Lorentz-Poincaré relativity and a scalar theory of gravitation

1. The Lorentz-Poincaré version of special relativity

2. Why is the Lorentz-Poincaré version of special relativity important for the theory of gravitation?

3. References quoted in parts 1 and 2

4. A naive introduction to the scalar ether-theory of gravitation   

5. A short guide to the new scalar theory of gravitation  

To a review on ether theories of gravity, with the scalar theory reviewed in more detail 

The currently accepted theory of gravitation is Einstein's general relativity (1916), which arose after a decade of research trying to make gravitation compatible with relativity. Once the heart of the theory that we now name special relativity had been found in 1905, it was indeed obvious that Newton's gravitation was not compatible with this theory. The first attempt to modify Newton's theory in order to make it compatible with special relativity was made by Poincaré (1905 , 1906): he formulated a Lorentz-invariant modification of Newton's attraction law. Later he found (Poincaré 1908) that this modification accounted for just one sixth of Mercury's then unexplained residual advance in perihelion, of 43'' per century.

1. The Lorentz-Poincaré version of special relativity

Poincaré's version of the relativity principle was different from Einstein's (1905), indeed Poincaré's version (1904) was this: "The laws of physical phenomenons must be the same, whether for a fixed observer, as also for one dragged in a motion of uniform translation, so that we do not and cannot have any mean to discern whether or not we are dragged in a such motion." This formulation does not preclude in any way the possibility that there might be an absolute meaning to the words "fixed" and "moving". Like Lorentz, Poincaré did not actually reject the concept of an "ether" [see e.g. Poincaré (1908)]. The concept of an ether, or physically preferred reference frame, had been dominating the 19th century physics after the discovery that light behaved as waves. (It was then considered difficult to think of waves without a medium to carry them: is it easier now?) Thus the Lorentz-Poincaré version of special relativity starts from the "ether", seen as an inertial frame E such that (i) Maxwell's equations are valid in E (in particular, light propagates at a constant velocity c with respect to E) and (ii) any material object that moves with respect to E undergoes a Lorentz contraction. One derives first the Lorentz transformation from these assumptions, and then the whole of special relativity follows. This theory (Lorentz 1904 , Poincaré 1905 , Poincaré 1906) has been thoroughly investigated by Jánossy ( 1957 , 1965 ), Builder (1958), Prokhovnik ( 1967 , 1993 ), Pierseaux (1999) , Brandes (2001), among others, who all showed that it is physically equivalent to the textbook version of special relativity. [The latter is, of course, derived from the work of Einstein (1905) starting from (i) the relativity principle and (ii) the invariance of the velocity of light.] Indeed both theories lead to the same equations: the Lorentz transformations for coordinates and for velocities, the relativistic dynamics with velocity-dependent mass, the transformation of the electromagnetic field, etc. The crucial equations are already explicitly written in Poincaré (1905), the other ones are in Poincaré (1906). It is immediate to see this for the Lorentz transformations, and for the rest it appears very clearly if one changes Poincaré's notations and adopts current notations, as was shown by Logunov (1995). Apart from the historical aspect, it is important for physics research to realize that the physics of special relativity was found by scientists who based their work on the concept of ether. Indeed today's physics makes it obvious that vacuum has physical properties.

The Lorentz-Poincaré version sees the key "relativistic" effects (space contraction and time dilation) as real physical effects of the motion with respect to the "ether": a moving ruler is just "really" smaller than a fixed one, a moving clock just has a "really" greater period than a fixed one. Here, "really" means: with the simultaneity defined in the preferred inertial frame or ether E, by using the standard synchronization procedure with light signals [first proposed by Poincaré (1900), discussed  in more detail in Poincaré (1904), and rediscovered (?) by Einstein (1905)]. The reciprocity of these metrical effects (when interchanging the respective roles of two inertial frames) is then seen as an illusion due to the necessity of still using the Poincaré-Einstein synchronization convention in a moving frame: As long as we don't know the velocity V of our inertial frame F with respect to E, we are obliged to admit that, also in F, light has the same velocity in both directions n and -n, for whatever n (that velocity must then be the constant c). What experiments like Michelson-Morley's ask for, and what the rod contraction and clock slowing say, is merely that the velocity of light on a to-and-fro path is the constant c. In summary: according to the Lorentz-Poincaré interpretation of special relativity, the metrical effects of uniform motion are seen as absolute effects due to the motion with respect to the preferred inertial frame, and the relativity of simultaneity is just an artefact. That artefact leads to the reciprocity of the "relativistic" effects, which reciprocity is thus itself an artefact.

