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A novel model of gravity is proposed and developed by modifying general relativity through propagating the gravitational field in an entirely analogous way to that of electromagnetic fields. It is therefore not a purely geometric model of gravitation, but is self-consistent, having clear causality and has the benefit of being inherently compatible with unified field theories. This model reproduces the observed almost constant rotational velocities of many galaxies as well as other large scale non-Keplerian motion. This is achieved without assuming the existence of dark matter and is made possible by modelling a rapidly rotating central star which with the inclusion of a velocity induced Doppler shift (of gravity) generates a highly anisotropic and intense, sheet like gravitational field. At extremely high gravitational fields this model remains real and finite i.e. does not generate a black hole, instead it asymptotically approaches a field limit below which light may escape. This is due to the inclusion of self-interaction of gravity in vacuum leading to a non-li nearity in the propagation of gravitational energy i.e. the effects of a gravitational field upon itself. This model is implemented computationally using an iterative finite element model. On the scale of our solar system these corrections are small and are shown not to be in obvious disagreement with high precision solar system tests.

Despite the success of general relativity in high precision experiments within the solar system there are many extremely troubling features of this theory when applied to larger spatial and mass scales. Specifically the failure to reproduce observed motion within and between galaxies. The occurrence of infinities is also normally taken as a sign of unphysical behavior (i.e. the prediction of black holes), with associated problems of causality and paradoxes. Additionally attempts to integrate general relativity into unified theories have failed. This has stimulated the proposal of theories competing with or modifying general relativity.

One of the most important observational features of galaxies (spiral, elliptical, lenticular, etc.) is their approximately constant rotational velocity which does not follow expected Keplerian motion (

In general relativity gravitational effects are modelled as a set of accelerated (accelerating) frames of reference and thereby as curvature of space time. It employs not just Galilean or weak equivalence, but furthermore assumes, “the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system”. i.e. accelerating frames are indistinguishable from rest frames. The basic components of general relativity are therefore modelling a gravitational field by the transport (in an accelerated frame) of electromagnetic radiation also including the effect of the gravitational field (curved space-time) upon the generation of gravitation. It is in this way non-linear (including the gravitational energy in the total energy-momentum tensor), however the gravitational field is propagated independently of the gravitational field. In general relativity it is assumed that; “The space-time geometry is entirely determined by the energy-momentum tensor of matter” i.e. it employs strong equivalence in a metric theory of gravity [

In the work presented here a gravitational field is assumed to propagate in an entirely analogous fashion to that of an electro-magnetic field. This imposes an additional non-linearity, not currently present in general relativity, which corresponds to the (vacuum) interaction of a gravitational field upon itself. This will be especially significant at high gravitational fields (g) and long distance (r) i.e. gr → c^{2}. Similarly the propagation of gravitational fields will also be susceptible to relative velocity of the emitting frame i.e. Doppler effects, these again are not present in general relativity and relative motion, only appearing in an indirect sense through variation of energy-momentum and thus the generation of the gravitational field. Clearly this relative velocity induced effect would become significant at high relative velocities of a gravitating body.

In this paper these modifications are quantified for a few physically significant cases. In this paper a simple finite element computational approach is taken to quantify the effect of gravitational propagation induced non-linearity (section 2) applicable in the case of massive/compact (non-rotating) objects. The effects of high velocity (relativistic Doppler effects) on the propagation of gravitational fields will be crudely modelled in section 3 and applied to a rapidly rotating star at the galactic center in order to (quantitatively) describe the observed rotation curves of galaxies (such as spiral, elliptical, lenticular, etc.) including the Milky-way.

Since this model is founded upon relativity theory it is not in principle in disagreement with many precision predictions made under general relativity within the solar system (e.g. gravitationally induced time dilation). Deviations from general relativity caused by the proposed model will be discussed (quantified) in section 4.

The aim here is to quantify the effect of including an additional non-linearity to the conventional general relativistic approach to gravitation. This non-linearity will be due to the effect of space time curvature (gravitational acceleration) upon the propagation of the gravitational field itself. In this work a computational approach was applied in which the problem was reduced to a discrete set of velocity transformations i.e. by integrating from each infinitesimally radially separated reference frame to the next.

