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Derivation of generalized Einstein’s equations of in some non-inertial reference frames based on the theory of

Xiao-Song Wang Institute of Mechanical and Power , Henan Polytechnic University, Jiaozuo, Henan Province, 454000, China (Dated: Dec. 15, 2020) When solving the Einstein’s equations for an isolated system of , V. Fock introduces har- monic reference frame and obtains an unambiguous solution. Further, he concludes that there exists a harmonic reference frame which is determined uniquely apart from a if suitable supplementary conditions are imposed. It is known that wave equations keep the same form under Lorentz transformations. Thus, we speculate that Fock’s special harmonic reference frames may have provided us a clue to derive the Einstein’s equations in some special class of non-inertial reference frames. Following this clue, generalized Einstein’s equations in some special non-inertial reference frames are derived based on the theory of vacuum mechanics. If the field is weak and the reference frame is quasi-inertial, these generalized Einstein’s equations reduce to Einstein’s equa- tions. Thus, this theory may also explain all the experiments which support the theory of . There exist some differences between this theory and the theory of general relativity. Keywords: Einstein’s equations; gravitation; general relativity; principle of equivalence; gravitational aether; vacuum mechanics.

I. INTRODUCTION p. 411). Theoretical interpretation of the small value of Λ is still open [6]. The Einstein’s field equations of gravitation are valid 3. The problem of the existence of is in- in all reference frames is a fundamental assumption in teresting [7]. Einstein believed that black hole cannot the theory of general relativity [1–3]. R. P. Feynman exist in the real [8]. Recently, the Horizon once said:”What I cannot create, I do not understand.” Telescope Collaboration (EHTC) reconstructed event- ([4], p. xxxii). New theories which can derive Einstein’s horizon-scale images of the can- field equations may be interesting. The reasons may be didate in the center of the giant elliptical galaxy M87 [9]. summarized as follows. EHTC reports that the observed image is consistent with 1. Many attempts to reconcile the theory of general rel- for the shadow of a Kerr black hole based on ativity and by using the techniques the theory of general relativity. in meet some mathematical d- 4. The existences and characters of dark and ifficulties ([5], p. 101). J. Maddox speculates that the dark are still controversy, refers to, for instance, of the familiar procedures to cope [10–14]. with Einstein’s equations may stem from two possible 5. The existence and characters of gravitational aether reasons. One possibility is that Einstein’s equations are are still not clear. Sir I. Newton pointed out that his incomplete. The other possible reason may be that some inverse-square law of gravitation did not touch on the underlying assumptions in Einstein’s theory about the mechanism of gravitation ([15], p. 28;[16], p. 91). He con- character of the or may be not suitable ([5], jectured that gravitation may be explained based on the p. 101). of an aether pervading the space ([15], p. 28;[16], p. 2. The value of the is a puz- 92). In the years 1905-1916, Einstein abandoned the con- zle [6]. In 1917, A. Einstein thought that his equations cepts of electromagnetic aether and gravitational aether should be revised to be ([3], p. 410) in his ([17], p. 27-61). However, H. A. Lorentz believed that general relativity could be rec- 1 onciled with the concept of an ether at rest and wrote a R − g R + Λg = −κT m , (1) µν 2 µν µν µν letter to A. Einstein ([17], p. 65). Einstein changed his view later and introduced his new concept of ether ([17], where gµν is the metric of a Riemannian space- p. 63-113). However, Einstein did not tell us how to de- µν time, Rµν is the Ricci tensor, R ≡ g Rµν is the rive his equations theoretically based on his new concept , gµν is the contravariant , κ is a of the gravitational aether. m constant, Tµν is the energy- tensor of a matter 6. Whether Newton’s GN de- system, Λ is the cosmological constant. pends on time and space is still not clear. It is known However, it seems that the cosmological constant Λ that GN is a constant in Newton’s and Einstein’s the- is unnecessary when Hubble discovered the expansion of ory of gravitation. P. A. M. Dirac speculates that GN the . Thus, Einstein abandoned the term Λgµν may depend on time based on his large number hypoth- in Eqs. (1) and returned to his original equations ([3], p. esis [18]. R. P. Feynman thought that if GN decreases 410). The value of the cosmological constant Λ is also on time, then the earth’s temperature a billion years a- related to the energy-momentum tensor of vacuum ([3], go was about 48◦C higher than the temperature

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([4], p. 9). in a fluidic substratum has been proposed by many re- Furthermore, there exist some other problems relat- searchers in the history, for instance, J. C. Maxwell ([15], ed to the theories of , for instance, gravitational p. 243), B. Riemann ([30], p. 507), H. Poincar´e ([31], p. waves [19], the of in vacuum [20], the defi- 171), J. C. Taylor ([32], p. 431-436). Thus, we suppose nition of inertial system, origin of inertial , the ve- that electric charges are sources or sinks of the Ω(1) sub- locity of the propagation of gravity [21], the of stratum [26]. Maxwell’s equations in vacuum are derived individual [22], unified field theory, etc. by methods of based on a mechan- The gravitational interaction seems to differ in char- ical model of vacuum and a source or sink flow model of acter from other interactions. The existing theories of electric charges [26]. The electromagnetic aether behaves gravity still face the aforementioned difficulties. Thus, it as a visco-elastic continuum [26]. Maxwell’s equations seems that new ideas about the gravitational phenome- approximate the macroscopic behavior of the Ω(1) parti- na are needed. Following Einstein [23], it may be better cles, in analogy to the way that classical elastic mechanics for us to keep an open and critical mind to explore all approximates the macroscopic behavior of the atoms of possible theories about gravity. solid materials. In 2012, M. J. Dupre and F. J. Tipler propose an aether Descartes interpreted the celestial of celestial theory of general relativity [24]. Recently, generalized E- bodies based on the hypothesis that the universe is filled instein’s equations of gravitational fields in inertial ref- by a fluidic vortex aether [15]. Since Newton’s law of erence frames are derived based on a sink flow model of gravitation was published in 1687 [33], this action-at-a- particles [25]. However, the cases of non-inertial refer- theory was criticized by the French Cartesian ence frames are not discussed in Ref. [25]. The purpose [34]. Sir I. Newton tried to obtain a derivation of his of this manuscript is to propose a derivation of the E- law based on Descartes’ scientific research program. At instein’s equations in some non-inertial reference frames last, he proved that Descartes’ vortex aether hypothesis based on the theory of vacuum mechanics [25–28]. could not explain celestial motions properly [33]. Newton himself even suggested an explanation of gravity based on the action of an aetherial medium pervading the space II. A BRIEF INTRODUCTION OF THE ([15], p. 28). Euler attempted to explain gravity based THEORY OF VACUUM MECHANICS on some hypotheses of a fluidic aether [34]. Following Descartes, we introduce the following assumption [27]. Ancient people believe that there is a kind of contin- Assumption 2 We suppose that vacuum is filled by an uously distributed substance which fills every corner of extremely thin medium which may be called the Ω(0) sub- the space. Ancient Egyptians called this substance as stratum, or the gravitational aether. nun ([29], p. 172). According to ancient Egyptians, the nun is a primeval watery darkness which is continuously Thus, two sinks in the Ω(0) substratum are found to distributed surround the world ([29], p. 172). Ancient attract with each other according to the inverse-square Greeks, such as Thales and Anaximenes, believe that ev- law of gravitation [27]. A feature of this theory is that the erything in the universe is made of a kind of fundamental gravitational constant depends on time and the location substance named aether. in space. The thought that our universe is an infinite hierarchy The particles that constitute the Ω(1) substratum may which has within universes without end is a be called the Ω(1) particles. Lord Kelvin believes that the charming idea. Astonished by this attractive idea and electromagnetic aether must also generate gravity [24, inspired by the aforementioned ideas, we propose the fol- 35]. Following Lord Kelvin, we introduce the following lowing mechanical model of the universe. assumption. Matter is composed of molecules. Molecules are con- structed by atoms. Atoms are formed by elementary par- Assumption 3 We suppose that there exist a kind of ticles. Modern experiments, for instance, the Casimir ef- basic sinks of the Ω(0) substratum, which may be called fect, have shown that vacuum is not empty. Therefore, monads after Leibniz. The Ω(1) particles and elementary new considerations on the old concept of aether may be particles are formed of monads. needed. Since monads, the Ω(1) particles, elementary particles Thomson’s analogies between electrical phenomena are sinks of the Ω(0) substratum, they attract with each and elasticity helped J. C. Maxwell to establish a me- other according to Newton’s law of gravitation [27]. chanical model of electrical phenomena [15]. Following There exists an universal drag force exerted on each J. C. Maxwell, we introduce the following assumption sink of the Ω(0) substratum [27]. Therefore, each mon- [26]. ad, each Ω(1) particle and each elementary particle, as Assumption 1 We suppose that vacuum is filled with sinks of the Ω(0) substratum, will experience the uni- a kind of continuously distributed matter which may be versal drag force. On the other hand, all the monads, called the Ω(1) substratum, or the electromagnetic aether. Ω(1) particles and elementary particles are undertaking stochastic movements. Based on this universal damping The idea that all microscopic particles are sink flows force and some assumptions, microscopic particles are