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2. Why is the Lorentz-Poincaré version of special relativity important for the theory of gravitation?

If all of physics is really Lorentz-invariant, we shall never be able to measure the "absolute" velocity V, so that the difference between the "true" simultaneity (that defined in the preferred inertial frame E) and the "artificial" ones (those defined in the other inertial frames F) will remain at the metaphysical level. In that case, one may indeed qualify the concept of ether as "superfluous", as Einstein (1905) put it. His own starting point, from the relativity principle, is then to be preferred. Moreover, the full equivalence between all inertial frames, obtained in that case, leads to give a physical meaning to the concept of space-time, which was Minkowski's (1907) assumption. [In his "Rendiconti" paper, Poincaré had already introduced the concept of the space-time, as a 4-dimensional space with coordinates x, y, z, ict ; but this looked more like a (very useful and clever) mathematical tool in his paper.] This means that past, present and future are relative notions, and that time travels are at least theoretically possible. It also enforces to consider this mixture of space and time as the true arena for physical theories. In particular, it may lead (though not in a compelling way) to Einstein's general relativity (GR), according to which gravitation is the curvature of space-time, and in which free test particles follow the geodesics of a pseudo-Riemannian metric on space-time. [If general relativity is correct, then, according to Bonnor (2002), time travels are indeed quite plausible.] In my opinion, this is no physical explanation of gravity. Moreover, GR's invariance under arbitrary coordinate changes is antinomic to quantum theory, in which, in particular, the choice of the time coordinate cannot be arbitrary. This leads to the longstanding difficulties with quantum gravity, which the mainstream researchers want to overcome by going into string- and M-theories with their many-dimensional manifolds. One should be allowed to explore simpler possibilities.

There is indeed another possibility, which seems to have been little explored before: that not all of physics is subjected to the relativity principle, and that the force which contradicts relativity is precisely gravitation. (Of course, the conflict with relativity has to be small in usual conditions.) This possibility exists only in the Lorentz-Poincaré version of special relativity (SR), that sees SR as a consequence of the Lorentz contraction: it obviously doesn't exist if the reason for SR is the universal validity of the relativity principle, as in Einstein's version. Why should gravitation be the range of a such violation? Simply because SR really doesn't include gravitation. After all, even in GR, the relativity principle of SR does not hold true.

The scalar theory investigates just this possibility. It does have a physical explanation for gravity, it is much simpler than GR, yet it seems to agree with observational facts. Rigorous work has been done, and further work is in progress, to check this in celestial mechanics, and the results of this work are, in my opinion, of importance also for GR.

3. References quoted in parts 1 and 2

I am grateful to the Institutions that implemented the online versions of the original works, and to Emili Bifet, who provided the links to those online versions.

W. B. Bonnor (2002) Communication at the Second British Gravity Meeting, Queen Mary University of London, June 10/11, 2002; http://arxiv.org/abs/gr-qc/0211051.

J. Brandes (2001) Die Relativistischen Paradoxien und Thesen zu Raum und Zeit, Verlag Relativistischer Interpretationen, Karlsbad.

G. Builder (1958) Austr. J. Phys., 11 , 279-297 and 457-480.

A. Einstein (1905) Ann. der Phys., (4) 17, 891-921 (Eingegangen 30. Juni 1905).

A. Einstein (1916) Ann. der Phys., (4) 49, 769-822.

L. Jánossy (1957) Usp. Fiz. Nauk, 62, N°1, 149-181.

L. Jánossy (1965) Acta Phys. Polon., 27, 61-87.

A. A. Logunov (1995) On the Articles by Henri Poincaré "On the Dynamics of the Electron" (English translation G. Pontecorvo), Publishing Department of the Joint Institute for Nuclear Research, Dubna. (1st Russian edition 1984.) This monograph includes an English translation of Poincaré 1905 and 1906.

