Andrew Worsley

UCL, University of London, Gower Street, London WC1E 6BT, UK.

A correction was made to the e-First version of this paper on 20 October prior to the final issue publication. The current online and print versions are identical and both contain the correction.

Received April 6, 2014. Accepted April 24, 2014.

Canadian Journal of Physics, 2014, 92(11): 1485-1488, https://doi.org/10.1139/cjp-2014-0184

In this paper we find that the equations for gravity can be adapted by defining the equations for the curvature of space–time in terms of geodesics. Using these equations, we translate this curvature back into equations for an advanced Newtonian force of gravity. Using worked examples, we can show that the advanced Newtonian equations give results that technically agree exactly with gravitational experiment. These equations also technically agree exactly with binary pulsar data. At the same time these gravitational equations resolve the difficulties with the formation of singularities. Importantly, advanced Newtonian gravity provides readily testable gravitational predictions, particularly in the vicinity of black holes.

Here we develop equations for high-mass-density conditions, such as with binary pulsars and black holes. These equations give results that technically agree exactly with experiment. These advanced Newtonian equations also resolve the problems associated with the formation of singularities and by the same token can be used to shed further light on “dark matter”, the missing cosmological matter. Importantly these gravitational equations offer readily testable predictions for gravity, particularly in the vicinity of black holes.

As gravity is represented by geodesics, the diminution of the circumference and in turn the radius can be seen in terms of a (straight-line) geodesic. In this case, from (1) , it is that gives the correct answer. Using this principle, in this paper we further develop an adaptation that can be applied to Newtonian mechanics that translates this extra reduction of radius (1) back into an advanced Newtonian force. These dynamic Newtonian advanced equations are readily usable and have been shown to yield equivalent results to gravitational experiment in the solar system [ 2 ] and the gravitational time dilation seen in global positioning system (GPS) data (see Appendices A–E ).

Newton’s standard equation for gravity is insufficient to account for a number of gravitational observations. In particular, it does not account for the observed advance in the perihelion of Mercury. However, it is possible that an advanced adaptation to Newton’s equations can be made, which can then account for a number of recent gravitational observations including the advance in the perihelion of Mercury. The perihelion advance, in Straumann [ 1 ], is given by (1)

All mathematical calculations follow strict standard algebraic and standard mathematical rules. The principal physics proofs are based upon standard physical formulae. The worked examples offer a high degree of agreement with currently known values from gravitational experiment ( Appendices A–E ) [ 2 ]. This paper also proposes observational methods for experimental verification, based upon an analysis of the data from black holes, as listed in Sect. 4.

In this paper Top of page 1. Introduction 2. Methods 3. Results: an advanced dynamic Newtonian adaptation of gravity « 4. Conclusions and discussion References

3. Results: an advanced dynamic Newtonian adaptation of gravity

In this paper, we address the following question: what are the problems related to Newtonian gravity, and can these be resolved by an advanced dynamic adaptation of Newtonian equations for gravity? Here we explore the technique of translating the equations of gravity from describing curved space–time into a radius reduction, and then back into describing a force.

Straumann’s calculations showed that (1) accurately reproduced the results of gravitational experiment, even when applied to binary pulsar data [1]. Thus, by using this algebraic equation, it is possible to reformulate the equations for gravity into a force. This approach has the added advantage of being much easier to use, and it obviates the infinitely dense black hole singularity. Marmet was able to develop this relation by using Straumann’s original equation, published earlier [3]. Using Straumann’s equation (1), we find a first approximation to the change in the circumference of the orbit of Mercury. The relativistic space–time radius reduction (R–) is given by (2)

where G is the gravitational constant, M is the gravitational mass, and c is the speed of light.

The value of R– differs from the gravitational matter radius reduction (r–) by a factor of nine [4]: (3)

As a result, (3) technically gives the same as the change in radius as calculated by Straumann [1]. Equations (2) and (3) are a first approximation for the radius reduction that occurs in gravitational objects such as in the solar system, giving results that agree closely with experiment. We can, using worked examples, calculate that the (average) radius of the orbit of the moon around the Earth is reduced by 1.323 cm compared with the value found using classical Newtonian gravitation (see Appendix A). Unfortunately, these equations still break down in objects like binary pulsars and black holes, so we must develop more advanced equations for these radii that do not break down in high-mass-density objects: (4) (5)

where R′ is the gravitational radius for space and r′ is the gravitational radius for matter.