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found to obey a generalized non-relativistic Schr¨odinger Eq. (7) shows that the inertial mip of a pro- equation [28]. ton equals its gravitational mass mgp. Similarly, we can For convenience, we may call these theories [25–28] as demonstrate that the inertial mass of another type of the theory of vacuum mechanics. Vacuum mechanics is microscopic particle equals its gravitational mass.  a physical theory which attempts to derive some basic This result is called the principle of equivalence in the physical laws based on a new mechanical model of vacu- theory of general relativity [1–3]. um and particles.

IV. THE DYNAMICAL GRAVITATIONAL III. EQUIVALENCE BETWEEN THE INERTIAL POTENTIALS IN INERTIAL REFERENCE MASS AND THE GRAVITATIONAL MASS FRAMES

Proposition 1 The inertial mass of a microscopic par- The purpose of this section is to review the mathe- ticle equals its gravitational mass. matical forms of the dynamical gravitational potentials in inertial reference frames. These results may provide Proof of Proposition 1. Newton’s law of gravitation us some clues to explore possible - can be written as ([36], p. 2) s of inertial potential and inertial force Lagrangian in non-inertial reference frames, which are introduced in the mg1mg2 next section. F12 = −GN ˆr12, (2) r2 We introduce a Cartesian {0, x, y, z} for a three-dimensional that attached to where F12(t) denotes the force exerted on the particle the Ω(1) substratum. Let {0, t} be a one-dimensional with gravitational mass mg2 by the particle with grav- time coordinate. For convenience, we introduce the fol- itational mass mg1, mg1 and mg2 are the gravitational lowing Galilean coordinate system x0 ≡ ct, x1 ≡ masses of two particles, GN is Newton’s gravitational 2 3 x, x ≡ y, x ≡ z. Let ηµν denotes the metric tensor constant, ˆr12 denotes the directed along the of the Minkowski . line from the particle with mass mg1 to the particle with We will use Greek indices α, β, µ, ν, etc., to denote the mass mg2, r is the distance between the two particles. range {0, 1, 2, 3} and Latin indices i, j, k, etc., to denote In 2008, we show that the force F12(t) exerted on the the range {1, 2, 3}. Einstein’s summation convention will particle with inertial mass mi2(t) by the velocity field of be used, i.e., any repeated Greek superscript or subscript the Ω(0) substratum induced by the particle with inertial appearing in a term of an equation is to be summed from mass mi1(t) is [27] 0 to 3. The definition of the strength g of a gravitational field m (t)m (t) F (t) = −γ (t) i1 i2 ˆr , (3) is ([37], p. 24) 12 N r2 12 F where g = g , (8) mtest ρ q2 γ (t) = 0 0 , (4) where m is the mass of a test point particle, F is the N 4πm2(t) test g 0 gravitational force exerted on the test point particle by a gravitational field. ρ is the density of the Ω(0) substratum, m (t) is the 0 0 According to Newton’s second law, we have inertial mass of a monad at time t, −q0(q0 > 0) is the strength of the monad. Fg = mtesta, (9) Suppose that GN = γN (t). Comparing Eq. (2) and Eq. (3), we have where a is the of the test point particle. Comparing Eq. (9) and Eq. (8), we have mi1mi2 = mg1mg2. (5) g = a. (10) Now we study a gravitational system of two . According to Eq. (5), we have The definition of the acceleration a is ([37], p. 24)

m2 = m2 , (6) d2xk ip gp a = γ , (11) i ik dt2 where mip and mgp are the inertial mass and gravitation- al mass of a respectively. where Noticing mip > 0 and mgp > 0, Eq. (6) can be written g0ig0k γik = −gik + , (12) as g00

mip = mgp. (7) gµν is the metric tensor of a Riemannian spacetime.

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Based on the time tracks of free particles described Assumption 4 The inertial force exerted on a matter by curves in Minkowski spacetime, we have the system in a non-inertial reference frame stems from the following results ([1], p. 279;[37], p. 26) interactions between the matter system and vacuum. √ ∂Π 2Π ∂γ Based on Assumption 4, we introduce the following a = − − c 1 + i , (13) i ∂xi c2 ∂t concepts for inertial , which are similar to those concepts for gravitational interactions. where iner − Definition 2 Inertial potential ψµν is an interaction 1 g00 2 gi0 Π = − c , γi = −√ . (14) potential between a matter system and vacuum resulting 2 g00 from the inertial force Finer exerted on the matter system by vacuum in a non-inertial reference frame Sn. If we suppose that ∂g00/∂t ≈ 0, then Eqs. (13) can also be written as ( ) Definition 3 Inertial force Lagrangian Liner is an inter- ∂ 1 − g ∂(−g ) action Lagrangian between a matter system and vacuum a = c2 00 − c2 i0 . (15) i ∂xi 2 ∂(ct) resulting from the inertial force Finer exerted on the mat- ter system by vacuum in a non-inertial reference frame Sn. V. INERTIAL POTENTIAL AND INERTIAL FORCE LAGRANGIAN IN NON-INERTIAL Now our task is to explore possible models of inertial iner REFERENCE FRAMES potential ψµν and inertial force Lagrangian Liner. Let ′ ηµν denotes the metric tensor of the non-inertial refer- ′ ′0 ≈ According to the theory of general relativity [2, 3], the ence frame Sn. Suppose that ∂η00/∂x 0. Then, fol- Einstein’s equations are valid not only in inertial ref- lowing similar methods as in the derivation of Eqs. (15), erence frames but also in non-inertial reference frames. we obtain the following relationship for the inertial ac- Thus, it is needed to explore the possibility to derive the celeration a of a test point particle in the non-inertial Einstein’s equations in non-inertial reference frames. reference frame Sn ( ) When solving the Einstein’s equations for an isolated ′ ′ ∂ 1 − η ∂(−η ) system of masses, V. Fock introduces harmonic reference 2 00 − 2 i0 ai = c ′i c ′0 . (16) frame and obtains an unambiguous solution ([38], p. 369). ∂x 2 ∂x Furthermore, in the case of an isolated system of mass- ′ If η are time-independent, the inertial acceleration a es, he concludes that there exists a harmonic reference i0 of the test point particle in Eqs. (16) simplifies to ([1], p. frame which is determined uniquely apart from a Lorentz 280) transformation if suitable supplementary conditions are ( ) imposed ([38], p. 373). It is known that wave equations − ′ 2 ∂ 1 η00 keep the same form under Lorentz transformations [1]. ai = c . (17) ∂x′i 2 From Eqs. (93) in Ref. [25], we notice that the field e- quations of gravitation in inertial reference frames in a Using Eqs. (17), the inertial force Finer exerted on the Minkowski spacetime are wave equations. Thus, we spec- test point particle can be written as ulate that Fock’s special harmonic reference frames may ( ) have provided us a clue to derive the Einstein’s equations 1 − η′ F = ma = mc2∇ 00 , (18) in some special class of non-inertial reference frames. iner 2 We introduce an arbitrary non-inertial coordinate sys- ′0 ′1 ′2 ′3 tem (x , x , x , x ) and denote it as Sn. It is known where m is the mass of the test point particle, ∇ = that a particle in a non-inertial reference frame will ex- i∂/∂x′1 + j∂/∂x′2 + k∂/∂x′3 is the Hamilton operator, perience an inertial force. Unfortunately, we have no i, j, k are three unit vectors directed along the coordi- knowledge about the origin of inertial forces. nate axes. For convenience, we introduce the following definition From Eq. (18), the inertial force Lagrangian of a sys- of matter system. tem of vacuum and the test point particle can be written as Definition 1 A matter system is a system of a number ( ) of elementary particles, or continuously distributed ele- 1 − η′ L = −mc2 00 . (19) mentary particles. iner1 2

