MOND and Newtonian Gravity

New Insights into Milky Way Satellite Galaxies Raise Awkward Questions for Newtonian Gravity
Posted by Guy Pirro on 5/27/2009 7:12 PM

This article was shared by starseeker on the theotretical_physics yahoo group

Do we have to modify Newton's theory of gravitation as it fails to explain so many observations? Voices are increasingly being heard that support this heretical hypothesis. Two new studies conducted by physicists at the University of Bonn, in collaboration with scientists from Austria and Australia, are likely to provide yet more grist for the mill. Their latest results related to satellite galaxies at the periphery of the Milky Way could rock the theoretical foundations of standard physics.

As modern cosmologists rely more and more on the ominous "dark matter" to explain otherwise inexplicable observations, much effort has gone into the detection of this mysterious substance in the last two decades, yet no direct proof could be found that it actually exists. Even if it does exist, dark matter would be unable to reconcile all the current discrepancies between actual measurements and predictions based on theoretical models. Hence the number of physicists questioning the existence of dark matter has been increasing for some time now. Competing theories of gravitation have already been developed which are independent of this construction. Their only problem is that they conflict with Newton's theory of gravitation. "Maybe Newton was indeed wrong", declares Professor Dr. Pavel Kroupa of Bonn University in Germany. "Although his theory does, in fact, describe the everyday effects of gravity on Earth, things we can see and measure, it is conceivable that we have completely failed to comprehend the actual physics underlying the force of gravity".

This is a problematical hypothesis that has nevertheless gained increasing ground in recent years, especially in Europe. Two new studies could well lend further support to it. In these studies, Professor Kroupa and his former colleague Dr. Manuel Metz, working in collaboration with Professor Dr. Gerhard Hensler and Dr. Christian Theis from the University of Vienna in Austria, and Dr. Helmut Jerjen from the Australian National University in Canberra, have examined so-called "satellite galaxies". This term is used for dwarf galaxy companions of the Milky Way, some of which contain only a few thousand stars. According to the best cosmological models, they exist presumably in hundreds around most of the major galaxies. Up to now, however, only 30 such satellites have been observed around the Milky Way, a discrepancy in numbers which is commonly attributed to the fact that the light emitted from the majority of satellite galaxies is so faint they remain invisible.

A detailed study of these stellar agglomerates has revealed some astonishing phenomena: "First of all, there is something unusual about their distribution", Professor Kroupa explains, "the satellites should be uniformly arranged around their mother galaxy, but this is not what we found". More precisely, all classical satellites of the Milky Way - the eleven brightest dwarf galaxies - lie more or less in the same plane, they are forming some sort of a disc in the sky. The research team has also been able to show that most of these satellite galaxies rotate in the same direction around the Milky Way - like the planets revolve around the Sun.

Contradiction upon Contradiction

The physicists believe that this phenomenon can only be explained if the satellites were created a long time ago through collisions between younger galaxies. "The fragments produced by such an event can form rotating dwarf galaxies", explains Dr. Metz. But there is an interesting catch to this crash theory, "theoretical calculations tell us that the satellites created cannot contain any dark matter". This assumption, however, stands in contradiction to another observation. "The stars in the satellites we have observed are moving much faster than predicted by the Gravitational Law. If classical physics holds this can only be attributed to the presence of dark matter", Manuel Metz states.

Or one must assume that some basic fundamental principles of physics have hitherto been incorrectly understood. "The only solution would be to reject Newton´s classical theory of gravitation", says Pavel Kroupa. "We probably live in a non-Newton universe. If this is true, then our observations could be explained without dark matter". Such approaches are finding support amongst other research teams in Europe, too.

It would not be the first time that Newton's theory of gravitation had to be modified over the past hundred years. This became necessary in three special cases: when high velocities are involved (through the Special Theory of Relativity), in the proximity of large masses (through the theory of General Relativity), and on sub-atomic scales (through quantum mechanics). The deviations detected in the satellite galaxy data support the hypothesis that in space where extremely weak accelerations predominate, a "modified Newton dynamic" must be adopted. This conclusion has far-reaching consequences for fundamental physics in general, and also for cosmological theories. Famous astrophysicist Bob Sanders from the University of Groningen declares: "The authors of this paper make a strong argument. Their result is entirely consistent with the expectations of modified Newtonian dynamics (MOND), but completely opposite to the predictions of the dark matter hypothesis. Rarely is an observational test so definite."

Modified Newtonian dynamics

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Jump to: navigation, search "MOND" redirects here. For other uses, see Mond.

In physics, Modified Newtonian dynamics (MOND) is a theory that proposes a modification of Newton's Second Law of Dynamics (F = ma) to explain the galaxy rotation problem. When the uniform velocity of rotation of galaxies was first observed, it was unexpected because Newtonian theory of gravity predicts that objects that are farther out will have lower velocities. For example, planets in the Solar System orbit with velocities that decrease as their distance from the Sun increases. MOND theory posits that acceleration is not linearly proportional to force at low values. The galaxy rotation problem may be understood without MOND if a halo of dark matter provides an overall mass distribution different from the observed distribution of normal matter.

