Reconsidering Gravity as Dynamic Phenomenon

Reconsidering Gravity as Dynamic Phenomenon Martin Mayer ma.mayer.ph[email protected] Aichach, Germany Rev 1.02, October 2025 Abstract The purpose of this paper is to explore gravity as a dynamic phenomenon where the gravitational effects described by the Schwarzschild metric are transferred to changes in the quantities of time, length, velocity, acceleration, mass, energy, force as well as the electric and magnetic field constant. These changes are assumed to be physically real and not just a superficial effect of the used coordinate system. Using these malleable quantities various calculations can be done in ordinary orthogonal space again which were regarded as requiring curved space, as demonstrated by applying those malleable quantities to the typical tests of general relativity. This endeavour also uncovers how the formula for Newtonian gravity has to be extended to properly explain the perihelion precession of planets. Moreover, formerly concealed connections between Newtonian gravity and the theory of general relativity are revealed, in particular gravitational potential energy is identified as a change in mass energy. Thereafter a quantum gravity approach is put forward, as originally proposed by John A. Macken, which provides clues to how gravity can be expressed on the quantum level with a classical wave based approach that uses the ’acoustic’ impedance of space. In that context the electrostatic and magnetic force are also reexamined, since they exhibit an unexpected mathematical similarity to gravity. Keywords: gravity; gravitational potential; space; general relativity; Schwarzschild metric; special relativity; time dilation; Planck units; Planck force; quantum physics; light speed; relativistic mass; refraction; waves; perihelion precession; light bending; redshift 1 Introduction Ever since the theory of general relativity has been seemingly affirmed beyond doubt by experimental evidence, the notion that an unified as well as curved space-time is responsible for gravity is not questioned any more by most physicists, particularly since special relativity and classical Newtonian gravity can also be derived from the equations of general relativity as limiting cases. The general theory of relativity is certainly a useful tool that brought up new ideas about space, time and the evolution of our universe, but its suppositions are not proven irrefutably. For example, time is phenomenologically distinct from space and there is no good conceptual justification for treating time like an additional spatial dimension, except that it works out nicely mathematically. Moreover, experimental evidence for the physical reality of curved space is still sparse since proving it requires very sensitive experiments and therefore the most compelling evidence for spacetime curvature still seems to be the perihelion precession of the planet Mercury, which could not be explained satisfactorily before general relativity, and gravitational light bending. There are, however, ways to explain the typical tests of general relativity in other ways as will be demonstrated later on. To achieve this it is necessary to derive dynamic quantities for time, length, velocity, energy and mass from the line element of the Schwarzschild metric, which here is considered to be applicable in general to static and spherically symmetric bodies, and not only to black holes. For those bodies the Schwarzschild radius is simply not reachable because it lies inside the object. Naturally, the question that follows from assuming malleable quantities is what mechanism allows those dynamic properties to arise? We definitely need to look for a framework that is already dynamical in nature and a wave based quantum gravity, which is embedded in some kind of medium that can also carry gravitational waves and light, might be what is needed. However, it should be emphasized that this paper does not disavow any physical effects predicted by general relativity theory. Before venturing deeper into this paper, though, it is sensible to first establish a few prerequisites. 1 2 Prerequisites The Planck units are used extensively in the quantum gravity sections of this paper, in particular the Planck mass ml, Planck length lland the Planck force Fl=c4/G, which may also be called the super force. This is a term coined by Salvatore Pais that emphasizes the extreme strength of the Planck force. The definitions of the Planck units, as well as the values of the natural constants used in this paper, can be found in appendix A. 2.1 Relativistic particle energy The total special relativistic energy for an object with rest mass mthat possesses a velocity vcan be stated as Eγ=p(mc2)2+ (γmvc)2, whereby the relativistic Lorentz factor is defined as γ= 1/p1−v2/c2and light speed is denoted as c. For spin ½ particles which possess a Compton wavelength λc=h/(mc)it is also possible to define their total relativistic energy using their Compton wavelength and their de Broglie wavelength λdB =h/(mv), but it is more sensible to use their frequency counterparts fc=c/λcas well as fdB =c/λdB and to combine those into a novel frequency fγ=pf2 c+γ2f2 dB , which I refer to as the Lorentz frequency, for describing total special relativistic energy, as originally shown in (1). Eγ=hqf2 c+γ2f2 dB =hfγ(2.1) The last equation suggests that the special relativistic energy of a spin ½ particle is stored intrinsically to the particle, contrary to the currently prevalent opinion in physics. This assertion might also be related to the notion of Doppler shifted waves, as proposed by John A. Macken (4)(5). Moreover, it is shown in (1), (3), (4) and (5) that spin ½ particles possess an equatorial perimeter speed of c, from which it follows that a spin ½ particle’s relativistic radius is given by rγ=c/(2πfγ)or that its size is at least proportional to that radius by a constant factor. This, in turn, leads to the controversial situation that electrons should be bigger than protons, a topic which is also discussed in the aforementioned papers. In case a spin ½ particle is at standstill rγ equals the so called reduced Compton wavelength λc/2π, since its de Broglie frequency will be zero, and its Lorentz frequency fγsimply equals its unchanging Compton frequency fc. Moreover, spin ½ particles should exhibit a Lorentz frequency increase as they move faster since then their total relativistic energy Eγis increasing, which in turn implies that they are shrinking as they move faster since their equatorial perimeter speed cannot exceed light speed c. 2.2 Properties of space The energy density of space itself is given by ρs=1 2 ml l3 l = 2.577 59 ×1096 kg/m3(2.2) as proposed in (1). This huge energy density is consistent with the so called zero point energy of quantum physics which presumably is caused by tiny Planck oscillators that space is comprised of. Subsequently, the ”acoustic” impedance of space itself, in which gravitational waves are propagating with light speed c, evaluates to Zs=ρsc=c 2 ml l3 l =c3 2G l2 l = 7.727 42 ×10104 s Pa/m(2.3) as originally