Journal of Modern Physics, 2012, 3, 14651478 http://dx.doi.org/10.4236/jmp.2012.310181 Published Online October 2012 (http://www.SciRP.org/journal/jmp) Geometrization of Radial Particles in NonEmpty Space Complies with Tests of General Relativity Igor E. Bulyzhenkov Moscow Institute of Physics and Technology, Moscow, Russia Email: inter@mipt.ru Received August 1, 2012; revised September 5, 2012; accepted September 13, 2012 ABSTRACT Curved spacetime 4interval of any probe particle does not contradict to flat nonempty 3space which, in turn, as sumes the global material overlap of elementary continuous particles or the nonlocal Universe with universal Euclidean geometry. Relativistic particle’s time is the chain function of particles speed and this time differs from the proper time of a motionless local observer. Equal passive and active relativistic energycharges are employed to match the universal free fall and the Principle of Equivalence in nonempty (material) space, where continuous radial densities of elemen tary energycharges obey local superpositions and mutual penetrations. The known planetary perihelion precession, the radar echo delay, and the gravitational light bending can be explained quantitatively by the singularityfree metric without departure from Euclidean spatial geometry. The flatspace precession of nonpoint orbiting gyroscopes is non Newtonian one due to the Einstein dilation of local time within the Earth’s radial energycharge rather than due to un physical warping of Euclidean space. Keywords: Euclidean Material Space; Metric FourPotentials; Radial Masses; EnergyToEnergy Gravitation; Nonlocal Universe 1. Introduction The ideal penetration of a static superfluid medium through a rotating drag one was observed in He3H e4 experiments well before the distributed Cooper pair ex plained the nonlocal nature of superconductivity. But does spatial distribution of paired superelectrons mean that two nonlocal carriers move one through another as overlapping continuous distributions of massdensities or do these densities bypass each other separately without mutual penetrations? Is there a principle difference be tween the superfluid motion of two paired electrons and the free, geodesic motion of any normal electron between drag collisions with energy exchanges? During the last fifty years the celebrated Aharonov Bohm effect is trying to dismiss doubts regarding the nonlocal nature of the electron, while the Classical The ory of Fields is persisting to accept a selfcoherent ana lytical distribution of the charged elementary density (instead of the point particle approximation for the elec tron). Fermions take different energies and, therefore, cannot exhibit one net phase even under the ideal (with out dissipation) motion. At the same time, each distrib uted electron may have a selfcoherent state of its own matter. Particles motion with drag collisions and heat release represents much more complicated physics than the superfluid motion with a selfcoherent state of dis tributed elementary mass. Such a nonlocal superfluid state is free from energy and information exchanges. Charged densities of drag and superconducting electrons in the same spatial point can move even in opposite directions, for example under thermoelectric phenomena where nonequilibrium superconductors exhibit up to five [1] different ways for heat release/absorption. Such a steady countermotion of drag and superfluid densities of elec trons may be a laboratory guiding for theories toward the global counterbalance of all material flows in the nonlo cal Universe with local energy dissipation. However, if there is a mutual penetration of extended bodies (with or without dissipation), then how can General Relativity (GR) address the laboratory nonlocality of each electron in order to incorporate the material spatial overlap of distributed carriers of massenergy? Below we accept the ideal global overlap of all elementary energy flows in all points of their joint 3D space, which is associated with the superposition of flat material 3sections of curved elementary 4D manifolds. We shall rely on superfluid, selfcoherent states of extended elementary particle (called the astroparticle due to its infinite spatial distribu tion [2]) between drag collisions and dissipation events. At the same time, 3D overlap of such selfcoherent radial distributions can rarely exhibit, due to drag collisions, C opyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1466 summary superfluid states of identical bosons, while 3D energy ensembles of extended fermions can exhibit only ideal summary flows without joint coherent properties. It is important to emphasize that strict spatial flatness is principal for reasonable Quantum Mechanics, say for the BohrSommerfeld quantization, and for reasonable Electrodynamics, which is based on constant Gauss flux through any closed surface. The author does not see clear experimental reasons why one should redesign Classical Electrodynamics for a curvedspace laboratory in ques tion. On the contrary, due to well established measure ments of magnetic flux quantization in superconducting rings, one may insist that would gravity contribute to length of superconducting contours, then SQUIDs could not be explained satisfactorily, for example [3]. Indeed, would spatial intervals depend on gravity or acceleration, working SQUID accelerometers were already created. In such a view, Einsteins metric gravitation, which started from the very beneficial 1913 idea of 4D geometrization of fields, should doublecheck its wide opportunities and overcome the current phase with unphysical metric sin gularities. There are no sharp material densities in reality like Dirac operator deltadensities and relativistic physics should try continuous particles prior to declare singulari ties and black holes in physical space. One may expect that advanced metric gravitation should be a selfcon tained theory of continuous energy flows which ought to derive analytical components of the metric tensor for spacetime dynamics of distributed astroparticles without references on the point matter paradigm in ques tion and the Newtonian limit for point masses. Advanced GR solutions for massenergy densities of moving mate rial space should provide Lorentz force analogs even in the nonrelativistic limit. Newtons gravitation cannot satisfactorily describe this limit for moving sources and, therefore, should not be used for relevant gravitational references for a rotating galaxy (that raised the dark mat ter problem). Recall that in 1913 Einstein and Grossmann published their Entwurf metric formalism for the geodesic motion of a passive material point in a gravitational field [4]. In October 1915, Einstein’s field equation [5,6] and the Hilbert variational approach to independent field and par ticle densities [7] were proposed in Berlin and Gottingen, respectively, for geometrization of gravitational fields “generated” by the energymomentum density of Mies continuous matter [8], which later failed to replace point masses of the prequantum