The problem with gravity

Of concern has been the inability to identify particulate dark matter. One of the two pillars of modern physics is not as secure as the other, and the identification of dark matter is key to shoring it up.

But dark matter in particle form might be a straw man. As long as extra mass can be shown to exist in some acceptable form, the situation might be resolved.

Standard cosmology with the Lambda CDM model (cosmological constant, cold dark matter) rests on the continuum of Newtonian gravity and general relativity. The observed stability of the Universe at various scales cannot be explained without the necessity or prediction of substantial unidentified extra mass/energy that somehow differs from visible matter.

To the contrary, a model has been specifically designed to mimic the observed motion of a spiral galaxy, for instance, with only the mass that is visible; it is referred to as MOND. It would not require one of the pillars of modern physics. Thus the challenge to physics at large. The historical name of the challenge is Modified Newtonian Dynamics, but this is a serious misnomer, and has nothing in common with classical empirical Newtonian gravity except a similar, yet distinct, mathematical form. MOND is the antithesis of the Newtonian gravity and general relativity continuum. While the attempted modification of standard gravitation is a broad and deep study [1][2], LCDM and MOND sufficiently represent opposites to frame the controversy -- dark matter, or no dark matter -- proposed continuity within standard gravitation, or discontinuity.

Dark matter and LCDM. The debate arises, for instance, because there is apparently not enough mass in a spiral galaxy to explain its stability. When such a galaxy rotates it acts like a rigid wheel, rather than a cyclone; in the latter the center rotates relative to the periphery. This unknown extra mass in the LCDM view is commonly believed to be particulate dark matter that has not yet been identified, which is adjacent to the visible matter.

No dark matter and MOND. On the opposite side of the debate is modified gravity, MOND. The principal contention is that dark matter does not exist in any form, and that only visible matter is present, and that neither Newtonian dynamics nor general relativity is necessary. This would remove one of the pillars of modern physics, not by empiricism or logic, but by the ad hoc introduction of another "natural" constant. Essentially MOND reads

a^2/a0 ≈ MGr^-2

where a0 = (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 a0 (a sub-0), the acceleration of a particle at distance r from a mass M satisfies approximately a^2/a0?≈ MGr^-2, where a0?is a constant of the dimensions of an acceleration." [3] [4] In this formulation the introduced constant, a0, is arbitrarily chosen, for instance, to mimic the observed rotation of spiral galaxies (as rigid wheels), and not derived from fundamentals. This is a marked departure from Newtonian gravity (and general relativity) in parameter interpretation as well as formal mathematics, despite any apparent mathematical similarity to classical Newtonian form.

Dark matter and classical (unmodified) Newtonian gravity,

F = Gmn/r^2 (1)

could just as well be written

F = G(-m)(-n)/r^2 (1a)

without changing the outcome. The only reason for not using Equation (1a) is economy. But with Equation (1a), there is a suggestion that it might be applicable unmodified to the Universe at large, because it is observed to be not only expanding but also accelerating, as in

F = (-n)A (2)

where A is acceleration of visible mass (-n) relative to but away from mass, -m. Since Equation (1a) equals Equation (2) and visible mass, -n, cancels,

-m/r^2 = A/G (3)

which has the same general mathematical form as the MOND relation, but where acceleration A (quantified below) replaces the arbitrary approximate a^2/a0 (≈ a), and where -m represents a sufficiently large region of accelerated expanding space of radius r. The principal difference, then, is in the interpretation of the mass, size (i.e. scale) and acceleration parameters; there is seen to be actual and sufficient extra mass/energy. Mass/energy, -m, then, is not necessarily distinct from space itself; according to general relativity space(time) is a real physical entity that is undergoing accelerated expansion and carrying along visible matter imbedded within it; also, space(time) is equivalent to the pure gravitational field. Then mass/energy, -m, could represent a gravitational field relieving itself of tension (such stressed fields have mass); recall, general relativity and Newtonian gravity merge under such conditions of low acceleration/gravitation. [Note a]

Elementary particle scale. Let r in Equation (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)

the order of electron mass magnitude. Therefore at the largest scale and that of the elementary particle Equation (3) seems to hold. But visible matter was seen to cancel, and the existence Equation (4) would seem contradictory -- unless particles are gravitational sinks instead of sources [5].

Solar system scale. Essential precision with the classical Newtonian form and interpretation, refined by general relativity.