In this computer model the gravitating object is assume to be spherically symmetric and modelled in (two dimensional) polar coordinates. The mass M and radius R of the object are variable, also the mass density is assumed to decrease with an adjustable scale height. The gravitational field is assumed to be emitted isotropically by each mass element within its emitting frame of reference. Integration is performed over all emitting angles and each radial mass element. Adjacent frames of reference (finite elements) are assumed to be held stationary with respect to each other and the gravitational field is propagated from each space-time frame to the next in an analogous manner to electromagnetic propagation, see Appendix 1.

By approximating the gravitational acceleration field to a discrete radial component of velocity difference;

where

To be clear the

This model employs the concept of ‘gravitons’ in the formal calculations, though it is important to stress that this should been seen as a generic form of gravitational theory with focus on gravitational propagation. The gravitons here should be seen only as representations of the transport of gravitational information (gravitational energy) in the same way that photons are employed in special relativity as a proxy for electromagnetic information transport. Specifically this model assumes that a volume of space-time containing gravitating mass M (energy-momentum) generates a spherically symmetric flux of ‘gravitons’(as seen in its rest frame)which can be defined arbitrarily as a total rate of 4πM gravitons/s. The flux at a distance of r from this mass will be; flux = M/r^{2} gravitons/s/m^{2} in that rest frame. The gravitational acceleration (gravitational field strength) is defined as ^{2} = 1 etc.

In the results presented here a (much simplified) model of a sun?like star is used (M = 2 × 10^{30} kg, R = 7 × 10^{8} m), it is assumed to be spherically symmetric and non-rotating. ^{8} m) under the Newtonian model (red) and the proposed model (blue). In this crude Newtonian calculation of gravity the mass is assumed to be a point source which probably accounts for the discrepancy seen within the star between the two models presented here. ^{−7} solar radii (R = 207 m). The vertical purple line shows the star surface. As can be seen the gravitational field is inhibited in the case of the more compact star. As shown in ^{2} dependence (though with an absolute reduction compared to the Newtonian model).

In ^{12} m. In ^{2}), which is conventionally the mass and radius limit for the creation of an event horizon (i.e. where light cannot escape and a black hole is generated).

As can be seen in this model the gravitational field remains real and finite, asymptotically approaching the g limit at which light propagation ceases. In

This limit is a direct result of the coupling of gravitation to itself during propagation i.e. the acceleration of gravity in the gravity field, of the form

At extreme mass and compaction in this model there becomes essentially a limiting range (d) of the gravitational field (

this crude approximation to the limit is shown in

An important and immediate consequence of the treatment of gravity as a propagating field (analogous to that of an electric field) as suggested in this paper is that the velocity of a gravitating mass will affect the propagated gravitational field. Relativistic effects will affect the gravitational field i.e. the gravitational field observed will

be distorted at velocities approaching that of light. A gravitating body moving towards the observer will gravitate more (have an enhanced observed gravitational field) and one moving away will have a reduced gravitational pull i.e. one must apply the relativistic Doppler shift also to gravitational fields. This Doppler shift in gravity allows for the generation of highly anisotropic gravitational fields and thereby immediately begins to explain some of the structure and non-Keplerian motion observed on large scales and in most galaxies. The aim in this section is to quantify the effect of high rotation velocity on the propagation of the gravitational field generated by a massive/compact star.

Ideally the effects of high velocity rotation upon the generation of a gravitational field would be (as in section 2) implemented into a finite element computational model with, for example, the addition of a relative velocity term (dv) between elements during integration. However this would involve additional dimensions and substantially increase integration (computational) times. Instead a simple model of a rotating star has been applied here in order to quantify the modification produced to the gravitational field of a rotating mass compared to a stationary mass. The rotation velocity is assumed to be constant and has been varied from a non-relativistic value (v = c) to a value as high as v = 99.999% of c corresponding to γ~ 223. The emitting angle to that of the rotation plane is varied from −90˚ to +90˚. Integration is made over all emitting angles within the plane of rotation. No consideration has been made here of the internal structure or stability of the star due to the simplicity of the model. The large relative velocity gradients expected within the star make it non-trivial to estimate its stability. However, the presence of intense magnetic fields could in principle readily provide necessary confinement of even weakly electrified plasma. Also the presence of internal slowly rotating mass could contribute additional gravitational confinement without detracting from the arguments presented here.