H. A. Lorentz (1904) Proc. Acad. Sci. Amsterdam, 6, 809- . German translation: H. A. Lorentz, A. Einstein, H. Minkowski, "das Relativitaetsprinzip", Teubner, Leipzig u. Berlin, 1913, pp. 6-26.

H. Minkowski (1907) Nachr. Kgl. Ges. d. Wiss. zu Goettingen, Math.-phys. Kl., Sitzung vom 21. Dezember 1907.

Y. Pierseaux (1999) La "Structure Fine" de la Relativité Restreinte (L’Harmattan, Paris).

H. Poincaré (1900) Arch. Néerland. Sci. Ex. Nat., (2) 5, 252-278 (1900). Reprinted in Oeuvres Complètes, tome IX (Gauthier-Villars, Paris, 1954), pp. 464-488.

H. Poincaré (1904) Bull. Sci. Math., (2) 28, 317-, November 1904. Reprinted in La Valeur de la Science (Flammarion, Paris, 1905), chapters 7,8 and 9. English translation: Bull. Amer. Math. Soc. 37 (2000), 25 - 38. (Several books of Poincaré may be accessed on line - in French - at  http://www.ac-nancy-metz.fr/enseign/philo/textesph/default.htm#poinca).

H. Poincaré (1905) C.-R. Acad. Sci. Paris (Séance du 5 Juin 1905) 140, 1504-1508. (The online version is due to the Bibliothèque Nationale de France.) Reprinted in Oeuvres Complètes, tome IX (Gauthier-Villars, Paris, 1954), pp. 489-493.

H. Poincaré (1906) Rendiconti Circ. Matemat. Palermo (Adunanza del 23 Luglio 1905), 21, 129-176. Reprinted in Oeuvres Complètes, tome IX (Gauthier-Villars, Paris, 1954), pp. 494-550, and also in La Mécanique Nouvelle (Gauthier-Villars, Paris, 1924), pp. 18-76. (The online version is due to the Bibliothèque Nationale de France.) Partial English translation available at

H. Poincaré (1908) Rev. Gén. Sci. Pures et Appl., 19, 386-402. Reprinted in Oeuvres Complètes, tome IX (Gauthier-Villars, Paris, 1954), pp. 551-586.

S. J. Prokhovnik (1967) The Logic of Special Relativity (Cambridge University Press, Cambridge, U.K.).

S. J. Prokhovnik (1993) Z. Naturforsch., 48a, 925-931.

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4. A naive introduction to the scalar ether-theory of gravitation

a ) The micro-ether

b ) Archimedes' thrust in the micro-ether

c ) The micro-ether as the universal fluid. The velocity of light and the velocity of "sound"

d ) The macroscopic nature of the fields pe and rho_e 

e ) The macro-ether or preferred reference frame of the theory

 f ) The limiting case of Newtonian gravity and the effect of the compressibility of the fluid

 g ) Provisional conclusion and acknowledgment

 a ) The micro-ether     

The basic idea of this theory is that gravity could be just Archimedes' thrust exerted on matter at the scale of elementary particles by a perfect continuous fluid, which I call the micro-ether (you will see at point e ) why "micro"). The fluid is assumed continuous at any scale. It means that its fluid nature is not the resultant of a microscopic substructure made of a myriad of molecules that move in all directions, but is intrinsic. In contrast, the liquids or gases that we know are made of molecules. The pressure exerted by them may thus be interpreted as resulting from shocks of those tiny particles. I.e., we now know that it just turns out to be so for the fluids we meet every day: water, air, wine, etc. However, mind that the molecular nature of these fluids is known since hardly more than a century. During several centuries, it was considered that all of these fluids are continuous, and that did not make any problem in interpreting their mechanical behaviour, which is the behaviour of interest here. I mean that from the mechanical point of view, fluids might be continuous as well. For sure, the intuitive notion of a fluid refers to something continuous, may be it is the most obvious representation of what "continuous" means. What makes the molecular structure of usual fluids necessary is their thermal behaviour. An usual fluid may be cold or hot, i.e., it may take energy from neighbouring bodies or bring energy to them, and we know that this is due to the random motion of the constitutive molecules: the temperature of the fluid is related to the mean kinetic energy per particle. By contrast, we may infer that a truly continuous fluid has no temperature and does not allow any heat exchange, hence it must be a perfect fluid, with merely two state variables: pressure and density, which are connected together by a one-to-one relationship -- the barotropic state equation.