The equivalence between inertial mass and gravitation- Therefore, the inertial force Lagrangian of a system al mass implies that to some degree gravitational forces of vacuum and continuously distributed particles can be behave in the same way as inertial forces ([36], p. 17). written as Thus, we speculate that inertial forces may originate from ( ) the interactions between matter systems and vacuum. 1 − η′ L = −ρ c2 00 , (20) Therefore, we introduce the following assumption. iner m 2

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′ ≡ where ρm is the rest mass density of the continuously where√ m is the rest mass of the point particle, dτη distributed particles. 1 ′µ ′ ′µν c dx dxµ is the infinitesimal interval, Let Tm denotes the contravariant energy-momentum ′ ′ µ µ tensor of the continuously distributed particles system. If u ≡ dx /dτη′ . Ignoring the contravariant energy-momentum tensor of the inertial force is small enough, then we may regard this ′ non-inertial reference frame S as an inertial reference the Ω(1) substratum, i.e., T µν ≈ 0 and using Eq. (28) n ′ Ω(1) 00 ≈ 2 frame approximately. Thus, we have Tm ρmc [25]. and Eq. (24), the total Lagrangian Lp of a system of Noticing η00 = 1, the inertial force Lagrangian Liner in the Ω(0) substratum, the Ω(1) substratum and the point Eq. (20) can be written as particle can be written as

′ ≈ iner 00 ′ 1 ′µ ′ iner ′µ ′ν Liner f0ψ00 Tm , (21) L = L + L = mu u + f ψ mu u . (29) p 0 iner 2 µ 0 µν where The Euler-Lagrange equations for the total Lagrangian 1 Lp can be written as ([36], p. 111) ψiner = − (η − η′ ). (22) 00 2f 00 00 0 ∂Lp − d ∂Lp ′µ ′µ = 0. (30) ∂x dτη′ ∂u Following Ref. [25], the parameter f0 is √ √ Putting Eq. (29) into Eqs. (30), we have 2 [ ] 2ρ0q0 8πγN ′ ′ ′ f0 = = . (23) ( ) ν iner α β 2 4 4 d dx ∂ψαβ dx dx m0c c iner − ηµν + 2f0ψµν f0 ′µ = 0. dτη′ dτη′ ∂x dτη′ dτη′ Inspired by Eq. (21) and Eq. (22), we introduce the (31) following assumption. Using Eq. (25), Eqs. (31) can be written as ( ) Assumption 5 Suppose that the inertial force La- ′ν ∂η′ ′α ′β d ′ dx − 1 αβ dx dx grangian Liner of a system of a free point particle and ηµν ′µ = 0. (32) dτη′ dτη′ 2 ∂x dτη′ dτη′ vacuum in the non-inertial reference frame Sn can be written as Eqs. (32) represent a geodesic line in a Minkowski s- ′ ′ ′ pacetime with a metric tensor η , which can also be L = f ψinermu µu ν , (24) µν iner 0 µν written as Eqs. (26) ([37], p. 50).  where m is the rest mass of the point particle, u′µ ≡ Eqs. (26) is a geodesic curve in a - ′µ time. It is known that a geodesic curve is a straight line dx /dτη′ , τη′ is the proper time, in a Minkowski spacetime ([40], p. 235). For instance, ac- cording to Newton’s first law, a free particle moves along iner − 1 − ′ ψµν = (ηµν ηµν ). (25) 2f0 a straight line in the Galilean coordinates. Therefore, Assumption 5 may be supported by some experiments. Following similar methods as in [25], we obtain the Inspired by the inertial force Lagrangian for a free point following result. particle in Eq. (24), we introduce the following assump- tion for a matter system and vacuum based on Assump- Proposition 2 Suppose that Assumption 5 is valid. tion 3. Then, the equations of of a free point particle can be written as Assumption 6 The inertial force Lagrangian Liner of a matter system and vacuum in the non-inertial reference 2 ′µ ′ν ′σ d x ′ dx dx µ frame Sn can be written as + Cνσ = 0, (26) 2 ′ ′ dτ ′ dτη dτη ′ ′ η iner µν µν iner 2 Liner = f0ψµν (Tm + TΩ(1)) + O[(f0ψµν ) ], (33)

where ′ ( ) ′µν µν ′ ′ ′ where Tm and TΩ(1) are the contravariant energy- ′ ′ ∂η ∂η ∂η ν ≡ 1 µν µα µβ − αβ Cαβ η ′ + ′ ′ (27) momentum of the matter and the Ω(1) substra- 2 ∂x β ∂x α ∂x µ iner 2 tum respectively, O[(f0ψµν ) ] denotes those terms which iner 2 are the corresponding Christoffel symbols in the non- are small quantities of the order of (f0ψµν ) . inertial reference frame Sn. Proof of Proposition 2. The Lagrangian of a free point VI. EQUATIONS IN A SPECIAL CLASS particle in Sn can be written as ([4], p. 57;[39]) OF NON-INERTIAL REFERENCE FRAMES

′µ ′ ′ 1 dx dxµ 1 ′µ ′ 1 ′ ′µ ′ν Suppose that the transformation equations between a L0 = m = mu uµ = mηµν u u , (28) ′0 ′1 ′2 ′3 2 dτη′ dτη′ 2 2 non-inertial coordinate system (x , x , x , x ) and