shown in (2). Moreover, the impedance of space can also be specified without the Planck units, in which case it takes on the form Zs= 1/(2ℏ) (c3/G)2. The gigantic value resulting from these formulas for Zsimplies that space has a huge stiffness, which is why even gravitational waves that were caused by black holes are barely detectable by highly sophisticated measurement equipment. Physics literature usually denotes the impedance of space as c3/G, for example in (11), which is also the expression that John Macken uses in his works (4)(5). This alternative expression for the impedance of space is equivalent to the one stated in equation 2.3 when using it in wave equations together with a dimensionless strain amplitude instead of an amplitude which is defined in an unit of length. In this paper, however, Zsis always used as expressed in equation 2.3 so that an ordinary amplitude, specified in an unit of length, can be used for wave equations that involve Zs. Interestingly, the impedance of space can also be specified in terms of the Hubble mass mHand Hubble radius rH, i.e. c3/G ∼ =2c mH/rHand Zs∼ =4π c2m2 H/(r2 Hh). Whether these relations are exact, or only approximate, remains to be seen. In any case, however, the various formulations for the impedance of space indicate that the microscopic quantum world, the macroscopic world as well as the cosmological world are connected through the impedance of space. 2 2.3 Scope This paper operates within a space-time as described by the so called Schwarzschild metric, from a general relativity point of view, i.e. the gravitational effects of rotating masses are not covered. Consequently, the appropriate line element is associated with a non-rotating non-moving spherically symmetric homogenous and dominant central mass Mwhich can be stated in polar coordinates as follows, ds2=−1 + 2Φ c2c2dt2+1 + 2Φ c2−1 dr2+r2dΘ2+r2sin2(Θ) dϕ2(2.4) whereby Φdenotes the classical Newtonian gravitational potential Φ = −GM/r (2.5) Using the following dimensionless factor to quantify gravitational effects Γ = 1/p1−2GM/(r c2)(2.6) the aforementioned line element can also be rewritten as follows ds2=−(1/Γ2)c2dt2+ Γ2dr2+r2dΘ2+r2sin2(Θ) dϕ2(2.7) whereby the factor Γpractically equals 1.0far away from the central mass Mand slowly increase in value towards that mass. In close vicinity to the Schwarzschild radius rs= 2GM/c2associated with a mass M the value of Γstarts to increase faster until it finally reaches positive infinity at rs. The region inside a black hole is not treated in this paper. 3 Dynamic gravitation Using the aforementioned line element it is possible to explore how the quantities of frequency, time duration, energy, length, speed, acceleration, mass and force are changing inside the gravitational field of a mass Mand to derive the appropriate scaling factors. This endeavour is motivated by the presumption that those changes are physically real, which in the case of gravitational time dilation is universally accepted by physicists and proven experimentally. Consequently, it is only logical to extend that notion to other physical quantities. Please note that in the following sections the subscript falways denotes a physical quantity infinitely far away from the gravitational field of Mand the subscript Γdenotes a physical quantity within its gravitational field, whereby the mass Malways denotes a non-rotating non-moving dominant spherical mass with a homogeneous density. Linear approximations are always preceded by a ∼ =sign hereafter. 3.1 Time As proven experimentally clocks run slower near a large mass than far away from it. The following table contains an example of this gravitational time dilation for an extreme black hole scenario with Γ = 10. Near ←Far Frequency 1 Hz 1/Γ 10 Hz Period 1 s Γ 0.1 s Table 1: Example with Γ = 10 This table shows that while a needle watch completes 10 turns far away from a black hole it only does complete 1 turn close to the black hole, as observed from far away, when Γ = 10. From this example we can infer that frequencies change as follows inside the gravitational field of a mass M: fΓ=ff/Γ∼ =1−GM r c2ff(3.1) Since a certain time period and its associated frequency are always inversely related a time duration ∆t, like a full clock period, changes inside the gravitational field of a mass Maccording to: ∆tΓ= ∆tfΓ∼ =1 + GM r c2∆tf(3.2) 3 3.2 Energy The factor Γcan also be formulated in terms of an escape velocity s=p2GM/r as follows for a scenario with a central non-rotating non-moving homogeneous spherical mass M. Γ = 1 p1−s2/c2(3.3) Interestingly, this equation looks very similar to the Lorentz factor γ, which is also used for special relativistic energy calculations. γ=1 p1−v2/c2(3.4) This congruence gives rise to the supposition that Γshould be relevant for describing the presumed change in the quantity of energy inside a gravitational field. Since energy terms of the form E=hf are applicable to spin ½ particles as well as photons, and Planck’s constant his assumed to be independent of gravity, it can be inferred using equation 3.1 that energy must in general scale as follows inside a gravitational field: EΓ=Ef/Γ∼ =1−GM r c2Ef(3.5) Extending equation 2.1 in the same way the mass energy of a spin ½ particle can be expressed as follows, EΓ=hfγ/Γ = hqf2 c/Γ2+ (γ2/Γ2)f2 dB =hq(1 −s2/c2)f2 γ=hfΓ(3.6) whereby fγ,fcand fdB keep their respective meaning and are therefore defined in the absence of a gravitational field. A decreasing mass energy due to gravity seems to be concerning at first glance, but this change is assumed to be undetectable locally, as discussed later on, and moreover lost mass energy is accounted for by a changing gravitational potential energy, as shown in section 5.1. 3.3 Length Since the scaling behaviour of length should be inverse to the scaling of a time duration, according to the line element described by equation 2.7, a reasonably small length element ∆dshould scale as follows inside the gravitational field of a mass Munder consideration of equation 3.2. ∆dΓ= ∆df/Γ∼ =1−GM r c2∆df(3.7) This result implies that a ruler which had a length of 1 m outside of a gravitational field shrinks to 0.1 m at a location where Γ = 10 over the extend of that ruler, as observed from outside of the gravitational field. In this interpretation for the effect of gravity on length there is no need for a curved space and thus the three dimensions of space can be regarded as remaining orthogonal inside a gravitational field. Consequently, from the perspective of an object falling into a gravitational field it appears as if the whole universe is expanding. 