2.1. Gravity as an Energy Density Difference
We will proceed to show that the rotation of galaxies can be understood without dark matter when gravitation is considered as a force field as Feynman proposed [
31]. When such a consideration is made, gravity can be understood as a force just like any other force, whose magnitude and direction is determined by the energy difference (
i.e., the free energy between the system and its surroundings) [
4,
6,
7,
32,
33]. Namely, when the potential within the system exceeds the surrounding potential, the system will emit quanta of actions to its surroundings in a quest for leveling off the energy gradient. Conversely, when the potential within the system is below the surrounding potential, the system will absorb quanta from its surroundings to level off the energy gradient. Either way, the energy difference between the system and its superior surroundings causes changes in momenta. Therefore the system is driven toward steady-state trajectories (
i.e., toward the paths (e.g., orbits) on which the resultant of forces vanish).
According to this general definition of a force, gravity is an attractive force within a system of bodies when the surrounding potential is lower than that within the system. To attain balance the bodies will accelerate toward each other by releasing quanta (
i.e., carriers of the gravitational force, also known as gravitons) from the potential associated with the bodies to the sparser surroundings (
i.e., to the vacuum). So, an apple falls straight down toward the ground (
i.e., in least time), just like a nearby galaxy is moving toward the Milky Way because quanta escape from the energy-dense potential associated with the system of galaxies to the surrounding sparser free space (
Figure 1). According to the same universal principle to consume free energy in least time, an exergonic chemical reaction will proceed forward from substrates toward products so that the system of reactants emits quanta of heat to its colder surroundings. The dissipative effect of gravity was recently demonstrated dramatically when propagating density perturbations, known as gravitational waves, were captured from a black hole binary merger [
34].
Conversely, gravity is a repulsive force within the system of bodies when the surrounding potential is higher than that within the system. In a case such as this, to attain balance the bodies will move apart by acquiring quanta from the richer surrounding potential to the sparser potential within the system. So, an apple can be lifted up from the ground by consuming free energy (
i.e., fueling the potential associated with the two bodies with quanta that are, for example, captured from insolation). Likewise, a distant galaxy is moving away from us because the vast Universe fuels the space between the two galaxies with fluxes of quanta (
Figure 1). By the same universal principle a chemical reaction will proceed backward from products to substrates when a system of reactants absorbs fluxes of quanta from its hot surroundings.
In short, if the surroundings are neglected from the analysis, one cannot understand why the system is changing from one state to another, and one does not properly understand either what governs a dynamic or quasi-stationary state, such as a rotating galaxy.
The whole Universe is the surroundings of a galaxy. It must be taken into account. When there are energy gradients between the galaxy and its surroundings, these are understood by the least-time principle to decrease as soon as possible. This natural process leads to the observed characteristics. Namely, the large scale distribution of mass is uniform and the expansion of the Universe is symmetrical about any galaxy’s center. From this perspective, it is no coincidence but a natural consequence that the vacuum’s energy density ρE, on the order of 10−9 J/m3, is in balance with the matter density ρm which is subject to the universal acceleration aR within the radius of the Universe (i.e., ρmaRR = ρm(c/T)R = ρmc2).
According to the general definition of a force as an energy density difference, there is a certain distance about a galaxy where the efflux of quanta from the gravitational potential of falling bodies equals the influx of quanta from sources in its universal surroundings. When the net flow of energy from the system to its surroundings vanishes, the distance between the two bodies is steady. By the same token, concentrations of reactants do not change at a thermodynamic balance. In other words, at a stationary state, the resultant force is zero. According to astronomical observations, this zone of dynamic steady state for our Local Group of galaxies resides at a radius
ro of 1.0–1.5 Mpc away from the Group’s center [
7,
35,
36,
37]. Obviously, only objects that are well within
ro of a given galaxy or a system of galaxies could be its orbiters. Naturally, the specific shape of a steady-state zone where inward and outward forces balance each other (e.g., for a group of galaxies) depends on the detailed distribution of mass, and hence the observed dynamics in clusters of galaxies is more intricate than that outlined simply by
ro for a single galaxy (
Figure 1).