Universe. This metric theory of gravitational fields around still localized particles, known today as General Relativity, can operate fluently with curved spatial displacement d= dd ij N Nij lxx of a point mass m 1 by accepting the Schwarzschild [9] or Droste emptyspace solutions [10] without specific restric tions on the space metric tensor NN NN ijoi ojooij gg g 222 ddd=dd N . GR solutions for dynamics of the considered probe parti cle N are related to its spacetime interval, NN gxx l , where the time element 12 2 1 dd d oi NN NooNoo oi gxggx N depends on the local pseudoRiemannian metric tensor and, consequently, on local gravitational fields. Hereinafter, 1, 2,3i , 0,1,2,3 , and the speed of light c = 1 in the most of equations. The author intends to revisit time, d , and space, ddd ij N Nij lxx , elements within the conventional GR fourinterval ddd N gxx in order to prove that the time element of the freely moving mass m dt de pends not only on the world time differential (with dδddd >0 oo o oo txxx di ) and gravitation, but also on space differentials or matter displacements in gra vitational fields. Then the ratio ddlv NN , called the physical speed in Special Relativity (SR), should nonlinearly depend on spatial displacement ddd ij N Nij lxx , called the space interval in SR. Non linear field contributions to such an anisotropic (Finsler type) time element d,d N x within the fourinterval 22 2 dd ,dd xxl x of Einstein’s Relativity may modify Schwarzschildtype metric solutions based on curved threespace around nonphysical point singulari ties for GR energysources [2]. Moreover, the calculated ratio dd = NN lvv may differ from a real speed ddl O measured by a motionless observer with local propertime dd=0dlv ON . This metrictype anisotropy of measured time rate was already confirmed by observations of the gravitational Sagnac effect when dd 0 i gx oi . Rigorous consideration of anisotropic physical time d,d d xv dl of each moving parti cle may preserve universal flatness of its 3space element . We shall start from the 1913 Entwurf metric formal ism for the geodesic motion of passive masses. Then, we shall employ the tetrad approach and analyze nonlinear relations in the anisotropic relativistic time for a passive mass under the geodesic motion. This will suggest to keep for physical reality Euclidean 3D subintervals in curved 4D intervals of moving probe particles. The first attempt to interpret GR in parallel terms of curved and flat spaces was made by Rosen [11], Ein stein’s coauthor of the unpublished 1936 paper about the non existence of plane metric waves from line singulari ties of cylindrical sources. Later, Sommerfeld, Schwinger, Brillouin and many other theorists tried to justify Eucli dean space for better modern physics. Moreover, the Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1467 original proposal of Grossmann (to use 4D Riemannian geometry for geometrization of gravitational fields in the 1913 Entwurf version of GR) relied exclusively on 3D Euclid ean subspace. Grossmann did not join further GR metric developments with curved 3D intervals. In 1913 Einstein clearly underlined that space cannot exist without matter in the Entwurf geometrization of fields. However, at that prequantum time there were not many options for geometrization of particles, because all (but Mie) considered them localized entities for local events. This might be the reason why in January 1916 Einstein promptly accepted Schwarzschild’s warping of 3D space around the point particle. Nonetheless, in 1939 Einstein finally rejected Schwarzschild metric singularities for physical reality. The well derived Schwarzschild’s solu tion has no mathematical errors in the emptyspace para digm. However, we tend to use the nonemptyspace paradigm for the global superfluid overlap of selfco herent elementary particles, when each continuous parti cle is distributed over the entire Universe together with the elementary field. This nonlocal approach to matter can avoid difficulties of the Entwurf geometrization of fields, proposed in 1913 without geometrization of parti cles, and, ultimately, can avoid nonphysical warping of the universal spatial ruler, which becomes the same for all local observers in the flat Universe. Contrary to nonmetric approaches to gravitation with spatial flatness, for example [12,13], we shall comply with the EinsteinGrossmann extension of Special Rela tivity (SR) to gravitation through warped spacetime with nonEuclidean pseudogeometry, founded by Lobachev sky, Bolyai and Riemann [11]. Inertia and gravitation keep the same metric nature in our reiteration of the Ein steinGrossmann approach. The proposed 4D geometri zation of matter together with fields will be made under six metric bounds for (called sometime intrinsic metric symmetries) in the GR tensor formalism for every physical object. In other words, the author is planning to revise neither Einstein’s Principle of Relativity nor the GR geometrization concept. On the contrary, I am plan ning further GR geometrization of continuous particles together with the already available geometrization of gravitational fields. Local nullification of the Einstein tensor curvature for paired densities of the distributed astroparticle and its field will be requested in their rest frame of references. I intend to prove, for example, that Schwarzschild’s solution for a central field is not “the only rotationally invariant GR metric extension of the SR interval”. One should admit nonempty (material) space or Newtonian stresses of the material mediumaether associated with continuous very low dense distributions of nonlocal gravitation/inertial massener gies. Then bound ensembles of elementary radial energies form so called “macroscopic” bodies with sharp visual bounda ries (observed exclusively due to experimental restric tions to measure fine energy densities). First, we discuss a local time element, dddlv , which should be considered as a chain function of speed ddvl or spatial displacement of a passive ma terial point in external gravitational field. Then, we dis cuss the electric Webertype potential energy dl 211 11 W ooNoooo UU vmUPUP for a point planet with mass mPmV and relativistic energy oNo in the Sun’s static field generated by the active energycharge E. Ultimately, this paper presents the selfcontained GR scheme with the energytoenergy interaction poten tial oo M UPGE r for Machian mechanics of non local astroparticles with an analytical radial density 2 2 4π o nrrrr r o instead of the Dirac delta den sity r . One should see arguments for the singular ityfree gravitational contribution oo UP to the smooth metric tensor component 2 1gUP P ooo o. The main challenge here was to keep the free fall universality and the GR Principle of Equivalence for all carriers of probe (passive, inertial) energies o in radial fields of the Sun’s gravitational (active) energy E . In the speeddependent time approach, the warped GR fourinterval ddd,d ll cannot be approximated in weak fields by pure time and pure space subintervals, like in Schwarzschildtype