3.4 Velocity A velocity v= Λfcan be calculated using a wavelength Λ, which qualifies as a length element, and a frequency f. Subsequently, considering equation 3.2 and 3.7 it can be said that any given velocity scales as follows in the gravitational field of a mass Mwhen assuming no net acceleration influence due to gravity or other forces. vΓ=vf 1 Γ 1 Γ=vf/Γ2∼ =1−2GM r c2vf(3.8) Since the Γfactors don’t cancel out here a velocity which should be constant according to classical physics is effectively getting reduced inside a gravitational field, as observed from outside the gravitational field. This also implies that a moving object which gets very close to a black hole horizon is freezing in its motion, as observed from far away. The scaling behaviour for velocity is universal and therefore also applies to light rays. Consequently, the scaling behaviour for the speed of light in a gravitational field of a mass Mis described as follows, whereby the letter c, without any subscript, still denotes the usual constant speed of light to avoid confusion and to retain the common convention, but cnow has to be defined in the absence of a gravitational field to be unambiguous, i.e. c=cf. cΓ=c/ Γ2∼ =1−2GM r c2c(3.9) 4 Distances measured using light rays will thus change, i.e. deeper inside a gravitational field they become shorter, in accordance with the findings of the previous section. However, since measurement devices themselves also experience changes in length and time inside a gravitational field it is not possible to measure the slowdown of light locally. For example, imagine a rocket flying at constant speed in a region of space with Γ=2, for which special relativistic time dilation is negligible (i.e. γ∼ =1), to traverse a certain distance measured in meters that is established using a light ray that travels for a certain time duration. The rocket’s velocity will be 1/4th compared to the speed it would have outside of the gravitational field, however, a clock inside the rocket is running at a halved rate, compared to outside the gravitational field, and the distance is also half as long comparatively. Thus these effects cancel out for an observer inside the rocket, which should also be the case locally for all other scaling behaviours caused by a gravitational field. Whilst the slowdown in the speed of light is not measurable locally the bent trajectory of light that passes by close to a large mass is likely the result of that change in the speed of light. To illustrate this purpose the last equation can be easily reformulated into a refractive index nwhich can be used to calculate said light bending - see section 4.2 for more details. n=c cΓ = Γ2∼ =1 + 2GM r c2(3.10) A higher refraction index nusually, but not always, denotes a denser medium and the linear approximation of nshows that the refractive index is increasing towards a gravitational source M. If this implies that the energy density of space is increasing towards a mass Mis not clear yet and we should be careful to not take the optical analogy too far, as that might be misleading. The knowledge about a changing speed of light now allows to consider what happens to the size of a spin ½ particle, whose radius inside a gravitational field, as observed from outside of it, must be given by rΓ=cΓ/(2πfΓ), which is a logical extension of the special relativistic radius rγ=c/(2πfγ). Consequently, the size of a spin ½ particle also scales like a length element as specified by equation 3.7, i.e. it shrinks as observed from outside of the gravitational field it is in. 3.5 Mass Since mass can be stated as m=E/c2the scaling behaviour for a mass minside the gravitational field of a mass Mis given as follows under consideration of equation 3.5 as well as 3.9. mΓ=mf 1 Γ Γ4 1=mfΓ3∼ =1 + 3GM r c2mf(3.11) Thus objects effectively gain mass as they move deeper into a gravitational field, as observed from outside of the gravitational field. 3.6 Local force Since force can be stated as F=E/ ∆dthe scaling behaviour for a force Finside the gravitational field of a mass Mis given as follows under consideration of equation 3.5 as well as 3.7. FΓ=Ff 1 Γ Γ 1=Ff(3.12) Interestingly, force is an unchanging quantity, whereas even energy is changing within a gravitational field. For example, when a space probe is leaving the gravitational field of a planet by using a chemical thruster that can only operate with an unchanging chemical reaction the force of that thruster will remain constant during the space probe’s journey, as observed locally as well as from outside of the gravitational field. However, there is something strange going on with the Newtonian gravitational force Fg=G M m/r2, which seems to violate the independence for a force from Γunless Ggets adapted too. Sticking to the convention that Γis linked to the mass Mand that force should be unchanging the adapted equation for the Newtonian gravitational force then looks as follows when writing the Γfactors explicitly. Fg=G Γ5 M 1 Γ3mf 1 Γ2 r2 f (3.13) This arrangement implies that the gravitational constant Gvaries with 1/Γ5. However, this simple approach could be flawed since the expression Γ/rfmight be an inappropriate measure for distance as Γvaries with distance from M. Therefore section 3.3 also stated that equation 3.7 only applies to reasonably small length elements over which Γis practically constant. 5 3.7 Acceleration Since acceleration can be stated as a=F/ m the scaling behaviour for an acceleration ainside the gravitational field of a mass Mis given as follows under consideration of equation 3.11 as well as 3.12 when assuming no net jerk influence due to changing gravity or other changing forces. aΓ=af/Γ3∼ =1−3GM r c2af(3.14) This equation, together with the gravitational scaling of mass, implies that objects get more inert as they go deeper into a gravitational field, as observed from outside of the gravitational field. 3.8 Electric & magnetic field constant Since the electric field constant ϵ0and magnetic field constant µ0are related to light speed cby 1/c2=ϵ0µ0 the electric and magnetic field constant must also change inside a gravitational field when considering equation 3.9. Subsequently, we get c2 Γ=c2/Γ4= 1/(Γ2ϵ0) 1/(Γ2µ0), when assuming a symmetrical distribution of the Γfactor, which in turn leads to the following equations: ϵΓ= Γ2ϵ0∼ =1 + 2GM r c2ϵ0(3.15) µΓ= Γ2µ0∼ =1 + 2GM r c2µ0(3.16) An identical result was obtained by Harold Puthoff using a polarized vacuum approach (13). 3.9 Alternative to Γ Kris Krogh devised an alternative for Γwhich uses Euler’s number and also applies to a non-rotating nonmoving dominant homogeneous spherical mass M(8). expGM r c2=exp−Φ c2∼ =Γ(3.17) This alternative to Γmight turn out to be interesting for further research on gravity, as its approximation differs from the approximation of Γfor powers of rbeyond one. 4 Tests 4.1 Redshift & blueshift The gravitational change of a length segment as specified in equation 3.7 matches with the equation for gravitational redshift, i.e. z+ 1 = λf λΓ = Γ (4.1) whereby λΓ= ∆dΓdenotes the wavelength as sent out by an emission source inside a gravitational field and λf= ∆dfdenotes the wavelength received far away from that gravitational field. A photon moving out of a gravitational field will thus obtain a longer wavelength during its journey, i.e. it will redshift. 