An explanation of dark matter and dark energy from unmodified Newtonian gravity*
Traditionally the gravitational field associated with a mass stems from the mass itself according to the classical interpretation of Newtonian gravity. In general relativity the mass, or mass/energy in general, and space are more closely related -- a coexistence. Such coexistence is suggested now in Newtonian terms, and offers, as well, an explanation of dark matter and dark energy without modifying the classical mathematical form. The format here is more formal than usual, but what little math there is remains accessible to the general reader.
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Abstract.?Newtonian gravity is applied to large-scale space such that locality might be established, and compared with Modified Newtonian Dynamics (MOND).?Principally, m/r^2= A/G, where A ≈ 6 x 10^-13 m/s^2 is a prediction for the acceleration of the local galactic supercluster.??At the elementary particle scale the relation?yields the order of electron mass, m^3 = k^2(A/G)(e/c)^4.?At the galaxy scale, dark mass/energy in field form, to one order of magnitude beyond the visible radius.?At the cluster of galaxies scale, dark mass/energy in field form, to about the visible radius.?Dark matter is seen as dark energy at smaller scales, and dark energy is seen as accelerated expanding space.?It is proposed that accelerated expanding space is, in essence, gravity, and that gravity is fundamentally repulsive.?If particles are gravitationally associated only with large-scale space and not with one another, the hierarchy problem might be resolved, since in the standard cosmology space would have been contracted in the early Universe, contracting space/particle interaction to the immediate vicinity of the other natural forces, removing the fine-tuning free parameter. This association of particles with large-scale space also recalls Mach's Principle.
1.?Introduction.?While Newtonian gravity is empirical, its long use has likened it to a law, a law where all normal visible matter is assumed fundamentally attractive. General relativity was configured to this attractive observation at the smaller scales, merging with Newtonian gravity at low accelerations, where the disparate (active and passive respectively) backgrounds merge.?This continuum of Newtonian gravity and general relativity comprises one of the two pillars of modern physics.?Yet Newtonian gravity is fundamentally empirical, and general relativity is fundamentally mathematical, reducing to an empirical relation.?This combination is the weaker of the two pillars of modern physics, regardless of selected experimental confirmation, particularly regarding dark matter and dark energy, most of the known Universe according to standard cosmology.????
From its outset, Newtonian gravity has had the "action at a distance" character of non-locality; there is no apparent physical means of one particle signaling another to initiate mutual attraction.?Hypothetical graviton exchange is not applicable to Newtonian gravity.
Establishing locality for Newtonian gravity could place the empirical relation on a firmer footing, and possibly place general relativity on a firmer base.??
Also of concern has been the inability to identify particulate dark matter, or more generally the observation of spiral galaxies, for instance, appearing as rigid discs or having flat rotation curves.?The identification of dark matter has been key to shoring up the standard gravitation of the Newtonian gravity and general relativity continuum, along with the Big Bang standard model or Lambda Cold Dark Matter cosmology, Lambda referring to the cosmological constant or the unknown dark energy in accelerated expanding space.???
But dark matter in hypothetical particle form might be a distraction. As long as extra mass/energy can be shown to exist in sufficient quantities in some acceptable form, the situation might be resolved.?Given the standard theory, the observed stability and structure of the Universe at various scales cannot be explained without the necessity of substantial unidentified extra mass/energy that somehow differs from visible matter.
To the contrary, Modified Newtonian Dynamics (MOND) has been specifically configured to mimic the observed motion of a spiral galaxy, for instance, with only the mass that is visible.?Ostensibly, neither Newtonian dynamics nor general relativity would be necessary.?
While general?relativity merges seamlessly with classical Newtonian gravity (with due consideration to formal background difference), general relativity is the antithesis of?MOND. The attempted modification of standard gravitation is a deep study [1][2], but the standard theory and MOND sufficiently represent opposites to frame the question -- dark matter, or no dark matter -- proposed extension of the standard gravitation continuum, or continued doubt.
1.1.?Dark matter and standard theory.?This unknown extra mass in the standard view of galaxy rotation, for instance, is commonly believed to be particulate matter, adjacent to the visible matter?that has not yet been identified.??
It is offered that an extra stabilizing means is present, and in sufficient quantities, but not in particulate form, and where the seat of this extra means is not adjacent to the visible matter, yet consistent with the standard view and gravitational lensing observations.??
1.2.?No dark matter and MOND. On the opposite side of the debate is MOND. The principal contention is that dark matter does not exist in any form, that only visible matter is present, and that neither Newtonian dynamics nor general relativity is necessary. This would undermine one of the pillars of modern physics by the ad hoc introduction of another?constant of nature. Essentially MOND reads
M/r^2≈ (a^2/a sub-0)/G?
where a sub-0 = (1.2 + or - 0.2) x 10^-10 m/s^2. "The basis of the modification is the assumption that in the limit of small acceleration a very low a sub-0, the acceleration of a particle at distance r from a mass M satisfies approximately a^2/a sub-0 ≈ MGr^-2, where a sub-0 is a constant of the dimensions of an acceleration." [3][4] In this formulation the introduced constant, a sub-0, is arbitrarily chosen to mimic the observed rotation of spiral galaxies, for instance, and not derived from fundamentals.??
