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General relativity as a unified fluid and field theory

Einstein’s dream for a unified field theory of Nature is attained with a classical fluid theory founded on space, time and spin, rather than on Einstein’s spacetime. An invariant quantum theory for the primordial fluid obeys the homogeneous Klein-Gordon equation, which is the same three-dimensional classical wave equation (CWE) initially tried by Schrödinger to formulate quantum mechanics (QM), but abandoned by linear superposition considerations. Primordial fluid pervades universe, obeys energy and momentum conservation, and is formed by sagions: energy-like, discrete, extended Planck-size objects of finite size, carriers of linear momentum and spin, moving in absolute 3D-curved space with speed C along straightest path. We briefly describe our novel non-harmonic and inherently quantized solutions for CWE in spherical coordinates, discovered in 1995. Solutions include a steady-state background field (possibly related to the CMB, to non-locality, and to action-at-a-distance), quantized helices, and inherently quantized functions exhibiting stable dynamic equilibrium, and isomorphism under many transformations, including the classical Doppler case, and the relativistic Lorentz, Poincaré, and Einstein transformations. Isomorphism pre-empts ab initio a few interpretative issues regarding relativistic and classical transformations. Mathematical fields represent the realistic physical temporal evolution of the primordial fluid in curved 3D-space.

predictive capability of Newtonian mechanics, defects (a) and (b) were pragmatically accepted or tolerated. Unfortunately, it is not generally known that in his private letters [3] Newton perseveringly and strongly argued for an ether (be it material, or immaterial) to explain gravity propagation [4]. An economic solution for issues (a) and (b) is provided by a Cartesian approach based on conservation of quantity of motion as a primitive notion [5][6] thus downgrading force and mass from the role they have in Newtonian mechanics. A consistent kinematic definition of mass was proposed by Barré de Saint Venant around 1850 [7][8]. Without mentioning him, the method of Saint Venant is implemented in some modern textbooks [9]: stroboscopic photography to measure average velocity of two interacting bodies before and after a collision upon a pneumatic table. In contrast, the betterknown operational definition of mass proposed by Mach around 1870 is based on mutually induced accelerations during the interaction of two bodies [8,10]. In criticizing the notions of instant velocity and instant acceleration fields, Russell [11] noted that to measure average velocity two observations at separate intervals of time are required, while an estimate for average acceleration requires at least three observations. From Ockham's razor, Saint Venant's more economical method is preferable to Mach's approach. Our energy-like primordial fluid (PF) is a modern version of ether, which inherently carries an indestructible quantity of motion, thereby explaining propagation of the derived notion of forcewhich merely is the average exchange of quantity of motion in a collision. Such PF provides a causal and physically objective basis for the mathematical notion of field [4,[12][13].
Passing now to open questions in QM, the completeness issue noted by Einstein still is there. Secondly, the Copenhagen interpretation of QM accepts the existence of undefined states of Nature between being and not-being -as the iconic Schrödinger's cat, and Wigner's friend. To stress the philosophical absurdity of the latter, Einstein used to pose the rhetoric question of whether the Moon existed when an observer was not looking at her! The Einstein-Podolsky-Rosen paradox and the ensuing Bell theorem are common knowledge. This writer agrees [14] with Einstein's views on causality and objectivity. Finally, there is Born's interpretation of the QM-wave-function as a probability. Since Jacob Bernoulli's Ars Conjectandi (circa 1713), it is well-known that probability is not a physical magnitude, but a mere mathematical construct of human mind [15]. Further, present writer posits that the rest of Natureas far as known on earthdoes not conjecture about outcomes of forthcoming events. If there is little information, probability is subjective and is a mere (non-) informed guess; probability is less subjective if there is symmetry, or when it is viable to enumerate the (hopefully) complete list of physical trajectories leading to an outcome, as in Feynman's continuous path integrals, or in discrete decision trees [16]. When long-term observations or controlled experimentation are viable, objective frequency-based probability is obtained. However, in any of the three previous scenarios probability is not a physical object. In 1952 Louis de Broglie recalled [17] a distinction he made in the mid-1920s between the continuous solutions of the linear QM-equations and its interaction with a localized (presumably material) corpuscle in such wave field, leading to an objective and causal "dual solution", that foreshadowed t'Hooft remarks [1] for eighty years. Since this mathematically coupled problem was difficult to solve, de Broglie settled for a simpler pilot-wave model: motion of a material corpuscle is controlled by the wave function Ψ . However, he also abandoned this proposal after realizing what we already said in previous paragraph: "Ψ certainly is not a physical reality... it only is a representation of probabilities dependent upon the state of our knowledge" [17] (our literal translation from French, page 266). As alternative to the Copenhagen view of QM, de Broglie suggested in the 1950s a stochastic interpretation, where Schrödinger's linear equation represents fluctuations of a fluid [18], but the fluid itself was proposed much earlier by . The five-dimensional universe suggested by Klein in 1926, led de Broglie in 1927 [21] to an alternative for Schrödinger's linear equation: the non-linear wave equation known today as Klein-Gordon equation (KGE). De Broglie also entered the long-standing controversy between Cartesians and Newtonians [3], and pointed out that a most appealing aspect of general relativity (GR) was to eliminate "the metaphysical concept of force from gravitational theory" [21]. In 1962 de Broglie insisted once again on non-linearity: "agreeing with some deep remarks by Einstein energy and linear momentum between corpuscles probably arise from transitory non-linear processes, whose description is beyond current linear theories" [22] (our translation from French, p 425). De Broglie also used a KGE in relativistic thermodynamics [23], soliton solutions of KGE are reported in [24], and a summary of de Broglie's pilot-wave theory is in [25].
De Broglie consistently noted that all waves in his theory had physical reality, thus requiring an underlying physical fluid, often identified with Dirac's ether [26]. In 1999 present author proposed a primordial fluid (PF) [27] similar to Dirac's, but with two conceptual differences: (a) an unidirectional arrow of time from past to future, thus excluding reversibility, and science-fiction travels to the past, and (b) the fluid constituents were material 3D-extended preons. From Leibniz continuity principle stating that a change from state A to B has to go through all intermediate states, and considering the controversy on impenetrability of matter [28][29][30], we concluded that matter should have structure at all scales, so that the smallest bit of matter is divisible, but its components cannot possibly be material. So, instead of material preons, it is postulated here that the primordial fluid is formed by energy-like sagions, described as discrete, 3D-extended, spherical and Planck-size objects of radius R, carrying quantity of motion P, spin S, and moving with speed C along the straightest path [12]. Our fluid theory agrees with de Broglie's relativistic QM and with the notion that matter has electromagnetic origin. In our theory the simplest entity of Nature is an energy-like sagion, rather than a material object as in the Leucippus and Democritus atom.

