A Theory of Gravity and General Relativity based on Quantum Electromagnetism

Based on first principles solutions in a unified framework of quantum mechanics and electromagnetism we predict the presence of a universal attractive depolarisation radiation (DR) Lorentz force ($F$) between quantum entities, each being either an IED matter particle or light quantum, in a vacuuonic dielectric vacuum. Given two quantum entities $i=1,2$ of either kind, of characteristic frequencies $\nu_i^0$, masses $m_i^0=h\nu_i^0 / c^2$ and separated at a distance r^0, the solution is $F=- G m_1^0 m_2^0/ (r^0)^2$, where $G= \chi_0^2 e^4/12 \pi^2 \epsilon_0^2 \rho_\lambda$, $\chi_0$ is the susceptibility and $\rho_\lambda$ is the reduced linear mass density of the dielectric vacuum. This force $F$ is accurate at the weak $F$ limit and resembles in all respects Newton's gravity; hence $G$ is the gravitational constant. The DR wave fields and hence the gravity is propagated in the dielectric vacuum at the speed of light $c$; these can not be shielded by matter. A test particle $\mu$ of mass $m^0$ at $r^0$ apart from a large mass $M$ is therefore gravitated by all of the building particles of M directly, by a total gravitational potential $V = -G M m^0/ r^0$. For a finite $V$ and hence a total Hamiltonian $H= m^0 c^2 +V$, solution for the eigenvalue equation of $\mu$ presents a red-shift in the eigen frequency $\nu= \nu^0 (1- GM/r^0 c^2)$ and accordingly other wave variables. The quantum solutions combined with the wave nature of the gravity further lead to dilated gravito optical distance $r=r^0/(1- GM/r^0 c^2) $ and time $t=t^0/(1- GM/r^0 c^2) $, and modified Newton's gravity and Einstein's mass energy relation. Applications of these give predictions of the general relativistic effects manifested in the four classical test experiments of Einstein's general relativity (GR), in direct agreement with the experiments and the predictions given based on GR.


Introduction
It has long been established in experiment that Newton's inverse square law of gravity, proposed by I Newton in his Principia (1686), is accurate for macroscopic objects and in weak gravitational (g) field. It has been further established in experiment since around the turn of the 20th century that Newton's law of gravity is deviated in strong g fields for matter objects and light, whereas light having no rest mass is implicitly subject to Newton's law only. The deviations mainly exhibit as general relativistic (g-r) effects manifested in the three or four classical-test experiments [1,2,3,4,5] of A Einstein's general theory of relativity (GR). Einstein's GR predicts all the phenomena successfully [6,7,8,5]. The cause of gravity yet remains unresolved up to the present, with or without a g-r effect. I. Newton proposed his inverse square law essentially on phenomenological basis. A. Einstein assumed in his GR that a geometric curvature is being produced in the empty-space and time about a material object and is the cause of gravity. GR remains a phenomenological theory. The geometric description of GR is incompatible with the particle and field based description of quantum mechanics (QM), electromagnetism, and the other three fundamental forces.
QM and GR, along with Newton's gravity in the weak g limit, have been experimentally corroborated to great extents. The two remain yet un-united [10]. A unification of the two, under quantum gravity (QG), has been the holly grill of modern physics. Superstring theory is a much acknowledged contender for QG [11]. Yet the super-partner particles of the theory remain to be observed in experiment; the strings and the six or so extra dimensions remain to be demonstrated in experiment, and supplied with physics foundations. As one of other aspects, the tiny strings ∼ 10 −35 m are just points to the particle waves; they do not ultimately resolve the outstanding difficulty related to wave-particle duality of QM. Loop quantum gravity is focused on quantising the space, rather than unifying gravity with the other fundamental forces. QG is desired in black hole and big bang researches.