According to the least action principle, as well as according to modern physics, galaxies do not whirl in emptiness, but in the vacuum whose potential is embodied by gravitons. The vacuum energy density
ρE =
c2/4π
GT2 ≈ 10
−9 J/m
3 is in balance with the gravitational potential
U =
GM2/
R due to all bodies, each of mass
mi in the Universe of total mass
M = Σ
mi. The energy balance
GM2/
R =
Mc2 [
31] follows from the summation of the mass density
ρm = 1/4π
GT2 within
R =
cT,
i.e.,
M = ∫
ρm4π
R2d
r =
c2R/
G. When this balance equation (
i.e., the virial theorem 2
K +
U = 0 for the entire Universe) is rearranged to
comparison of Equation (3) with Equation (1) relates the numerical value of the asymptotic acceleration per cycle
at =
aR/2π =
c/2π
T =
cH/2π ≈ 10
−10 ms
−2 to the age of the Universe
T = 13.8 billion years [
38]. The value of
at agrees with those values that have been obtained from fitting the asymptote velocity formula (Equation (2)) to the data [
39]. This agreement means to us that the orbital motion of a body with velocity
v at a radius
r from the galaxy center balances the tiny acceleration by virtue of the curvature 1/
R =
aR/
c2 of the huge, yet (here assumed) finite-size Universe. The length quantity
R =
cT =
c/
H can be also viewed as the horizon size, defining the largest volume with which can be causally connected to us and from which the gravitons now arriving can possibly originate.
Gravitation as a manifestation of the curvature is, of course, also at the heart of general relativity. Likewise, our reasoning about gravity applies equally to both a local and the universal curvature. Since the Universe is expanding, the asymptotic acceleration is time-dependent, and the proposed explanation of at could, at least in principle, be falsified by astronomical observations of the early Universe.
In the same way as the orbital velocity asymptote (Equation (2)) characterizes a galaxy with mass
Mo, the recessional velocity asymptote of the expansion characterizes the Universe with total mass
M This relation is obtained from Equation (3) by multiplying with aR = c2/R. The universal velocity asymptote (Equation (4)) can be rearranged to give the force of expansion F = MaR = Mc2/R = GM2/R2 = c4/G and the corresponding (negative) pressure p = F/4πR2 that powers the expansion. Likewise, the contribution of a single galaxy to the universal energy gradient (i.e., force) is obtained after rearranging Equation (2) to Fo = Moat = v4/G.
Gravitation when understood as the energy difference between the system of bodies and its surroundings, be it either way, displays itself also in Hubble’s law
u =
Hr, which serves to determine the distance
r to a body that is receding with velocity
u. The law can be rearranged by
cH =
c/
T =
aR to a scaling relation
u/
r =
c/
R. According to the general principle, the scaling relation holds likewise for an approaching body, since the gravitational force is understood, like any other force, merely as the energy difference per distance. According to this holistic tenet, the space as the physical vacuum [
7,
32] between galaxies is emerging, not only when the distant galaxies are moving away from us, but also when the nearby galaxies and other close-by bodies are moving toward us. Thus, to account for the zone out there
r′ ≈
ro, where the body is neither receding nor approaching the scaling relations for velocity and acceleration can be rewritten as [
7]
Consequently, when the difference between the surrounding vacuum potential and the potential within the system is negative (i.e., r′ < ro, in Equation (5)) the body will accelerate toward the galactic center because the sparser surroundings will accept the quanta that are released in the process. The magnitude of universal acceleration is the same for the approaching objects as it is for the receding ones, with only the sign of acceleration within ro being opposite from that of beyond ro.