2.?Application of Newtonian gravity to expanding space; scale invariance examples.?Newtonian gravity applied to accelerated expanding space reads
F = Gmn/r^2?..................................................................(1)
where m is the total mass/energy in a sufficiently large region of radius r, consisting of visible mass/energy, dark matter and dark energy; and where n is a normal visible particle at radius r; also?
F = nA ..........................................................................?(2)
where A is the acceleration of n away from the center of mass/energy m, and where n is embedded in this space, possibly establishing locality for Newtonian gravity.?Since these equations are equivalent and visible mass, n, cancels,
m/r^2 = A/G .............................................................?(3)
which has the same general mathematical form as the MOND relation, where acceleration A is in place of?the arbitrary approximate a^2/a sub-0, and where mass/energy, m, represents that of a sufficiently large region of accelerated expanding space of radius r.?The principal functional difference, then, is in the scale.
2.1.?Calibrating A at the galactic supercluster scale.?Since eq. (3) was derived at this scale, acceleration A is calibrated: given observed accelerated expansion here, let m ≈ 10^15 Solar masses, the local supercluster [5]; this mass encompasses some 100 million light years (ly) -- again, to include visible mass/energy, dark matter, and dark energy -- so that let r ≈ 50 x 10^6 ly; then
?A ≈ 6 x 10^-13 m/s^2 ........................................?(prediction)?
which is the acceleration of said visible particle, n, relative?to the center of mass of m, which acceleration is equivalent to that of the local supercluster; as mentioned, the visible particle is embedded in this space, as clusters of galaxies are embedded in expanding space and observed to move away from one another.
2.2.?Elementary particle scale. Let r in eq. (3) be the classical electron radius, ke^2/mc^2, and m be electron mass; substituting,
m^3 = k^2(A/G)(e/c)^4?............................................... (4)
which is the order of electron mass magnitude.??
2.3. Galaxy scale.?Let r in eq. (3) now be proposed galaxy radius,
r = (mG/A)^1/2.?........................................................ (3a)
Given total mass, m, of the Andromeda galaxy as between 8 x 10^11 Solar masses [6a] and 1.1 x 10^12 Solar masses [6b],?then using the average value, and according to eq. (3a),
r ≈ 1.5x 10^22 meters.?............................(prediction)
On the other hand, observed visible matter gives r ≈ 1.04 x 10^21 meters [7], so that dark mass/energy is proposed to extend one order of magnitude beyond the visible radius.?This enhanced field might be verified by the pattern of light bending around a galaxy in a gravitational lens experiment.?In the proposed the pattern would be broader and less distinct, than for hypothetical particulate dark matter adjacent, to perhaps double the radius of the visible disc.??
2.4.?Cluster of galaxies scale (local group), where total mass is about 5.27 x 10^12 Solar masses [8]; from eq. (3a),
r?≈ 3.4 x 10^22 meters;?..............................(prediction)
while r ≈ 2.96 x 10^22 meters visible radius [9], so that dark mass/energy appears near the visible size at this scale.??
2.5. Galactic supercluster scale.?Acceleration, A, being derived at this scale, dark mass/energy overlaps the visible size.?See sect. 2.1.??
3. Galaxy scale and MOND. On the other hand, MOND is defined at the galaxy scale, with the assumption that gravity is fundamentally attractive, patterned on "action at a distance" empirical Newtonian gravity, compiled at the Solar system scale. Thus MOND is an empiricism of an empiricism of the Solar system scale, and not a firm basis for the assumed absence of dark matter.?
4. Gravity as fundamentally repulsive.?Sect. 2 implies that gravity is fundamentally repulsive, as illustrated here:??
?
At sufficiently small scales, in the upper visible particle pair, gravity is seen as attractive, and in the lower pair, repulsive, where the physical interaction is between a representation of large-scale?accelerated expanding space?and the visible particles. In the proposed repulsive case there is no gravitational interaction between the visible particles, and the particles are completely unaware of one another gravitationally. They appear to sense and be attracted to one another at sufficiently small scales because there is less accelerated expanding space between than about them. When there is sufficient accelerated expanding space between the particles they are mutually repelled, as observed, i.e. galactic superclusters and above.?In place of two unknowns -- dark matter and dark energy -- there is gravity in its proposed essence as accelerated expanding space, where a semblance of the cosmological constant is implicit.??
Observations that the Universe is generally accelerating, due to dark energy, have been moderated by more recent observations where some data is consistent with no acceleration -- and no dark energy [10].??But while acceleration is generally accepted, and the recent isolated study confirmed,?dark energy and general acceleration is assumed to be present.
5.?The hierarchy problem.?That particles might not be directly active physically with one another in a gravitational sense, but only physically associated with large-scale space in this regard, could resolve the hierarchy problem, where gravity is much weaker than the other natural forces.?The forces might approach one another in strength at the outset of space expansion, regarding the standard cosmological model, because space would have been more localized, to effective disappearance in the extreme, possibly avoiding the fine tuning free parameter, and serving as a commonality for quantum theory and general relativity.
6.?In closing.?Mathematically unmodified Newtonian gravity is posed sufficient to explain the stability and structure of visible matter combinations, where dark matter is proposed to exist and in sufficient quantity, but in the form of an incoming enhanced gravitational field from large-scale accelerated expanding space (suggesting?a localizing basis and quantification of Mach's Principle). Then it might no longer be necessary to identify particulate dark matter, nor dark energy, to support the standard continuum of Newtonian gravity and general relativity, along with the standard cosmology.
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