It is clear that at a high (relativistic) velocity each mass element of the object will generate essentially a beam of gravitational field which when integrated around the entire object will sweep out a sheet like emission of intense gravitation. ^{5} Msol and radius of 10^{12} m. An angle of 0 degrees here is in the plane of rotation as observed by a stationary observer not rotating. Several orders of magnitude enhancement in g are seen within the rotation plane. Towards the axial direction there are several orders of magnitude reduction in g.

In ^{5} M_{sol} and radius of 10^{12} m. The galactic disc thickness is taken as 2 kpc. Within the disc the radial gravitational field as a function of radial distance is taken as the average over the disc thickness. This corresponds to an increasingly narrow angular range with distance from the star. It should be pointed out that in this model the variation of the gravitational field strength averaged over all angles is not strictly being modified from the conventional 1/r^{2} dependence. Rather that the angular dependence is being highly modified which may be

observed as a non-Keplerian space-time variation i.e. that averaging over a fixed area around the rotation plane (angle = 0) leads to a 1/r dependence. Again in

Note that the rotation velocity curve shown in

Firstly precession of the star would affect the angular distribution of gravitation. Even a precession as low as 3 degrees could reduce the peak gravitational field by more than an order of magnitude. This is illustrated in

Secondly in this simplified model non-linear effects have been ignored which in principle could be significant at these high gravitational fields. This effect will depend upon the geometry (radius) of the star and will be a second order effect compared to that of the Doppler blue shift in g. At a high enough surface gravitational field it would impose a maximum field intensity to _{surf} < c^{2}/R = 10^{5} m/s^{2}).

Thirdly depending upon the structure and (radial) velocity profile the geometry of the star might be expected

to be non-spherical. In reality there will be various rotational velocities within the star and possibly also other rapidly rotating stars within the central region. This may help to explain the complex structure and variation in galactic rotation velocities. The possible effects of magnetic fields have not been considered here, but may also contribute.

Several features expected for a rapidly rotating massive star are qualitatively observed in the central regions of galaxies (e.g. in active galactic nuclei). As shown in

For the Milky-way galaxy a possible candidate for this star might be Saggitarius. A*. The reasoning here is that it generates high gravitation, it is a radio and x-ray source, it is close to central. However, the existence of stable orbiting stars close to Saggitarius. A* (e.g. S2, S1, S14, etc.) seems not to support this as a viable choice for this central highly rotating object. Unfortunately the simple model presented here breaks down at radial distances comparable to the disc thickness and a more sophisticated approach is required for the inner galactic region. It is beyond the scope of this study to conclude whether Saggitarius A* is a likely candidate for such a rapidly rotating massive star or not. This would require detailed modelling and comparison with astronomical observations.

Clusters of gravitationally bound galaxies are observed which cannot be easily explained by conventional models of gravity (without the need to assume dark matter). In this model the gravitational ‘sheet’ produced by aM = 10^{5} Msol star with rotation velocity 99.999% c and radius R = 10^{12} m has been calculated (see

Within the solar system it is possible to perform high precision tests of general relativity, typically studying pla-

netary/satellite orbits and also performing tests involving gravitationally induced time dilation. It is also possible to test aspects of equivalence. The success of general relativity is in large part due to its success in such tests. These effects of (gravitationally induced) time dilation and electromagnetic field propagation in general relativity will not be strongly affected by this new model which only treats (modifies) the propagation of gravitational fields.

Another way of re-stating the model presented here is that general relativity as it is applied conventionally is a weak field approximation which accounts for time dilation effects and propagation effects of electromagnetic energy-momentum which are not significantly affected in this model the focus here is on the propagation effects of the gravitational field. Similarly the equivalence of inertial and gravitational mass is assumed here (the weak equivalence principle). However the assumption of an accelerating (accelerated) reference frame being indistinguishable from a rest frame is only valid in a weak field approximation, but not a viable assumption (especially at high field and high velocity).