An Internally Electrodynamic (IED) model of particles, along with a polarisable vacuuonic dielectric vacuum, has been developed by the author since 2000 using overall experimental observations for particles and vacuum as input information. Based on first principles solutions for the IED particles in a unified framework of classical, quantum and relativistic mechanics, it has been possible to predict a range of the well-established basic properties and relations of particles (see [12] for a review and the original papers). One of the solutions is a depolarisation radiation (DR) Lorentz force acting between IED matter particles in a polarisable dielectric vacuum [13], which resembles in all respects Newton's gravity. In this paper, we extend the solution of gravity to including light quanta (Secs 2, 3), and the effect of gravity on the dynamical variables of the interacting entities (Sec 4) within the same unified theoretical framework, which for gravity essentially consists in quantum electromagnetism. We thereby obtain a generalised quantum-electromagnetic theory of gravity and general relativity. The basic solutions within the theory are then applied to predict the g-r effects manifested in the four classical test experiments of GR (Sec 5). The basic solutions include quantum gravitational waves (composed of DR fields) from individual matter particles or light quanta; the natural extension of this to an accelerating macroscopic object is a macroscopic gravitational wave, which formal solution we shall describe in a separate paper.

Quantum electromagnetic descriptions of matter particles and light quanta. Dielectric theory of the vacuum
We shall work in a three dimensional flat space; this space has a one to one correspondence to a polarisable dielectric vacuum. Let there be in this space a charge q, of charge density ρ q , at the origin of a co-ordinate system x, y, z fixed therein. q is in accelerated motion, forming a current density j q , and generates thereby electromagnetic radiation fields E, B according to the Maxwell equations where c(= 1/ √ ǫ 0 µ 0 ) is the speed of light in the vacuum; ǫ 0 is the permittivity and µ 0 permeability of the vacuum. In the empty space (∅ ) left over after removal of the dielectric vacuum, we directly measure the applied radiation fields E ∅ and B ∅ . An applied E ∅ at a location r will polarise the vacuum in the vicinity, and induce a vacuum polarisation P and a depolarisation (radiation) field E p relative to ∅. Applying the usual dielectric theory [14,15] here, these are given in terms of the macroscopic fields E, B measured in the vacuum as κ 0 (> 1) is the dielectric constant and χ 0 (> 0) is the susceptibility of the dielectric vacuum against ∅; ǫ 0 = κ 0 ǫ ∅ , and ǫ ∅ is the permittivity of the empty space ∅. We have implicitly assumed no significant magnetisation of the vacuum by B ∅ . B p is thus a magnetic field purely induced by the time rate of E p (Maxwell-Ampere's law). In terms of the vacuuonic vacuum model ‡ [12,15,16], (a) the vacuum is filled of densely packed, electrically neutral but polarisable entities called vacuuons separated at a mean spacing b; b ∼ 10 −18 m estimated based on the observational shortest wavelength (1 ∼ 2 × 10 −17 m) of electromagnetic radiation; and (b) each vacuuon is composed of a pair of spinning entities called p-and n-vaculeons of a charge +e and −e, that are located at the core and on a spherical shell about the core, and are strongly bound electromagnetically. Assuming q (= +e or −e) is oscillating sinusoidally about r = 0 along z direction, with a frequency ν and amplitude A, Eqs (1) have the usual solutions for E at a distance r(r, θ, φ) sufficiently far from q, which combined with (2) further yield solutions for E p , given as Here ω = 2πν, k = 2π λ , λ = c/ν, α 0 is an initial phase, and a is a linear dimension defined after (5) below. q may implicitly also be in linear motion, at velocity υ in +x direction here. The fields (ϕ's) generated oppositely say in +x, −x directions, as ϕ † , ϕ ‡ , are thus in general Doppler effected, differently. For the DR force in question in Sec 2, υ affects E 0 through ω; ω = √ ω † ω ‡ = γΩ, where γ = (1 − υ 2 /c 2 ) −1/2 and Ω = lim υ 2 /c 2 →0 ω from IED solution (see [12]). υ does not affect the integration of ϕ j 's which are written as ϕ in (3c).