Galaxy scale. Solving Equation (3) for galactic radius, r,

r = (mG/A)^1/2. (3a)

where mass may be defined positive if all mass/energy is negative.[5] Given mass m of the Andromeda galaxy as between 8 x 10^11 Solar masses and 1.1 x 10^12 Solar masses [6], then using the average value, according to this relation,

r ≈ 3.85 x 10^22 meters. (proposed)

On the other hand, the observed visible matter gives r ≈ 1.04 x 10^21 meters (Wikipedia), an order of magnitude less than that proposed -- leading to the question, Does this extra size indicate "dark matter" as possibly the dark energy repulsive field pointing at galaxies at these smaller scales?[Note b] This might be verified by the pattern of light bending around a galaxy in a gravitational lens experiment. In the proposed the pattern would be more dispersed and less distinct, than for hypothetical particulate dark matter adjacent, to perhaps double the radius of the visible disc.

Galaxy scale alternative. Solving for mass in Equation (3),

m = A/G r^2

where mass m is that of a galaxy, and mass is defined positive as discussed. This mass increases with the square of the distance, in conformance with mass/energy of the expanding gravitational (dark energy) field seated in the large-scale voids directed at the smaller scales, as mentioned. Assuming this mass/energy is contained in a sphere of radius r about the galaxy, volume, V, of this sphere (galactic halo) is 4/3 π r^3. Density of the mass/energy would be m/V = ρ,

ρ = 3/4π (A/G) r^-1 (prediction) (5)

so that mass density falls off linearly with galaxy radius (to an order of magnitude beyond the visible radius), where A is quantified below. This would be subject to the same gravitational lensing experiment as above, to distinguish field dark matter from particulate dark matter (limited to perhaps twice galaxy visible radius).

Cluster of galaxies scale (local group), where mass is about 5.27 x 10^12 Solar masses; [7] from Equation (3a),

r = 9.08 x 10^22 meters; (proposed)

while r ≈ 9.26 x 10^22 meters visible radius (Wikipedia). Apparently, there is sufficient enhanced gravitational mass/energy without exceeding the visible radius of the system at this scale.

Galactic supercluster scale. Since Equation (3) was derived at this scale, calibrating acceleration A in Equation (3): Given observed accelerated expansion here, let m ≈ 10^15 Solar masses, the local supercluster [8]; this mass encompasses some 100 million light years (ly), so that let r ≈ 50 x10^6 ly, then A ≈ 6 x 10^-13 m/s^2, shown subsequently to be closer to 8.5 x 10^-14 m/s^2. [5]

Concluding. Thus, at these scales mathematically unmodified, but reinterpreted Newtonian gravity is sufficient to explain the stability of visible matter combinations, where dark matter is proposed to exist in sufficient quantity, but in the form of an incoming enhanced gravitational field from large-scale space -- a quantification of Mach's Principle? Then it might no longer be necessary to identify particulate dark matter to support the standard LCDM model.

References and Notes

[1] https://www.sciencedirect.com/science/article/pii/S0370157312000105

[2] https://kavlifoundation.org/news/modified-gravity

[3] Milgrom, M., Astrophysical J. 270, 365 (1983a)

[4]https://www.scholarpedia.org/article/The_MOND_paradigm_of_modified_dynamics

[5] W. Frisina, Particles from Gravity, https://lnkd.in/dDihkf2 https://lnkd.in/dzpp_tu

[6] https://en.wikipedia.org?? wiki ? Andromeda_Galaxy

[7] Mass of the local group, https://academic.oup.com?? article-pdf ? mnras0384-1459

[8] 5.?M. Einasto, E. Saar, L. J. Liivam?gi, J. Einasto, E. Tago, V. J. Martínez, J.-L. Starck, V. Müller, P. Hein?m?ki, P. Nurmi, et al., The richest superclusters - I. Morphology, Astronomy?and Astrophysics, 476 (2), 697–711 (2007)

[Note a] This mass/energy, -m, under this condition might identify and quantify dark energy, commonly said to be responsible for the accelerated expansion of the Universe (positive mass/energy here would result in accelerated contraction), although, dark energy and gravitational energy are not commonly considered identical (while dark matter and dark energy are commonly considered distinct). But since dark matter and dark energy are not yet commonly identified, it may be proposed that these and gravitational energy are identical. Recall, the dark energy repulsive fields of the large-scale voids point at surrounding galactic supercluster shells and therethrough to clusters of galaxies and galaxies, etc., so that this proposition is consistent in this regard.

[Note b] The Universe at the largest scale resembles an open cell sponge where formation is primarily due to the accelerated expanding large-scale voids that are surrounded galactic superclusters shells, and appear to compress them into sheets and filaments.