4.2 Light deflection The refraction index ncan also be calculated by considering the motion of a light ray in a gravitational field by using the line element as specified in equation 2.4 or 2.7 to describe that motion. To simplify the calculation a straight motion can be chosen with an angle Θsuch that sin2(Θ) is zero and dϕ= 0, i.e. a radial movement with Θ = 0◦. Moreover, considering that for a light ray ds2= 0 and using velocity v= dr/dtthe line element can be reshaped into a refractive index n. Due to the spherical symmetry of the Schwarzschild metric the obtained result must hold true everywhere outside of the Schwarzschild radius associated with a mass M. 1 + 2Φ c2c2dt2=1 + 2Φ c2−1 dr2 1 + 2Φ c22 c2=dr2 dt2 n=c v= 1 1−2GM r c2= Γ2 (4.2) 6 This result matches exactly with equation 3.10, which means that the Schwarzschild metric can be regarded as a description of an optical medium with a changing refractive index nand therefore any gravitational light bending calculation based on the refractive index nhas to produce results that are equivalent to other calculation approaches which use the framework of general relativity theory. Calculations using the refractive index nthus have to result in the typical gravitational light bending angle θfor light rays moving past a nonrotating stationary homogeneous spherical mass Mat a distance r, which is given by the following equation. θ=4GM r c2(4.3) The refractive index nwas first used by Robert Dicke to correctly calculate the deflection angle for light passing by our sun (6). This is a topic that is covered extensively by Alexander Unzicker, for example in (9). There Unzicker explains that Albert Einstein also attempted to use a refraction index to calculate the amount of light deflection around our sun before he developed his theory of general relativity (12). Einstein’s result using a refraction index was exactly half of the correct value that he predicted later on by using general relativity, which was because Einstein only took light’s gravitational frequency change into consideration but missed the change in wavelength. Thus his expression for the reduced speed of light in a gravitational field was slightly off, i.e. [1 −GM/(r c2)] c. Today, however, it is pretty much forgotten that Einstein was taking the notion of a variable speed of light due to gravity very seriously. There are also lots of other papers available which calculate the refractive index using general relativity, like (14) which treats gravity as a purely refractive phenomenon. However, these derivations are often more complicated than the derivation done here in equation 4.2 and their respective authors usually refrain from interpreting the refractive index as evidence for space being a physical medium which constitutes a preferred frame and has substance to it. 4.3 Perihelion precession Calculating the precession of Mercury’s perihelion still stands as one of the most significant tests of Einstein’s general theory of relativity. This section will present a classical approach which doesn’t require curved space and still achieves the same result. The mass of our sun is designated here as M, Mercury’s mass is denoted as mand ris specifying the distance between our sun and Mercury. Mercury’s kinetic energy Texpressed in polar coordinates is given by the following equation since Mercury moves in a plane and subsequently ˙ Θ = 0 can be used. T=m( ˙r2+r2˙ ϕ2)/2(4.4) Usually, a calculation of Mercury’s perihelion precession based on Newtonian physics is not giving the same result as a general relativity based calculation and the reason behind that failure is a wrong gravitational potential energy term, as shown hereafter. From black hole calculations it is known that the gravitational potential limit of a Schwarzschild black hole is −c2/2whereas the gravitational potential limit of an extreme Kerr black hole which spins at the speed of light is given by −c2(2). A similar doubling approach can be tried by constructing a gravitational potential energy formula which incorporates the motion of Mercury around our sun as follows, whereby an expression is sought which only relies on rfor practicality. −V=GMm r1 + v2 t c2=GMm r+GMm rr˙ ϕ c2 =GMm r+GMm 1 r˙ ϕ2 c2=GMm r+GMmh2 c2r3(4.5) Here Mercury’s tangential speed is vt=r˙ ϕ, its specific angular momentum is h=r2˙ ϕand the minus sign in front of Vindicates an attractive force. Moreover, Vshould not be confused with the ordinary effective potential Veff =−GMm/r +mv2 t/2which has another meaning. The expressions for Tand Vcan then be used to setup the relativistic Lagrangian as follows. L/Γ = T/Γ−V/Γ L=m( ˙r2+r2˙ ϕ2)/2 + GMm/r +GMmh2/(c2r3)(4.6) Here the 1/Γterms account for any changes of the involved physical quantities due to the gravitational field of M, but these terms can simply be cancelled out. The remaining equation is identical to one that is treating the situation as perceived from outside of the gravitational field of our sun. Applying the Euler-Lagrange equation with respect to ϕgives the following result, d dt ∂L ∂˙ ϕ=∂L ∂ϕ d dtmr2˙ ϕ= 0 (4.7) 7 which means that Mercury’s angular momentum mr2˙ ϕis constant because the product of these variables doesn’t change with time. Applying the Euler-Lagrange equation with respect to rgives the following equation of motion. d dt ∂L ∂˙r=∂L ∂r d dtm˙r=∂ ∂r m˙r2+mr2˙ ϕ2/2 + GMm/r +GMmh2/(c2r3) ¨r=∂ ∂r r2˙ ϕ2/2 + GM/r +GMh2/(c2r3) ¨r=r˙ ϕ2−GM/r2−3GMh2/(r4c2) (4.8) Mercury’s mass mwas assumed to be constant here and could therefore be cancelled out. Special relativistic effects will be discussed later on in this section. Using u= 1/r,¨r= (d/dt)(du−1/dt), specific angular momentum h=r2˙ ϕ(which also characterizes Kepler’s second law since dA/dt=h/2), the chain rule in the form of d/dt=˙ ϕ(d/dϕ) = hu2(d/dϕ)and the quotient rule in the form of du−1/dt=−u−2(du/dt) = −h(du/dϕ)it is possible to obtain a standard differential equation from equation 4.8 which is not dependent on time and thus avoids any time dilation issues of special relativity or general relativity. −h2u2(d2u/dϕ2) = h2u3−GMu2−3GMu4h2/c2 −u2(d2u/dϕ2) = u3−GMu2/h2−3GMu4/c2(4.9) d2u dϕ2+u=GM h2+3GMu2 c2(4.10) This is the same result as obtained from a derivation that is directly using the Schwarzschild metric, which shows that the perihelion precession of Mercury can, in principle, be calculated without a curved space. It should be noted though, that equation 4.10 is only valid for a weak gravitational field, i.e. when GM/(rc2)≪1, according to physics literature. Stronger gravitational fields would require additional terms on the right side of that equation. Moreover, a modified gravitational potential energy term was necessary, as specified by equation 4.5, to achieve the desired result. This suggests that the gravitational potential energy for an orbiting planet also depends on its tangential velocity, not just its position, and that an increasing tangential velocity is deepening the gravitational well a planet is in. In case of Mercury the added v2 t/c2 term results in values around 2.6×10−8as Mercury moves along its orbit, which is a small number but still the additional contribution to Vamounts to approximately 2.0×1025 Jon average. But what is the deeper physical cause for the extra 1 + v2 t/c2term? Finding an explanation that doesn’t depend on space curvature is obviously crucial for