MOND was proposed by Mordehai Milgrom in 1981 to model the observed uniform velocity data without the dark matter assumption. He noted that Newton's Second Law for gravitational force has only been verified when gravitational acceleration is large.

Contents
[hide]
1 Overview: Galaxy dynamics
2 The MOND Theory
2.1 The change
2.2 Predicted rotation curve
3 Consistency with the observations
4 The mathematics of MOND
5 Discussion and criticisms
6 Tensor-vector-scalar gravity
7 In-line references
8 See also
9 References
10 External links

Overview: Galaxy dynamics

Observations of the rotation rates of spiral galaxies began in 1978. By the early 1980s it was clear that galaxies did not exhibit the same pattern of decreasing orbital velocity with increasing distance from the center of mass observed in the Solar System. A spiral galaxy consists of a bulge of stars at the centre with a vast disc of stars orbiting around the central group. If the orbits of the stars were governed solely by gravitational force and the observed distribution of normal matter, it was expected that stars at the outer edge of the disc would have a much lower orbital velocity than those near the middle. In the observed galaxies this pattern is not apparent. Stars near the outer edge orbit at the same speed as stars closer to the middle.

Figure 1 - Expected (A) and observed (B) star velocities as a function of distance from the galactic center.

The dotted curve A in Figure 1 at left shows the predicted orbital velocity as a function of distance from the galactic center assuming neither MOND nor dark matter. The solid curve B shows the observed distribution. Instead of decreasing asymptotically to zero as the effect of gravity wanes, this curve remains flat, showing the same velocity at increasing distances from the bulge. Astronomers call this phenomenon the "flattening of galaxies' rotation curves".

Scientists hypothesized that the flatness of the rotation of galaxies is caused by matter outside the galaxy's visible disc. Since all large galaxies show the same characteristic, large galaxies must, according to this line of reasoning, be embedded in a halo of invisible "dark" matter as shown in Figure 2.

The MOND Theory

In 1983, Mordehai Milgrom, a physicist at the Weizmann Institute in Israel, published two papers in Astrophysical Journal to propose a modification of Newton's second law of motion. This law states that an object of mass m, subject to a force F undergoes an acceleration a satisfying the simple equation F=ma. This law is well known to students, and has been verified in a variety of situations. However, it has never been verified in the case where the acceleration a is extremely small. And that is exactly what's happening at the scale of galaxies, where the distances between stars are so large that the gravitational acceleration is extremely small.

The change

The modification proposed by Milgrom is the following: instead of F=ma, the equation should be F=mµ(a/a₀)a, where µ(x) is a function that for a given variable x gives 1 if x is much larger than 1 ( x≫1 ) and gives x if x is much smaller than 1 ( 0 <x≪1 ). The term a₀ is a proposed new constant, in the same sense that c (the speed of light) is a constant, except that a₀ is acceleration whereas c is speed.

Here is the simple set of equations for the Modified Newtonian Dynamics:

$\vec{F} = m \cdot \mu\!\left( { a \over a_0 } \right) \ \vec{a}$

$\mu (x) = 1 \mbox{ if } |x|\gg 1$

$\mu (x) = x \mbox{ if } |x|\ll 1$

The exact form of µ is unspecified, only its behavior when the argument x is small or large. As Milgrom proved in his original paper, the form of µ does not change most of the consequences of the theory, such as the flattening of the rotation curve.

In the everyday world, a is much greater than a₀ for all physical effects, therefore µ(a/a₀)=1 and F=ma as usual. Consequently, the change in Newton's second law is negligible and Newton could not have seen it.
Since MOND was inspired by the desire to solve the flat rotation curve problem, it is not a surprise that using the MOND theory with observations reconciled this problem. This can be shown by a calculation of the new rotation curve.

Predicted rotation curve

Far away from the center of a galaxy, the gravitational force a star undergoes is, with good approximation:

$F = \frac{GMm}{r^2}$

with G the gravitation constant, M the mass of the galaxy, m the mass of the star and r the distance between the center and the star. Using the new law of dynamics gives:

$F = \frac{GMm}{r^2} = m \mu{ \left( \frac{a}{a_0}\right)} a$

Eliminating m gives:

$\frac{GM}{r^2} = \mu{ \left( \frac{a}{a_0}\right)} a$

Assuming that, at this large distance r, a is smaller than a₀ and thus $\mu{ \left( \frac{a}{a_0}\right)} = \frac{a}{a_0}$ , which gives:

$\frac{GM}{r^2} = \frac{a^2}{a_0}$

Therefore:

$a = \frac{\sqrt{ G M a_0 }}{r}$

Since the equation that relates the velocity to the acceleration for a circular orbit is $a = \frac{v^2}{r}$ one has:

$a = \frac{v^2}{r} = \frac{\sqrt{ G M a_0 }}{r}$

and therefore:

$v = \sqrt[4]{ G M a_0 }$

Consequently, the velocity of stars on a circular orbit far from the center is a constant, and does not depend on the distance r: the rotation curve is flat.

The proportion between the "flat" rotation velocity to the observed mass derived here is matching the observed relation between "flat" velocity to luminosity known as the Tully-Fisher relation.