The energy density of the E, B fields, as In the dielectric vacuum of an induced elasticity, E, B correspond to a transverse elastic wave, of a transverse (tensile) displacement u = (A/E 0 )E and amplitude A. So accordingly ρ ε = 1 2 ρ l ω 2 |u (c) | 2 , where ρ l is the linear (sub-vacuum) mass density of the perturbed vacuum. Suppose that the E, B comprising one energy quantum hν extend through a spherical volume V of radius r 0 ; ℓ = 2r 0 = N 1 λ = n 1 b contains N 1 wavelengths λ's and n 1 vacuuon spacings b's. The one energy quantum, or the Hamiltonian of the one quantum, is thus equal to the integration of ρ ε in V -, with dV -= r 2 drdΩ -, dΩ -= sin θdθdφ, represents a cross sectional area through which ε is transported. ρ λ = ℓρ l /λ = N 1 λρ l /λ = N 1 ρ l is the reduced linear mass density of the vacuum.
E, B may be the internal components of a simple, single charged matter particle such as a proton or an electron, generated by its charge q = +e or −e; q is of zero rest particle mass but is oscillating with a frequency ν specified by (5) (the IED model). Its ε is in general conveyed a fraction a q by its oscillatory charge, and a r by its radiation field, with 0 ≤ a q , a r ≤ 1. So a q ε + a r ε = a r ε = ε(= hν = mc 2 ), given for the extreme case a r = 1 and a q = 0 in (5), is the total (rest) mass energy and m is the (rest) mass of the resultant IED particle (at rest). Or, E, B may comprise a light quantum, (originally) emitted by a matter particle of charge q = +e or −e here. m then represents the dynamical mass, or simply mass, of the light quantum. (5) and (3b) give a few further relations to be used:

Newtonian gravity between light quanta and matter particles
We shall in this section derive a general equation of Newton's gravity between a light quantum µ and an IED matter particle i (Secs 3.1-2) or j (Sec 3.3), their mean mutual gravity (Sec 3.4), and finally between µ (designating thereof a light quantum or IED matter particle) and an object composed of many IED matter particles (Sec 3.5). We shall disregard a g-r effect until Sec 4.
3.1 Action of µ on i µ and i are specified as follows: (a) µ is one of N µ light quanta that are being constantly (re-) generated by an array of point sources in the vicinity of the origin on the yz plane, at x = 0, in the co-ordinate system x, y, z. Its source has an oscillation frequency ν µ and amplitude A A A µ . Let specifically the sources be pairs of dipole charges q µ = ±e of the vacuuons comprising the vacuum here, referred to as virtual sources. They are induced by the plane wave E µ fields propagated to x = 0 at time t = 0, that were generated say at −x 1 at time −t 1 earlier by some natural sources of unit charges ±e. (b) i is one of N i IED matter particles located at a mean position x = r on the x axis. i has a mass m i , and charge q i = +e or −e oscillating with a frequency ν i = m i c 2 /h (Eq 5) and amplitude A A A i . (c) A A A µ and A A A i are along the z direction during a brief time under immediate consideration. So both sources η = i and µ generate fields E η , B η and E pη , B pη according to (3)-(4) in their local co-ordinate systems, x, y, z for µ, and a parallel set of axes And, (d) i and µ are in dynamical equilibrium.
The fields E pµ = E µ − E ∅ µ and B pµ = B µ − B ∅ µ (Eq 4) generated by q µ at x = 0 at time t = 0 will propagate to q i at x = r at time t, and act on q i a (momentary) Lorentz force F totµi = q i E pµ + F µi . q i E pµ drives q i into a transverse motion at velocity υ υ υ pi given as where given after (3a), etc.; α µ = k µ · r − ω µ t; we have set α 0µ = 0 for the initial phase.
is the difference between a Lorentz magnetic force acted by B µ in the vacuum, and one by B ∅ µ in ∅, which we know to each obey the right hand rule. Setting E ′ µ = 0, B µ = 0, we obtain the final Lorentz magnetic force acted by E ′ pµ , B pµ on q i , referred to as DR (Lorentz) force, f (α µ ) = | sin(α µ ) cos(α µ )|. (7a) for υ υ υ pi , (3b) for E 0µ , θ = π 2 (i is essentially a point on the x axis), q i = q µ = ±e, and the identity relations (5) have been used. Based on (8), F µi is pointed from i to µ, hence an attraction, irrespective of the signs of q i , q µ ; and F µi in relation to υ υ υ pi and B pµ obeys a left hand rule.