) Archimedes' thrust in the micro-ether

Let us assume that the void between the molecules of ordinary matter, or rather (since molecules are composite objects) between the elementary particles: electrons, quarks, etc., is not empty in fact, but is filled by a such continuous fluid. Because a such fluid should be a perfect fluid, the motion of the particles is not braked by the presence of this fluid (this is well-known in fluid mechanics as "d'Alembert's paradox", but it is not a paradox actually). At least, this is so for a uniform motion. As to accelerated motion of a particle, it does feel the fluid, in a way that might give insights on the relativistic mass increase with velocity -- but that is another story. Also, the particles are not point particles (a point particle does not make any sense, it seems to me), instead each has a finite size. In that case, if there is a gradient of the fluid pressure, say more pressure upwards, then any particle will be pushed downwards, with a force proportional to the volume occupied by the particle -- Archimedes' thrust. (Think of a ball immersed in the sea; in the sea, the pressure gradient is downward, so the ball wants to go upwards.) Moreover, assume that the mass density inside a particle [the ratio (mass of the particle)/(volume of the particle)] is the same for all particles. Then each particle will be attracted downwards (in fact, precisely in the opposite direction to the pressure gradient), with a force proportional to the particle's mass: Newton's attraction. It turns out that Newton's gravity had been interpreted in that way ca. 1746 by Euler (I learned this several years after I had this idea).

) The micro-ether as the universal fluid. The velocity of light and the velocity of "sound"

We know from accelerators experiments that existing particles, even the elementary ones (electrons, photons, quarks, ...), can be transformed into other ones, e.g. electron-positron annihilation giving rise to photons. We also know from quantum mechanics that there is some fuzziness in their position and velocity, and that there is some indiscernability between identical particles. These facts would be easier to understand if actually all particles were just local organizations of the hypothetical fluid which I assumed above to be between them. Thus, an elementary particle would be a kind of localized flow in the micro-ether, something like a vortex. (A vortex may be everlasting in a perfect fluid; however, note that particle physics says that there are much more unstable particles or "resonances" than stable ones: this also is easy to understand with this assumption.) In other words, the micro-ether would be the universal, "constitutive" fluid, of which any matter should be made. If we assume this, it becomes just natural that the mass density inside a particle is the same (or nearly so) for all particles -- the more so, if this hypothetical universal fluid has a very low compressibility:  the common density inside the particles would then be just the nearly uniform density rho_e in the universal fluid. The mathematical form of the relation between the (field of) ether pressure pe and the gravity acceleration g is then fixed (Ref. [A18], Eq. (5), or Ref. [A28], Eq. (2.3)):  

 g = - (grad pe)/rho_e.                        (1)

Actually, relativity suggests strongly that the compressibility of the micro-ether should be K = 1/c2 with c the velocity of light, which is indeed an extremely small compressibility. The latter equality means exactly that the "velocity of sound" (the propagation speed of the pressure waves) in this fluid, say ce, is equal to c. This is indeed suggested by relativity together with this assumption of a "constitutive ether": we know from special relativity that mass particles cannot be accelerated up to reaching the velocity c. On the other hand, if any elementary particle is a kind of organized flow in the micro-ether, the velocity of the pressure waves should set an upper limit to the velocity of elementary particles, because, on exceeding that velocity, a particle (a vortex or so) should be destroyed by shock waves. Hence, we must assume that ce = c, because there cannot be two different upper limits. This is equivalent to assuming that the state equation connecting the pressure pe and the density rho_e in the micro-ether is just pec2 rho_e. This relation, which was postulated by Lucien Romani in his 1975 book (see again [A18] for the reference and for a discussion of Romani's work in connection with this one), is quite crucial in the investigated theory. Note that current cosmological models often assume a similar relation for the "vacuum": pvacuum = -c2 rho_vacuum. Of course, "vacuum" is nothing else than another name for "ether" (people avoid using the original name, because we are told that the concept of ether has been discarded by special relativity; I hope that part 1 and part 3 may contribute to make the reader change his mind on this point). Putting the minus sign: pe = -c2 rho_e, would leave the interpretation of gravity as Archimedes' thrust formally unchanged, but would make the "sound" velocity ce imaginary!

 d ) The macroscopic nature of the fields
pe and rho_e 

Gravitation is a macroscopic force, in the sense that the gravity acceleration g varies significantly only over macroscopic distances, and due to the presence of massive, macroscopic bodies (although g is felt by small particles). This is related to the fact that gravity is a very small force as compared with the other forces (electromagnetic and nuclear), which enormously dominate at small distances. According to the concept of the micro-ether, gravity is indeed a kind of correction: the fields pe and rho_e, which are relevant to gravitation, are merely the "smoothed-out", spatially averaged fields, conceptually obtained from the true microscopic fields of pressure and density in the micro-ether. According to the concept of the micro-ether, these microscopic fields should have much larger variations, and this over small distances, and they -- together with the velocity field of the micro-ether, including its local arrangements as particles of matter... -- should be responsible for the other interactions. Clearly, it would be an ambitious program to try a reconstruction of micro-physics along this line! However, we do not need this to describe gravitation. 

 e ) The macro-ether or preferred reference frame of the theory

The ether should account for the inertial frames which are revealed by Newton's mechanics. The inertial frames of classical mechanics lead to Newton's concept of the absolute space: as is well-known, there is just one inertial frame, up to a uniform translation. (Rotation, in particular, is detectable by experiments such as Foucault's, but it is also the case for any kind of deformation of the selected reference frame with respect to an inertial frame [A8].) For Newton, this could be true only if one of the inertial frames had a privileged status. Thus the ether should define that preferred inertial frame, for otherwise we would have two independent privileged spaces: that of Newton's theory, plus the ether. The universal fluid which I postulate (following Romani), and which I name micro-ether, is assumed to have a complex microscopic motion, involving in particular those motions which should define the elementary particles of matter. Hence the micro-ether, whose motion thus includes that of any kind of matter, cannot define an inertial frame.

However, the average motion of matter at a very large scale defines the best approximation of an inertial frame. Note, for instance, that the astrometrical reference frames are kinematically defined from distant "astronomical pointers". These are material objects, including very distant quasars which have no detectable motion, and also including usual stars, which are closer objects, whose motion is indeed averaged. And those kinematical reference frames are excellent dynamical reference frames, i.e., they are very-well adapted to write and solve the equations of celestial mechanics -- in other words, they are good inertial frames. (The plural is to account for the successive improvements of the astrometrical reference frames.) Now, in the line of the concept of the constitutive micro-ether or universal fluid, it is natural to assume that the average motion of the micro-ether coincides with the average motion of matter. Therefore, the average motion of the micro-ether should define an inertial frame, which I call the macro-ether. It is in that reference frame that the equations of the theory, most of which are indeed merely space-covariant, are assumed to be true.

 f ) The limiting case of Newtonian gravity and the effect of the compressibility of the fluid

Thus, gravitation would result from the macroscopic part of the pressure gradient in the micro-ether. We expect then that, if that fluid is (macroscopically) compressible, a disturbance in the (macroscopic) ether pressure should propagate with the "sound" velocity,

ce = [dpe/d(rho_e)]1/2.               (2)

On the other hand, we know that Newton's gravitation propagates instantaneously. Therefore, it should correspond to the limiting case of an  incompressible fluid. The latter should be an excellent approximation in many cases, because Newton's gravity is already very accurate in certain situations (in particular at the solar-system scale!). Newton's gravity is characterized by 
Poisson's equation for the gravity acceleration g:

div(g) = -4 pi G rho                        (3)

with pi the usual number of trigonometry, G Newton's constant and rho the mass density of matter.
(div is the divergence operator.)  Together with Eq. (1) for g, the requirement that Poisson's equation (3) is recovered in the incompressible case, i.e., rho_e = Const., leads immediately to an equation for the field pe :

Delta (pe) = 4 pi G rho rho_e,                     (4)

where Delta = div grad  is the Laplace operator. It is natural to assume that this equation remains valid for the "real" case with a compressibility, in the particular situation where the effects of propagation can be neglected -- i.e., in the static situation (e.g. one unique massive body, possibly with "test particles" that do not affect the attraction field). Thus, in that case, rho_e  in Eq. (4) becomes a function of the local ether pressure, rho_e = rho_e (pe). However, in the general situation for the compressible case, there should be pressure waves, thus gravitational waves, propagating at the speed (2). (This may indeed be shown as a consequence of natural assumptions about the "disturbed" and "undisturbed" pressure and density fields, that leads to substituting the d'Alembert (wave) operator for Delta on the left of Eq. (4), in the general situation [A8].) Since, as explained at point c) above, relativity plus the concept of the "constitutive ether" lead to admit that ce = c, that theory naturally predicts gravitational waves propagating at the velocity of light.

 g ) Provisional conclusion and acknowledgment

I have tried to expand on the heuristic concept of the theory that I am investigating. It seems to me that this concept makes gravitation understandable. I could go further, but the development of the theory naturally leads to the equations playing a more and more important role. Equations and their derivable consequences are the most important thing in a theory, from the point of view of physical science (as opposed to "intuitive physics"). From this point of view, the heuristics developed in points a) to e) is just a way to the beginning of the scalar theory, in particular to the expression of g. (There is indeed another, phenomenological, way to the expression of g : see [A16], § 3.) The interested reader might have a look on the guide below. Although most papers are rather technical, several ones contain an important amount of non-technical comments, especially [A18], [A28], [B13], and [B17].

Many persons expressed encouragements to this theory and/or made interesting remarks about it. They are too numerous to be quoted here. However, I wish to express my  particular gratitude to the late Eugen Soos and to Pierre Guélin, both of whom gave me a lot of their time, and to my wife Marie-Alix Arminjon, who is enduring since quite a lot of years my devotion to this curious activity.

5. A short guide to the new scalar theory of gravitation

a) To get an idea
b) The development of the theory; the dynamics in the theory
c) Asymptotic post-Newtonian approximation (PNA) and equations of motion
d) Celestial mechanics
e) Gravitational radiation and pulsar energy loss
f) Cosmology
g) Link with other parts of physics: electromagnetism, quantum theory

a) To get an idea

In short, this is a preferred-frame bimetric theory, in which the scalar gravitational field both determines the gravity acceleration vector and influences the metric: there is a gravitational dilation of physically measured distances, as compared with distances evaluated with the Euclidean space metric; and there is a gravitational contraction of measured time intervals, as compared with the intervals of the "absolute time" (the preferred time coordinate of the theory). This gives a "curved space-time", as in GR, but here with a simple origin, and together with a flat space-time (depending on which of the two metrics is chosen). The dynamical equations are based on a consistent formulation of Newton’s second law in a curved space-time.

Ref. [B25] is a short, readable summary of the construction of the theory, in the context of some modification whose necessity appeared rather recently [see point c) iii) below]; a more extended summary of that modified theory is Ref. [B26]. A quick technical presentation of the main equations of the initial theory and its first test in celestial mechanics (spring 2002) can be found in Ref. [B21]. The first part of Ref. [A28] gives a rather detailed summary of the motivation for the theory and the construction of it (Sect. 2; a more incisive introduction is Ref. [B17]); the status (at summer 2001) of the observational confrontation is then reviewed (Sect. 3 of [A28]); afterwards the application of the theory to cosmology is discussed. In those three papers, reference is made to the earlier work.

b) The development of the theory; the dynamics in the theory

The bulk of the theory was presented in Ref. [A18], including the interpretation of gravity as a pressure force and the derivation of the field equation, and together with some applications - that to the spherically symmetrical case includes the gravitational collapse of a dust sphere. (Ref. [A8] presented an extension of Newtonian mechanics to fluid inertial frames and described in more detail the beginning of the construction of the theory ; Ref. [A9] described in detail a further step of this construction and the first test of the theory.) In Ref. [A15], the question of the equations of motion in the theory (geodesics or Newton's 2nd law?) is reexamined in the context of obtaining a conservation law for energy, and a simpler form of the gravitational equations is got, moreover the r.h.s. of the equation for the scalar gravitational field is precised. (Due to the review process, Ref. [A18] was published after [A15].)