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the Galilean coordinates (ct, x, y, z) are substratum, the Ω(1) substratum, vacuum and matter in SF can be written as x′α = f α(x0, x1, x2, x3). (34) 1 ′ ′ ′ ′ ′ L′ = ∂′ ψ′ ∂′λψ µν − 2∂′ ψ′ ∂ µψ λν − 6∂ µψ′ ∂ ν ψ′ Following V. Fock ([38], p. 370-373), we introduce the tot 2 λ µν λ µν µν following definition of a special class of reference frames. 3 ′ ′ ′ ′ ′ ′ ′ ′µν − ∂ ψ ∂ λψ + L + f ψ (T µν + T ) 2 λ more 0 µν m Ω(1) ′ Definition 4 Suppose that a coordinate system iner ′µν µν iner 2 ′ ′ ′ ′ +f ψ (T + T ) + O[(f ψ ) ] (x 0, x 1, x 2, x 3) satisfies the following conditions: (1) 0 µν m Ω(1) 0 µν ′α ′ 2 every coordinates x satisfies the d’Alembert’s equation +O[(f0ψµν ) ], (40) ([38], p. 369), i.e., ′ ( ) where ψµν is a tensorial potential of the gravitational √ ′α ′ ′α 1 ∂ ′ ′µν ∂x field, L denotes those terms involving more than t-  ′ ≡ √ − more η x ′ η0η ′ = 0, (35) ′ ′ 2 − ′ µ ν wo derivatives of ψ , O[(f0ψ ) ] denotes those terms η0 ∂x ∂x µν µν ′ 2 which are small quantities of the order of (f0ψµν ) . ′ ′ where ηµν is the metric of the reference frame Sn, η0 = ′ Proof of Proposition 3. Based on some assumptions, Det ηµν , DetAµν denotes the value of the corresponding ′α of the tensor Aµν ; (2) every coordinates x the total Lagrangian Ltot of a system of the Ω(0) sub- converges to the Galilean coordinates (ct, x, y, z) at large stratum, the Ω(1) substratum and matter in an inertial enough distance, i.e., reference frame can be written as [25]

′α α 1 lim x = x , (36) L = ∂ ψ ∂λψµν − 2∂ ψ ∂µψλν − 6∂µψ ∂ν ψ r→∞ tot 2 λ µν λ µν µν √ 3 µν 2 2 2 ′µν ′µν − λ µν where r = x + y + z ; (3) η −(η )∞ are outgoing ∂λψ∂ ψ + Lmore + f0ψµν (Tm + TΩ(1)) ′µν ′µν 2 waves, i.e., η − (η )∞ satisfy the following condition 2 → ∞ ′ +O[(f0ψµν ) ], (41) of outward radiation: for r , and all values of t0 = t+r/c in an arbitrary fixed interval the following limiting where ψµν is a tensorial potential of the gravitational field conditions are satisfied ([38], p. 365) in the inertial reference frame, Lmore denotes those terms [ ] 2 ′µν ′µν ′µν ′µν involving more than two derivatives of ψµν , O[(f0ψµν ) ] ∂[r(η − (η )∞)] 1 ∂[r(η − (η )∞)] lim + = 0, denotes those terms which are small quantities of the r→∞ ∂r c ∂t 2 order of (f0ψµν ) . (37) The total Lagrangian L′ can be written as ′µν ′µν tot where (η )∞ denotes the value of η at infinity. Then, we call this coordinate system (x′0, x′1, x′2, x′3) as a ′ Ltot = Ltot + Liner. (42) Fock coordinate system. Similar to the case of inertial reference frames ([36], p. We use SF to denote a Fock coordinate system. The 59-60, 63), we also have the following results in the Fock Galilean coordinate system (ct, x, y, z) is a Fock coordi- coordinate system SF nate system. V. Fock points out an advantage of Fock coordinate system ([38], p. 369):”When solving Einstein’s ′ σ ′λ λ σ ∂λ = aλ ∂σ, ∂ = a σ∂ , (43) equations for an isolated system of masses we used har- ′µν µ ν αβ monic coordinates and in this way obtained a perfectly ψ = a αa βψ , (44) ′ α β unambiguous solution.” Here the harmonic coordinates ψµν = aµ aν ψαβ. (45) called by V. Fock are Fock coordinate systems. According to a theorem of Fock about Fock coordinate The first term on the right hand side of Eqs. (40) can systems ([38], p. 369-373), the transformation equations be written as (34) can be written as a Lorentz transformation, i.e., 1 ′ ′ ′λ ′µν 1 σ α β ∂λψµν ∂ ψ = (aλ ∂σ)(aµ aν ψαβ) x′µ = aµ xν , (38) 2 2 ν · λ σ µ ν αβ (a σ∂ )(a αa βψ ). (46) µ where a ν are coefficients of a Lorentz transformation. For convenience, we introduce the following notations We have the following result ([36], p. 60) ( ) µ β µ ′ ′ a a ν = δν , (47) ′ ≡ ∂ ∂ ∂ ∂ µ ≡ µν ′ β ∂µ ′ , ′ , ′ , ′ , ∂ η ∂ν . (39) ∂x 0 ∂x 1 ∂x 2 ∂x 3 µ where δν is the Kronecker . Using Eq. (47), Eqs. (46) can be written as Proposition 3 Suppose that the reference frame SF is a Fock coordinate system and Assumptions 6 is valid, 1 ′ 1 ′ ′ ′ ′λ µν σ αβ ∂ ψ ∂ ψ = ∂σψαβ∂ ψ . (48) then the total Lagrangian Ltot of a system of the Ω(0) 2 λ µν 2

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Similarly, we can verify the following results Similarly, the second term on the right hand side of Eqs. (57) can be written as − ′ ′ ′µ ′λν − α σβ 2∂λψµν ∂ ψ = 2∂σψαβ∂ ψ , (49) ′ ′ ′ ′ ′ ′ λ µν ′ − µ ν − α β ′ ′ ∂(∂ ψ ) σ ′ 6∂ ψµν ∂ ψ = 6∂ ψαβ∂ ψ, (50) ∂λψµν ′ ′αβ = ∂ ψαβ. (61) ∂(∂σψ ) −3 ′ ′ ′λ ′ −3 σ ∂λψ ∂ ψ = ∂σψ∂ ψ, (51) 2 2 Using Eqs. (57), Eqs. (60) and Eqs. (61), we have ′ ′µν αβ f0ψµν Tm = f0ψαβTm , (52) [ ] ′ ′ ′λ ′µν ′ ′ ∂ ∂(∂λψµν ∂ ψ ) ′ σ ′ Lmore = Lmore, (53) ′σ ′ ′αβ = 2∂σ∂ ψαβ. (62) ′ 2 2 ∂x ∂(∂σψ ) O[(f0ψµν ) ] = O[(f0ψµν ) ]. (54) Putting Eq. (41) and Eq. (33) into Eq. (42) and using Similarly, we can verify the following results Eqs. (48-54), we obtain Eq. (40).  [ ] ′ ′ ′µ ′λν Applying similar methods as in Ref. [25], we have the ∂ ∂(∂ ψ ∂ ψ ) ′ λ µν = ∂ σ∂′ ψ′ following result. ′σ ′ ′αβ α βσ ∂x ∂(∂σψ ) ′ Theorem 1 If we ignore those terms which are smal- + ∂ σ∂′ ψ′ , (63) ′ 2 iner 2 [ ] β ασ l quantities of the order of (f0ψµν ) and (f0ψµν ) and ′ ′ ′ µ ′ ν those terms involving more than two derivatives of ψ ∂ ∂(∂ ψµν ∂ ψ) ′ ′ ′ µν = ∂ ∂ ψ ′ 2 ≈ iner 2 ≈ ′σ ′ ′αβ α β in Eq. (40), i.e., O[(f0ψµν ) ] 0, O[(f0ψµν ) ] 0 ∂x ∂(∂σψ ) ′ ≈ and Lmore 0, then the field equations for the total La- ′ ′ ′ ′σλ ′ + ηαβ∂σ∂λψ , (64) grangian Ltot in Eq. (40) can be written as [ ] ′ ′ ′λ ′ ′ ′ ∂ ∂(∂ ψ ∂ ψ ) ′ ′ ′σ ′ − σ ′ ′ σ ′ ′ λ ′ ′ σ ′ ∂σ∂ ψαβ 2(∂ ∂αψβσ + ∂ ∂βψασ) ′σ ′ ′αβ = 2ηαβ∂σ∂ ψ , (65) ∂x ∂(∂σψ ) − ′ ′ ′ ′σλ ′ ′ ′ − ′ ′ ′σ ′ 6(ηαβ∂σ∂λψ + ∂α∂βψ ) 3ηαβ∂σ∂ ψ ′ ( ) ∂L ′ ′Ω(1) ′ ′ tot m m Ω(1) ′αβ = f0(Tαβ + Tαβ ). (66) = f0 Tαβ + Tαβ . (55) ∂ψ