In this case, R′ gives values that technically agree exactly for the advance in the perihelion for Mercury (42.98 arcsec/century) and the other planets in the solar system. Importantly, this mathematical agreement with experiment is not just a coincidence, but a constant relationship. For further proof, we can do the same calculation for the change in the perihelion of the Earth around the Sun, and we get 3.84 arcsec/century, the same answer as the experimentally determined value of 3.84 ± 0.1 arcsec/century [4]. A similar calculation may be performed for any gravitational body in this mass-density range. Recent experiments have been able to estimate the advance in the perihelion of Mars, and we again get a result that agrees with the experimental value of 1.35 ± 0.1 arcsec/century [4].

Indeed, in higher-mass-density objects, advanced Newtonian equations (4) and (5), are also applicable to binary pulsars and black holes and are readily usable. These equations give better results than other models of gravitation, in particular with regard to gravitational radiation damping [1, 5, 6]. Additionally, as will be discussed later, in high-mass objects these equations also potentially give very accurate results while avoiding the formation of black hole singularities.

Now we can develop equations for the change in the force of gravitation, by taking into account the extra radius reduction. This approach again gives answers that technically agree exactly with experiment. We must translate (4) and (5) into equations for the force of gravity, as follows: (6)

which, averaged for elliptical orbits, gives (7)

where M is the larger mass and m is the smaller mass.

In (6) and (7), the second term for the radius reduction involving the smaller mass m gives far greater accuracy but makes little difference in low-mass systems with one major mass. The principal difference in this advanced Newtonian gravitation enters the equations on the large scale. That scale is important in our treatment of binary pulsars and in the gravity of black holes. Firstly we used data for the binary pulsar PSR B 1534 + 12 from Straumann [1]. Using (7), we compared this with the DD model (for Damour and Deruelle) [6, 7] and other models. The models in general gave results that were largely indistinguishable from that of observation for binary pulsars. However, the observed secular decrease in the pulsar orbital period, as a result of gravitational radiation damping, was significantly different: = –0.137 × 10−12 s/s, in agreement with the DD model and in exact agreement with the model presented here [8].

In very high-density objects, conventionally black holes form an infinitely dense singularity, and there is effectively an infinite force at the event horizon [9, 10]. The advantage of this new approach is that we can resolve the concept of singularities using the dynamic Newtonian equations. It is possible to use the advanced Newtonian equivalent, which gives exactly the same answers as gravitational experiment. However, not only is the force at the event horizon calculable, but the equations also automatically resolve a number of gravitational anomalies.

At the event horizon, distance is taken to be the Schwarzschild radius (R s ), where (8)

The advanced Newtonian force of gravity (F S ) is given by (9)

and the acceleration due to gravity in the vicinity of the black hole (a S ) is given by (10) (11)

In other words, according to (11), found using the advanced Newtonian equation, the acceleration at the event horizon is the normal acceleration due to gravity, multiplied by a factor of 6.25.

Using the advanced Newtonian equations, it becomes possible to estimate the forces exerted at the Schwarzschild radius. Additionally, the Schwarzschild radius now defines the radius for the escape velocity of light. Importantly, because modern physics tells us the speed of space–time itself is allowed to exceed the speed of light [9], the presumed singularities that appeared in general relativity [10] do not appear in these advanced Newtonian equations.

Moreover, using advanced Newtonian gravity, we are able to achieve far greater accuracy than with Newton’s laws of gravity. The addition of an extra term, which takes into account the extra radius reduction in the gravitating mass, can then be translated to correct for the radius of orbits (4). With high-mass objects, particularly with black holes, the advanced Newtonian equations show how the force of gravity can also be estimated, even at the Schwarzschild radius (11). Table 1 shows the force of gravity in the proximity of black hole binary systems.

»View table Table 1.The force of gravity, F = GMm/R2x(1 + 3Gm/Rc2)2 — where R is the orbital radius of the larger mass M, G is the gravitational constant, m is the smaller mass, x is the gravity increase coefficient, and c is the speed of light — in the vicinity of black hole binary systems, with distance measured as a multiple of the Schwarzschild radii (R S ) from the major black hole.

Notably, the forces of gravity increase significantly as the radius of orbit approaches 15 Schwarzschild radii or less. In the case where the event horizon has been reached at 1 R S , the increase in acceleration due to gravity (11) is equivalent to the expected acceleration due to gravity from standard Newtonian gravitation multiplied by a factor of 6.25.

These observations may also be used to estimate the missing dark matter of the Universe at the cosmological event horizon, as described in Sect. 4. Importantly the equations allow accurate tests of gravitation, particularly in the vicinity of black holes.