On the three laws of rotationally supported galaxies: the observed flattening of rotation curves, baryonic Tully-Fischer relation and the mass discrepancy-acceleration relation

In this paper we will find that, according to holographic principle [1] and thus considering Universe as the ensemble of א information bits or minimum particles of mass mg, the contribution to galactic rotation curves can be due the rest of the visible Universe through a non-local collective gravitational interaction of all particles within the Universe’s horizon, as a consequence of which all particles are gravitationally entangled and form a unified statistical ensemble. Therefore, we can to describe this global effect in terms of standard local Newtonian gravity within galaxies for the explanation of flatness galactic rotation curves as a possible alternative to the dark matter or MOND hypothesis. We will find a solution for the baryonic Tully-Fischer relation: Mb = Av 4 f ⇐⇒ A = [a0G] −1 with a0 = cH0 2π , where H0 is the Hubble constant at present Time t0 and Mb, G and c are the galaxy baryonic mass, gravitational constant and constant speed of light in vacuum respectively. Also we will find the mass discrepancy-acceleration relation, thus obtaining a possible solution for each of the three laws of rotationally supported galaxies proposed in ([2],[3]).


Introduction.
According to Einstein's theory of general relativity spacetime has no intrinsic properties other than its curved geometry: it is merely a stage, albeit a dynamical one, on which matter moves under the influence of forces. There are well motivated reasons, coming from theory as well as observations, to challenge this conventional point of view. From the observational side, the fact that 95% of our Universe "consists of mysterious forms of energy or matter" gives sufficient motivation to reconsider this basic starting point. And from a theoretical perspective, insights from black hole physics and string theory indicate that our "macroscopic" notions of spacetime and gravity are emergent together from the entanglement structure of an underlying microscopic theory, requiring taking into account the entropy and temperature associated with the cosmological horizon [4].

Cosmological part.
Let's consider universe as the ensemble of ℵ information bits or minimum particles of mass m g within volume V , with area A, radius R and ℵ ∈ Z + . Thus, we can express universe mass M as: Thus, universe mass M or universe energy E U = M c 2 are divided evenly over the bits ℵ. The holographic principle states that the description of a volume of space can be thought of as ℵ bits of information or minimum masses, encoded on a boundary to that region, a closed surface of area A. The information is distributed on the surface with each bit requiring an area equal to l 2 P , the so-called Planck area, from which ℵ can thus be established as: Where l P = G c 3 is the Planck length, G is the universal gravitational constant, c is the speed of light, and is the reduced Planck constant. When substituted ℵ in Eq.(1) we find: 2.1 Universe mass as a function of cosmological Time.
The age, cosmological Time or universe Time t with regard to any relative reference system, is the elapsed time from the origin of the universe, according to the current theory of the Big Bang which may be considered until now as the most prudent and widely accepted model of the formation of the universe, to any Time t. Thus, if we include universe Time t in Eq.(3), equation doesn't change and we find: Where we obtain universe mass at Time t by option a) in Eq.(4), which is in perfect agreement to M = c 3 GH0 according in [5] by approximation of t 0 1 H0 , where H 0 is the Hubble constant at present Time t 0 .

Universe gravitational potential.
If now, we include in Eq.(3) the universe radius R, equation doesn't change, and we can now find: Thus, the universe area A can be expressed as the compton wavelength of minimum mass m g multiplied by universe radius R. Furthermore, if we now equate option b) of Eq.(5) and Eq.(4) respectively for universe area, we obtain universe radius at Time t as: However, according to Eq.(1) we can express any material particle or body of mass m with (m ∈ M ) as: m = n · m g , where (n ∈ ℵ; n ∈ Z + ). Therefore, by option a) of Eq.(5) we can now obtain the Universe gravitational Potential Φ as: there exists a non-local collective gravitational interaction of all particles within the Universe's Horizon, as a consequence of which all particles within the Universe are gravitationally entangled and form a unified statistical ensemble. Thus, we can relate the origin of inertia of any material particle to its interactions with the whole universe, according to the non-local potential of the whole universe, Φ, acting on any material particle of the world ensemble as: Where E = mc 2 is the total energy of the particle. However, according to Eq.(7) and Eq.(6) for universe radius, we can obtain another consequence of non-local collective gravitational interaction of all particles within the universe's horizon due universe gravitational Potential Φ, as: That is, the existence of the Universe gravitational force F G = − GM m R 2 that causes a negative acceleration a P = − c t on any material particle of the world ensemble.

Particles creation process: entropy and temperature.
We consider with regard to a relative reference system any universe Time interval ∆t = t − t i according to option a) in Eq.(4). Thus, the universe mass must increase as Time progresses, that is, if the universe mass at some initial universe Time t i is M (t i ) and the universe mass at any later universe Time t is M (t), then the increase of universe mass ∆M = M (t) − M (t i ) during elapsed proper time interval ∆t = t − t i will be: Where E U = M c 2 is the universe energy at Time t. However, by Eq.(1) we can now express Eq.(9) as: Where ℵ t can be considered as the creation rate of minimum particles/bits. Therefore, universe increases its mass/energy as Time progresses by the creation of ℵ t new minimum particles of mass m g or information bits and energy m g c 2 at every universe Time interval ∆t. However as we know, according to the first law of thermodynamics, near the equilibrium any thermodynamic system at temperature T experiences a state change according to equation: Where ∆E is the energy variation of the system, T ∆S is the work done by internal forces generating a change of entropy ∆S and ∆W is the quantity of energy lost by the system due to work done by the system on its surroundings. However, Universe can be considered a whole, there no surroundings out of it, thus ∆W = 0. Therefore, any universe state change is due only to internal forces and we can consider universe as a entropic system, that is, the entropy as a function of its internal energy. Applying now the first principle, the universe energy increase ∆E U Eq.(9) can be expressed then as: Now, introducing the mass m g for the minimum particle or information bit, the reduced Planck's constant and the Boltzmann constant k B which relates the thermodynamic energy and temperature, equation doesn't change and we can now express the universe energy increase ∆E U during any elapsed universe Time interval ∆t as: And finally, we obtain by equalizing: 1. Temperature T: The temperature or thermal radiation T in the horizon associated to the universe energy increase ∆E U by the creation of ℵ t new minimum particles or information bits of mass m g and energy m g c 2 during elapsed universe time interval ∆t, Eq.(10).