The link between Newton's second law and Einstein's geodesic motion has then been found in a general context [A16]. In Ref. [A20], the equation for continuum dynamics (replacing the equation  of GR) is derived, and applied to show the occurrence of matter creation or destruction by a reversible exchange with the gravitational field. References [A16 , A20] are valid for the modified version of the theory [point c) iii) below] as well, because the dynamical equations are unchanged.

The whole construction of the theory was reviewed in Ref. [B13], together with its observational status as of 1998.
However, there was in fact a bifurcation in the construction of the theory, and it happens that the initially-chosen branch was the wrong one, see point c) iii) below -- but the main features of the theory are preserved.

c) Asymptotic post-Newtonian approximation (PNA) and equations of motion

i) The PNA is initiated for test particles in Ref. [A19] and applied there to light rays, with the consequence that the predictions of the scalar theory for light rays are the same as those of general relativity. (This remains true [A35] for the modified version of the scalar theory.) A systematic study of that approximation is presented for extended bodies in Ref. [A23], which derives the expanded local equations and the expanded boundary conditions. It is also shown there that the standard (Fock-Chandrasekhar) PNA does not pertain to the usual method of asymptotic expansions. However, in the particular case of a test particle in a Schwarzschild field, the asymptotic and standard PNA's are equivalent [A29]. In Refs. [A25] and [A26], the local equations are integrated inside the volume of the bodies, to get global translational equations of motion for the mass centers. However, it turns out to be necessary to introduce a definite asymptotic framework for the small parameter eta that quantifies a good separation between the gravitating bodies [A32]. As a result, the final, tractable equations of motion depend explicitly on structure parameters.

ii) The same has been found for general relativity in the harmonic gauge, by applying the same method: first, using the asymptotic PNA, one expands the exact local equations for a perfect-fluid system; then one integrates the local PN equations in the bodies; and finally one expands these integrated equations w.r.t. the separation parameter eta. Apart from corrections that cancel when there is exact spherical symmetry at Newtonian order, there is in the final equations of motion one additional term, as compared with the Lorentz-Droste (Einstein-Infeld-Hoffmann) acceleration [A36 , B27]. This term depends on the spin of that body whose the acceleration is calculated (self-acceleration term) and on its internal structure (universality-violating term), and does not seem negligible for the giant planets.

iii) Moreover, for the scalar theory, it has been found [A33] that the equation of motion for a test particle does not coincide with the point-particle limit of the equation of motion for extended bodies. This violation of the weak equivalence principle is due to the anisotropy of the spatial metric and, therefore, might also be true for GR, depending on the gauge [A33] (see a summary in [B23]), but this would be more difficult to check (due to the greater complexity of GR). For the scalar theory, this problem is not difficult to solve: one modifies the spatial metric to get it isotropic, which is nearly as natural as the initial assumption of a unidirectional space contraction [O3 , B25]. This leaves the dynamical equations of the theory unchanged (apart from the different metric), but, to keep the energy conservation law, it is necessary to modify the equation for the scalar gravitational field: now its right-hand side is just the (flat) wave or d'Alembert operator. This does solve the problem with the weak equivalence principle [A35]. However, the final, tractable equations of motion for the mass centers of a system of extended bodies, according to the modified version, are yet to be published.