At the same time, there is a clear relationship between the velocity and the constant a₀. The equation v=(GMa₀)^1/4 allows one to calculate a₀ from the observed v and M. Milgrom found a₀=1.2×10^-10 ms^-2. Milgrom has noted that this value is also

"... the acceleration you get by dividing the speed of light by the lifetime of the universe. If you start from zero velocity, with this acceleration you will reach the speed of light roughly in the lifetime of the universe."^[1]

Retrospectively, the impact of assumed value of a>>a₀ for physical effects on Earth remains valid. Had a₀ been larger, its consequences would have been visible on Earth and, since it is not the case, the new theory would have been inconsistent.

Consistency with the observations

According to the Modified Newtonian Dynamics theory, every physical process that involves small accelerations will have an outcome different from that predicted by the simple law F=ma. Therefore, astronomers need to look for all such processes and verify that MOND remains compatible with observations, that is, within the limit of the uncertainties on the data. There is, however, a complication overlooked up to this point but that strongly affects the compatibility between MOND and the observed world: in a system considered as isolated, for example a single satellite orbiting a planet, the effect of MOND results in an increased velocity beyond a given range (actually, below a given acceleration, but for circular orbits it is the same thing), that depends on the mass of both the planet and the satellite. However, if the same system is actually orbiting a star, the planet and the satellite will be accelerated in the star's gravitational field. For the satellite, the sum of the two fields could yield acceleration greater than a₀, and the orbit would not be the same as that in an isolated system.

For this reason, the typical acceleration of any physical process is not the only parameter astronomers must consider. Also critical is the process's environment, which is all external forces that are usually neglected. In his paper, Milgrom arranged the typical acceleration of various physical processes in a two-dimensional diagram. One parameter is the acceleration of the process itself, the other parameter is the acceleration induced by the environment.

This affects MOND's application to experimental observation and empirical data because all experiments done on Earth or its neighborhood are subject to the Sun's gravitational field, and this field is so strong that all objects in the Solar system undergo an acceleration greater than a₀. This explains why the flattening of galaxies' rotation curve, or the MOND effect, had not been detected until the early 1980s, when astronomers first gathered empirical data on the rotation of galaxies.

Therefore, only galaxies and other large systems are expected to exhibit the dynamics that will allow astronomers to verify that MOND agrees with observation. Since Milgrom's theory first appeared in 1983, the most accurate data has come from observations of distant galaxies and neighbors of the Milky Way. Within the uncertainties of the data, MOND has remained valid. The Milky Way itself is scattered with clouds of gas and interstellar dust, and until now it has not been possible to draw a rotation curve for the galaxy. Finally, the uncertainties on the velocity of galaxies within clusters and larger systems have been too large to conclude in favor of or against MOND. Indeed, conditions for conducting an experiment that could confirm or disprove MOND can only be performed outside the Solar system - farther even than the positions that the Pioneer and Voyager space probes have reached.

In search of observations that would validate his theory, Milgrom noticed that a special class of objects, the low surface brightness galaxies (LSB), is of particular interest: the radius of an LSB is large compared to its mass, and thus almost all stars are within the flat part of the rotation curve. Also, other theories predict that the velocity at the edge depends on the average surface brightness in addition to the LSB mass. Finally, no data on the rotation curve of these galaxies was available at the time. Milgrom thus could make the prediction that LSBs would have a rotation curve which is essentially flat, and with a relation between the flat velocity and the mass of the LSB identical to that of brighter galaxies.

Since then, many such LSBs have been observed, and some astronomers have claimed their data invalidated MOND. There is evidence that a contradiction exists.^[2]

An exception to MOND other than LSB is prediction of the speeds of galaxies that gyrate around the center of a galaxy cluster. Our galaxy is part of the Virgo supercluster. MOND predicts a rate of rotation of these galaxies about their center, and temperature distributions, that are contrary to observation.^[3]^[4]

One experiment that might test MOND would be to observe the particles proposed to contribute to the majority of the Universe's mass; several experiments are endeavoring to do this under the assumption that the particles have weak interactions.^{[citation needed]} Another approach to test MOND is to apply it to the evolution of cosmic structure or to the dynamics and evolution of observed galaxies.^{[citation needed]}.

Lee Smolin and co-workers have tried unsuccessfully to obtain a theoretical basis for MOND from quantum gravity. His conclusion is "MOND is a tantalizing mystery, but not one that can be resolved now."^[5]

The mathematics of MOND

In non-relativistic Modified Newtonian Dynamics, Poisson's equation,

$\nabla^2 \Phi_N = 4 \pi G \rho$

(where Φ_N is the gravitational potential and ρ is the density distribution) is modified as

$\nabla\cdot\left[ \mu \left( \frac{\left\| \nabla\Phi \right\|}{a_0} \right) \nabla\Phi\right] = 4\pi G \rho$

where Φ is the MOND potential. The equation is to be solved with boundary condition $\left\| \nabla\Phi \right\| \rightarrow 0$ for $\left\| \mathbf{r} \right\| \rightarrow \infty$ . The exact form of μ(ξ) is not constrained by observations, but must have the behaviour $\mu(\xi) \sim 1$ for ξ > > 1 (Newtonian regime), $\mu(\xi) \sim \xi$ for ξ < < 1 (Deep-MOND regime). In the deep-MOND regime, the modified Poisson equation may be rewritten as

$\nabla \cdot \left[ \frac{\left\| \nabla\Phi \right\|}{a_0} \nabla\Phi - \nabla\Phi_N \right] = 0$

and that simplifies to

$\frac{\left\| \nabla\Phi \right\|}{a_0} \nabla\Phi - \nabla\Phi_N = \nabla \times \mathbf{h}.$

The vector field $\mathbf{h}$ is unknown, but is null whenever the density distribution is spherical, cylindrical or planar. In that case, MOND acceleration field is given by the simple formula

$\mathbf{g}_M = \mathbf{g}_N \sqrt{\frac{a_0}{\left\| \mathbf{g}_N \right \|}}$

where $\mathbf{g}_N$ is the normal Newtonian field.

Commentary by John Shadow:

MOND gives the definition F=GMm/r² = m.f(x=a/a_o)a with a_o of the order 1.2x10^-10 acceleration units as the proposed formalism to 'change the Newtonian Law for Gravitation .

The equation works out because :

$a = \frac{v^2}{r} = \frac{\sqrt{ G M a_0 }}{r}$

and therefore:

$v = \sqrt[4]{ G M a_0 }$

Velocity becomes independent from the radius.

But Milgrom does not include G in the modification of F=ma?

So MOND says that instead of G one uses this modification of acceleration as a²/a_o.

Now QR claims variation of G. G changes as function of time (H_o=dn/dt) and so G(n)=G(H_o.t)=G_o.Xⁿ, with G_o=1/k.

And X=0.618033.. with a n defining 16.9 Billion years cycles, i.e. a H_o=1.88x10^-18 1/s.

For a present time t_p=n_p/H_o then, G(n_p)=G_o.Xⁿ and G_o=G(n_p)/Xⁿ as the presumed constant G(n) in the galactic application.

So writing Newton's Gravitational Law with that reproduces MOND.

F=GMm/r² = G_o.Xⁿ. Mm/r²=G(n_p)Mm/r² = m.f(x=a/a_o)a.

Therefore MOND's expression a²/a_o=G(n_n)M/r², and the MOND acceleration does in fact NOT modify Newtonian Mechanics but indicates the G-variance through the cosmic history.

MOND's (a/a_o) so maps the G_o applicable at the galactic scale to the G(n_p) as measured in the laboratory; and as a percentage less than 1, namely the expression (0.618033...)ⁿ with n>0 for all n as linear positive time parameter.

This holds for a<<a_o, meaning we are moving backwards in time towards the G-maximum which IS the Milgrom parameter a_o as G_o=1.111x10^-10 m³/kgs² in a veiled acceleration unit for the (Area/Mass)(Acceleration) dimensionalities.

The actual value of a_o is a_o= -2cH_o/(n+1)³=-1.127x10^-9 m/s² as a deceleration maximum (for n→0) from the expansion parameter of General Relativity in the Einstein-Friedmann-Walker cosmology. Presently a(n_p)=-1.16x10^-10 m/s²; which is Milgrom's value.

The variation of G(n) as a function of its string-parametric constant value of G_o is not linear but changes by about 62% every Hubble-oscillation of M-space, i.e every 16.9 billion years.
This in a sense resurrects Fred Hoyle's Steady State cosmology of 'mass creation' in the constancy of the G(n)M_im_j=G_oM_o.m_o productation for finestructering 'Black Hole masses' in their microquantum and macroquantum eigenstates m_im_j and M_closureM_o and a gravitational mass constancy in G_om_c²=G_oXⁿ.m_c²Yⁿ=constant for all n in Euler Identity: XY=X+Y=-1=i²=e^iπ.

An expression for the local measurement for G(n_p)=G_o{m_c/m_{neutron(np )}}²Y^np~6.67478x10^-11 (m³/kgs²)* for a present cycletime n_p=1.1324... (or 19.11 Gyears).

{The perturbation factor for the cosmological density deceleration parameter q_o= ½Ω_o=M_o/2M_critical is §²=2M_closure.M_Hawking^/M_o²=(2/Ω_o)(M_Hawking^/M_o)=1.00109... ½m_Planck.T_Planck=2M_HawkingMin.T_HawkingMax for the wormhole Temperature modulus derived by Stephen Hawking:
T_HawkingMax=E/k=hc/λ_mink~1.41671x10²⁰Kelvin=m_Planck.T_Planck/2M_HawkingMin for a minimum Hawking Black Hole mass of about M_min=(λ_min/2π)c²/2G_o~6,445.8 kg.
The Hawking maximum for a minimum Black Hole temperature T_{Hawking-GibbonsMinimum}~5.035x10^-28Kelvin then modulates
½m_Planck.T_Planck=hc³/4πG_ok=M_oT_{Hawking-GibbonsMinimum}=M_HawkingMaxT_Min[c²/4π²] by the sourcesink string modular coupling E_psE_ss via f_psf_ss=1 to the maximum Hawking-Black Hole mass M_HawkingMax=2.5447x10⁴⁹ kg by modulation factor [c/2π]² for modulated T_min=hcλ_min/4π²k~3.59x10^-26Kelvin.
For the critical holofractal spacetime quanta counter E: M_o=√[E.m_c²m_Planck²/ m_e²]~1.8137..x10⁵¹ kg* with M_critical=R_Hubblec²/2G_o~6.4706x10⁵²kg*}.