For a large N µ number of µ's being constantly (re-)generated at x = 0 over a long time ∆t(>> N 1µ /ν µ , N 1i /ν i ), firstly the α 0µ 's of different µ's are independent random variables. These reduce their relative radiation intensities by a factor f 0 = cos 2 α 0µ , rather than producing coherent waves. Secondly, under the influence of random environmental fields, the amplitude of each original natural source, A A A µ1 at −x 1 , hence A µ = a |x1| |A A A µ1 | at x = 0, would explore in ∆t all possible orientations Ω -(θ, φ). Its projection in the yz plane, becoming A µ yz = A µ (sin 2 θ sin 2 φ + cos 2 θ) 1/2 at x = 0, is now responsible for producing a DR force given otherwise by (8). Including the two features, F µi is written as Its average over Ωin (0,4π), and α 0µ , α µ in (0, π) is where The vacuum about x = 0 is electrically and dynamically characterised by the µ-disturbance as follows: (i) E µ polarises the vacuuons in the volume V µ = n 1 bσ = n 1 b · (n y b) · (n z b) occupied by the µ (wave) train. We find n 1 n y polarised vacuuons, and hence n 1 n y units of polarisation charges +e's (or −e's) on each enclosing Gauss surface perpendicular to E µ along the z axis. And (ii) the n 1 n y effective oscillators are driven by E µ into oscillations at the frequency ν µ of E µ , and hence imparted with a dynamical mass ∆m µ = mµ n1ny each; m µ = hν/c 2 (Eq 5). (n 1 n y e) × ∆m µ = em µ (∝ √ ρ λ µ ) returns correctly the e and m µ of the original natural source. The last identity relation renders an equivalent single vacuuon (virtual source) v µ representation to be used: v µ is instantaneously located at x = 0; it carries all the mass m µ and polarisation (dipole) charges (q µ =)q + , q -= +e, −e, oriented along E µ direction. E pi , B pi generated at x i = 0 (x = r) and t = 0, will propagate to µ at . F iµ ± is a DR Lorentz force acted by i on q + or qof v µ . This force on both charges, hence on v µ , is given as, applying directly the left hand rule (Sec 3.1), and f (α i ) = | sin(α i ) cos(α i )|, where α i = k i ·r i −2πν i t given for α 0i = 0. And r i = r +δx=r in the denominator for r >> δx = cN 1i /ν i , (3b) for E 0i θ i = π 2 , and (5) for m i , m µ have been used. For the N i IED particles measured over long time ∆t, with A A A i and α 0i being random variables similarly as of µ, the average DR Lorentz force acted by i on µ is similarly given as 3.3 Action of µ on j j is identical to i but is located at y = r on the y axis perpendicular to the µ path along the x axis. Each point on the µ train (of length ℓ µ ) serves as a new virtual source re-generating (spherical) radiation fields; a point x on it (say in [− 1 2 ℓ µ , 1 2 ℓ µ ]) is at a distance r ′ = xx + rŷ + δzẑ to j. Here the E ′ pµ and B pµ along the z and x directions (the ψ pµ waves) regenerated at time t = 0 will propagate to j at y = r, across a distance r ′ at time t ′ = |r ′ |/c, and act on j a DR Lorentz force given similar to (8) as, with m j = m i , where υ υ υ pj = aE0 µ cos α ′ µ r ′x ; K µi is as given by (8) , and α ′ 0µ = k µ x−ω µ ·0+α 0µ . The last of Eqs (12) is given for r >> ℓ µ , δz, so r ′= rŷ and all the ψ pµ wave fields from different points x on each µ train (µ quantum) will arrive at j at y = r at essentially the same time t = r/c. α ′ 0µ is thus inconsequential and may be set to α ′ 0µ = 0. For the E µ , B µ originally emitted from a (large) N µ natural sources (at −x 1 ), the