Entropy increase ∆S:
The entropy increase on every minimum particle or bit of mass m g during elapsed universe time interval ∆t.

Minimum mass/information bit.
Eq.(13) can also be expressed according to Eq.(1) and Eq.(4) by option a) for universe mass at Time t, as: Thus, Eq.(12) for universe energy increase ∆E U during any elapsed universe Time interval ∆t can be now expressed as: That is, temperature or thermal radiation T in the horizon associated to the universe energy increase ∆E U by the creation of new minimum particles or information bits equals to associate on every minimum particle m g a temperature: T g = kBt .
However, by Eq.(1), the energy E U = M c 2 is divided evenly over the bits ℵ, then temperature can be determined by the equipartition rule as the average energy per bit/minimum mass: By simplifying previous equation according to Eq.(15), we find: Where we obtain wave-particle duality as a function of universe Time. Furthermore, energy of minimum particle/bit m g c 2 = 1 2 w is equivalent to Zero-point energy (ZPE), that is, the lowest possible energy that a quantum mechanical system may have, as expected, since w = 1 t as inverse of universe Time, can be considered as the minimum frequency. Therefore, Universe can be considered as a quantum mechanical system of ℵ information bits or minimum particles of mass m g .
However, the minimum energy m g c 2 can be also expressed, according to the known relation w = 2πν and = h 2π , as: Which can be approximated as: Thus, approaching t 0 ≈ 1 H0 , the value H 0 = (74.03 ± 1.42) kms −1 M pc −1 in [6], we can obtain approximately: m g ∼ 10 −68 kg which can be in agreement with the range of values m g ∼ (10 −68 − 10 −66 )kg for the entropic minimum mass found by the Prof. J. R. Mureika and R. B. Mann in [7] where this range represents the smallest non-zero mass for any particle quanta in the entropic gravity framework. Now, according to Eq.(1), by option a) in Eq.(4) and Eq.(17), we can calculate the total number of minimum particles or bits of mass m g as: Which, approaching t 0 ≈ 1 H0 , we can obtain approximately: ℵ ≈ 10 122 .

Universe as a black hole.
The first indication of the emergent nature of spacetime and gravity comes from the laws of black hole thermodynamics [8]. As we know, a central role herein is played by the Bekenstein-Hawking entropy and Hawking temperature on the event horizon of a black hole given by: Here A denotes the area of the horizon, is the reduced Planck constant, c is the speed of light, k B is the Boltzmann constant, G is the gravitational constant and M is the mass of the black hole. However, it's remarkable that Eq.(15) and Eq.(16) for universe energy increase ∆E U during any elapsed universe Time interval ∆t can be now expressed as: Now, if we insert Eq.(2) for universe area as a function of total number of information bits/minimum masses, that is: A = ℵ · l 2 p in Bekenstein-Hawking entropy Eq.(20), we find: Thus, ℵ 4 is equivalent to the a black hole entropy with area A = ℵ·l 2 p equal to the universe area. Therefore, Eq.(21) can be expressed now as: That is, it's as if the universe could be considered as a black hole emitting black body radiation where the temperature: is the Hawking-Unruh temperature T HU and where a U = 8πc t can be considered as the acceleration at the surface of the universe. This expansion acceleration a U at the surface will be considered in section §4 for baryonic Tully-Fisher relation.
3 First law: the observed flattening of rotation curves in galaxies.
The concept of "Dark Matter" was proposed by Prof. F. Zwicky to explain the anomalous rotation curves of the galaxies. The problem was that, according to Newtonian dynamics, the velocities of any body of mass m at a distance r from the center of the galaxy, must be expressed by assuming a circular orbit, as: Where M b is the baryonic mass of the Galaxy, that is: the sum of its stars and gas: M b = M * + M g , m the mass of the body, G the gravitational constant, r the distance of the body to the center of the galaxy and v its tangential velocity. However, the astronomical observations indicate that velocity of the rotation curves are flattened tending to a certain limit, instead of complying with the previous law where velocity must decrease as the radius r increases.

Radius upper limit.
Now, we consider any gravitational local system (i.e., Galaxy) within the Universe of baryonic mass: M b = ℵ b · m g according to Eq.(1) where (M b ∈ M ) and (ℵ b ∈ ℵ).We will also consider that a material particle or body of mass m is located at a distance r from the center of the gravitational local system at Time t. Now, we apply Newtonian Mechanics to our gravitational local system (Galaxy), this is: F i = 0, then by Eq.(24) and according to Universe gravitational force Eq.(8) which acts on every particle or body of mass m causing a negative acceleration a P = − c t , assuming a circular orbit, we obtain at Time t the equation: