The Formulation of Dynamic Newtonian Advanced Gravity. The adaptation of ... gravitational observations and the advance in the perihelion of Mercury. The perihelion .... can do the same calculation, for the change in the perihelion of the Earth around the Sun and we .... the ratio of circumference to arc second: 3600. 360 ×.
The Formulation of Dynamic Newtonian Advanced Gravity
The adaptation of Newtonian dynamics.
Author :
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Abstract 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 it is possible to show that the advanced Newtonian equations, can 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.
Key Words: Black holes, Cosmology theory, dark matter, gravitation.
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1.Introduction: 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 and the advance in the perihelion of Mercury. The perihelion advance, in Straumann [1] is given by:
tan where m GM c 2 and L2/m = a(1- e2),
6 m 2 L2
(1)
a is the semi-major axis and e is the eccentricity.
This equation describes the extra curvature of space-time in terms of an extra reduction in the orbital circumference of Mercury. 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 Eq. (1), it is tan 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 the radius [Eq. (1)], back into an advanced Newtonian force. These dynamic Newtonian advanced equations are readily usable, and have been shown to equivalent results to gravitational experiment in the solar system [2], and the gravitational time dilation seen in GPS data (see Appendix A & B). Here the equations are developed, for high mass density conditions, such as with binary pulsars and black holes, these equations give results which are again technically in exact agreement with experiment.
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These advanced Newtonian
equations also resolve the problems associated with the formation of singularities and by the same token can also 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.
2. Methods:
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 & B) [2]. The paper also proposes observational experimental methods for experimental verification, based upon an analysis of the data from black holes, as listed in the conclusions.
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3. Results: An advanced dynamic Newtonian adaptation of gravity
In this paper, we address the 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 an algebraic equation [Eq. (1)], was a very accurate representation 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 has the added advantage of being much easier to use, and obviates the infinitely dense black hole singularity. Marmet was able to develop this relation, by using Straumann’s original equation, published earlier [3]. We find a first approximation to the change in the circumference of the orbit of Mercury, using Straumann’s equation (Eq.1),
Orbital (Space) Radius Reduction (R) R- = 3GM/c2
(2)
where M is the gravitational mass, c is the speed of light and G the gravitational constant, R is the relativistic space-time radius reduction.
In turn this also relates to the change in the radius of the gravitating mass, by a factor of nine (Eq. 3) [4].
Gravitational Matter Radius Reduction (r) r- = GM/3c2
(3)
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where M is the gravitational mass, c is the speed of light and G the gravitational constant, r is the gravitational radius reduction for matter.
As a result, Eq. (3), technically gives the same as the change in radius as calculated by Straumann [1]. These Eqs. (2 & 3) are a first approximation for the radius reduction that occurs in gravitational objects such as in the solar system giving results which are very closely in agreement 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 to Newton’s gravity, (see Appendix A). Unfortunately, these equations still break down in objects like binary pulsars and black holes. It is necessary to develop more advanced equations for these radii, which do not break down in high mass density objects [Eqs. (4 & 5)], such as binary pulsars and black holes.
Advanced Gravitation, Space Radius (R’) and Matter Radius (r’) Reduction
R’ =
R
,
(1 +3GM/Rc2) r’ =
r
(4) ,
(1 +GM/3rc2)
(5)
where M is the gravitational mass, c is the speed of light and G the gravitational constant, r’ is the gravitational radius for matter and R’ is the gravitational radius for space.
In this case, R’ gives answers that are technically in exact agreement in the advance in the perihelion for Mercury 42.98 arcsec/cy, and the other planets in the solar system †. Importantly, this mathematical agreement with experiment is not just a
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coincidence, but it is 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 again, technically get the same answer as experiment, 3.84 arcsec per century. Recent evidence confirms that this is the same as the experimentally determined advance in the perihelion of Earth, 3.84 0.1 arcsec/cy. [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 which agrees with the experimental advance in the perihelion of Mars,
1.35 0.1 arcsec/cy [4]. Indeed in higher mass density objects, these advanced Newtonian Eqs. (4-5), are also applicable to binary pulsars and black holes alike, and are readily usable. Indeed, these equation 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, whilst avoiding the formation of black hole singularities.
Now we can go on to develop equations for the change in the force of gravitation, by taking account of the extra radius reduction. This again gives answers that are technically in exact agreement with experiment. The equation has just been derived from the translation of describing the curvature of space-time back into describing the effective force of gravity. What is required is to translate the Eqs. (4 & 5), into equations for the force of gravity to give Eqs. (6 & 7). Dynamic Newtonian Advanced gravitational (DNAg), Force Equations (F) F = GMm . (1 + 3GM /Rc2)2 .(1 + 3Gm /Rc2)2 , R2
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(6)
averaged for elliptical orbits F = GMm . [1 + 3GM/ a (1- e2) c2]2.[1 + 3Gm/ a (1- e2) c2]2 ,
(7)
R2 where M is the larger mass, m is the smaller mass, c is the speed of light and G the gravitational constant, R is the radius, and a(1- e2) where a is the semi major axis and e is the eccentricity.