projection of A A A µ , A A A µ yz , hence E µ in the yz plane about x = 0 here is randomly oriented at an angle ϑ (and B µ at π 2 − ϑ) to the z axis; and so are E ′ µ and E ′ pµ . The z-projection of the E ′ pµ E ′ pµ,z = −E ′ pµ cos ϑẑ, and the induction magnetic field along the x direction, B pµ,x = − |Ep µ,z | cx = |Ep µ | cos ϑ cx , are directly responsible for producing a DR force on j at time t: where f ϑ = cos 2 ϑ. For a large N µ plane-waves propagated to x = 0 at time t = 0, ϑ may assume all possible values in (0, 2π). The average of F µj over N µ quanta over long time ∆t is therefore wheref = 1 π as before;f ϑ = 1 π π 0 cos 2 ϑdϑ = 1 2 . From (14) and (9) we have F µj = F µi . The same features underling the actions j on µ and i on µ inevitably yield F jµ = F iµ . We shall hereafter refer to the µ-i interaction only. If µ is also an IED particle, we could have obtained F iµ , say, directly from interchanging the subscripts of F µi , to be given exactly as (11). Unless otherwise specified we shall hereafter make no distinction between a light quantum and (IED) matter particle.
3.4 Average mutual DR Lorentz force F µi and F iµ are (attractive) action and reaction forces between µ and i. Under condition (d), the two forces must be equal in amplitude (and opposite in direction), and hence in turn equal to their mean, and N 1 = N 1i N 1µ ; ρ λi , ρ λµ are as given by (6b). The negative sign indicates that F is an attraction, as F µi and F iµ are. F given by the inverse square formula (15a) resembles directly Newton's gravitational force, acting between two quantum entities (a light quantum or IED matter particle each) µ, i here; hence G corresponds to the gravitational constant G, G = G. Based on (4), E p , B p are propagated at the speed of light c, and so is the gravity F . Assume as in typical applications that M is uniform and spherical, of a radius R; its mass centre is at r. µ is a light quantum or IED matter particle located at r = 0. E pµ , B pµ of µ are the de-polarisation and induction magnetic fields produced internal of (the polarised vacuuons comprising) the dielectric vacuum, as contrasted to density fluctuations of the vacuum; assume no work has been down by the resultant F . As such, E pµ , B pµ can not be absorbed, hence nor be shielded by the matter particles on their path. Each particle i of M therefore sees directly µ, and vice versa. The total DR Lorentz force between µ and M is thus G for each µ, i pair involves ρ λ(µi) , χ 2 0(µi) (Eq 15b) which are separately dependent of m µ , m i (Eqs 16a,b), hence indicated by (µi) here. Based on (16c), the ratio χ 2 0(µi) /ρ λ(µi) is independent of the masses, and so should be that of χ 2 0 , ρ λ for M and µ: . For the vacuum occupied by M , ρ λ say satisfying the equalities above may be expressed by

Generalised theory of gravity and relativity
We shall in this section generalise the theory of gravity of Sec 3 by including the effect of gravity on the dynamical variables of a test particle µ (Secs 4.1-2) and test macroscopic object (Sec 4.3). We refer to this effect as the general relativistic (g-r) effect in this work. The g-r effect manifestly coincides with the additional content of Einstein's GR over Newton's gravity; the usual light quantum and the IED particle employed in Sec 3 are already intrinsically special relativistic and governed by the Lorentz transformations. To facilitate the discussion, we