solutions [9,10] with their formal time and space metric split without chain relations. In order to justify the indivisible nonlinear involvement of space displacements into physical time ddl N of a probe particle under the geodesic motion, one should clarify how the already known gravitational tests of GR can be explained quantitatively without departure from spatial flatness. Then we discuss our energytoenergy attraction under the EinsteinGrossmann geodesic motion in metric fields with flat 3section (i.e. without Schwar zschild singularities). The author also accepts the Ein steinInfeldHoffmann approach (but under flat 3space) to nonpoint slowmoving gyroscopes in order to de scribe the Gravity Probe B quantitatively. In 1913, Einstein and Grossmann put weak Newtonian field only into the temporal part of the Entwurf 4D inter val. Today, one tends to justify that strongfield GR met ric may also admit for reality six metric bounds ijij which preserve universal 3D interval in specifically curved spacetime for any elementary particle N. Then the metric tensor for curved 4D with flat 3section depends on four gravitational potentials o GUP 2 =1 oNoo Pmg v for the particle energycharge . This finding matches 6 metric bounds for spatial flatness under any gravitational fields and their gauges. Since 2000, this postEntwurf metric scheme with warped spacetime, but strictly flat threespace, became consis tent with the observed Universe’s largescale flatness, Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1468 confirmed at first by balloon measurements of the Cos mic Microwave Background and then by all ongoing Wilkinson Microwave Anisotropy Probe (WMAP) Ob servations of the flat Universe [1720]. This new reading of curved 4D geometry with nonlinearly dilated anisot ropic time and flat nonempty space, explains quantita tively all GR tests, the known planet perihelion preces sion, the radar echo delay, and the gravitational light bending, for example [2123]. Speeddependent time corrections to postNewtonian dynamics in Sun’s flat material space lead to computa tion results similar to numerical computations of other authors who traditionally correct Newton in empty, but curved 3space. Observable dynamics of matter in mod erate and strong static fields provides, in principle, an opportunity to distinguish our metric solutions with iso tropic flat space and speeddependent time from Sch warzschild’s solutions, based on curved 3space and di lated time. Alternative emptyspace and nonempty space paradigms can also be distinguished through different probe body dynamics in stationary fields of rotating as trophysical objects. 2. Warped FourSpace with Intrinsic Metric Symmetries for Flat ThreeSpace To begin, we employ the GR tetrad formalism, for ex ample [24,25], in covariant expressions for an elementary restmass mi in order to justify the mathematical op portunity to keep a flat 3D subspace in curved fourspace with a pseudoRiemannian metric tensor N g (for short). First, we rewrite the curved fourinterval, dd d d, xx x x 2 ddd dd N NNN sgxxg ee xx in plane coordinates dd ex d and d ex , g1,1, 1, d with 1. One can findia o eg bb ; oo oo i gg and 0, i ee from the equal ity 2 dd i ij 2 dd d oi oo ij gxgxxx , ioioo gg . irst glance, thad i i At f e spatial tri bb ee(a, b = 1, 2, 3 and , = 0, 1, 2, 3) should apend essentially on thvitational fields of other particles because this triad is related to components of lways de e gra . However, this might not be the case when there are ernal metric rela tions or bounds in the general pseudoRiemannian metric with the warped tensor int . Shortly, a curved mathe matical 4D manifold does necessarily mean a curved 3D section for real matter (warped 2D paper in 3D trash, for example, keeps parallel Euclidean lines due to steady metric relations between neighboring points of paper). It is not obvious that physical restrictions for fourv not e lo rved pseud cities of real matter, like 1gVV , might require to keep flat 3D sections of cuoRiemannian 4D manifolds. Therefore, let us look at three spatial compo nents i V of the fourvector ddVgxs by using the conventional tetrad formalism, 12 1i gg vvv 12 1. oo i ii ob iii i ob bb oo iibb Ve VeV eV gg evvv Here, we used = o iooi egg and 12 1 bb b Vv vv . Now one can trace that the considered equalities ii VeV 12 1; bb v v admit trivial relations ib vvvv and bb ii i bb vevv d velocities, b i between the curve ddd joi iij ooi vxgxgxdd j ij x , and the plane velocities, dd a ab b vx . All spatial triads for t ations may be c N hese “trivial” relonsidered as universal Kro necker delta symbols, = i bb i e , and, consequently, the threespace metric tenrelevant to gravitation fields, i.e. 1N sor is ir ijoi oj ooijijijij ggg g . All com ponents , involved in these six relations gra N , may de pend on vitations fields or system accelerations but their combination should always keep spatial flatness under admissible coordinate transformations. One could, surely, ignore flat 3space option within curved 4D ma nifold, as was suggested by the above tetrad analysis, by trying curved 3D solutions in i V when i bb i e . But we do not see much physical sense in suchtions and, therefore, restrict GR geometrical constructions by a partial case with six metric relations 1 oi oj ooijij ggg g complica . Applications of pseudoRiemannian t 3sections will quantitatively describe all known gravita tional experiments plus magnetic flux quantization. The latter and the AharonovBohm effect require only flat 3space for satisfactory interpretations. Again, we shall read Kv spacetime with fla ee through =; oggg and bb oooo i e 0, b i e for all physical cases we arested in describine interg. This means for our consideration that oo ooo o ee, oo oio i ee, and oo o ij ijij gee e ab o abij ij ee e . And Euclidean spatial geometry, e applied to pseudo Riemannian 4intervals of 2 ddddd ij ij K Kijij lxxxx , will b all particles (due to intrinsic metric relations 1 KK KK oi ojooijij gg gg ). Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1469 Contrary to univeements variant fourinterpe rsal spatial acdl, in vals have differen ward metrics for pa displ tly rticles K and N, because N g and dd N s in different external fields (for example, in the twobody problem). The GR fourintervelected ergy carrier, al for a s massen 2 22 dd 2d d ddd, i oij oi oo ij oo gx lgxxx g (1) is defined for only one selectedobe mass pr m despite notifications dd N s and dd N x are regularly l s m e used for brevity. This geometrical 4intervahould be physically co mented in terms of tim 2 dd and space 2 ddddd ij ij ij ij lxxxx elements, albeit 3 space differentials di contribute to particsical time le’s phy dd . We prove below that particles proper time d depends on dlen in constant gravitational fields (wher is a first integral of motion o Pconst ev e there ). h an anisotropic time element Suc ddd d oi Noo i gx xgx of the moving mss a m always counts its spatial ented gravitational field, des imm e definition r gravi dis pite placement dl in a ori the fact that it is not ediately obvious from the physical tim for metrics with 0 oi g. This postNewtonian phe nomenon, related to the energy nature of anisotropic time, appears in nonlineatational equations through the energy (velocity)dependent potentials. Our interpreta tion of the warped fourinterval (1), based on warped anisotropic time in isotropic nonempty flatspace rather than in empty warped space, may be considered as a prospective way for further developments of the 1913 metric gravitation through joint geometrization of dis tributed fields and distributed elementary particles. Now we return to components of the fourvector dd NN Vgxs . Notice that bo b V eV 1 o bo oo bo o N Ve eV eVV eV mU with the bo bo V V fourvelocity eV 0 and bb i e 22 ; 11 NNNi mmv vv 22 1; 11 , No oNioo NN P mg mg g vv KU (2) where , because b eoi . Flat t to introduce l pot hreespace ge gauge invarianometry is a promising wayt gravitatio entials,naoN PG GU with oo ooN Ue mVUP , for the passive (probe) mass m, in close analoglectro al electric charge. The point is that a fourmomentum y to fourc tials for the classic omponent e magnetic poten N N PmV of the se lected scalar mass m (without rotation) can be rigor ously decomposed into mechanical, , and gravita tional, U , parts onunder strict spatial flatness, ly iij vv , 2i i vvv, dd ii vx , 12 dddsxx , dd gx , dd x , ioioo gg ; j ooijij ggg ij iij g . Again, we use a timelik worldline witdd> o tx and e h 0 12 ddd>0 i oo i gxgx onal energy ofor the passiveinertial mN > 0. The gravitatimomentum part U is de fined in (2) for a selected mass m and its positively defined passive energy 0 oNo PmV, associated with the global distribution of all other sses ma m. This gra vitational part, UGP o , is not a full fourvector in pseudoRiemannian spacetime, like P , r is the me chanical summanN no d Km . Because 0, bbb i e and dd ex , the tetrad with the zeo r (i.e. time) label takes the following corom (2): mponents f 21 21 11 ;1 o evUmvUm 21 1. oN iN oN vUm Ultimately, the tetrad e for the selected particle N and the metric tensor N ee , with gg , depends in Cartesiaoordinates only on the gravi tational fourpotential n c o UPG (ind for t e e troduceh rlativistic energycharge ooN cP cP [26]), 1 11 22 21 1 21 21 1 22 2 2 1 1 11 11 11 1 1 1 oo oo oo N oooooNo o oo N oioio Ni N N ooi o oo ab N ijijab ijijNij N ooij oij oo ij Noo i m UP UP geevUm UP geevUm vUm gUP gee eevUUm gUUP gUPUU 21 ,,, oiij ij jo NioN Pg UPg 2 1 oN evU (3) where we used 2 2 11 oo ooo oo gee vU and 22 1 ooo Vg v to prove that Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1470 11 11 o oo o 1 oo oo gU e, the paP UP . Therefors siveinertial GR energy, 21 2 1 1 oo v UP mvU 2 11 , ooo o Pmgv m takes a linear superposition of kinetic and potential ener gies in all points of pseudoRiemannian spacetime warped by strong external fields. Note that we did not assign spin S or internal angular mechanical momen tum to the EinsteinGrossmann “material point” or the probe mass m with the energymomentum (2). The affine connecions for the metric tensor (3) depend only on four gravitational potentials t o UP in our space time geometr which is not relevant to warped mani folds with asymmetrical connections and torsion fields, for example [2729]. Every component of the metric tensor in (3) depends on the gravitational part y, No UmVm GP of the probe carrier energymomentum P . At the same time, all the components of the threespace metric tensor, 1 ijoi oj ooijij ggg g , aom the gravitational potential re always independent fr o from (3 the inhar GUP or its gauge. Such inherent metric symmetries for 3D subspace may be verified directly). In fact, our tetrad, and the metric tensor, depends formally onmonic Weber type potentials, 211 11 No oo UvmUP UP , as sociated with the particle speed 222 ddvl . In 1848 Weber introduced [18] the nonCoulomb potential 2 2 212 12qq vr ac 12 easurements of ch q perime of based on lab m celerating forces between movin1 and 2 q with the relative radial velocity 2 12 1v . This might was the first extal finding that mechanical inertia and acceleration depend on the kinetic energy or speed interacting bodies. By substituting the metric tensor (3) into the interval 222 ddddd g arges n gxx l , one can rewrite (1) and submit the chain relation for the proper time dd of the probe massenergy carrier N in external gravita tional fields, 12 2 1 dd ddd o oi NN oooooi N lgxggxex 2 2 d dd 1 . dd N o N Ul xx ml (4) Notice that the propertime differential, 1 1 oK OoK xUm , of the local observer K, with e element (4) ergycharge dd i d0x of K and d0 K l, differs from the tim the moving mass m with the GR en 2 1 ooo g v. ing masits pro Pm The proper interval ds of the mov s and per time element (4) depends, in general, on all four components of U . Therefore, the observable threespeed dd O l , of a moving particle always differs in relativistic gravitomechanics from the nonlinear ratio dd dllv , called the particle’s physical speed (1). The chain relation ddf in the physical time (4) of a moving particle changes the GR interpretation of the geodesic motion and allows to apply flat 3D space for gravitational tests. The metric tensor (3), the interval (1), and the local time element (4) are associated with warped spacetime specified by external fields for one selected mass m or, to be precise, for the passive energycharge o P. We may employ common threespace for all elementary par ticles (due to universal Euclidean geometry for their spa tial displacements), but we should specify arped spacetimes with differently dilated times for the mutual motion of gravitational partners. The particle’s time ele ment w ddd, Nlv in (4) may depend on the parti cles velocity or displacement. Ultimately, a nonlinear time rate dd oo exx (hereinafter dd tf , dd o txc) of moving material objects in (4) depends on the ratio 22 2 lv . This nonlinear chain relation can be simplified in several subsequent steps through the following equalities to (4): 12 11 12 1 1d dd 1 11 1 Ni ioo oi iN i vt tUP PUv vUm v UP x 1 1 d. 1 o io oo Um tUP (5) Such anisotropic time dilatation in (5) by the ex fourpotential ternal NN oo GUP results in the gravitational Sagnac effect when an observer compares the dynamics of different elementary energycharges o P in fields with 0 i U . Now, one may conclude that the anisotropic time ele ment d in the metric interval (1) and, consequently, in the physical speed dd,vl depends only on univer ur sal fopotentials G for positive probe charges 0 o P. The potential energy part NNN mUP mV contributes to GR energymomentum of the probe body and, therefore, to its passive energycharge, oo mV P . The universal ratio o UP should be tried gravitation as a metric field fourpotential (which is not a covariant fourvector) of active gravitational charges for passive energycharges. Contrary to Newton’s gravita tion for masses, Einstein’s gravitation is the metric the ory for interacting energies. The static Sun, with the ac tive energycharge 2 M EMc, keeps the universal po tential in Einstein’s 1;0 mM UE GEr in the Sun’s frame of reference for the passive, inertial energy content 2 om cPEconst mc of the probe mass m. Below, we employ the unof the Sun’s potential, iversality Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1471 NN ooMo UP r all planets in our com putations for gravitational tests of General Re (5) and flat material spce filled everywhere by 2 r gravitational fields and the 4 r GEr rr , fo lativity with dilated time (4)a 3. Flatspace for the Planetary Perihelion Precession Now we consider extended masses. the metric tensor (3) a central gra 0 for vitational field with a static fourpotential, 1 io UP , 11 oo M UPGE r , where 2 EMcrGcons active gravitaMo of the “mo ment with sp t is tionless” Sun n’s cente ricalatial flatn th of e sphe tional energy ymm (in the moving Solar system). We use Euclidean geome try for the radial distance 1 ru from the Sur etry in agreeess maintained by (3) for any gravitational fourpotential G s and its gauge . Let us denote the energy con tent of a probe mass m in tc central field as a pas sive energycharge he stati 2 1 oNoNoo m PmVmgv E . n, the interval (1) for the passive energy carrier in a central field with 0 i U takes two equivalent presenta The ), tions due to (4) and (5 2 2 222 2 d d1 1dd dd Mm GE El 2 22 d1 d, M tl l (6) rm GE tl r where iterations 2 22 1dd dd d Mm tGEErm lll ion 22 d1 over the chain funct 2 ddl in the Lorentz factor resun lt i 2 d1 M tGEr for the SunMercury po 2 ten ergy tial enoMm UGEEr . In other words, t e depend iterations common for all p h specific, Weber velocityent potentials exhibit after chainrobe particles local time, 22 ddd d K sl static metric solution (6 oes not coincide with the Schwarzschild me herefore, the Schwarzschild exten si static fields. Spherical coordinates can be equally used in (6) for the Euclidean element 2222222 dd dsinddd ij ij lrrrxx in flat laboratory space. , in The) for probe elementary en ergycharges in nonempty space of the radial energy charge dtric [9] in empty space. T on of the SR interval is not the only rotationally in variant solution which GR’s tensor formalism can pro pose for tests of spacetimeenergy selforganizations. Ultrarelativistic velocities, dd1vl and 2 10v, in the Webertype energytoenergy inter action in (6) revise the Schwarzschild singularity. The latter is not expected at the finite radius in the energy charge formalism of Einstein. Einste n’s gravitatioin, “the reluctant father of black holes”, very strictly expressed his final opinion regarding the Schwarzschild solution: “The essential result of this investigation is a clear un derstanding as to why Schwarzschild singularities do not exist in physical reality” [31]. In authors view, Schwarz schild’s metric solution, and all Birkhoff class solutions for the empty space dogma, originates with ad hoc mod eling of matter in the 1915 Einstein equation in terms of point particles. However, Einstein anticipated extended sources for his equation and for physical reality. Below, we prove that the static metric (6) corresponds to the 4 r radial energycharge or the extended source of gravity. Therefore, our analysis denies the empty space paradigm. Nonempty material space is in full agreement with Ein stein’s idea of continuous sources and Newton’s “ surd” interpretation of distant attractions through stresses in an invisible material ether (called in 1686 as “God’s sensorium”). Our next task is to derive integrals of motion for the passive (probe) massenergy in a strong central field from the geodesic equations ab 22 2 dd ddd pxxp . Nonzero affine connections for the metric (6) take the following components: rr , 2 sin rr , d2d, r ttoo r ΓΓΓΓ1 rrrr r , Γsin cos , ΓΓctg , and d2d tr rtoo oo tt gr, wh ere oo is the function next 2 dt in the interval (6), 222 ddd oo to gt l. wing thpoach withBy folloe verified apr π2const for the ispic ceral field, for exam ple [25], and by substituting flatspace connections otro nt into GR’s geodesic eq tric write the following grav uations, one can define the para me differential dp andita tional relations, 22 2122 2 d ddconst dd const,ddconst d dconst oooo m m oo m psgtsEm rpJrsJEmL rpJr gmE 2 2 22 dd 1,d ddd d1, mo o gtp rsrs Emg (7) with the first integrals ,, m Em and of the relativis tic motion in strong static fields. The last line in (7) is the interval equation 222 ddd oo gt l with two integrals of motion 22222 dd moo Em gt s and π2 . Therefore, the lly the n in sca lar invariant (6) is actuaequatio of motion for the constant energy charge m Econst a central field with the static Webertype potential 2 1 , omo MM m vUEU GErGE W oo UU Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1472 which is inharmonic for t 20 W o U. Re call that Schwarzschild’s curved 3D he Laplacian, solution not only differs from (6), but results in conceptual inconsistencies [32,33] for the Einstein equation. We can use (6)(7) for relativistic motion in strong central fields in order to re inforce the ignored statement of Einstein that Schwarzs l reality. child singularities do not exist in physicaThere are no grounds for metric singularities either in the in terval (6), or in the radial potential W o Ur for 0r, because dd oo tgrrGM is a smooth func tion. One can verify that the nonempty space metric tensor (3), as well as 20 W o U, does correspond to the continuous energysource in the 1915 Einstein equa tion. The strong field relations (6)(7) can be used, for ex ample, for computations of planetary perihelion preces sion in the solar system. The planet’s gravitational en ergy for the GR energytoenergy attraction, 1 oMmom UGEEr rEu , where 2const o rGMc and 1ur, is small compared to the planet’s energy, 1const om urU E, that corresponds to the non relativistic motion of a planet N (with const1 m Em, mM EE, and 222 dd 1vl ) in the Sun’s rest frame, with 0U. The Gement for i s from (6) or (7) as R time elt read the plane 1 12r 2 2 22 22 2 22 d ddddd1 dd d d, m o oo El sllt ru ml utru l (8) where we set 1 o ru , 1 m Em, 22 dddll , 2 dt . with spatial displacement 2 d o rul the right hand side of (8) belothe physical time element withinvariant and dd The 22 dtl field term on ngs to 2 d the in . Th u is displacement nlire of anisotropiccorresponds to the near chain nat time no ddlddlf origiom the Weber ,  ty nating fr pe energy potential 2 1Uvm in (3). Ths no departure from Euclidean space geometry with the flat metric ere i 22 224222 d=π2d dddlrruuu in the chain reading of geometrical intervals (6) or (8). Againnear time with chain spatial dis, a particle’s nonli placement ddl differom the propertime rs in (8) f 12 d12 d Oo ru t of the local (motionless) observer. Displacement corrections, 22 ddrult, for the nonrela tivistic limit are very small compared to the main gravi tational corrections, 2o ru, to Newtonian time rate o 222 12 dd o trurult . However, the chain de pendence of a icle’s time element 2 d o part from spatial displacement dl accounts for the reverse value of this time element, 2 22 dd o rul , that is ultimately a way to restore strict spatial flatness at all orders of Einstein’s metric gravitation. Hee is some kind of analogy re small contributions of Maxwell’s displacement currents restore strict charge conservation in Ampere’s quasistationary magnetic law. Two integrals re ther with electrodynamics whe e th of motion12 dd om ru t sE m and 2ddrsL result from (7) and (8) for weak fields in a rosette motion of planets, 222222 1213=, oo ruLruuuE Lm (9) where dduu and 1ru . Indeed, (9) may be o fferentiated with respect todi the polar angle , 2 2 93 oo r u uru2 3, 22 oo ru uru L (10) by keeping only the largest gravitational terms. Thi ay be solveo step s equ ation md in tws when a noncorrected Newtonian solution, 21cos oo urL , is substi tuted into the GR correction terms at the right hand side of (10). The most important correction (which is summed over century rotations of the planets) is related to the “reso nance” (proportional to cos ) GR terms. Therefore, one may ignore in apart from 224 2cosuL (10) all corrections and 24 cos o uu rL . Then the approximate equation for the rosette motion, u234 6cos oo urLrL , leads to the well known perihelion precession 222 Δ6π6π1 oo rLr a , which may also be derived through Schwarzschild’s met ric with w, as in [21 25]. approximations arped threespace It is important to empha reces size that the observed result sion Δ for a planet perihelion p (in the Solar non ated time by Sun’s energy densi  empty flatspace with dil ties) has been derived here from the invariant fourin terval (1) under flat threespace, , ij ij rather than under empty but curved threespace. 4. The Radar Echo Delay in Flatspace The gravitational redshift of light frequency can be considered a direct confirmation that gravity couples to the energy content of matter, includithe massless photon’s energy Eng , rather than to the scalar mass of the 2 c for inverse particle. Indeed, Einstein’s direct statement Em all restmass particles is well proved, but the reading, 2 mEc, does not work for electromagnetic waves (with 0m ) and requires a new notion, the wave energycharge 20Emc or the relativistic mass 0 m . In 1907, Einstein introduced the Principle of Eq lence for a uniformly accelerated body and concluded that its potential energy depends on the gravitationally passive (“heavy”) mass associated with the inertial mass [34]. This correct concluf Einstein was generalized ron uiva sion o in a wg way that any energy, including light, has a Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1473 “relativistic mass” (the gravitational energycharge in our terminology) for Newtons mechanics. Proponents of this generalization in question proposed that photon’s “rela tivistic mass” is attracted by the Sun’s mass in agreement with the measured redshift 12 S EEmGMR mc . Nonetheless, the coherent application (in the absence of the correct EM wave equations in gravitational fields) of the “rela tivistic mass” to zeromass waves promptly resulted in the underestimated light deflection, 2 22 S GMR crR , for the al field [35]. In 1917, when Schwarzschild’s option [9] for spatial curva ture had been tried for all GR solutions, the new non Newtonian light deflection, oS “mechanical free fall” of photons in the Sun’s gravitation 4oS rR , had been pre dicted due to additional contributions on. Later, all measurements supported this curvespace modification for the “relativ istic mass” deflection by the Sun that provided false “experimental evidences” of nonEuclidean threespace in contemporary developmenravitation. Below, we prove that Einstein’s GR for the Maxwell wave equation firmly maintains the flatspace concept for interpretation of light phenomena in gravitational fields if one coherently couples the Sun’s rest energy to the pho ton’s wave energy E from the supposed spatial curvature in questi ts of metric g . We consider both the radar echo delay and the gravitational deflection of light by coupling its imp energycharge with local gravitational potentials. Our purpose is to verify that Euclidean space can match the known measurements [2123,36,37] of light phenomena in the Solar system. Let us consider a static gravitational field (0 i g, for slicity), where the physical slow ness of photons, 1 nvc , can be derived directly from the covariant Maxwell equations [24], 1oo ng . Recall that a motionless local observer associates oo with the gravitational potential oo UP at a given point. The light velocity dd O vl , measured by this observer, as well as the observed light frequency dd oO t , is to be specified with respect to the observer’s time rate dd Ooo t . This consideration complies with Ein steins approach, where the light’s redshift is associa with different clock rates (of local observers) in the Sun’s gravitational potential [34]. Compared to the physical speed of light, ted 1 dd o vl cn , its coordinate speed 2 dd 2 1 oo oo O oo tn rr cc d d= O lc gcg 1 l rr (11) is doubleshiftedtial by the gravitational potenoo UP o rr, where 2 oS rGMc oS rrR 1.48km and . Notice that both the local physical slowness 1oo ng and the observer time dilation dd O t sponsible for the double slowness of the coordinate ve locity (11), which is relevant to observations of light co ho reads through relation (11) as oo g are re ordinates or rays under gravitational tests. A world time delay of Mercury’s radar ec 22 o 2 44 ln 220μs, oM SES S 2d 11 2 Δ2d MM EE lx lx rx tl cc l y and 7 5.7910 km MS r are the EarthSun and MercurySun distances, respectively. Notice that in flat space we use the Euclidean metric for spatial distance, rr r cR (12) where 5 710km S yR is the radius of the Sun, while 8 1.49510 km ES r 12 22 rxy , between th ter (0,0) and any point (x, y) on the photonic ray. One can measure in the Earth’s laboratory only the physical time e Sun’s cen delay E Eoo t , which practically coincides with the world time delay t in the Earth’s weak field, i.e. 220 Ets . From here, the known experimental results [2425,37] the radar echo delay (12), based on strictly flat threespace and dilated time as in correspond to A coordinate angu 1913 Entwu rf metric scheme. 5. Gravitational Light Bending in NonEmpty Flatspace lar deflection of a light wave front in the Sun’s gravitational field can be promptly derived in flat space geometry by using the coordinate velocity (11) for observations, 00 2 2 2d2d o r l lx ycy 2 32 022 d4 41 .75. So oS S y r (13) at principle to light waves. This basic principle of physics should also justify spatial flatness under suitable applications [3840]. In agreement with Einstein’s original co [34], one may relate the vector component o Rx rR xR The most rigorous classical procedure to derive the ray deflection (13) is to apply the verified Ferm nsideration in the scalar wave equation 0KK to the measured (physical) energyfrequency of the photon (dd , oooo cK Etconst ). Recall that o P is also the measured particle’s energy in the similar equation, 24 PP mc , for a restmass particle. The scalar wave equation 0 N KKgKK has the Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1474 following solution for the electromagnetic wave, 22 22 222 dd dd dddd ddd d, oi oo i ij ooo ooo i o iioo iooo Ktc gK KKgKgK g tcg Ktxcgl Kxlggtcg oo i ijoo io i oo gK K (14) with 1 di =, d j ij oo EExg lg c . The Fermattype variationspect to with res and u (1 ru , , and π2 aree spherical c nates) f in a static gravitational field are thoordi or photons 2 222 d dd d dd1 0, j oij ii ioo o o x Kx x cg l uu ru (15) 2 cu (where 1 oo o 2 ru , 0 i g ,ij ij , 222ij dddx ij lx rr for o ru ) resulting in a couple of 1, 22 const oo u (16) 2 1 cos light ray equations 2 2 (1 4) 2oo ru uu uuru Solutions of o ru (16), 1 sin 2 oo ur u and 1 ooo S rur R, may n of r be used for the Sun’s weak field. The propagatio light from , π to r , corresponds to the angular deflection 1 arcsin41 cos4=1.75 oSo S rRrR fr l direction. This deflection coin with (13) and 2325]. We may that this no nee t the “nonNe ne strictly follows Einstein’s original approach to light in gravitational fields [34]. In fact, the massless electromagnetic energy exhibits an om the light’s initia cidesis in agreement with the known measurements 1.660.18 , for example [ concludeere do warp Eucli dean threespace for the explanation of wto nian” light deflections if o inhomogeneous slowness of its physical velocity, dd Ooo vl cg , and, therefore, a double slowness of the coordinate velocity, dd oo ltcg. This coordinate velocity slowness is related to the coordinate bending of light measured by observers. In closing, the variational Fermat’s principle supports Entwurf physics of Einstein and Grossmann with dilated time and strict spatial flat 6. Geodetic and FrameDragging Precessions of Orbiting Gyroscopes ness for light in the Solar system. Precession of the has been co utions of masses. This original GR approach practically coincides in the weak ph orbiting gyroscopes in the Gravity Probe B Experiment [41] mpared only with Schiffs formula [42,43] based on the Schwarzschildtype metric for curved and empty 3D space. Here the author plans to criticize the point spin model for GPB compu tations in favor of the regular EinsteinInfeldHoffman approach to slowly rotating distrib Earths field with our flatspace reading of Einstein’s ysics. Recall that our Entwurftype space interval is strictly flat due to the intrinsic metric bounds in the GR fourinterval (1) with the metric tensor (3). However, the GR tensor formalism can be universally applied to any warped spacetime manifold with or without intrinsic metric bounds. By following Schiff and many other point particle proponents in gravitation, one has to assume for a mo ment that the vector geodesic equation, dd ddSp Sxp , in pseudoRiemannian four space with only symmetrical connections, , may be applied to the point spin “fourvector” S with “invariant” bounds 0VS or i oi SxS for or thonormal fourvectors, ΓΓ ΓΓ . oj okj jjk ioikio ikj dΓΓ d oj i SSx Sx tio i j xxx S (17) Our flatspace for a strong static field with (3) and 0 oi g , 2 1 11 oo oo oo UP g , and ij gij d formally maintain an inertial conservation, , woul 22const, ooij ij i ooooi ji xg xg SS ij oo gg SS SS SSS vS S in agreement with Einstein’s teaching for a freefalling bo (curved space) tends to suggest [25,42,43] the n dy. At the same time, Schwarzschild’s metric option d oncom pensated Newtonian potential GMr “free fall” equation, 2 const gSS even in the 21 2 Sch vS S. Therefore, formal applications of the Einstein Gross mann geodesic relations (derived for spatial translations of material points) to localized spins S (which are not fourvectors in 4D manifolds with symmetrical affine connections) contradict the spirit of GR inertial motion and, ultimately, the Principle of Equivalence. Our affine connections ΓΓ , related to the met ric tensor (3), depend only on four field potentials 111 ,GUP UPUP . This postEntwurf metric oooio Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1475 tensor has been introduced for ergymomen tum (2) without an Moreover, neither the mechanical part, the local en y rotational or spin components. , nor the gra vitational part, o PG , in (2) are separately covariant fourvectors in warped spacetime with the metric tensor (3). Therefore, there are no optimistic grounds to believe that four spin components S might accidentally form a covariant four vector in spacetime with symmetrical cof thennections for translatio energymomentum fourvector, PKPG n o o l con k 12 k k o oo UPg oo UP . Nonetheless, we try by nections for the point spin avenue (17) in question in constant fields (when 0 og , for simplicity), 2 2 111 122 11 1 211 11 1 2 21 2 1 j ioj oijjoooij ooo o ioooioioo oi j ioooijooo j ikjooo ijo io ookjo j o oi UPgUP gUP g UPUP g PU UPgUPg UU PgU PgUP UP gUP UP UU (18) chance t 2 ik One h i U could start ese sym i k oo ji UP P UU metrica 2 2 ij ooo UUPg 11 M GE r and oo o oo oi j o P 11 2 ko ooiooo i jook ko gUPg U P g with 13 Ir ss M rotating with l 2G mio UP ai rω f o or th w a e homogeneous sphe ngular i.e. rrical 1 velocity, , 21, iio UUP M EM, and 2 25MR nn n Im keeping only n for E14]. Then, by linear terms to E xv Rr [ with respect io UP, one owly rotating gravitational field: can rewrite (17) for a sl 11 1 dln d . 2 ioo kiooo ooo jjo j SSx g t UPg g SU xxSg vanish forotating cewhen 0 1 2 ik ooo jk j ikoooki jk oo UP g S UP gUP P j ji i g r non oo (19) The last three terms on the righthand side of (19) are responsible for frame rotation and frame dragging, which nters and 0UPecessions of magnitude vec tor io . Pr the constant 1 o JSv UP , obtained for the weak field limit of 22 12 1 1 2const, j ooi ioj iijij jojii ji j gSSUPUUPxS UP SxSSSJJ when 22 vS 1, 1 ii oo UP xx , and iiioo 1 vUP in (19), 11 11 1 2 , jk jik o kio jjoioo iojoo JUP UP JUP UP UPUP 11 1 d d2 jji iioo joo J JvUP vUP t (20) mayf’s nonrelativistic predic tion be compared with Schif geo fd ddt J = for Gravity Probe B. The second summand at the right hand side of (20), 11 2 jk jikokio fd i UP UPJ , takes e Schiff’s answer [42,43] for the framedragging precession, xactly 332 12 . 2 GI rr r r r 3 fd GI r (21) The first and third precession terms in (20) depend on the Earth’s radial field 1 ioo UP and they count to gether geodetic and frame phenomena. These terms pro vide 11 1 2. gfoo o P vU U Such a prec for a point spin model, formally borrowed from the Ein steinGrossmann theory for the probe mass without rota ession tion, fails to reiterate the already well verified de Sitter geodetic precession, 13 323 2 geoo o UP GM vrvr , of the EarthMoon gyroscope in the Sun’s field, where 122 ,, 0UUU U. Why does the EinsteinGrossmann geodesic point mass fail for physics of spins and mass rotations? First of all, there is a clear mathematical reason to re je roach t ct point spins from the EinsteinGrossman metric for malism. The point spin appo GR matter cannot justify that S is a covariant fourvector in pseudo Riemannian spacetimetric tensor is de fined exclusively for matter without selfrotations or for the fourmomentum of a probe particle without spin. Therefore, one cannot place S e where the m into the Einstein Grossmann geodesic equation with symmetrical connec tions. RiemannCartan geometries with the affine torsion and asymmetrical connection [2729] are still under dis 19 e cussions for proper applications. In 38 Einstin already answered the point spin ques tion by developing with Infeld and Hoffmann relativistic dynamics of slowly moving distributions of active and passive masses. It is well known (Weyl in 1923 and Ein steinInfeldHoffmann in 1938 for example [24]) that the inhomogeneous GR time dilation (or inhomogeneous Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1476 oo r for mass elements rotating over a joint axis) defines a relvistic Lagrangian for the classical non point gyroscope. Therefore, Einstein’s relativity quanti tatively explains the de Sitter precession through local nonNewtonian time rates for distributed rotating sys tems. The nonNewtonian (threetimes enhanced) pre cession originates exclusively from different GR time rates in neighboring material points, rather than from a local space curvature in question for the illdefined GR spin of a point mass. The autho ati r does not understand Sc all gra Eu metric scheme [2,26]. In order to achieve this main goal, warping around the localized gravitational source (including the e contrary, our chain analysis of par allows us to infer that curved 4in hiffs reasons to ignore EinsteinInfeldHoffmann physics and Weyl results for relativistic gyroscopes prior to testing General Relativity through rotation of masses. The EinsteinHilbert tensor formalism for energy den sities of a gravitational source (rather than for a point source) requires nonSchwarzschildian interpretation of vitational tests, including LunarLaserRanging and Gravity Probe B data. In authors view, the 1913 Ein steinGrossmann geodesic motion in pseudoRiemannian spacetime with flat space can provide a physical basis for translational dynamics of only point particles, but not for selfrotations of distributed relativistic matter. Point spin models for geodetic and framedragging angular drifts of freefalling gyroscopes cannot be reasonable for GR physics even under formal success of pointspin ap proximations for the observable geodetic precession. Possible speculations that the de Sitter geodetic preces sion of the EarthMoon gyroscope or that the Mercury perihelion precession have already confirmed non clidean space geometry are against proper applications of the welltested GR time dilation by gravitational fields, and, therefore, against EinsteinInfeldHoffmann’s phys ics of slowly rotating systems having finite active/passive masses at finite dimensions. In fact, the available GPB releases (einstein.stanford.edu) of the processed geodetic precession data perfectly confirmed time dilatation for EinsteinInfeldHoffmann rotating distributions of masses. Lunar laser ranging of the EarthMoon gyroscope and the GPB geodetic precession are irrelevant, in fact, to ex perimental proofs of space warping by the missing inch. These tests are equally irrelevant to experimental proves of black holes existence. On the contrary, all known pre cision measurements in gravitation confirms the strong field metric (3) with time dilation and continuous gravi tational masses in nonempty Euclidean 3space. 7. Conclusions There are a lot of disputes in modern gravitation and as troparticle physics. Our main goal was to reinforce spa tial flatness for real, nonpoint matter in a line of the original En twur f geometrization of fields, rather than to discuss other consequences of the selfcontained SRGR we derived quantitative geodesic predictions for Mer cury’s perihelion precession, Mercury’s radar echo delay, and the gravitational light deflection by the Sun in strict ly flat threespace without references on the 1915 GR equations at all. The numerical results are well known from the Schwarzschild emptyspace approximation of reality. Recall that the conventional interpretation of postNewtonian corrections relies on space “point” Sun). On th ticles physical time terval can keep strict spatial flatness and the Entwurf metric scheme for strongfield gravitation. The GR dis placement dl may be referred as a space interval (like in Special Relativity) in flatspace relativity of nonlocal superfluid masses with mutual spatial penetrations. Con sequently, the integral dl along a space curve does not depend anymore on gravitational fields and takes a well defined meaning. Such a Machiantype nonlocality of superfluid astroparticles reconciles 3D space properties with the relativistic Sommerfield quantization along a line contour. Indeed, these are no reasonable explana tions for quantized magnetic flux in laboratory SQUIDs, unless one accepts 3D spatial flatness for any 2D surface [3]. GR physics may attach all field corrections within the GR invariant 2 d to the time element 2 ddl with chain relations. Gravity indeed curves elementary space time intervals (therefore d and d are specific for each moving particle), but their space subintervals dl are always flat or universal for all particles and observers. It is not surprising that our approach to relativistic cor rections, based on the strongfield Equation (7), resulted in Schwarzschildtype estimations, which are based on very close integrals of motion in the Sun’s weak field. However, strong fields in (7) will not lead to further co incidences with emptyspace Schwarzschildtype solu tions for dynamics of probe particles. Both the Euclidean space interval ddd>0 i i lxx and the Newtonian time interval dtddd>0 oo xxx are independent from local ery par n o fields and prop parameters of elementarticles. This absolute universality of world space and time rulers is a mandatory requirement for these otions in their ap plications to different particles and their ensembles. Oth erwise, there would be no way to introduce for different observers one universal ruler to measure threeintervals and to compare dynamics of particles in common 3space under the common time parameter. For example, it is impossible to measure or to compare differently warped fourintervals ddd N N gxxx of different parti cles. In other words, there is no universal, nonspecific Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1477 pseudoRiemannian geometry for all world m Th carriers can be ob when they maintain cs. Spacetimeenergy selfo t 10−15 m, then this ma atter. erefore, joint evolution of energy served only in common subspaces universal (for all matter) submetri rganization of extended mat ter can be well described without 3D metric ripples, which have no much sense in strictly flat material space. Laboratory search of observable chiral phenomena for paired vector interactions in flat material space is worth to be performed before expansive projects to find 3D metric ripples in cosmic space. Record measurements of flat material space beyond the present limit 10−18 m might not be required for confirmation of the residual EM nature of elementary masses under their Einstein type geometrization. Once chiral symmetry for hadrons was violated assforming symme try was equally violated in the entire nonlocal structure of the superfluid astroparticle [2] or in its infinite mate rial space. 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