The corrections suggested here for general relativity include specifically; a Doppler shift (relative velocity) and non-linearity due to gravity-gravity interaction. At high velocity (section 3) and high field/large distance (section 2) these effects are extremely significant. However in the weak field/low velocity seen in the solar system the effects are small and difficult to isolate [

Gravitational non-linear effects can be crudely quantified for the orbits of solar system planets by estimating dg/g = gdr/c^{2}, using dr ~ 10^{12} m and g ~ 10^{−4} m/s^{2} this is of the order dg/g ~ 10^{−9} or dg = 10^{−13} m/s^{2}. This is a high precision. This effect could also be seen as a variation in G, i.e. a deviation from GM/r^{2} or a gradient in G within the solar system, however the accuracy of direct (laboratory and satellite) determinations here are dG/G around 10^{−4} [

A more stringent test of this model could be the observed velocity induced dependence in g (g_{Doppler}). A prediction of this model which could in principle be experimentally verified is that an object moving towards a gravitating body, for example a planet or satellite, would experience a blue shifted (enhanced) gravitation compared to one travelling away from the body i.e. an object moving upwards would experience g/(γ(1 + v/c)), downwards g/(γ(1 − v/c)) or horizontally (γg). For such an experiment at the surface of the Earth would require a precision of the order v/c which is extremely challenging, requiring either high velocity and/or high precision. Several other possible solar system probes of this effect will be discussed here, specifically satellite trajectories e.g. flybys, planetary rotation and planetary orbits.

Flyby

In order to probe the Doppler induced shift in gravity a satellite with a relative mean velocity in the direction of a massive object (planet) of for example v_{rad} = 2 × 10^{3} m/s would require a precision in measuring the acceleration of 1/(γ(1 − v/c)) ~ (v_{rad}/c) < 10^{−5}. In fact there are several observed (so called) flyby anomalies. An ROM of this effect is around; dv_{Dopp} = (v_{rad})^{2}/c = 13 mm/s, where dv_{Dopp} is the deviation in velocity attained during the flyby compared to that expected conventionally. This is the same order of magnitude of some observed flyby anomalies (several mm/s).

Planetary rotation

For the rotation of planets (in our solar system) one crude method of evaluating the Doppler effect in gravitational propagation is to compare the magnitude of the change in gravitational acceleration with that of the ‘tidal’ gradient i.e. the decrease due to the spatial variation in solar gravitational field. For the case of the Doppler shift for both Earth and Jupiter a crude approximation shows them be of the order of;

This can be compared to respectively the tidal gradients across the planets given by;

Thus giving; Earth;

The gravitational Doppler effect is therefore comparable, but significantly smaller than the tidal gradient. The Doppler effect in gravity is unlike the tidal effect in that it will seek to accelerate the rotational velocity instead of reducing it. It may therefore contribute a small additional source of internal heating. This effect would be difficult to isolate observationally given uncertainties in planetary internal structure and dynamics.

Planetary orbits

In the case of the orbits of planets the rotation of the sun would be expected (through a gravitational Doppler effect) to lead to a systematic shift in the center of gravity and therefore a shift in the gravitational barycenter. This could lead to a retrograde (with respect to the rotation of the sun) precession of the planets orbit. In the case of Mercury this would be in addition to, for example, the effect of other planets which corresponds to around

532"/century i.e.

A crude quantification of this effect can be made by estimating the average radial velocity component in the planets direction (for the entire sun). This was determined by numerically (computationally) integrating the velocity and mass within the sun in order to determine the average speed in the direction of Mercury. The internal mass distribution of the sun was assumed based upon the ‘standard model’, specifically a mass density distribution of;

where x is the relative radial position i.e. x = r/Rsol, (see [

the barycenter of around

where ^{−}^{9} or 1.3"/century. This is comparable to the estimated observational uncertainty i.e. around 0.65"/century or

≈ 1.2 × 10^{−}^{9} (e.g. [

In summary several potential tests of this model have been investigated using (ROM) estimates e.g. of; planetary precessions, flyby anomalies, spacecraft anomaly. They seem not to be in obvious disagreement with observations at the level of accuracy of these estimates [

In this study a model is proposed in which general relativity is modified by propagating the gravitational field in an analogous manner to that of an electromagnetic field. Specifically this entails the inclusion of velocity induced changes in gravitational field propagation (relativistic Doppler effects) and non-linearity (the effect of gravity upon itself in vacuum). In the work presented here a simple computational approach has been taken to quantify these effects.