re-locate M at r = 0 and µ at a distance r from it in the coordinate system x, y, z. And we re-express Newton's gravity here using "proper" dynamical variables r 0 , m 0 measured at the limit g · r 0 → 0, indicated by a superscript 0, acting along r direction as, 4.1 Effect of gravity on the wave and particle-dynamics variables of a single µ µ may be an IED matter particle or light quantum, and is in stationary state in a g field as specified by (19). Let firstly be no applied non-gravitational force present. At r 0 → ∞, g(r 0 ) · r 0 = −V /m 0 → 0 which is a maximum. Accordingly µ has an inertial mass m 0 = m(∞), mass energy ε 0 = m 0 c 2 , and a capacity to work ε 0 = m 0 c 2 , which are maximum each. When brought from infinity to a finite separation r = r(r 0 ) under F , F has done a negative work to µ, ∆V = − r 0 Assume that the process is (quasi) static and hence in general non-adiabatic, so no kinetic energy (T ) has been gained by µ; T would be lost, to such as heat. The total mechanical energy or Hamiltonian of µ, hence also its capacity to do work, is thus reduced by the amount −∆V = −V to µ is dually a quantum wave. For V = 0, µ has a usual total eigen plane wave function ψ(r 0 , t 0 ) = Ce i(k 0 d ·r 0 −2πνt 0 ) (the same ψ is given by the IED solution through ψ(r 0 , t 0 ) = j [ϕ j (r 0 , t 0 ; π 2 ) + iϕ j (r 0 , t 0 )]) and an eigen frequency ν 0 = m 0 c 2 /h, where k 0 d = ( υ c )k 0 , υ is particle speed and k 0 = 2πν 0 /c. For a finite V , and H as given in (20), we where N is the number of wavelengths contained in r 0 . The relationship (25a) firstly means that if ψ p is emitted by M at r 0 = 0 at time t = 0, it then arrives to µ at position r 0 after a time t 0 . Alternatively, it also means that if the first wave front of ψ p enters µ at position r 0 at time t 0 1 , it takes a further time t 0 , i.e. at an absolute time t 0 1 + t 0 latter, for the N wavelengths to pass through µ. The second meaning connects t 0 , r 0 directly with the local variables τ 0 , λ 0 at the location r 0 , and it is the so-signified t 0 , and r 0 = ct 0 , that directly characterise the magnitude of F in (19.1). In this latter sense, r 0 = N λ 0 stands for a gravitational optical, or simply "gravito optical" distance traversed by the gravity wave ψ p ; and t 0 for the time.
For a finite V , the red shifted ν from ν 0 in (22b) directly describes the particle fields ϕ's of µ located at r 0 and hence, based on (Eq 4), the ψ p wave emitted by µ here. For the ψ p wave transmitted to µ (from M ), we can be led to the same red shifted ν by arguing simply based on Newton's law for action and reaction, assuming M and µ are in dynamical equilibrium. So, the local wave variables for characterising the magnitude of gravity transmitted either to or from µ are the red shifted ν of (22b), and λ given in (26a) below. Accordingly, the "gravito optical" distance r and time t for the N number of wavelengths to pass through (either into or out of) µ are defined by the local λ and τ = 1/ν at r(r 0 ) given by (26b,c) below: (26) hold irrespective of the variant GM r ′0 c 2 with r ′ ≈ r ′ 0 (< r) before ψ p arrives at µ or after ψ p has left µ. Based on (26b,c), it takes elongated gravito optical distance and time to transmit the same N wavelengths of gravity F . µ is thus acted by a reduced F , hence able to move further apart, and re-equilibrated at the dilated r. r gives then the observational distance.