The protonucleon stringmass m_c=Alpha⁹.m_Planck~9.924725...x10^-28kg* derives from the heterotic stringclass HO(32) as a bosonic stringmembrane via the XL-Boson at the 1.9x10¹⁵ GeV energy level.

This elucidates the MOND model with additional details for thre above analysis published below.

John Shadow

NEWTON'S GRAVITATIONAL CONSTANT MEASUREMENTS

The speed of light 'c' has been measured to an accuracy of 8 decimal places and Planck's Constant 'h' is known with an error not exceeding one part per million.

This is not so for Newton's Gravitational Constant 'Big G'.

The National Bureau of Standards (NBS) in Gaithersburg, Maryland, US began measuring 'G' in the 1930's to establish the Luther-Towler-Number LTN=6.67259x10^-11 G-units (m³/kg.s²).

So it stood until 1994, when the renowned PTB in Braunschweig, Germany's Standards Laboratory measured G much higher, differing in the 3rd decimal place.

Then New Zealand's Measurements Standard Laboratory published a value significantly below the LTN and the University of Wuppertal derived a value in between the NZ one and the LTN.

Notwithstanding the ever improving technological advances and measuring techniques; using torsion pendulums, tungsten cylinders or suspended or accelerating testmasses; 'Big G' has proven to be intractable to conformity. Two of the latest measurements are 6.67327x10^-11 and 6.6742(10)x10^-11 G-units and values by no means definitive.

What is going on?

Shifting heavy objects in the vicinity of the test apparatus seems to influence the atomic structure of the testmasses, irrespective of the isolated environment created for the testing conditions.

The following treatise shall resolve the conundrum and illustrate the unruly behaviour of 'G' as a consequence of the initial boundary conditions for the universe's subsequent evolvement.

It shall indicate that even a 'massless' universe would contain a diminished G-component as the electric permittivity of a massless macroquantised (Hawking) BlackHole and that the present dilemma derives from a finestructure of the nucleonic constituents, which, by definition, must comprise the testmasses.

A precise measurement so would rely on an unambiguous calculation for two neutronic restmasses, a condition which we shall show to be unachievable, because of the nature and interrelationship between the parameters of inertial mass and those of electromagnetic charges.

There are actually TWO G-Constants, one constant as say G_o in the quantum gravity models defining the Planck-Scale and another one used for agglomerated masses, say two masses M_1,2 being comprised of say N_1,2 neutrons.

We shall find a maximised neutron mass mnmax (or nucleon mass via the beta-minus decay of the weak nuclear interaction giving the quark-lepton content of the Standard Model in say protons, electrons and antineutrinos) and a minimum neutron mass mc, and the latter being a direct consequence of the Planck-Mass m_P=√(hc/2πG_o) from the gravitational finestructure G-alpha=2πG_oM²/hc.

So setting G-alpha to unity gives the Planck-Mass.

But setting M=m_c gives the G-alpha as the force-interaction ratio between the electromagnetic alpha, defined as: alpha=2πke²/hc with k=1/4πε_o.

One can so immediately calculate the minimum neutron mass as the expression:

m_c=√{G-alpha.hc/2πG_o}.

Now the ratio between the electromagnetic- and the gravitational interaction strengths is measured and of the order of alpha/G-alpha~10^-39 and one can actually define the G-alpha as a function of alpha and as G-alpha=alpha¹⁸, using the string parameters of Quantum Relativity.

This defines the minimum neutronmass mc explicitely as: mc=√{ke².alpha¹⁷/G_o}.In string parameters, the unification condition for the interactions at the string energy scale demands kG_o=1 for a m_c=[e/G_o].alpha^8.5=9.9247246..x10^-28 kg*. This represents so 58% of the neutron (or nucleon) mass as measured today and is the actual minimum neutron mass.

Now the truly CONSTANT GM² structure in say Newton's Law, is given by the product G_om_c²=1.094446..x10^-64 Nm².

This however is finestructured in introducing a maximum neutron mass given in a unification condition, known as the Euler Identity: X+Y=XY=-1=i²=℮^iπ and applying the absolute value of unitised 1.

We write: G_om_c²={G_oX^n+k}.{m_cYⁿ}.{m_cY^k}=Gm.m_nmax.m_nmin and where G_m is the actual G value as measured and which has proved difficult to do so in the laboratories.

So the applied G value is: G_m(n)=G_o.X^n+k and where n is a cycletime n=H_ot for a nodal universe with dn/dt=H_o the nodal Hubble Constant H_o=c/R_max for a Hubble radius R_max.

The applied G_m so ALWAYS engages a maximised neutron mass (calculated as{m_cYⁿ}~ 1.7115x10^-27 kg in string parameters for a present cycletime coordinate n_p=1.1324..) AND a minimised neutron mass (calculated as {m_cY^k}).