Reinterpretation of empirical evidence usually cited to support Einsteinian relativity
Next question: where does the primordial fluid exist? In a Newtonian context the sagion moves with speed C in absolute space, described in 1590 by Francesco Patrizi as physical space Σ [31][32]. For Gauss, the geometrical nature of space was an empirical question, tackled by measuring the angles of a triangle defined by three mountain peaks in Germany. Gauss found a locally flat space up to his empirical accuracy. However, the observation of light-bending by Eddington in the 1919 solar eclipse suggests that, at a larger cosmological scale, space is positively curved.

On the two different principles of equivalence
In contrast, for Einstein light-bending was a strong support for GR [33]. In his correspondence during the Swiss years [34], Einstein mentions three possible methods to confirm GR: (1) examination of existing solar photographs, (2) [34]). According to Ginzburg [36][37] EP-A was formulated in 1907 [38]. In the opinion of the present writer, EP-A contains the deepest physical and philosophical implications of Einstein's GR, and has everyday manifestations easily seen in the laboratory [39]. Hence, there is no need to invoke lightbending as empirical proof for EP-A. There is a different, although related, question: is the ratio between inertial mass and gravitational mass a constant, independent of the composition of matter? This is the equivalence principle for mass  [3]). For Einstein EP-M was also an empirical question, and in July 10/1912 he asked help from Prof. Wilhelm Wien to carry out two "quite easy" experiments involving (a) two pendulums, and (b) a torsion balance (L413, p 319-320 in [34]). It seems that Einstein was unaware of the earlier experiment by Eötvös involving torsion balances carried out in Hungary in the 1890s [40], and published around 1910 [41]. A re-analysis of Eötvös data by the Fischbach group in 1986 [42][43][44] shows that gravitational mass depends on the number and distribution of protons and neutrons in the atomic nucleus. Fischbach fitted the dashed line in Figure 1 to Eötvös observations with their Yukawa-type gravity model, and the correlation was poor. The anomaly remained unexplained [41] until in 2011 present writer separately analyzed the Ptseries and the Cu-series in Eötvös experiments (Figure 1a). Taking into account orbital electrons plus the binding energy of protons and neutrons in the nucleus, an atomic-like pushing gravity model was obtained [45][46], which is consistent with Eötvös data at 92% correlation coefficient (Figure 1b). The only adjustable parameter turned out to be, as expected, the electron/proton mass ratio. Summarizing, EP-A and EP-M are two different questions; this may clarify some controversies regarding Einstein's equivalence principle [36][37][47][48][49]; also note that Einstein suggested different and separate empirical methods to verify EP-A and EP-M. In our view, EP-A is locally valid, but EP-M is not. We also depart from Einstein's interpretation of star light-bending; instead, it is curvature of absolute 3D-space in the local region from star to earth.