By virtue of the underlining second meaning of (25a), the effect of gravity on the space and time co-ordinates manifests exclusively along a g field line. Given a µ at a distant point (x, 0, 0) on the x axis. Then in a small local region such that every point in it is connected to M by a g line parallel with the x axis, the general relativistic transformations from the proper co-ordinates x 0 , y 0 , z 0 , t 0 to the x, y, z, t affected by the g field, are given as In spherical polar co-ordinates and for a point r on the x axis, these become r = γ g r 0 , t = γ g t 0 ; φ = φ 0 ; θ = θ 0 . Using the above for r, keeping t as a dependent variable, the transformation for the (invariant) squared shortest line element (or geodesic) ds of light is Based on experiment, Newton's law (19) holds accurately in the g · r → 0 limit. This suggests that, by retrieving from r the g · r → 0 value r 0 based on (26b), and using this in (19), we can obtain the correct gravity, the ratios F/m 0 and V /m 0 here, as: From a more basic consideration, the G value in Newton's law (19) is experimentally determined on the earth (or between two Cavendish balls) which mass to r ratio is small on an astronomical scale. This G thus represents the g · r → 0 value. (19) is thus expected to hold exactly if the g · r → 0 values of all other variables (m, r) are consistently used in it; this is done in (28). Using (28a) for F , (26b) for r 0 , the Newtonian equation of motion in the g · r → 0 limit (which we are certain to be correct) is thus expressed using the observational r; the proper t 0 is kept, assuming this may be theoretically estimated. In spherical polar co-ordinates (r 0 , θ 0 , φ 0 ), dr 0 = dr 0r0 +r 0 dθ 0θ0 +r 0 sin θ 0 dφ 0φ0 , d 2 r 0 d(t 0 ) 2 = d 2 r 0 d(t 0 ) 2r + ( dr 0 dt 0 + r 0 d dt 0 ) dθ θ θ 0 dt 0 + (sin θ 0 dr 0 dt 0 + r 0 cos θ 0 dθ 0 dt 0 + r 0 sin θ 0 d dt 0 ) dφ φ φ 0 dt 0 . If an applied non-gravitational force F ap also presents along r 0 direction, similarly we can firstly write down the eom in the g · r → 0 limit, F (r 0 ) + F ap (r 0 ) = m 0 d 2 r 0 d(t 0 ) 2 . Using the g-r transformations in it then gives the eom expressed in r, etc. The g · r → 0 eom is a statement of the "(weak) equivalence principle". Namely, the m 0 acted by F (r 0 ) or by F ap (r 0 ) is the same mass in nature; the accelerations produced by F (r 0 ) and F ap (r 0 ) are accordingly equivalent.

4.3
Effect of gravity on the space and time co-ordinates of a macroscopic test object m a m a is composed of a (large) N simple single charged matter particles µ's of masses m µ 's; it has a mass m a = µ m µ . m a is (i) spherical, of a radius R, and (ii) in internal thermal and hydrodynamic equilibrium. Within m a , each µ, µ ′ act on one another a central force F F F µ,µ ′ . In addition, each µ is subject to the g potential V (r µ ) = −g g g(r µ ) · r µ m µ of a large mass M at r = 0 at a separation |r| from m a , where r µ = r + ζ ζ ζ µ and ζ ζ ζ µ (≤ R) are the expectation values of the distances of µ to the CM's of M and m a . No other external force presents. Assume also r >> 2R (condition iii), so lim 2R/r=0 g g g(r 0 µ ) = g g g ≡ g g g(r 0 ) = GM (r 0 ) 2r , i.e. all µ's are subject to the same g g g. In terms of r µ , each µ moves according to Newton's eom (the correspondence principle), m µrµ = m µ g g g + µ ′ =µ F F F µ,µ ′ . Sum over all µ, with m a = µ m µ and µ µ ′ =µ F F F µ,µ ′ = 0 under condition (ii): where O = r 2 s 4Rc sec −1 r R . The total extra time, or Shapiro time delay, for µ in a round trip between the earth (at r e = 1.496 × 10 11 m) and mars (at r p = 2.28 × 10 11 m) is, omitting O, τ round = 2(τ 1 | R→re + τ 1 | R→rp ) = (2r s /c) ln[(r e + r 2 e − R 2 )(r p + r 2 p − R 2 )/R 2 ] = 247 µs. The measured value is 250 µs [5].
The author thanks Professor C Burdik for providing the opportunity to present this research at the ISQS25 in Prague, June, 2017, where the author has enjoyed communications with Dr Dahm, Professor S Catto, and other ISQS-25 participants. Professor B Johansson has given valuable moral support for the unification research. P-I Johansson has privately financed this research and given moral support.