The value of k is so determinative for G_m and differs over the evolution of the universe with respect to cycletime n and as finestructured for an AVERAGE G-value (G_av) obtained in using the geometric mean for the neutron masses in extremum (minmax productation).

One can easily calculate G_av=G_o.Xⁿ=6.44317..x10^-11 G-units for a geometric neutron mass product of m_nmax.m_nmin=m_c².Yⁿ =1.69861...x10^-54 kg² for the constancy condition of G_om_c²=1.094446..x10^-64 Nm² and omitting the k-factorisation.

But this averaged G value applies for a massless universe under the initial unification condition of the finestructures described in G_o.k=1 or G_o=4πε_o (using Stoney Units for the Planck-Scaling of the chargequantum e).

So BECAUSE an initial mass seedling M_o={m_c.m_P/m_e}√E ~ 1.8137..x10⁵¹ kg* became transformed in the de Broglie phase inflation from its preinertial state as gravitational mass into the state of inertia (this is called the Big Bang for a spacetime quanta counter E); this 'Principle of Equivalence' introduced the hitherto massless 'ylemic' 'neutron bosons' as dineutronic states, which under the Higgs mechanism became fermionic and established the mass seedling M_o as the primordial neutron matter, then decaying via beta minus decay into the observed matter in the universe (there was no antimatter).

Subsequently the EMERGENCE of inertial mass under c-invariance also introduced a finestructure for 'G' as described in the above.

One can determine the value of k from finestructuring the critical masses M_o, M_∞ and M_Hawking as boundary Black Hole masses coupled to the quantum minmax neutron masses.

For curvature radius R_maxand the critical density ρ_c=M_∞/V_max=3H_o²/8πG_o the Schwarzschild metric gives M_∞=R_max.c²/2G_o=c³/2G_oH_o= ~ 6.47058..x10⁵² kg*.

For the curvature radius R_Sarkar=2G_oM_o/c², we have the deceleration parameter q_o=½Ω_o=M_o/2M_∞=2G_oH_oM_o/c³ ~0.014015.. and which so determines the 'missing mass' in the universe to be a consequence of the initial boundary conditions set by the de Broglie inflation and the overall Black Hole evolution of the stringed parameters.

The Mass-Temperature modulus of Stephen Hawking determines M_Hawking=Constant/T_Hawking for a boundary condition of maximised Black Hole Mass for a minimised Black Hole Temperature in M_HawkingT_Hawking=constant for Constant=hc³/4πG_ok and k the Stefan-Boltzmann constant.

The relationship is given in superstring (Planck) parameters by

M_min.T_max={c/2π}².M_max.T_min=hc³/4πG_ok= ½m_P.T_Pand T_P the Planck Temperature T_P=m_P.c²/k.

This sets the Hawking-Gibbons thermodynamic temperature minima for T_o=constant/M_o ~ 5.03..x10^-28 K* and T_∞=constant/M_∞ ~ 1.41..x10^-29 K*.

As the minimum macro Black Hole has Schwarzschild metric λ_min/2π=2G_oM_min/c² for T_max=hf_max/k=hc/λ_mink; and modular duality requires the unification condition for the minimum curvature to relate to a maximum curvature in R_min=λ_min/2π=1/R_max or R_max=2πλ_max, as R_min.R_max=1.

In gauge bosonic string parameters, this modular duality then is given in E_max=hc/λ_min=m_max.c²=kT_max and E_min=hc/λ_max=m_min.c²=kT_minand in the invariance of the lightspeed parameter c as c=f_maxλ_min=1/f_minλ_max or the dimensionless unification conditions: E_max.E_min=h² and

E_max/E_min=f_max²=1/f_min²={c/λ_min}²={c/2πR_min}²=={cR_max/2π}²={cλ_max}².

This gives a proportionality: m_max.T_min=m_min.T_max for the gauges, which is however modified in the dimensionless factor {c/2π}² for the Black Hole masses for the given temperatures, as bosonic masses describe bosonic Black Holes via E=kT and not the cosmological Black Holes of the Schwarzschild metric.

The c-invariance so uses modular duality in the quantum Black Hole limit c=f_maxλ_min=2πf_maxR_min for f_min=c/λ_max=c/2πR_max as an unmodulated frequency in T_min=E_min/k=hc/2πkR_max=hc.λ_min/4π²k=3.58856...x10^-26 K* and a temperature above the Hawking-Gibbons limit as required.

This differs in a factor {2π/c}² from the lightspeed inversion in T_min=hf_min/k and so 1.574..x10^-41 K*, which violates the Hawking-Gibbons boundaries in NOT using the modular duality and with f_min=1/f_max in frequency units and NOT inverted time units.

And so M_min.T_max=hc³/4πGok= ½m_P.T_P =M_Hawking.hc.λ_min/4π²k and the Hawking Mass is determined as M_max=M_Hawking=πc²λ_max/G_o ~ 2.544690...x10⁴⁹ kg*.

We can see, that this modulation closely approximates the geometric mean of the seedling mass in 1/§²=M_o²/2M_∞.M_Hawking=3.2895..x10¹⁰²/3.2931..x10¹⁰² ~ 0.9989...

This also circumscribes the actual to critical density ratio in the omega of the general relativistic treatment of the cosmologies.