In Eqs. (6 & 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 Eq. (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. Using the data for (
b)obs,
then (
b)obs
= - 0.137 x10-12 sec/sec, in agreement with the DD model, and
in exact agreement with the model presented here [8]. In very high density objects such as black holes, conventionally these 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. This can be achieved using the same principal dynamic Newtonian equations. It is now possible to use the advanced Newtonian equivalent, and importantly this technically gives the same answers as gravitational experiment.
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But, not only is the force at the event horizon calculable, but the equations automatically resolve a number of gravitational anomalies.
Advanced Newtonian Force of Gravity at the Event Horizon (Fs) RS = 2GM
,
c2
(8)
and as FS = GMm . (1 + 3GM/RS c2 )2 .(1 + 3Gm/RS c2 )2 ,
(9)
RS 2 for the acceleration due to gravity in the vicinity of the black hole (aS). aS = GM . (1 + 3/2)2 ,
(10)
RS 2 aS = GM x 6.25 ,
(11)
RS 2 where M is the larger mass, m is the smaller mass, c is the speed of light and G the gravitational constant, RS is the distance, (taken as the Schwarzschild radius)
If we do the calculation using the advanced Newtonian equation, by inserting the Schwarzschild radius in to the term R, the actual acceleration at the event horizon is: aS = GM/RS2
x 6.25 , which would be the normal acceleration due to gravity,
multiplied by a factor of 6.25, [Eq.(11)]. Utilising 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
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of light [9], then 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 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 make the correction for the radius of orbits [(Eq. (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 [Eq. (11)]. Here we tabulate the force of gravity in the proximity of black hole binary systems (see Table 1). 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 RS, the increase acceleration due to gravity is equivalent to a factor of 6.25 multiplied by the expected acceleration due to gravity from standard Newton [Eq. (11)]. These observations may also be utilised to estimate the missing dark matter of the Universe at the cosmological event horizon, as described in the discussion and conclusions. Importantly the equations allow accurate tests of gravitation, particularly in the vicinity of black holes.
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3: Conclusions and Discussion
The principal findings in this paper are, that using Dynamic Newtonian advanced gravitational equations, we can formulate the equations for gravity in a way, which technically agrees exactly with gravitational experiment. By such means we can readily calculate the advance in the perihelion of Mercury and other planets and astronomical objects in the solar system, to a very high degree of accuracy [2].† These results suggest a straight line correlation between dynamic Newtonian advanced gravitation, for low and medium mass density gravitational bodies. A reformulation of the equations shows that advanced Newtonian equations are also very accurate for binary pulsars, using recent data for PSR B 1534+12 from Straumann [1]. Additionally, in extreme cases, such as the force at the Schwarzschild radius of black holes, these advanced Newtonian laws still hold. Equation (11), indicates that the force exerted at the Schwarzschild radius is rises by a factor of 6.25 multiplied by that expected by Newton’s original gravitational equation.
Using this Equation (11), in the advanced Newtonian model the radius reduction of matter, does not result in a black hole singularity. Moreover, using the
same principles of advanced Newtonian gravity for event horizons, it is also possible to accurately account for the dark matter of the observable Universe at the cosmological event horizon. The advanced Newtonian equations can be interpreted as a reduction in the radius, or because of an increase in the force exerted, they can be regarded as an effective increase in the mass. Either way the results, specifically the increased force of gravity are the same. The mass of the observable Universe is conventionally divided into three main components, the most recent data from supernova observations indicate that visible or baryonic matter Ωb = 4.5%, dark
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matter Ωdm = 24%, and “dark energy” ΩΛ = 71.5% [11]. These observations suggest that the Schwarzschild event horizon is similar to the cosmological event horizon [9]. The result is, that using advanced Newtonian gravity (Eq. 11), we can estimate the amount of dark matter in the Universe from the equations for the additional force of gravity exerted at the cosmological horizon, (aS = GM/RS2 x 6.25). These equations accurately account for the total amount of both ordinary and dark matter ΩM = 28%. These results are in keeping with the latest observations of these parameters [11]. The remaining ~0.5 % is most likely due to primordial black holes present in the galactic halo [12]. At the same time, the black holes themselves would be exerting a greater force of gravity than predicted by Newton. Thus, the advanced Newtonian equations, can also help explain the apparent missing mass of the galaxy, due again to the presence of black holes in the galactic halo [12], and to the indirect effects that an increased force of gravity has on the galactic rotation curve. Modified Newtonian Dynamics (MOND) has been surprisingly successful at explaining the missing mass of the galaxy, by predicting an increase in the force of gravity [13]. Advanced Newtonian gravity agrees with the results of MOND at the galactic level because of the increase gravity that is predicted by the dynamic Newtonian advanced gravity. In this paper, it has been demonstrated that advanced Newtonian gravity can explain the physical phenomena of gravity, and technically agrees accurately with gravitational experiment. In particular, the more recent binary pulsar data for PSR B 1534+12 [1, 8], tends to favour the results of advanced Newtonian gravity presented here. From the force equations (6, 7) it is clear that the radius reduction can also be viewed as an increase in the “apparent mass” of an object. At the cosmological or black hole event horizon the apparent mass increases by a factor of 6.25. Advanced
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Newtonian gravity can thus additionally account for the “dark matter” of the Universe, as well as the missing mass of Galaxies. It explains the recent findings of increased gamma rays emanating from the supermassive black hole at the centre of the galaxy [14]. It also resolves the difficulties that arise from the formation of infinite density singularities. Overall, Dynamic Newtonian advanced gravity (DNAg), de facto technically gives exactly the same answers as gravitational experiment, but it is also offers a solution to a number of contemporary gravitational anomalies, including the apparent presence of dark matter in the Universe. Importantly it offers a number of verifiable predictions for gravity, particularly in the vicinity of black holes.
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Appendix A 1). Averaged reduction in the radius of the orbit of the Moon (worked example)
RM
3GM E c2
= 1.323 cm
Equivalent general relativistic value. = 1.323 cm where RM is the reduction in the radius of the orbit of Moon, G is the gravitational constant, ME the mass of the Earth, and c the speed of light.
2). Advance in the Perihelion of Mercury (worked example). change in orbital circumference of Mercury:
circ
3GM S c a 1 e2 2
7.987 10 8
multiplied by the number of Mercury orbits in a century:
3.316 10 5 the ratio of circumference to arc second:
360 3600 1.296 10 6 calculated advance in the perihelion of Mercury per century:
3.316 10 5 1.296 10 6 42.98 arcsec/cy. Equivalent general relativistic value per century
42.98 arcsec/cy Experimentally estimated advance in the perihelion of Earth per century [5]
43 0.1arcsec/cy
circ is the change in circumference of the orbit of Mercury, G is the gravitational constant, M S the mass of the Sun, c the speed of light, a is the semi major axis of Mercury ‘s where
orbit (in meters), e is the eccentricity.
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3). Advance in the Perihelion of Earth (worked example). change in the circumference of orbit of the Earth:
circ
3GM S c a 1 e2 2
2.961 10 8
multiplied by the number of Earth orbits in a century
2.961 10 6 the ratio of circumference to arc second
360 3600 1.296 10 6 calculated advance in the perihelion of Earth per century
2.961 10 6 1.296 10 6
3.84 arcsec/cy.
Equivalent general relativistic value per century
3.84 arcsec/cy Experimentally estimated advance in the perihelion of Earth per century [5].
3.84 0.1 arcsec/cy
circ is the change in circumference of the orbit of Earth, G is the gravitational constant, M S is the mass of the Sun, c the speed of light, a is the semi major axis of Earth ‘s where
orbit (in meters), e is the eccentricity.
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4). Advance in the Perihelion of Mars (worked example). change in the circumference of Mars orbit:
circ
3GM S c a 1 e2 2
1.9595 10 8
multiplied by the number of Mars orbits in a century:
1.0416 10 6 the ratio of circumference to arc second
360 3600 1.296 10 6 calculated advance in the perihelion of Mars per century
1.0416 10 6 1.296 10 6
= 1.35 arcsec/cy.
Equivalent general relativistic value per century = 1.35 arcsec/cy
Experimentally estimated advance in the perihelion of Mars per century [5]. = 1.35 0.1 arcsec/cy
circ is the change in circumference of the orbit of Mars, G is the gravitational constant, M S is the mass of the Sun, c the speed of light, a is the semi major axis of Mars where
orbit (in meters), e is the eccentricity.
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1). Calculation of Gravitational Time Dilation relative to GPS satellites (worked example). Δt = GM c2 R
GM E = 4.436 x 10-3 c2 The total radius R is given as the altitude of GPS = 20,184 km, plus the radius of the Earth RE = 6,372 km. Thus for the GPS satellite R = 26,556 km: Δt = GM E = 1.670 x 10-10 c2 R
For Earth bound observers: Δt = GM E = 6.959 x 10-10 c2 RE The time difference ratio for the satellite is: 6.959 x 10-10 - 1.670 x 10-10 = 5.289 x 10-10 Multiplied by the number of seconds in a day 86,400 Δt = 45.7 μ sec/day. Observed gravitational time dilation relative to GPS, Δt 45μ sec/day. where Δt is the change in time, G is the gravitational constant, ME is the mass of the Earth, RE is the radius of Earth, R is the total orbital Radius, c the speed of light.
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