Proof of Theorem 1. We have the following Euler- Putting Eq. (40) into Eqs. (56) and using Eqs. (62-66), Lagrange equations [41] we obtain Eqs. (55).  ( ) Following Ref. [25], we introduce the following nota- ∂L′ ∂ ∂L′ tot − tot tion in the Fock coordinate system SF . ′αβ ′σ ′ ′αβ = 0. (56) ∂ψ ∂x ∂(∂σψ ) ′ ′ ′ ′ ′ ′ ′ µν ′ λ µν − ′ µ νλ − ′ ν µλ Ψ = ∂λ∂ ψ 2∂λ∂ ψ 2∂λ∂ ψ We have the following results ′ ′ ′ ′ ′ ′ − µν ′ ′ σλ − µ ν ′ − µν ′ λ ′ 6η ∂σ∂λψ 6∂ ∂ ψ 3η ∂λ∂ ψ .(67) ′ ′ ′λ ′µν ′ ′ ∂(∂λψµν ∂ ψ ) ∂(∂λψµν ) ′λ ′µν ′ ′ = ′ ′ ∂ ψ For convenience, we introduce the following definition ∂(∂ ψ αβ) ∂(∂ ψ αβ) ′ σ σ of the total energy-momentum tensor T µν of the sys- ′λ ′µν ′ ′ ∂(∂ ψ ) tem of the matter, the Ω(1) substratum and the Ω(0) +∂λψµν ′ ′αβ , (57) ∂(∂σψ ) substratum in the Fock coordinate system SF

′ ′ ′ ′ T µν = T µν + T µν + T µν , (68) ′ ′ ′ ′ρτ m Ω(1) Ω(0) ψµν = ηµρηντ ψ . (58) ′µν Using Eqs. (58), we have where TΩ(0) is the energy-momentum tensor of the Ω(0) ′ ′ ( ) substratum in the Fock coordinate system SF . ∂(∂ ψ ) ∂ ′ λ µν = η′ η′ ∂′ ψ ρτ We introduce the following notation in SF ∂(∂′ ψ′αβ) ∂(∂′ ψ′αβ) µρ ντ λ σ σ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ µν λ µν − µ νλ ν µλ ′ ρτ Θ = ∂λ∂ ψ (∂λ∂ ψ + ∂λ∂ ψ ) ′ ′ ∂(∂ ψ ) = η η λ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ µρ ντ ′ ′αβ +(∂ µ∂ ν ψ + η µν ∂ ∂ ψ σλ) − η µν ∂ ∂ λψ . (69) ∂(∂σψ ) σ λ λ ′ ′ λ ρ τ = ηµρηντ δσ δαδβ Following Ref. [25], we introduce the following defini- ′ ′ λ ′µν = ηµαηνβδσ . (59) tion of the contravariant energy-momentum tensor Tω of vacuum in SF Using Eqs. (59) and Eqs. (58), the first term on the ′ right hand side of Eqs. (57) can be written as ′µν 1 ′µν 1 ′µν µν Tω = Ψ + Θ + TΩ(0). (70) ′ ′ f0 f0 ∂(∂λψµν ) ′λ ′µν ′σ ′ ′ ′αβ ∂ ψ = ∂ ψαβ. (60) ∂(∂σψ ) Using these definitions, we have the following result.

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Corollary 1 The field equations (55) can be written as Definition 6

√ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ µν µν µν µν λ µν − µ νλ − ν µλ g˜ ≡ −g0g ≡ η − 2f0ϕ , (80) ∂λ∂ ψ ∂λ∂ ψ ∂λ∂ ψ ′µ ′ν ′ ′µν ′ ′ ′σλ +∂ ∂ ψ + η ∂σ∂λψ where g0 = Det gµν . − ′µν ′ ′λ ′ − ′µν − ′µν η ∂λ∂ ψ = f0(T Tω ). (71) Applying similar methods of V. Fock ([38], p. 422- 430), Fock’s theorem of the Gµν in the We can verify that the field equations (71) are Galilean coordinates ([38], p. 429) can be generalized to under the following gauge transformation non-inertial coordinate systems (x′0, x′1, x′2, x′3). ′ ′ ′ ′ ψ µν → ψ µν + ∂ µΛν + ∂ ν Λµ, (72) Proposition 4 The contravariant Einstein tensor Gµν in a non-inertial coordinate systems (x′0, x′1, x′2, x′3) µ where Λ is an arbitrary vector field. can be written as We introduce the following definition 2 µν 1 ∂ g˜ ′ ′ 1 ′ ′ Gµν = g˜αβ + Π µ,αβΠ ν − y µy ν ′ ′ 1 ′ ′ ′α ′β αβ ϕ µν = ψ µν − η µν ψ . (73) 2g0 ∂x ∂x 2 2 1 ′ + gµν (L′ + B′) − B µν , (81) Using Eqs. (73), the field equations (71) can be written 2 as where ′ ′ ′ ′ ′ ( ) ′ λ µν ′ µ νλ ′ ν µλ µβ µα αβ ∂ ∂ ϕ − ∂ ∂ ϕ − ∂ ∂ ϕ ′ λ λ λ µ,αβ ≡ 1 αλ ∂g˜ βλ ∂g˜ − µλ ∂g˜ ′ ′ ′ ′ ′ ′ Π g˜ ′λ +g ˜ ′λ g˜ ′λ , µν σλ − µν − µν 2g0 ∂x ∂x ∂x +η ∂σ∂λϕ = f0(T Tω ). (74) (82) We introduce the following Hilbert gauge condition [39] ′ν ≡ ′ν,λσ ( ) Παβ gαλgβσΠ , (83) ′ ′µν − 1 ′µν ′ ∂µ ψ η ψ = 0. (75) 2 ′ ′ α ≡ σλ α Γ g Γσλ, (84) Using Eqs. (73), the Hilbert gauge condition Eqs. (75) ( ) simplifies to ′ν 1 µν ∂gµα ∂gµβ ∂gαβ Γ ≡ g ′ + ′ − ′ (85) ′ αβ β α µ ′ µν 2 ∂x ∂x ∂x ∂µϕ = 0. (76) ( ) If we impose the Hilbert gauge condition Eqs. (76) on ′ν ′µ µν ′ 1 ∂Γ ∂Γ ∂g ′ the fields, then the field equations (74) simplify to Γ µν ≡ gµα + gνα − Γ α , (86) 2 ∂x′α ∂x′α ∂x′α ′ ′λ ′µν ′µν ′µν ∂ ∂ ϕ = −f0(T − T ). (77) λ ω √ ∂(lg −g ) ′ The field equations (77) can also be written as y′ ≡ 0 , y α ≡ gαβy′ , (87) β ∂x′β β 2 ′µν ′αβ ∂ ϕ − ′µν − ′µν η ′α ′β = f0(T Tω ). (78) αβ ∂x ∂x ′ ≡ − √1 ′ν ∂g˜ 1 ′ ′ν L Γαβ ′ν + yν y , (88) 2 −g0 ∂x 2 VII. GENERALIZED EINSTEIN’S EQUATIONS IN A SPECIAL CLASS OF NON-INERTIAL ′ ′ 1 ′ ′ ′ ′ ′ B µν ≡ Γ µν + (y µΓ ν + y ν Γ µ),B′ ≡ g B µν . (89) REFERENCE FRAMES 2 µν A proof of Proposition 4 can be found in the Appendix. Definition 5 The Einstein tensor Gµν is defined by Theorem 2 In a Fock coordinate system S , we have 1 F G ≡ R − g R, (79) the following field equations µν µν 2 µν ( ) √ √ ′ 2 − µν µν 1 αβ αβ ∂ ( g0g ) where gµν is a metric tensor of a Riemannian space- G − −g0g − η ′ ′ µν µν 2g ∂x α∂x β time, Rµν is the Ricci tensor, R ≡ g Rµν , g is 0 2 ′µν the corresponding contravariant tensor of gµν such that 1 ′αβ ∂ η ′µ,αβ ′ν 1 ′µ ′ν λν ν − η − Π Π + y y g g = δ ([37], p. 40). ′α ′β αβ µλ µ 2g0 ∂x ∂x 2 2 Similar to Ref. [25], we introduce the following defini- −1 µν ′ ′ ′µν f0 ′µν − ′µν g (L + B ) + B = (T Tω ). (90) tion of a metric tensor gµν of a Riemannian spacetime. 2 g0