Now recall our applied G value in G_m(n)=G_o.X^n+k and apply our just derived Black Hole Mass modulation coupled to that of the quantum micro-masses.

We had: G_om_c²={G_oX^n+k}.{m_cYⁿ}.{m_cY^k}=G_m.m_nmax.m_nmin and where G_m is the actual G value as measured and which has proved difficult to do so in the laboratories.

G_m(n)=G_o.X^n+k=G_om_c²/m_nmax.m_nmin =G_om_c²/({m_cYⁿ}{m_nmin}) and where we have m_nmin={m_cY^k} for the unknown value of k.

So G_m(n)=G_o.X^n+k=G_oXⁿ[m_c/m_nmin]=G_o{m_c²/m_cYⁿ}.{M_o²/2M_∞.M_Hawking.m_av}} and where now {m_nmin}={m_cY^k}={2M_∞.M_Hawking.m_av/M_o²}=1.0011..m_av.

m_av={M_o²/2M_∞.M_Hawking}{m_nmin}={M_o²/2M_∞.M_Hawking}{m_cY^k}=0.9989..{m_cY^k} and obviously represents a REDUCED minimum mass m_nmin=m_cY^k.

But the product of maximum and 'new' minimum now allows an actual finetuning to a MEASURED nucleon mass m_N by: m_N² = m_avYⁿ.m_cYⁿ=m_av.m_nmax.Yⁿ.

So substituting for m_av in our G_mexpression, will now give the formulation:

G_m(n)=G_o.X^n+k=G_oXⁿ[m_c/m_nmin]=G_o{m_c²/m_cYⁿ}.{M_o²/2M_∞.M_Hawking.m_av}}

G_m(n)=G_o{m_c²/m_cYⁿ}.{M_o²/2M_∞.M_Hawking}.{m_cY²ⁿ/m_N²}

G_m(n)=G_o{m_c²/m_N²}{M_o²/2M_∞.M_Hawking}Yⁿ=G_o.X^n+k

The average nucleon mass m_N is upper bounded in the neutron mass and lower bounded in the proton mass, their difference being an effect of their nucleonic quark content, differing in the up-down transition and energy level.

For a Neutron Restmass of: m_n=1.680717x10^-27 kg* (941.6036 MeV*) the substitution (and using calibrations m=0.9983318783m*; s=0.9990230094s*; kg=0.99626135kg* and C=0.997296076C*) gives: G(n_p)=6.678764x10^-11 (m³/kgs²).

A perturbation corrected m_n=1.681100563x10^-27 kg* (941.818626 MeV*) gives:
G(n_p)=6.675715x10^-11 (m³/kgs²).

The perturbation upper limit is given in the m_n=1.681335x10^-27 kg* (941.9506 MeV*) and gives:
G(n_p)=6.6738445x10^-11 (m³/kgs²). The average for the last two values then approximates as a 'best fit' for:

G_m(n_p)=6.6747798x10^-11 (m³/kgs²).

This is a best-fit approximation, considering the uncharged nature of the testmasses.

This then gives the value of k from G_m(n)=G_o.X^n+k as k=ln(G_mYⁿ/G_o)/lnX and which calculates as k= -0.073387..

Two protons (m_p=1.6789x10^-27 kg* (940.56 MeV*) would give:

G(n_p)=6.6936x10^-11 (m³/kgs²) and a proton-neutron pair would yield:

G(n_p)=6.6791x10^-11 (m³/kgs²); both of the latter values unsuitable because of the electrocharges increasing the intraquarkian Magnetocharge coupling between the two mesonic rings of the neutron and the single mesonic ring in the proton's down- or KIR-quark.

The best approximation for 'Big G' hence depends on an accurate determination for the neutron's inertial mass, only fixed as the base nucleon minimum mass at the birth of the universe.

A fluctuating Neutron mass would also result in deviations in 'G', independent upon the sensitivity of the measuring equipment. The inducted mass difference in the protonic-and neutronic restmasses, derives from the Higgs-Restmass-Scale and can be stated in a first approximation as the groundstate.

Basic nucleon restmass is m_c=√Omega.m_P=9.9247245x10^-28 kg*.

(Here Omega is a gauge string factor coupling as:

Cuberoot(Alpha):Alpha:Cuberoot(Omega):Omega for Omega=G-alpha).

KKK-Kernelmass=Up/Down-HiggsLevel=3x319.62 MeV*=958.857 MeV*, using the Kernel-Ring and Family-Coupling Constants.

Subtracting the Ring-VPE (3L) gives the basic nucleonic K-State as 939.642 MeV*. This includes the electronic perturbation.

For the Proton,one adds one (K-IR-Transition energy) and for the Neutron one doubles this to reflect the up-down-quark differential.

Proton (mp=u.d.u=K.KIR.K=(939.6420+1.5013-0.5205)MeV*=940.6228 MeV*. Neutron (mn=d.u.d=KIR.K.KIR=(939.6420+3.0026-1.0410)MeV*=941.6036 MeV*.

This is the groundstate from the Higgs-Restmass-Induction-Mechanism and reflects the quarkian geometry as being responsible for the inertial massdifferential between the two elementary nucleons. All groundstate elementary particle masses are computed from the Higgs-Scale and then become subject to various finestructures.