The ratio of measured galactic to universal asymptotic velocities gives the ratio of a local mass
Mo to the universal mass
M, which in turn is available from the virial theorem for the Universe at the age of
T (Equation (3)). By acknowledging
aR our estimates for the Milky Way
Mo = 4 × 10
10 solar masses and for the Andromeda Galaxy
Mo = 4 × 10
10 solar masses parallel those that are based on luminous matter in the Milky Way [
40] and the Andromeda Galaxy [
41]. Thus, our analysis of the flat orbital velocities curve (Equation (2)) by the general action principle leaves no room for dark matter. Likewise, we understand that escape velocities of the Milky Way [
42] build up to high values because the universal potential, not the putative potential due to dark matter, has to be also compensated. By the same token, high velocity dispersion of galaxies in clusters [
43] can be obtained from the ratio of local to universal asymptotic velocities without more mass than has been deduced from the luminosities
However, if one applies the virial theorem to deduce masses in the clusters from velocities, but ignores from this equation of balance the universal gravitational potential due to the total mass of the Universe, erroneous estimates of the local masses will follow invariably [
44]. Therefore, the universal gravitational potential due to all matter, communicated via the energy density of the vacuum, has to be included in the analysis of galactic rotation, just as it has to be acknowledged in all accurate accounts of gravity.
2.2. Velocity Asymptote
We understand that an orbiter at a distance
r′ <
ro from the galactic center is on a stable trajectory when its orbital velocity
v(
r) compensates both the galactic acceleration
ao =
GMo/
r2 due to the central mass
Mo within
r (e.g., at the orbital radius of the Sun) and the universal acceleration
aR = 2π
at =
GM/
R2 due to the centrally distributed total mass
M = Σ
mi of the expanding Universe,
i.e.,
Far away from the galaxy’s luminous edge where
at >>
ao (
Figure 1), the approximation
v2ao/
r ≈
atGMo/
r2 of Equation (6) is excellent. Therefore Equation (6) can be rearranged using
v2 =
aor for the well-known asymptotic form (Equation (2)).
The flat tail of the orbital velocity curve indicates that the distant orbiter with velocity
v at
r′ <
ro is on a least-time trajectory (
i.e., on a bound geodesic whose curvature 1/
r =
a/
v2 is dominated by the universal curvature 1/
R =
aR/
c2 =
c2/
GM (
Figure 1)). Conversely, when
r′ >
ro, the body is receding with velocity
u along an open geodesic whose curvature is also 1/
R =
aR/
c2. So, any one body in the Universe is always subject to the tiny universal acceleration due to all other bodies, so that no body will move exactly along a straight line, which exists only in an ideal flatness without bodies.
At this point it is worth clarifying that Equation (6) is only a simple model without detailed mass distribution for the actual rotation curves. In other words, we acknowledge recent observations that reveal the flatness by Equation (2) as an oversimplification. A more matching phenomenology of rotation curves is available by including detailed mass distribution of luminous matter and halo [
45].
Obviously the proposed insight to the rotation of galaxies prompts one to ask: Does the universal surroundings (
i.e., the gravitational potential due to all bodies in the Universe) display itself also in the orbits of planets? It does. Anomalously advancing perihelion precession, customarily attributed to the curved space-time of general relativity, has been found also by the least-action principle as a manifestation of the universal gravitational potential [
4,
5,
6]. The planet’s precession tallies the acceleration due to all matter in the Universe.
Yet, one may wonder how could the centrally distributed mass that resides outside of a galaxy possibly exert any net effect? It does because according to the virial theorem, the kinetic energy of a system is in a dynamic balance also with the universal gravitational potential due to the total mass of the Universe. At any moment on such a stable orbit this detailed balance of forces (
i.e., Newton’s third Law) becomes apparent by differentiating the virial theorem
where it is implicit that momentum
p and acceleration
a are orthogonal (
i.e.,
p ×
a = 0). It is worth emphasizing that although the large distribution of mass about the galactic center is symmetric, the energy density of the Universe increases from the current position at
r = 0 toward the nascent Universe at
R =
cT, and hence there is indeed a gradient to be balanced by the orbital motion within
ro.