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Proof of Theorem 2. Using Eqs. (80), Eqs. (81) can Eqs. (96) are only valid approximately in a quasi- be written as inertial Fock coordinate system SF . Now we consider ′ ′ weak fields. (√ ) 2 µν µν 1 ′ ′ ∂ (η − 2f0ϕ ) µν − αβ − αβ αβ ′ G = g0g η + η ′α ′β µν 2g0 ∂x ∂x Definition 8 If ϕ and their first and higher deriva- ′ ′ ′ ′ ′ tives satisfy the following conditions µ,αβ ν − 1 µ ν 1 µν ′ ′ − µν +Π Παβ y y + g (L + B ) B 2 2 ′µν ( ) ′ ′ 2f0ϕ ≪ 1, (97) √ ′ 2 µν − µν 1 αβ αβ ∂ (η 2f0ϕ ) = −g0g − η ′ ′ ′ α β j+k µν 2g0 ∂x ∂x ∂ (2f0ϕ ) ′ ′ ′ ′ ≪ 1, j + k = 1, 2, 3, ··· (98) ′ 2 µν ′ 2 µν α j β k 1 αβ ∂ η − f0 αβ ∂ ϕ ∂(x ) ∂(x ) + η ′α ′β η ′α ′β 2g0 ∂x ∂x g0 ∂x ∂x ′µν ′ ′ 1 ′ ′ then we call this field ϕ weak. +Π µ,αβΠ ν − y µy ν αβ 2 Following similar ideas as in Ref. [25], we introduce 1 ′ + gµν (L′ + B′) − B µν . (91) the following assumption. 2 Assumption 7 For weak fields, the following relation- Using Eqs. (80) and Eqs. (78), Eqs. (91) can be written ships are valid as Eqs. (90).  We need to study the relationships between Eqs. (90) ′ ′ ′ ′ T µν − T µν ≈ T µν + T µν . (99) and the Einstein’s field equations. In a harmonic coordi- ω m Ω(1) nates system, we have ([38], p. 254) Similar to Ref. [25], we have the following result. ′ ′ ′ ′ Γ ν = Γ µν = B µν = B = 0. (92) Corollary 4 Suppose that (1) the Fock coordinate sys- tem SF is quasi-inertial; (2) the field is weak. Then, the Using Eqs. (92) and Eqs. (90), we have the following field equations (96) reduce to result. 2 ( ) − 1 ≈ f0 ′m ′Ω(1) Corollary 2 In a Fock coordinate system SF the field Rµν gµν R Tµν + Tµν . (100) equations (90) can be written as 2 g0 ( ) √ Proof of Corollary 4. According to Definition 8, √ ′ 2 − µν ′ µν 1 αβ αβ ∂ ( g0g ) µν − − − f0ϕ and their first and higher derivatives are small G g0g η ′α ′β 2g0 ∂x ∂x quantities of order ε, where |ε| ≪ 1 is a small quantity. 2 ′µν ′ ′ ′ Since the reference frame is quasi-inertial, Eqs. (94) are − 1 αβ ∂ η − µ,αβ ν η ′α ′β Π Παβ 2g0 ∂x ∂x valid. Using Eqs. (94), Eqs. (80) can be written as 2 √ ′ 1 ′µ ′ν − 1 µν ′ f0 ′µν − ′µν − µν ≈ µν − µν + y y g L = (T Tω ). (93) g0g η 2f0ϕ . (101) 2 2 g0 Since the field is weak, Eqs. (97) and Eqs. (98) are Definition 7 If the following conditions are valid valid. Thus, applying Eqs. (97) and Eqs. (101), we have ′ the following estimations of the order of magnitude of the η µν ≈ ηµν , (94) following quantities 2 ′µν 1 ′ ∂ η ′ ′ αβ ≪ 2 µν − µν √ ′ η ′ ′ f0 (T Tω ) , (95) − αβ − αβ ∼ 2 ∂x α∂x β g0g η ε. (102) Using Eqs. (98), we have the following estimations then we call this reference frame quasi-inertial. ∂g ∂gµν Using Eqs. (95) and Eqs. (93), we have the following µν ∼ ∼ ′α ′α ε. (103) result. ∂x ∂x Applying Eqs. (101) and Eqs. (98), we have the follow- Corollary 3 If the reference frame S is quasi-inertial, F ing estimations then, the field equations (93) can be written as √ √ 2 − µν 2 − ′µν (√ ) 2 µν ∂ ( g0g ) ∂ ( 2f0ϕ ) 1 ′ ∂ ( −g0g ) ∼ µν − − αβ − αβ ′α ′β = ′α ′β ε. (104) G g0g η ′α ′β ∂x ∂x ∂x ∂x 2g0 ∂x ∂x ′ ′ 1 ′ ′ 1 Thus, using Eqs. (102) and Eqs. (104), we have the −Π µ,αβΠ ν + y µy ν − gµν L′ αβ 2 2 following estimations 2 f ′ ′ ( ) √ ≈ 0 µν − µν √ ′ 2 − µν (T Tω ). (96) αβ αβ ∂ ( g0g ) 2 g −g g − η ′ ′ ∼ ε . (105) 0 0 ∂x α∂x β