Overall, the MEASURED gravitational constant 'G' can be said to be decreasing over time. The ratio given in k is G_mYⁿ/G_o~0.60073... and so the present G-constant is about 60% of the one at the Planck Scale.

G decreases nonlinearly, but at a present rate of 0.60073/19.11x10⁹ year, which calculates as 3.143..x10^-11 G-units per year.

So gravity appears stronger when one 'looks back in time' or analyses cosmological objects at large distances.

The expansion parameter (a) in the Friedmann-Einstein standard cosmology can be rewritten as a curvature ratio R(n)/R_max={n/(n+1)} and describes the asymptotic universe in say 10 dimensions evolving under the inertial parameters of the c-invariance.

This 'lower dimensional universe' is open and expands under hyperbolic curvature under the deceleration parameter q_o=½Ω_o=M_o/2M_∞=2G_oH_oM_o/c³ ~0.014015... This open universe is bounded in the 'standing wave' of the Hubble Oscillation of the 11D and 'higher dimensional universe'.

The boundary is given in the omega of the 'missing mass' of the volumes, which differ in a factor of V₁₁/V₁₀=nR_max³/(n/(n+1))³R_max³=(n+1)³/n²=DIM-Factor (and which assumes its minimum for one complete oscillation for n=2 as DIM=27/4=6.75 so 14.7 Billion years from the present).

Presently, for n=1.132419.. DIM=7.561.. and so the 'missing mass' will be measured as a 'dark matter' distribution of 'dark haloes' etc. around the luminous matter given in the ylemic mass seedling M_o of the baryonic matter.

As M_o is just 2.8% of M_∞, but is subject to a 'growth' in the maximising factor Yⁿ=1.724.. for the present epoch, one can take the factor M_av=M_o.√Yⁿ=1.313.. for a 'dark matter' percentage upper bounded in 2.8%(1.724)~4.83% and lower bounded in 2.8%(1.313)~3.68%.

But so 7.56 open universes are contained within the closed and spherical universe given in the Hubble bound. And the 'dark matter' will be 7.56 times the luminous baryonic matter in the interval {27.82%, 36.51%} as percentage of the total energy of closure for Ω_o=1 and the critical density ρ_c=M_∞/V_max=3H_o²/8πG_o.

Our Big Bang happened at the modular time 1/f_max=t_min=f_min=3.33..x10^-31 seconds*, coinciding with the end of the stringed inflation epoch of the standard cosmology.

The 'de Broglie' inflation established the crucial boundary parameters as say given in the M_o and M_∞ Black Hole masses described.

As the baryonic mass seedling M_o sets the Sarkar Scale for the cosmic architecture in the size of galactic superclusters as the limit for the gravitationally interacting systems before cosmic homogenuity; there must be a Black Hole evolution superposed onto the expansion of the 10D universe and the oscillation of the 11D universe which 'adds' a 'electromagnetic' volume of 2π²R_max³ at the Hubble nodes every 16.9 Billion years.

In terms of the dimensional 'intersection' this can be described as a 'Strominger Brane' evolution with the Sarkar Scale set at the instanton, decreasing as a 'shrinking' Black Hole until it becomes massless at the wormhole scale defined in the minimum macro Black Hole λ_min/2π=2G_oM_min/c²=1.591549..x10^-23 meters*.

This then resets the bosonic micro Black Holes with their macro counterparts under the modular duality.

This Black Hole evolution is higher dimensional and purely electromagnetic, not being observable due to its noninertial nature, except the so called 'dark matter' and 'dark energy' scenarios of the boundary- and initial conditions. This can lead to a feasible model for the phenomenon of consciousness.

The process will take place in a DIM factor of about 457 as: M_min.√Y^N=M_o and for
N=2ln(M_o/M_min)/lnY~454 and so in 16.9x454 Billion years, which are about 7.673 Trillion years. The gravitational constancy of G_om_c²=1.094446..x10^-64 Nm² will then be effected by a very small G_av=G_oXⁿ~1.463x10^-105 G-units, but compensated with a 'mass-evolved' universe with m_cYⁿ~7.535..x10⁶⁷ kg* and where this 'evolution' energy can be physically modelled as 'cosmic consciousness' defined in the 'awareness' df/dt minimised in f_min² and maximised in f_max² and as a form of radial displacement independent angular acceleration acting on spacetime volumars defined in the classical electron diameter (2R_e) times c² defining the magnetocharge e* as inversion of the Big Bang base parameter of the wormhole energy quantum E_max=1/e*=1/2R_ec² for a Planck Constant finestructure h=λ_min/e*c.

This "Strominger brane' evolution avoids the so called 'heat death' of the universe in a form of 'recharging' and coincides with the projected 'running out of stellar nuclear fuel of the transformation of the elements within stars in the stellar evolution scenarios.

The entire cosmology is underpinned by a Black Hole evolution, which incorporates the quantum geometric microcosmos and the geometric relativistic macrocosmos simultaneously - all for the 'cosmic purpose' to manifest 'evolved mass' as 'consciousness' or 'dark light' or antiradiation.