the hypothesis of this paper. The gravitational force which corresponds to equation 4.5 can also be written as follows Fgo =GM r2m1 + v2 t c2 (4.11) and according to general relativity theory all kinds of energy are gravitating, as embodied in its stress – energy – momentum tensor. Thus the kinetic energy of Mercury should indeed increase its gravitational attraction to our sun. Extending Newtonian gravity to include the special relativistic mass for one of the masses, i.e. using mγ instead of m, implements this line of thinking and gives the following result. GM r2mγ =GM r2mr1−v2 c2∼ =GM r2m1 + 1 2 v2 c2 (4.12) The resulting approximation is quite similar to Fgo when assuming that Mercury’s total velocity vsatisfies v∼ =vtas its tangential velocity vtaccounts for most of its total velocity v(between 97.8% and 100%). However, even with a relativistic mass there is still a missing factor of ½ v2/c2, which means that the extra energy is twice as large as the special relativistic mass accounts for, i.e. mneeds to scale with γ2instead of γ. Consequently, the generalized sought-after gravitational force is given by the following formula: Fgr =G r2m1γ2 1m2γ2 2∼ =G r2m11 + v2 1 c2m21 + v2 2 c2 (4.13) A possible candidate for the extra γfactor is special relativistic time dilation, in case that the slower local time on Mercury, compared to a stationary observer in our solar system, is increasing the gravitational attraction between our sun and Mercury. Admittedly, this notion is speculative, since special relativity does not deal with gravity, which is why equation 4.13 should be checked experimentally, maybe through the use of a cyclotron, to clarify if special relativity is responsible for the extra term in V. In any case, however, the kinetic 8 energy of Mercury definitely increases its gravitational attraction to our sun. Interestingly, the kinetic energy T, as described by equation 4.4, did not require a special relativistic extension to calculate Mercury’s perihelion precession correctly. The presumed reason is that for T, contrary to the classical expression for a gravitational potential energy, the influence of Mercury’s motion is already accounted for, at least approximately, and Mercury’s practically non-relativistic speed of around 0.016% of light speed cis varying less than 2.2%. A simple approach to reduce the calculation error in Twould be to correct Mercury’s mass mby the average applicable Lorentz factor, i.e. by γ(v= 47.4 km/s) = 1.0000000125. However, the resulting mass difference would be well below the precision to which Mercury’s mass is known as of now. For very strong gravitational fields the mass difference might become relevant, though, in case that the affected planet survives the associated gravitational tidal forces. Using the already mentioned expression G∼ =c2rH/(2 mH)from (2) it is also possible to express Vin a form that does not exhibit G, which is a desirable feature for moving towards an alternative theory of gravity. V∼ =−1 2 rH r M mHmc2+mv2 t(4.14) In case that our observable universe qualifies as a Schwarzschild black hole this expression for Vwould be exact. A somewhat similar expression can be found using the escape speed s=p2GM/r. V=−1 2 s2 c2mc2+mv2 t(4.15) Equation 4.14 should be appealing to friends of Mach’s principle since it relates the gravitational potential energy of Vto all the masses in our universe. However, that equation may also be written in terms of the Planck mass mland Planck length ll, i.e. V=−(ll/r) (M/ml) (mc2+mv2 t). 5 Newtonian gravity reexamined 5.1 Gravitational potential energy The following equation quantifies the classical gravitational potential energy that is released when a test mass mt= 1.000 kg is moved inside the gravitational field of earth, with mE= 5.972 ×1024 kg being the mass of earth and rnear = 6371 km being the radius of earth, from a position rfar =rnear + 10.000 km to the surface of the earth. ∆E=−G mEmt rfar −−G mEmt rnear =1 rnear −1 rfar G mEmt= 98.043 kJ (5.1) So roughly 100 kilojoules of energy are released according to Newtonian gravity as the test mass is moved to a lower gravitational potential, i.e. the surface of earth. To make a connection to general relativity the last equation can also be written as follows: ∆E=G mE rnear c2−G mE rfar c2mtc2=1−G mE rfar c2−1−G mE rnear c2mtc2(5.2) Using the following linear approximation 1/Γ = p1−2GM/(r c2)∼ =1−GM/(r c2)(5.3) it is possible to rewrite ∆Eas ∆E=1 Γfar −1 Γnear mtc2= 98.046 kJ (5.4) whereby Γnear denotes Γfor r=rnear with M=mEand Γfar denotes Γfor r=rfar with M=mE. It can be seen that the results of equation 5.2 and 5.4 are quite close and they get even closer for larger separation distances. Moreover, the results also get closer when treating mtas a variable mass in the Newtonian calculation. The remaining deviation between the result of equation 5.1 and 5.4 is presumably largely due to the used linear approximation as stated in equation 5.3. The calculation presented in this section is based upon a related example in John A. Macken’s work where he claims that time dilation is responsible for the gravitational potential energy (4), which can be said rightfully so since frequency fas well as energy Eboth scale with 1/Γand are directly related via E=hf. This effectively means that when a spin ½ particle is moving towards the earth’s surface it looses mass energy, due to the reduction in its frequency fΓ, and since this change in its mass energy is not directly 9 for in his works. His approach is also suggesting a wave phenomenon behind these mysterious gravitational anomalies that, according to McCulloch, arise because there is a limit for gravitational wavelength between a mass in space and the cosmic horizon, which in turn has some effects on inertial mass and gravity. According to McCulloch there is an upper limit to the wavelength between an accelerated object and the horizon of the observable universe, which is given by 2c2/a for an acceleration a. This wavelength limit might be the lowest possible harmonic of the gravitational waves which were proposed before. As a consequence, McCulloch claims, inertial mass must be modified and the equivalence principle mi=mgfor the inertial mass miand gravitational mass mgof an object gets violated. The inertial mass correction factor, that I denote here as η, is stated in (10) as η=1−2c2 a d (6.27) whereby dis the distance between a cavity’s confines, which in the absence of an artificial cavity is twice the Hubble radius, i.e. d= 2c/H0, and adenotes a mass’s acceleration. This correction factor is usually irrelevant since it is extremely close to 1, but it becomes relevant as it goes lower for artificial cavities, like two closely placed parallel metal plates, and ultra low accelerations, like at the edges of a spiral galaxy. Moreover, an interesting side effect of the inertial mass correction is that any gravitational attraction should exhibit a residual acceleration of 2c2/d = 2c2/(2c/H0) = cH0, as claimed in chapter 7.1 of (10). Calculating this residual acceleration with a Hubble radius of 1.25 ×1026 mgives a value of 7.2×10−10 m/s2. The same result can be obtained using the Hubble mass mHand Hubble radius rH, i.e. cH0= 2GmH/r2 H, when presuming that the observable universe is a Schwarzschild black hole, or at least close to being one. Moreover, there is also the alternative expression cH0=c2/rHwhich follows from H0=c/rH. In closing it should be remarked that the result for the residual acceleration seems to be related to the acceleration constant a0∼ =1.2×10−10 m/s2of the modified Newtonian dynamics hypothesis (MOND) since these acceleration values are quite close and their proportionality constant is seemingly an integer factor, i.e. a0=cH0/6.0 = c2/(6.0rH) = GmH/(3.0r2 H). 7 Discussion Several Γbased factors were provided in section 3 that quantified how various physical quantities scale inside a gravitational field compared to being so far away from the gravitational field that its effects are negligible. The presented supplementary linear approximations of these scaling factors were also obtained by Dehnen, H¨ onl and Westpfahl in (7), by Krogh in (8) and by Puthoff in (13), for those relations that were derived by them using different approaches than the one applied here. Clearly, these agreements support the scaling factors put forward in this paper. Macken, however, derived different factors in his work (4) which presumably are not able to correctly quantify gravitational light bending. The presented scaling factors should work quite well even for stronger gravitational fields, when the respective Γfactor is utilized instead of the associated linear approximation. In case the gravitational field becomes so strong that its influence on Γeven varies over a fundamental particle’s confines, the presented scaling factors become inappropriate. That situation should be rather unattainable, though, since it was shown in section 3.2 that gravity should shrink fundamental particles, so that they become substantially smaller in a strong gravitational field, but in such circumstances general relativity would also not be able to make correct predictions for particles anymore. This paper also analysed a classical calculation of Mercury’s perihelion precession. However, Mercury’s equation of motion, as described by equation 4.8 or 4.10, wasn’t obtained by adding Γfactors to any of the relevant equations as these factors cancelled out in equation 4.6. Instead the associated Newtonian gravitational potential energy had to be extended (see equation 4.5) to give a potential function that is also known from general relativity theory. It was shown that the extra potential term, which features Mercury’s tangential velocity vt, might be caused by special relativistic mass and special relativistic time dilation, which both seem to increase the gravitational attraction of Mercury to our sun. However, the associated gravitational force equation 4.13 would need to be checked experimentally to verify that line of thinking. The effect that a rotating mass has on space and gravity, which wasn’t considered in this paper, would likely also result in additions to the Newtonian gravitational force. Moreover, according to general relativity literature the obtained equation of motion for Mercury is not sufficiently correct for strong gravitational fields where GM/(rc2)≪1isn’t true anymore. Possible causes for issues with strong gravitational fields in the calculation presented in section 4.3 are that the difference between using vor vtwas not fully explored, as v∼ =vtwas assumed, and that Tdid not feature a correction for special relativistic mass increase. Moreover, even the time derivate d/dtitself may become an issue if the meaning of time varies noticeably over the size of a planet. 16 As it was already mentioned in section 3.4, the propagation speed of light is claimed to slow down slightly near a substantial mass but a local observer would not be able to measure that effect since any used measurement device is affected too by gravity. This clandestine behaviour of light also gives rise to questions about the validity of special relativity, which too has some inherent issues that are usually blanketed. Special relativistic time dilation, for example, may actually result from a Doppler effect in a medium which constitutes a unique absolute reference frame that is obscured by the circumstance that we can’t measure the one way speed of light. This inability is not merely a technical issue, but a conceptual problem. However, Prof. Unnikrishnan claims to have measured the speed of light experimentally (15) with the result that it travels with a different velocity away from an observer and towards him, i.e. it adheres to Galilean transformation instead of Lorentzian transformation. Moreover, special relativity rests on the arbitrary claim that the proper length of an object is measured in any inertial frame, but the measurement of proper length may be reserved for the already mentioned unique absolute reference frame. Special relativistic space-time, i.e. Minkowski space-time, may thus not be physically real, which in turn indicates that curved space-time might not be physically real too, since Minkowski space-time is a special case of it. From an energy conservation point of view it is also not feasible to uphold a purely relativistic view of our universe, since there is a huge difference in total energy if a single object moves with near light speed through a static universe or if the remaining universe moves with the same speed past that object at rest. Intelligible lectures on the issues with the theory of special relativity can also be found on the YouTube channel ”dialectphilosophy”. One of the main questions addressed in this paper was if space curvature can be avoided in calculations, as curved space might not be physically real. Arguably, this paper showed that various calculations can be made without using the concept of space curvature which yield the same results as a treatment with the framework of general relativity theory. In addition to this evidence a number of conceptual arguments can also be brought forward against the existence of a curved space. • The Schwarzschild radius can be calculated using classical Newtonian escape velocity when also assuming an universal speed limit c, i.e. c=p2GM/r gives the Schwarzschild radius when rearranging for radius r. This is a smoking gun that is hard to dismiss on technical grounds. • If gravity really means that an object in free fall just follows its geodesic equation, then obstructing that object in its path should eliminate the gravitational effect on it, because there should be no force or pressure that pushes it against the obstruction. Since this is not the case gravity has to be a force, despite what is universally proclaimed nowadays. • Why and how exactly mass bends space is an unresolved mystery. General relativity theory is incomplete regarding that subject. • Shrinking particles are conceptually equal to proper volume decreasing inside a gravitational field, because rulers are made of particles. • The theory of general relativity does not explain the gravitational constant Gitself, which may be an emergent constant since it seemingly can be explained in terms of all the masses present in our observable universe (2). Moreover, section 3.6 suggested that even the gravitational constant Gis changing inside a gravitational field. • The observable universe turns out to be a black hole, or at least it is close to being one, when calculating the mass of the observable universe using ordinary Cartesian space and the measured average energy density (2). The fact that we do not observe strong space curvature in our vicinity thus speaks against the notion of a curved space. • A refractive index for space, which implies a variable speed of light, can be used to explain gravitational light bending and refraction usually implies a physical medium with changing properties. • The so called Shapiro delay can also be calculated using a refractive index for space, since it has the same causes as gravitational light bending. • The perihelion precession of a planet can be calculated in a classical fashion, as shown in section 4.3. • A number of gravitational effects only require gravitational time dilation to be explained, like the gravitational redshift of light and GPS time delay. These effects therefore don’t provide support for the notion of a bent space since time dilation can exist independent of curved space. • The rotation curves of spiral galaxies, and other anomalies, indicate that something is not right with the framework of general relativity theory. The issue at hand might not be to find dark matter or dark energy, but a systemic problem with the theory of general relativity itself. • The similarity between Macken’s wave equations for classical Newtonian gravity and the electric force are indicating a medium of space whose forces are based on waves. • Newtonian gravity can be obtained for the surface of a Schwarzschild black hole using the BekensteinHawking entropy, the Unruh temperature and the equipartition theorem (1), as originally shown by Eric Verlinde using a model of a holographic and entropic gravity. 17 • The equivalence principle, which general relativity rests on, may not hold true in all situations, as discussed in section 6.5. • There seems to be a link between the strong force and gravity. A side effect of this is that the proton’s true mass appears to be the Planck mass mlfrom the perspective of classical Newtonian gravity. • The incompatibility of contemporary quantum physics with general relativity theory may indicate that bent space is not a sensible physical concept. That list can probably be prolonged, which in turn reinforces the idea that it is sensible to question the underlying assumptions of general relativity theory and to explore how the gravitational effects derived from general relativity theory can be described without having to assume space curvature or a unified space-time. Unbeknownst to many people Albert Einstein did not dismiss the notion of an aether, even after introducing general relativity. In a talk that he gave at the university of Leiden in 1920 Einstein focused on the history of the notion of an aether. During the final remarks of that talk he said that special relativity even demands the existence of an aether to enable the existence of light clocks and rulers. General relativity, even according to Einstein himself, can also be interpreted as describing an aether. Einstein back then only insisted on the limitation that an aether must not be composed of flowing corpuscles, like it was common with the notions of an aether at that time. The wave based quantum gravity concept presented in this paper fulfils that requirement since the proposed aether waves are not producing an overall net flow of the medium of space. In this paper spin ½ particles are considered to be dynamic self sustaining entities made of space, or made of the substance of space itself, instead of them being something that is merely positioned in space and fundamentally different in its essence from space (1)(3)(4)(5). This notion was not discussed in this paper, as it was not the focus here. However, this unified view on particles and space should fit nicely with the notion of dynamically changing physical properties inside what is commonly called a gravitational field as there must be an immediate relation between particles and space to enable such dynamics. How that mechanism works exactly, and what really constitutes time, is not clear yet, though. In case space is filled with oscillators and/or waves, as it was discussed in this paper, space is inherently energetic and subsequently it also makes sense to assume that some kind of gravitational temperature exists. Such a measure already exists for the horizon of Schwarzschild black holes in the form of the Hawking temperature and interestingly it is possible to extend that notion to spin ½ particles and gravitational fields, as already discussed in (1) and (3). 8 Conclusions From what was presented in this paper it can be concluded that it should be possible to find an alternative theory for gravity that does not require a curved space-time if such theory transfers the effects that general relativity describes into the physical quantities of time, length, energy, mass, etc. Subsequently, these quantities become relative in the sense that they are dependent on a position in a gravitational field. It was shown that for a local observer inside a gravitational field the shifts in the physical quantities mostly remain clandestine and that the physical laws stay invariant. The aforementioned gravitational shifts only become readily apparent when physical processes take place over larger distances or in the vicinity of massive objects, like with gravitational redshift of light or gravitational time dilation. One local manifestation, though, is the gravitational potential energy that was shown to be an effect of varying mass due to gravity, something that Newtonian physics lacked a concrete physical mechanism before. Moreover, the equation for Newtonian gravitational force must be extended to account for special relativistic effects, which in turn allows to obtain Mercury’s perihelion precession correctly using a classical calculation approach. Considering special relativistic mass and kinetic energy for gravitational attraction makes sense since every form of energy should be gravitating as learnt from general relativity theory. Thus, for situations with two nonspinning masses that are moving with a velocity vthe equation of Newtonian gravity for flat space has to be extended as follows. The dynamic notion of gravity which was presented in this paper ultimately calls for a dynamic foundation to provide a mechanism for the required variability in the various physical quantities, which might be realizable through the presented wave based quantum gravity, and it also calls for the comeback of some kind of aether which can be the substrate for microscopic quantum effects and ordinary macroscopic behaviour as well as cosmological evolution. For convenience the Γscaling factors which were derived in this paper for various physical quantities are summarized in the following table. 18 Quantity Near ←Far Approximation Force (local) 1 1 Time period Γ 1 + 1GM/(rc2) Electric constant ϵ0Γ21+2GM/(rc2) Magnetic constant µ0Γ21+2GM/(rc2) Mass Γ31+3GM/(rc2) Frequency 1/Γ 1 −1GM/(rc2) Energy 1/Γ 1 −1GM/(rc2) Length element 1/Γ 1 −1GM/(rc2) Velocity (incl. c)1/Γ21−2GM/(rc2) Acceleration 1/Γ31−3GM/(rc2) Table 2: Scaling factors Since the effect of gravity on time is physically real the other gravitational effects must be physically real too. 19 Appendix A Natural Constants A.1 Classical expressions Light speed c= 2.9979 ×108m/s Planck constant h= 2πℏ= 6.6261 ×108J/Hz Gravitational constant G= 6.6743 ×10−11 m3/(s2kg) Electric field constant ϵ0= 8.8542 ×10−12 F/m Magnetic field constant µ0= 1.2567 ×10−6mT /A Fundamental charge e= 1.6022 ×10−19 C Sommerfeld constant α=e2/(2chϵ0)∼ =1/137 Magnetic flux quantum ϕe=h/2e Planck length ll=pℏ/c ×pG/c2=pℏG/c3= 1.6162 ×10−35 m Planck mass ml=pℏ/c ×pc2/G =pℏc/G = 21.765 µg Planck time tl=pℏ/c ×pG/c2×p1/c2=pℏG/c5=ll/c = 5.3912 ×10−44 s Planck charge ±ql=±e/√α=±1.876 ×10−18 C Planck force Fl=c4/ G = 1.21 ×1044 N A.2 Expressed using the Planck force Gravitational constant G=Fll2 l/m2 l Electric field constant ϵ0=Fl/(4πα)e2/l2 l=Fl/4π q2 l/l2 l Magnetic field constant µ0= 4πα/Fll2 l/(e2c2) = 4π/Fll2 l/(q2 lc2) A.3 Expressed as emergent properties A.3.1 Expressed in base units of quantized spacetime Light speed c=ll/tl Gravitational constant G=c2ll/ml Sommerfeld constant α=e2/q2 l Electric field constant ϵ0=e2/(2αch) = q2 l/(2ch) Magnetic field constant µ0= 2αh/(e2c) = 2h/(q2 lc) Planck force Fl=cℏ/l2 l A.3.2 Expressed using cosmological values Hubble constant H0∼ =74.3 km/s/Mpc Hubble radius rH=c/H0∼ =1.25 ×1026 m Hubble mass mH∼ =8.4×1052 kg Gravitational constant G∼ =(c2/2) rH/mH(note: this is effectively a rearranged Schwarzschild mass equation) Planck length ll∼ =pℏ/c ×prH/(2mH) Planck mass ml∼ =pℏ/c ×p(2mH)/rH Planck time tl∼ =pℏ/c ×prH/(2mH)/ c 20 Appendix B Forces in terms of the Planck force The main four fundamental forces reveal the following pattern when expressed in terms of the Planck force Fl, aka the super force, for the special case of two spin ½ particles. Force Formula Strong Fll2 l/r2 Electric αFll2 l/r2 Magnetic αFll2 l/r2v1v2/c2 Gravitational Fll2 l/r2m1m2/m2 l Table 3: Forces of spin ½ particles The following notes apply to this table: • The Planck force seems to be the reference for the main four fundamental forces and to be the strongest possible force. This gives rise to the speculation that the Planck force was the first force in our universe from which the other forces descended in some way. • The depicted formula for the strong force is an approximation that is applicable in close vicinity to the border of a proton or neutron. Equation 6.2 denoted the special case of r=rc. The strong force may exist for electrons too, but due to a substantially different particle size its effect seems to be practically irrelevant for electrons. • The relevance of the Sommerfeld constant αfor electromagnetism is immediately obvious. • The stated formulas for the electric and magnetic force only apply to electrically charged particles. • The stated magnetic force is for the special case of two fundamental charges emoving in parallel or anti-parallel to each other, whereby the sine term of any involved cross product from the Lorentz force law evaluates to 1. This implies that the relative separation vector ˆris orthogonal to both trajectories. • Distance ris relevant in multiples of the Planck length ll, i.e. 1/(r/ll). This circumstance supports the notion that the Planck length is the smallest possible length in our universe. • If the Planck units were not fundamentally meaningful the equations in the table above should contain arbitrary correction factors. Appendix C Newtonian gravity in terms of escape speed Newtons gravitational force Fg=GM1M2/r2for two homogeneous spherical masses Mwith a distance r between their centres can also be expressed in terms of escape velocity s=p2GM/r as follows. Fg=1 4Fl s2 1 c2 s2 2 c2(C.1) This equation highlights the following points: • Light speed cis the velocity limit in our universe, since s/c normalizes swith respect to c. • The Newtonian gravitational force can never reach the Planck force Flbecause of the factor 1/4. A gravitational force stronger than Fl/4requires additional gravitational effects not taken into account by Newtonian gravity, like a (rapidly) rotating mass. • The term s2/c2can be seen as a comparison of gravitational potentials, because the units for squared velocity and a gravitational potential are equal, i.e. m²/s² = J/kg. • Subsequently, the gravitational potential limit in our universe is given by −c2, as described in more detail in (2). • The repeated division by c2makes it immediately obvious why the gravitational force is relatively weak when used in scenarios which don’t involve black holes. In case that space is interpreted as a flowing medium with actual currents, as some aether theories propose, the escape velocity sshould denote the flow of that medium towards gravitational sources, which makes sense since this is the velocity that an object would need to overcome to fully leave the region of influence of a gravitational source. Moreover, a flowing medium could also account for the meaning of ∆ß as presented in section 5.2. For rotating masses the escape speed should be greater than the static case as described by the presented formula for s. 21 Acknowledgments The development of this paper has been greatly aided by Andrey Ivashov’s SMath Studio (https://smath.com) and Wolfram Alpha’s online computation facility (https://www.wolframalpha.com/input). References [1] Mayer, Martin. (2020). Compton Particles and Quantum Forces in a holo-fractal universe. http://vixra.org/abs/1906.0490 [2] Mayer, Martin. (2021). The c2Gravitational Potential Limit. https://doi.org/10.5281/zenodo.4475113 [3] Mayer, Martin. (2023). Physics Recombined. https://doi.org/10.5281/zenodo.8127612 [4] Macken, John (2015). The Universe Is Only Spacetime. http://dx.doi.org/10.13140/RG.2.1.4463.8561 [5] Macken, John (2023). A Single Field Model of the Universe. http://dx.doi.org/10.13140/RG.2.2.30242.61125 [6] Dicke, Robert Henry. (1957). 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Foundations of Physics, Volume 32, Pages 927–943 https://www.doi.org/10.1023/A:1016011413407 [14] Evans, James and Alsing, Paul and Giorgetti, Stefano and Nandi, Kamal Kanti. (2001). Matter waves in a gravitational field: An index of refraction for massive particles in general relativity. American Journal of Physics, Volume 69, Issue 10, Pages 1103–1110 http://dx.doi.org/10.1119/1.1389281 [15] Unnikrishnan, C. S. (2023). New Relativity in the Gravitational Universe: The Theory of Cosmic Relativity and Its Experimental Evidence. ISBN 978-3031089374 ————————————————————————————————————————————————————— ©2025 Martin Mayer This document is shared under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0) which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited. 22