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Applying Eqs. (82), Eqs. (83) and Eqs. (103), we have Corollary 5 Suppose that (1) the Fock coordinate sys- the following estimations tem SF is quasi-inertial; (2) the field is weak; (3) g0 ≈ ′ ′ −1. Then, the field equations (100) reduce to Π µ,αβ ∼ Π ν ∼ ε. (106) αβ ( ) − 1 ≈ − 2 ′m ′Ω(1) Using Eqs. (87), we have the following relationship Rµν gµν R f0 Tµν + Tµν . (117) ([38], p. 143) 2

′ ′ν We introduce the following notation yβ = Γβν . (107) 8πγ We also have ([38], p. 143) κ = f 2 = N . (118) 0 c4 ′ 1 ∂g Γ ν = gµν µν . (108) βν 2 ∂x′β Using Eq. (118), the field equations (100) can be writ- ten as Applying Eqs. (107), Eqs. (108) and Eqs. (103), we ( ) 1 κ ′ ′ have the following estimations R − g R ≈ T m + T Ω(1) . (119) µν 2 µν g µν µν ′ ∼ 0 yβ ε. (109) Using Eq. (118), the field equations (117) can be writ- Using Eqs. (87) and Eqs. (109), we have the following ten as estimations ( ) ′ 1 ′ ′ y α ∼ ε. (110) R − g R ≈ −κ T m + T Ω(1) . (120) µν 2 µν µν µν Similar to the case of the Galilean coordinates, we have We notice that the field equations (120) are the E- ([38], p. 430) ′ Ω(1) instein’s equations [1–3] if we suppose that Tµν ≈ 0. αβ 1 ′ ∂g ′ L′ = − Γ ν − Γ αy′ . (111) Therefore, the field equations (90) are generalizations of 2 αβ ∂x′ν α the Einstein’s equations in some special non-inertial ref- Applying Eqs. (92), Eq. (111) can be written as erence frames. Thus, all known experiments of gravita- tional phenomena which support the theory of general αβ 1 ′ ∂g relativity may also be explained by this theory of gravity L′ = − Γ ν . (112) 2 αβ ∂x′ν based on the theory of vacuum mechanics [25–28]. Using Eq. (112), Eqs. (85) and Eqs. (103), we have the following estimation VIII. DISCUSSION L′ ∼ ε2. (113) Although the field equations (90) are generalizations of Applying Eqs. (105), Eqs. (106), Eqs. (110) and Eq. the Einstein’s equations, there exist at least the following (113), we see that the second to the fifth term on the differences between this theory and the theory of general left side of Eqs. (96) are all small quantities of order ε2. 2 relativity. Ignoring all these small quantities of order ε in Eqs. (96) 1. In the theory of general relativity, the Einstein’s and using Eqs. (99), we obtain equations are assumptions [1–3]. Although A. Einstein ( ) 2 ′ introduced his new concept of gravitational aether ([17], f0 ′ µν Gµν ≈ T µν + T . (114) p. 63-113), he did not derive his equations theoretical- g m Ω(1) 0 ly based on his new concept of the gravitational aether. Applying the rules of lowering the indexes of tensors, In our theory, the generalized Einstein’s equations (90) ′ µν µσ νλ ′µν µσ νλ ′m µν are derived by methods of special relativistic continuum i.e., G = g g Gσλ, Tm = g g Tσλ , TΩ(1) = ′ mechanics based on some assumptions. gµσgνλT Ω(1), Eqs. (114) can be written as σλ 2. Although the theory of general relativity is a field ( ) 2 ′ theory of gravity, the definitions of gravitational fields are f0 ′m Ω(1) Gσλ ≈ T + T . (115) not based on continuum mechanics [1–3, 42]. Because of g σλ σλ 0 the absence of a continuum, the theory of general rela- Putting Eqs. (79) into Eqs. (115), we obtain Eqs. (100). tivity may be regarded as a phenomenological theory of  gravity. In our theory, gravity is transmitted by the Ω(0) ′ Using Eq. (23), the field equations (100) can be written substratum. The tensorial potential ψµν of gravitational as fields are defined based on special relativistic continuum ( ) mechanics. 1 1 8πγN ′m ′Ω(1) Rµν − gµν R ≈ T + T . (116) 3. In Einstein’s theory, the concept of Riemannian s- 2 g c4 µν µν 0 pacetime is introduced together with the field equations Similar to Ref. [25], we have the following result. [1–3]. The theory of general relativity cannot provide a

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physical definition of the metric tensor of the Riemanni- are derived based on the Euler-Lagrange equations. Ap- an spacetime. In our theory, the background spacetime plying the generalized Fock’s theorem, generalized Ein- is the Minkowski spacetime. However, the initial flat stein’s equations in Fock coordinate systems are derived. background spacetime is no longer physically . If the field is weak and the reference frame is quasi- According to the equation of motion of a point particle in inertial, these generalized Einstein’s equations reduce to gravitational field in inertial reference frames [25], to the Einstein’s equations. first order of f0ψµν , the physically observable spacetime is a Riemannian spacetime with the metric tensor gµν . The metric tensor gµν is defined based on the tensorial Acknowledgments ′ potential ψµν of gravitational fields. 4. The masses of particles are constants in the theory This work was partly supported by the Doctor Re- of general relativity [1–3]. In our theory, the masses of search Foundation of Henan Polytechnic University particles are functions of time t [27]. (Grant No. B2012-069). I would like to thank an anony- 5. The gravitational constant GN is a constant in the mous reviewer for his important remarks and opinions. theory of general relativity [1–3]. The theory of general relativity cannot provide a derivation of GN . In our the- ory, the parameter γN is derived theoretically. From Eq. Appendix (4), we see that γN depends on time t. 6. In our theory, the parameter γN in Eq. (4) depends Proof of Proposition 4. The definition of the covari- on the density ρ0 of the Ω(0) substratum. If ρ0 varies ant second rank curvature tensor Rµν is ([38], p. 422) from place to place, i.e., ρ0 = ρ0(t, x, y, z), then the space αβ dependence of the gravitational constant γN can be seen Rµν ≡ g Rµα,βν , (121) from Eq. (4). 7. The Einstein’s equations [1–3] are supposed to be where valid in all reference frames. It is known that gener- ( 1 ∂2g ∂2g ∂2g al relativity is a pure theory and contains no adjustable R ≡ µν + αβ − να µα,βν 2 ∂x′α∂x′β ∂x′µ∂x′ν ∂x′µ∂x′β constants. So the predictions of general relativity are ) 2 fixed. Thus, every experimental test of the theory is im- ∂ g ′ ′ − µβ − g Γ σ Γ λ portant. For more than 100 years, experimental tests of ∂x′ν ∂x′α σλ µβ να general relativity are carried out only in the solar sys- ′σ ′λ tem [43, 44]. However, the solar system can be regarded +gσλΓµν Γαβ, (122) approximately as an inertial reference frame. Therefore, is the fourth rank curvature tensor. it is still not clear whether the Einstein’s equations are The contravariant curvature tensor Rµν can be ob- valid in all non-inertial reference frames or not. How- tained by raising the indices ([38], p. 156) ever, in our theory the generalized Einstein’s equations (90) are valid only in Fock coordinate systems. Whether µν µσ νλ R = g g Rσλ. (123) it is possible for us to derive generalized Einstein’s equa- tions in other non-inertial reference frames is an interest- Following similar methods of V. Fock ([38], p. 425), we ing question. V. Fock said ([38], p. 394):”But physically have the ’general ’, in the sense that cor- 2 µν 1 ∂ g ′ ′ ′ responding processes exist in arbitrary reference frames, Rµν = g − Γ µν + Γ µ,αβΓ ν . (124) does not hold at all. Therefore Einstein’s conclusion that 2 αβ ∂x′α∂x′β αβ all reference frames are physically equivalent, is without The definition of the invariant of the curvature tensor foundation.” is ([38], p. 425) 8. The Einstein’s equations are rigorous [1–3]. How- ever, in our theory, Eqs. (120) are valid approximately R ≡ g Rµν . (125) under some assumptions. µν It is interesting whether it is possible for us to detect Following similar methods of V. Fock ([38], p. 428), we some of these differences by experiments. have αβ ′ − ′α ′ − ′ − ′ R = g yαβ Γ yα Γ L , (126) IX. CONCLUSION where √ Inspired by the mathematical forms of the dynamical ∂2 lg −g y′ ≡ 0 , (127) gravitational potentials in inertial reference frames, we αβ ∂x′α∂x′β establish mathematical models of inertial potential and inertial force Lagrangian in non-inertial reference frames. ′ ′µν Field equations of gravitation in Fock coordinate systems Γ ≡ gµν Γ , (128)