The inclusion of a Doppler shift in gravitational fields allow the rotation velocities of galaxies to be quantitatively reproduced simply by assuming a central rapidly rotating, moderately high mass (<10^{6} Msol) star. This also (quantitatively) explains the gravitational coupling of galaxies within clusters and supports observations of non-Keplerian (non-Newtonian) motion for gravitational systems on large scales. This results from the highly anisotropic gravitational field emitted at high (relativistic) rotation velocities. To re-iterate in this model it is suggested here that a massive extremely rapidly rotating star can generate an intense sheet of gravitation and thereby explain the almost constant rotation velocity component observed in many galaxies. A similarly (or even much greater) mass star at rest would not produce such an effect. The many varied forms of galaxies may reflect the distribution in spin velocity of the central massive star or even be due to a combination of more than one such high velocity rotating star.

The non-linearity resulting from the self-interaction of the gravitation field (in vacuum) has several important consequences. It prevents the anomalous and paradoxical generation of infinite space-time curvature (singularity) or infinite gravitationally-generated red-shift (an event horizon) i.e. black holes will not be possible. In this model the gravitational field will always be real and finite since a red-shift in the gravitational field prevents reaching the point at which light (or gravitational field) cannot escape. Increasing the density of a massive object will ultimately lead to reduction in the gravitational field compared to that expected under general relativity (or Newtonian gravitation). In this case there becomes an effective ‘penetration’ depth of gravity due to its self- coupling. This depth is crudely around;

In this model a specific ‘causality’ of gravitation is made analogous to electro-magnetism, i.e. the assumption (and one which is central to relativity theory) is that gravitational energy exchange is analogous to that of a photon. This allows gravity to be compatible with other field theories and unified with all other forces i.e. compatible with unified theories and/or quantum theories. The work here imposes few restrictions on the form of the carrier of gravitational information other than it resembles a photon/electromagnetic field perturbation.

Although this model is not a purely geometric model of gravitation, general relativity is assumed to be an accurate approximation at low fields and low velocities, specifically where time dilation and/or transport of electromagnetic fields are the dominant factors. Therefore many precision tests of general relativity within the solar system are also valid in this model. However this model proposes that gravitational fields must be propagated in the same way as electromagnetic fields and therefore in addition to time dilation effects and those of electromagnetic propagation, the propagation of gravity fields must also be included. Typically within the solar system these effects are small compared to measurement precision and/or other dominant physical processes. It seems that there is no definitive evidence to disprove the model proposed here. It is hoped that this work stimulates further experimental/observational tests of gravitational models, specifically more detailed calculation using more extensive computing power.

Jonathan Peter Merrison, (2016) A Modified Relativistic Model of Gravitational Propagation. International Journal of Astronomy and Astrophysics,06,312-327. doi: 10.4236/ijaa.2016.63026

Equation (1) (section 2) is the basis for each of the computational calculations, i.e. the observed gravitational field in a frame of reference having a small velocity shift relative to the adjacent frame.

In the case of the calculation in section 2 the effect of the gravitational field upon its own propagation is included.

In the case of calculation in section 3 the velocity of the gravitating mass relative to the observer is included.

Computation 1 (section 2)

The model assumes a gravitating, spherical, symmetric, non-rotating gravitating mass with a radial (r) mass distribution M(r) = Mexp(−r/h) to a maximum radius rmax of from 10^{−7} Rsol to 1 Rsol and scale height h = rmax/8, the total Mass varied between 1 Msol and 10^{7} Msol.

The first iteration of the gravitational field is based upon the Newtonian gravitational field.

The goal of the calculation is to obtain a new gravitational field g(r) which is stable against repeated propagation i.e. differs little from the previous iteration when propagating the gravitational field. Field propagation consists of;

・ Integration over discrete mass elements at different radius; r = 0 to rmax.

・ Integration over emission angle with respect to the axial direction (polar angle)

・ Integration over emission angle with respect to the radial direction (azimuthal angle)

・ Integration over time as the gravitation field is propagated (at c) from frame to frame.

This constitutes four nested loops plus repeated iterations.

Computation 2 (section 3)

This model assumes constant rotational velocity (v) of the gravitating mass, this is made only for simplicity.

The gravitational field at radius r from each gravitating mass element g(r) is computed.

・ Integration over emitting angle is made i.e. integration over azimuthal angle i.e. around the rotating axis.

・ Averaging is made over the thickness of the galactic disc, this contributes an additional angle to that of the rotation plane.

This consists of two nested loops.

Note that clearly one side of the star is rotating towards the observer and has a blue shift whereas the other side of the star is rotating away from the observer and has a redshift.

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