d) Celestial mechanics

To check the theory, detailed calculations of solar system ephemerides are being done, and compared with ephemerides of the Jet Propulsion Laboratory. To this end, an adjustment program has been built, that loops on the numerical integration of the equations of motion for the mass centers, in order to optimize the parameters [A31]. This code has been tested by investigating in which measure one may reproduce (over one century) the predictions of the DE403 ephemeris, by using purely Newtonian equations of motion [A31]. It has also been applied to adjust over 60 centuries a less simplified model, in which the PN corrections in the Schwarzschild field of the Sun are also considered [A30].When one implements in this code the equations of motion obtained with the asymptotic PNA of the scalar theory, one finds [O1] a small, but significant deviation from the JPL ephemeris, that is based on the standard PNA of GR. This difference comes in the first place from using the asymptotic PNA instead of the standard PNA. The difference is larger if the spins of the planets are accounted for [O2, § 4.6.2]. Ref. [O1] contains also a summary of the "asymptotic" PNA used for getting the equations of motion. The "asymptotic" PNA is compared with the standard PNA and it is conjectured that, if a general asymptotic PNA can be built also for general relativity (which seems difficult), its equations and numerical results should also differ from the standard PNA. This has now been verified specifically for the case of GR in the harmonic gauge (which is the gauge used in relativistic celestial mechanics), see  point c) ii) above.

The test of the theory in celestial mechanics should be redone with the modified version of the theory [see point c) iii)]. My guess is that it should be at least as good as with the initial version. There is a lot of specialized parameter adjustment in celestial mechanics [O2, Sect. 4.6]. Therefore, I now believe that the question of the link with quantum theory may be a more urgent work for a theorist [point g) ii) below].

e) Gravitational radiation and pulsar energy loss [A34 , B24]

An asymptotic scheme of post-Minkowskian (PM) approximation is built by associating a conceptual family of systems with the given weakly-gravitating system. It is more general than the post-Newtonian scheme in that the velocity may be comparable with c. This allows to justify why the 0PM approximation of the energy rate may be equated to the rate of the Newtonian energy, as is usually done. At the 0PM approximation of this theory, an isolated system loses energy by quadrupole radiation, without any monopole or dipole term. It seems plausible that the observations on binary pulsars (the pulse data) could be nicely fitted with a timing model based on this theory.

f) Cosmology [A28]

An analytical cosmological solution is obtained for a general form of the energy-momentum tensor. According to that theory, expansion is necessarily accelerated, both by the vacuum and even by matter. In one case, the theory predicts expansion, the density increasing without limit as time goes back to infinity. In the other case, the Universe follows a sequence of (non-identical) contraction-expansion cycles, each with finite maximum energy density; the current expansion phase will end by infinite dilution in some six billions of years.

g) Link with other parts of physics: electromagnetism, quantum theory

i) Electromagnetism: Section 7 of the review paper [B13] summarizes the derivation of gravitationally-modified Maxwell equations and the proof of the consistency of these equations with photon trajectories in the theory.

ii) Quantum theory: In Ref. [B15], Schrödinger's original wave mechanics is analyzed from the viewpoint of the modern theory of linear wave equations and their dispersion relations. This allows to extend the Klein-Gordon relativistic wave equation to the case where a constant gravitational field is present. It is argued that Schrödinger’s wave mechanics can be extended to the case with a variable gravitational field only if one accepts that the wave equation is a preferred-frame one. (Recent work suggests that a preferred-frame equation is indeed a possibility, but not the only one [A39].) From this viewpoint, generally-covariant extensions of the wave equations of quantum mechanics seem rather formal. Finally, it is conjectured that there is no need for a quantum gravity.

But the formulation of quantum mechanics in a gravitational field remains a problem, and I am pursuing its study along this line of research; in particular, I am studying the formulation of the Dirac theory in a curved space-time [A37, A38 , A39]. Thus, the Dirac equation has been derived directly from wave mechanics, i.e., from the classical Hamiltonian [A37]. This derivation applies either in the free case or with an electromagnetic field, as also in a gravitational field, be it a static one [A37] or a general one [A39]. However, two distinct gravitational Dirac equations may be thus derived, none of which does coincide with the standard gravitational Dirac equation, that is due to Weyl and to Fock [A39]. Moreover, the transformation behaviour of the Dirac equation has to be changed: instead of leaving the Dirac matrices unchanged and the wave function follow the (counterintuitive) spinor transformation, one may transform the set of the Dirac matrices as a third-order tensor and transform the Dirac wave function as a (spacetime) vector [A37A39]. This makes the Dirac equation covariant as well, and in fact for general coordinate changes.

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Copyright Mayeul Arminjon, 25 September 2002. Last updated 30 May 2007.