Similar to planets that are bound in the solar system, stars in globular clusters that are bound in a galaxy also do not display excessive velocities [
46]. That is to say, the clusters of stars within a galaxy present no notable evidence of dark matter. We find this only natural because the surroundings of star clusters are dominated by the galactic potential, just like the planetary surroundings are dominated by the potential associated with the Sun. In contrast, dwarf galaxies, which have stellar contents comparable to the clusters of stars in galaxies, do display the galaxy-like rotational curves [
47,
48]. In fact, the dwarfs’ velocity profiles, when interpreted by the contemporary consent, imply astonishingly high amounts of dark matter. This oddity also signals to us that dark matter is only a conjecture that follows from interpreting observations by an inaccurate tenet. Furthermore, there is no paralleling observation that a ray of light would bend astonishingly much when passing by a dwarf galaxy. Also mass distributions of early-types of galaxies are hard to model by lambda cold dark matter (ΛCDM) [
49].
Consistently with conclusions derived from the least-action principle, clusters of galaxies do display high velocity dispersion [
36,
43,
50] because these systems are exposed to the universal gravitational potential. Consequently, these systems are hard to model by localized dark matter [
51] or by adding a tiny term to the law of gravitation [
44]. Specifically, ΛCDM model does not account for the observations that dwarfs co-orbit the Milky Way in a plane as do those dwarfs about the Andromeda Galaxy. In contrast, the planar motion of dwarfs, as any other planar motion, appears to be a natural consequence of the central force, in this case
Fo =
Moat due to the tiny universal acceleration. The force generates a torque
τ =
r ×
F = d
tL (i.e., angular momentum
L) that is invariant over the orbital period. In other words, any action that displaces a body away from the center will be followed by a reaction taken by the rest of the Universe to restore the energetic balance. All in all, we conclude that the general virial theorem, also in the specific form of Kepler’s third law, holds for the rotation of galaxies as well as for motions of galaxies in the clusters, but obviously only when all potentials, notably including that of the whole Universe, and associated energy differences are acknowledged in the balance with the kinetic energy.
Equation (6) is the renowned modification of the gravity law obtained when the acceleration
a is multiplied with
μ = (1 +
at/
ao)
−1 [
26,
27]. Obviously, when the galactic acceleration
ao alone is used in Kepler’s law, it is a very poor approximation for the galactic rotation. Likewise, velocities of bodies that are chiefly exposed to the universal energy density, such as velocities of galaxies in clusters, tally primarily the universal potential. Conversely, when the local acceleration is strong, it alone is a very good approximation (e.g., for the planetary motion). When the universal acceleration is tiny relative to a local potential, it can, of course, be omitted from a practical calculation, but still
not from the explanation of how nature works. By today, the universal radius
R has grown so huge that the corresponding tiny curvature is easily masked by a local curvature.
It is worth emphasizing that the virial theorem 2
K +
U = 0 itself, even when including all potentials, is the special stationary-state case of the general principle of least action. It is easy to see that this special non-dissipative (
dtQ = 0) equation of state follows from the general evolutionary equation [
4,
52]
that equates changes in kinetic energy 2
K with changes in scalar
U and vector
Q potentials. Clearly, galaxies are not exactly stationary systems, but dissipative,
dtQ ≠ 0. Stars are burning, and other celestial mechanisms, most notably black holes, are also devouring matter. It is this combustion of matter-bound quanta to freely propagating quanta that propels the expansion of the Universe. According to the least-time imperative, space is not an immaterial abstract geometry, but a substance that is embodied in quanta [
3,
32,
53].
Moreover, according to the general principle, not only stationary motions but also dissipative processes pursue along geodesics (
i.e., least-time paths). For example, the orbital period of a binary pulsar decays with time along a parabola [
54]. The quadratic relationship between the change in the period and the consumption of energy (
i.e., mass) follows from Equation (7). In other words, at any moment, the rate of evolution could not be any faster, and hence it is accounted for by a constant. Finally, at a free energy minimum state, the constant is zero.