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√ αβ ′ 1 ′ ∂g ′ ∂(lg −g0) Comparing Eqs. (132) and Eqs. (131), we have ([38], L ≡ − Γ ν − Γ α , (129) 2 αβ ∂x′ν ∂x′α p. 429) The second derivative ofg ˜µν is ([38], p. 428) ( ∂2g˜µν √ ∂2gµν ∂g ∂g = −g + y′ µν + y′ µν ∂x′α∂x′β 0 ∂x′α∂x′β β ∂x′α α ∂x′β ) ′ ′ ′ +y gµν + y y gµν . (130) 2 αβ α β − 1 √1 αβ ∂ g˜µν Rµν gµν R = g ′α ′β 2 2 −g0 ∂x ∂x Multiplying gαβ, Eqs. (130) can be written as ([38], p. 1 ′ 428) + gµν (y′ y′α + Γ αy′ + Γ′ + L′) 2 α α ( µν 2 µν 2 µν µν ′ ∂g ′ ′ ∂ g˜ √ ∂ g ′ ∂g µν ′α µ,αβ ν αβ α −Γ + y ′ + Γ Γ .(133) g ′ ′ = −g ′ ′ + 2y ′ α αβ ∂x α∂x β 0 ∂x α∂x β ∂x α ∂x ) µν αβ ′ µν ′ ′α +g g yαβ + g yαy . (131) Using Eqs. (124) and Eqs. (126), we have ([38], p. 428) ( ) 1 1 ∂2g R − g R = gαβ µν + gµν gαβy′ µν 2 µν 2 ∂x′α∂x′β αβ 1 µν ′α ′ ′ ′ Using the notations defined in Eqs. (79), Eqs. (82), + g (Γ yα + Γ + L ) 2 Eqs. (83) and Eqs. (89), Eqs. (133) can also be written − ′µν ′µ,αβ ′ν  Γ + Γ Γαβ (132) as Eqs. (81) ([38], p. 429-430).

[1] C. Moller, The theory of relativity, second edition (2016). (Clarendon Press, Oxford, 1972). [20] J. R. Sambles, 347, 828 (2015). [2] S. Weinberg, Gravitation and (John Wiley & [21] N. Cornish, D. Blas, and G. Nardini, Phys. Rev. Lett. sons, New York, 1972). 119, 161102 (2017). [3] C. Misner, K. Thorne, and J. Wheeler, Gravitation (Free- [22] D. Giovannini, J. Romero, V. Potocek, G. Ferenczi, man, New York, 1973). F. Speirits, S. Barnett, D. Faccio, and M. Padgett, Sci- [4] R. P. Feynman, Feynman lectures on gravitation, fore- ence 347, 857 (2015). word by John Preskill and Kip S. Thorne (Addison- [23] A. Einstein, Letters to Solovine 1906-1955 (Citadel, Wesley Publishing Company, New York, 1995). 2000). [5] J. Maddox, What remains to be discovered (The Free [24] M. J. Dupre and F. J. Tipler, International Journal of Press, New York, 1998). 21, 1250011 (2012). [6] D. Marsh, Phys. Rev. Lett. 118, 011302 (2017). [25] X.-S. Wang, Reports on 84, 35 [7] A. Cho, Science 355, 560 (2017). (2019). [8] A. Einstein, Ann. Math. 40, 922 (1939). [26] X.-S. Wang, Progress in Physics 2, 111 (2008). [9] EHTC, The Astrophysical Journal Letters 875, 1 (2019). [27] X.-S. Wang, Progress in Physics 4, 25 (2008). [10] P. Hamilton, M. Jaffe, P. Haslinger, Q. Simmons, [28] X.-S. Wang, Physics Essays 27, 398 (2014). H. Muller, and J. Khoury, Science 349, 849 (2015). [29] G. Pinch, Handbook of Egyptian Mythology (ABC-CLIO, [11] D. Spergel, Science 347, 1100 (2015). Inc., Santa Barbara, California, 2002). [12] J. Kim, P. Sikivie, and S. Weinberg, 541, 464 [30] B. Riemann, Collected Papers, Translated from the 1892 (2017). edition by Roger Baker, Charles Christenson and Henry [13] A. Bosma, Nature 543, 179 (2017). Orde (Kendrick Press, Heber, 2004). [14] E. Gibney, Nature 551, 153 (2017). [31] H. Poincare, The science and hypothesis, translated from [15] E. Whittaker, A History of the Theories of Aether and the French edtion, in Chinese (Shangwu Press, Beijing, Electricity, Revised and enlarged edtion, vol. 1 (Thomas 1997). Nelson and Sons Ltd., London, 1951). [32] J. Taylor, Hidden unity in Nature’s laws. Appendix A [16] T. Hirosige, , in Chinese, translated (Cambridge University Press, 2001). from the Japanese edtion published in 1968 (Qiushi Press, [33] I. Newton, Mathematical Principles of Natural Philoso- Beijing, 1968). phy and His System of the World (Univ. of Calif. Press, [17] L. Kostro, Einstein and the Ether (Apeiron, Montreal, Berkeley, 1962). 2000). [34] E. Whittaker, A History of the Theories of Aether and [18] P. A. M. Dirac, Direction in Physics (Wiley, New York, Electricity, The Modern Theories 1900-1926 (Thomas 1978). Nelson and Sons Ltd., London, 1953). [19] B. P. Abbott and et al., Phys. Rev. Lett. 116, 061102 [35] L. Kelvin, Philosophical Magazine Series 6, 2, 161 (1901).

12 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 February 2021 doi:10.20944/preprints202102.0157.v1

[36] H. C. Ohanian and R. Ruffini, Gravitation and Space- [41] S. V. Babak and L. P. Grishchuk, D 61, time, 3rd edition (Cambridge University Press, New Y- 024038 (1999). ork, 2013). [42] A. C. Eringen, The Elements of Continuum Mechanics [37] L. Liu, The theory of general relativity, in Chinese, 2nd (Robert E. Krieger Pub. Co., Huntington, 1980). edition (Advanced Education Press, Beijing, 2004). [43] C. M. Will, Theory and Experiment in Gravitational [38] V. Fock, The Theory of Space Time and Gravitation, 2nd Physics, Revised Edition (Cambridge University Press, Revised Edition, Translated from the Russian by N. Kem- Cambridge, 1993). mer (Pergamon Press, London, 1964). [44] C. W. Will, Living Rev. Relativity 17, 4 (2014). [39] W. Thirring, Annals of Physics 16, 96 (1961). [40] V. Faraoni, (Springer, New York, 2013).

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