On the evidence against absolute space
Einstein rightly rejected the far-fetched properties of electromagnetic ether models in the second half of the 19 th century, but he erred in his 1905 general dismissal of ether, including Newton's [2] (possibly immaterial) fluid-like ether [4]. enable acceleration or rotation to be looked upon as something real" [51]. Surprisingly, Einstein did not distinguish between the container (absolute space) and its contents (ether).
In the mid-1990s present author uncovered several weaknesses in Michelson-Morley experiment (MMX) [52]. Most significant was that only the fractional part of the fringe-shift was recorded, leading to under-estimation of fringe-shift amplitude [53], which should be of several fringes as calculated with modern values of solar velocity [54]. This prompted us to repeat the MM experiment at CIF in Bogota (Colombia) in a continuous manner during two years (2003)(2004)(2005). As usual in second-order experiments, two velocities of sun relative to a preferred frame were obtained from our data: (a) CIF-S: VS = 500 km/s, R.A. = 16h-40m, Dec = -75º (southern celestial hemisphere) [55], and (b) CIF-N: VS = 365 km/s, R.A. = 5h-24m, Dec = 79º (northern celestial hemisphere) [56]. Our solar velocities are compatible with other observations supporting absolute motion [57]. Of significant interest is the compatibility of our results with two contemporary experiments: (1) de Haan [58] experiments involving standing waves and a Fabry-Perot interferometer, and (2) Lipa vertical interferometer at Stanford in 2002 [59], temperature controlled to a micro-Kelvin accuracy, that was interpreted as a "null" MMX. Applying procedure described in [54], absolute velocity of surface of earth at Stanford University (Palo Alto, California) was calculated for dates of Lipa's experiment using our CIF-S value for solar velocity. Figure 2 shows projection of our calculated absolute velocity upon Lipa's laboratory floor for May 30/2002. The observed frequency variations recorded by Lipa on same date [59] exhibit correlation coefficients at 99.8% with absolute speed, and 99.1% with absolute direction of motion, which imply that EM frequency may be affected by absolute motion, contrary to Lipa's interpretation. The paradox is easy to explain: Lipa attributed observed frequency variations to unexplained "mechanical disturbances", subtracted them, and obtained a noise of zero amplitude. Hence, they got the reported "null" result!  [59] are highly correlated with absolute motion.

New solutions for the homogeneous Klein-Gordon equation or classical wave equation
Empirical evidence sketched in previous section implies absence of evidence against absolute space; in Einsteinian mood, the Michelson-Morley experiment no longer puts us in any predicament. It is postulated that a primordial fluid entirely pervades the curved 3D-absolute space Σ , and that it obeys conservation of total energy, and linear and angular momentum. In regions of universe devoid of matter, the fluid is locally described by the homogeneous Klein-Gordon equation (1), which is the same classical wave equation; the physical potential field A Π ( , )   contains a scalar potential  (conservation of total energy) and a vector potential A (conservation of linear momentum), whose dimensions ML -1 T -2 correspond to density of total energy, and to flow of linear momentum per unit area (or, equivalently, density of angular momentum flow) respectively. Equation (1) represents De Broglie's pilot wave [21][22][23], and also complies with classical electrodynamics [60]: Time is treated as length w with C* being the local average speed of primordial fluid. In presence of matter, right hand side is non-zero due to elastic and inelastic sagion-matter interactions; a complete  [12,[61][62][63][64]. There are two families of helices spiraling inward/outward on the surface of quantized cones shown in Figure 3. Inward growth has been observed in circular micro-pyramids of YBa2Cu3O7 at the 125 nanometer scale [65], and silicon carbide crystals at the 100 micrometer scale [66]; at a larger scale, swirls, tornados and hurricanes are everyday examples. On the contrary, snail-and sea-shells display outward spiral growth. Whether spiral galaxies grow inwards or outwards is left open.