2.3. Velocity Profile
A detailed account of the entire rotation curve of a galaxy requires detailed knowledge of the mass distribution. Earlier studies, where the mass distributions have been deduced from surface photometry and radio measurements, have proven that many velocity profiles follow Equation (6) [
55]. The agreement is, in fact, impressive in comparison with dark matter halo models when considering that the only adjustable parameter is the stellar mass-to-luminosity ratio. Moreover, fine features in the observed profiles tend to get smeared out when curves are modeled by dark matter [
56]. In some sense though one could say that the universal background potential due to all matter could be regarded as the omnipresent halo. Although space is dark, its substance, as we will shortly explain, is not mysterious; the vacuum is embodied with tangible quanta.
Thus, mathematically we have nothing to add to the functional form of Equation (6), but we are able to give physical meaning to this model using the least-time principle. In general, not only is the galactic rotation curve a sigmoid from the center to outskirts, but similar cumulative curves, also with damping oscillations, are found everywhere in nature [
22]. These curves sum up from skewed nearly log-normal distributions [
57] and appear on a log-log scale approximately as comprising pieces of straight lines. Also, the rotational curve, when modeled by the Sérsic profile [
20] ln
I(
r) ∝
r1/n for the surface brightness
I vs. distance
r from the galactic center, is a power law [
58]. Sérsic index
n = 4 corresponds to de Vaucouleur’s profile for elliptical galaxies [
59]. For spiral disks and dwarf elliptical galaxies
n = 1 is a good model [
60].
In any case, the slope
of brightness
I vs. distance
r is a straight line on a log-log plot. Eventually the whole profile compiles from a series of straight lines (
i.e., brightness follows a broken power law when the index
n varies over a range starting from the central bulge to the luminous edge). Since brightness equals integrated luminosity, and luminosity, in turn, relates to mass, we conclude that the mass distribution also accumulates along a broken power law. Hence the orbital velocity
v vs. radial distance
r given by Equation (6) can be regarded as a profile comprising pieces of straight lines on the log-log plot.
In general, oscillatory behavior is common both in space and time when a system faces a sudden change in free energy (i.e., a potential step). For example, laser light oscillates for a while when switched on. Likewise, chemical concentrations and animal populations tend to fluctuate when exposed to rich resources, before settling to a steady state. Moreover, the intensity of coherent and mono-chromatic light builds up in an oscillatory manner as a function of distance from an obstacle’s edge. On astronomical scales, the change in potential from the dense active galactic nucleus to the sparse universal surroundings is a brisk change in energy density. Therefore, we expect the most massive and compact galaxies, as well as those that have been recently perturbed by mergers with other galaxies, to display velocity profiles with pronounced oscillations and asymmetry.
It is worth emphasizing that the power law is not merely a phenomenological model (e.g., for the velocity profile
v(
r) and mass distributions), but a consequence of the least-time free energy consumption. According to the principle in its original form by Maupertuis, the galaxies are regarded as powerful machinery for free energy consumption. These celestial engines (
i.e., stars, black holes,
etc.) transform matter-bound quanta to free quanta (
i.e., photons). This characteristic action manifests itself in the mass-to-light ratio that is constant over a broad range, at least over seven magnitudes in luminosity [
61].
According to the least-time principle, galaxies evolve and merge to attain and maintain maximal free energy consumption in the changing and ageing universal surroundings. When a galaxy increases in mass by mergers, its realm
ro contained within the Universal curvature will extend even further out for it to devour even more matter to institute even more powerful machinery of free energy consumption, such as a gigantic black hole. Apparently by this powerful celestial mechanism baryonic matter is broken down into quanta that jet out in free propagation [
62]. Star formation from gas clouds can also be regarded likewise (
i.e., as evolution in the quest of free energy consumption).