the gravitational mechanism (just as he kept his alchemical and religious studies secret).

**Update:**

John Rennie has commented on Woit’s blog:

“I think you’re being a bit harsh when you say:

I guess a precise mathematical expression of a theory is somehow undesirable, much better to have a vague description in English about how it’s all due to some mysterious entropy.

This is a valid point: finding a way to make predictions with quantum gravity doesn’t mean “abandoning” general relativity, but supplementing it by giving additional physical insight and making quantitative, falsifiable predictions. Although Professor Halton Arp (of the Max-Planck Institut fuer Astrophysik) promotes heretical quasar redshift objections to the big bang which are false, he does make one important theoretical point in his paper *The observational impetus for Le Sage Gravity:*

‘The first insight came when I realized that the Friedmann solution of 1922 was based on the assumption that the masses of elementary particles were always and forever constant, m = const. He had made an approximation in a differential equation and then solved it. This is an error in mathematical procedure. What Narlikar had done was solve the equations for m= f(x,t). This a more general solution [to general relativity], what Tom Phipps calls a covering theory. Then if it is decided from observations that m can be set constant (e.g. locally) the solution can be used for this special case. What the Friedmann, and following Big Bang evangelists did, was succumb to the typical conceit of humans that the whole of the universe was just like themselves.’

The remainder of his paper is speculative, non-falsifiable or simply wrong, and Arp is totally wrong in dismissing the big bang since his quasar “evidence” has empirically been shown to be completely bogus, while it has also been shown that the redshift evidence definitely does require expansion, since other “explanations” fail. But Arp is right in arguing that the Friedmann *et al.* solutions to general relativity for cosmological models are all based on the implicit assumption that the source of gravity is not an “emergent” effect of the motion of masses in the surrounding universe. The Lambda-CDM model based on general relativity is typical of the problem, since it can be fitted in *ad hoc* fashion to virtually any kind of universe by adjusting the values of the dark energy and dark matter parameters to force the theory to fit the observations from cosmology (the opposite of science, which is to make falsifiable predictions and then to check those predictions). That’s a religion based on groupthink politics, not facts.

**Update**

Copy of comment to:

http://scienceblogs.com/builtonfacts/2010/02/failing_at_gravity.php

“But there’s problems, too. There ought to be “air resistance” from the particles as the planets move through space. Then there’s the fact that the force is proportional to surface area hit by the particles, not to the mass. This can be remedied by assuming a tiny interaction cross-section due to the particles, but if this is true they must be moving very fast indeed to produce the required force – many times the speed of light. And in that case the heating due to the “air resistance” of the particles would be impossibly high. Furthermore, if the particle shadows of two planets overlapped, the sun’s gravity on the farther planet should be shielded. No such effect has been observed.

“For these and other reasons Fatio’s theory had to be rejected as unworkable.”

Wikipedia is a bit unreliable on this subject: Fatio assumed on-shell (“real”) particles, not a quantum field of off-shell virtual gauge bosons. The exchange of gravitons between masses in the universe would cause the heating, drag, etc., regardless of spin if the radiation were real. So it would dismiss spin-2 gravitons of attraction, since they’d have to be everywhere in the universe between masses, just like Fatio’s particles. But in fact the objections don’t apply to gauge boson radiations since they’re off-shell. Fatio didn’t know about relativity or quantum field theory.

Thanks anyway, your post is pretty funny and could be spoofed by writing a fictitious attack on “evolution” by ignoring Darwin’s work and just pointing out errors in Lamarke’s theory of evolution (which was wrong)…

“This can be remedied by assuming a tiny interaction cross-section due to the particles, but if this is true they must be moving very fast indeed to produce the required force – many times the speed of light.”

Or just increasing the flux of spin-1 gravitons when you decrease the cross-section …

Pauli’s role in predicting the neutrino by applying energy conservation to beta decay (against Bohr who falsely claimed that the energy conservation anomaly in beta decay was proof that indeterminancy applies to energy conservation, violating energy conservation to explain the beta decay anomaly without predicting the neutrino to take away energy!), and in declaring Heisenberg’s vacuous (unpredictive) unified field theory to be “not even wrong”, is well known, thanks to Peter Woit.

There is a nice anecdote about Markus Fierz, Pauli’s collaborator in the spin-2 theory of gravitons, given by Freeman Dyson on p. 15 of his 2008 book *The Scientist as Rebel:*

“Many years ago, when I was in Zürich, I went to see the play

The Physicistsby the Swiss playwright Friedrich Dürrenmatt. The characters in the play are grotesque caricatures … The action takes place in a lunatic asylum where the physicists are patients. In the first act they entertain themselves by murdering their nurses, and in the second act they are revealed to be secret agents in the pay of rival intelligence services. … I complained about the unreality of the characters to my friend Markus Fierz, a well-known Swiss physicist, who came with me to the play. ‘But don’t you see?’ said Fierz. ‘The whole point of the play is to show us how we look to the rest of the human race’.”

copy of a comment to:

http://scienceblogs.com/builtonfacts/2010/02/failing_at_gravity.php

“But there’s problems, too. There ought to be “air resistance” from the particles as the planets move through space. Then there’s the fact that the force is proportional to surface area hit by the particles, not to the mass. This can be remedied by assuming a tiny interaction cross-section due to the particles, but if this is true they must be moving very fast indeed to produce the required force – many times the speed of light. And in that case the heating due to the “air resistance” of the particles would be impossibly high. Furthermore, if the particle shadows of two planets overlapped, the sun’s gravity on the farther planet should be shielded. No such effect has been observed.

“For these and other reasons Fatio’s theory had to be rejected as unworkable.”

Wikipedia is a bit unreliable on this subject: Fatio assumed on-shell (“real”) particles, not a quantum field of off-shell virtual gauge bosons. The exchange of gravitons between masses in the universe would cause the heating, drag, etc., regardless of spin if the radiation were real. So it would dismiss spin-2 gravitons of attraction, since they’d have to be everywhere in the universe between masses, just like Fatio’s particles. But in fact the objections don’t apply to gauge boson radiations since they’re off-shell. Fatio didn’t know about relativity or quantum field theory.

Thanks anyway, your post is pretty funny and could be spoofed by writing a fictitious attack on “evolution” by ignoring Darwin’s work and just pointing out errors in Lamarke’s theory of evolution (which was wrong)…

“This can be remedied by assuming a tiny interaction cross-section due to the particles, but if this is true they must be moving very fast indeed to produce the required force – many times the speed of light.”

Or just increasing the flux of spin-1 gravitons when you decrease the cross-section …

Posted by: Nige Cook | February 11, 2010 11:55 AM

I was just suspended from Physics Forum for supporting the spin-1 graviton. It seems spin-1 gravitons are impossible because gravity is only attractive. So there is no such thing as anti-gravity, as with antimatter?