Derivation of Potentially Important Masses for Physics and Astrophysics by Dimensional Analysis

Email: valev@gbg.bg Abstract: The Hubble constant has been added in addition to the three fundamental constants (speed of light, gravitational constant and Planck constant) used by Max Planck, for derivation of the Planck mass by dimensional analysis. As a result, a general solution is found for the mass dimension expression m = γ m0, where m0 ≡ mp is the Planck mass, γ = 1.23×10 is a small dimensionless quantity and p is an arbitrary parameter in the interval [–1, 1]. The Planck mass mp = 2.17×10 -8 kg, mass of the Hubble sphere MH ~ 10 53 kg, minimum quantum of mass/energy mG = 2.68×10 -69 kg, Weinberg mass mW = 1.08×10 -28 kg, mass of hypothetical quantum gravity atom MG = 3.8×10 12 kg, Eddington mass limit of stars ME = 6.6×10 32 kg and some more masses potentially important for the physics and astrophysics represent particular solutions for values of p, expressed as fractions with small numerators and nominators.


Introduction
The dimensional analysis is a conceptual tool often applied in physics and astrophysics to understand physical situations involving certain physical quantities (Bridgman, 1922;Kurth, 1972;Bhaskar and Nigam, 1990;Petty, 2001). It is routinely used to check the plausibility of the derived equations and computations. When it is known with which other determinative quantities a particular quantity would be connected, but the form of this connection is unknown, a dimensional is composed for its finding. In the left side of the equation is placed the unit of this quantity q 0 with its dimensional exponent and in the right side of the equation is placed the product of units of the determinative quantities q i raised to the unknown exponents n i , where n is a positive integer and the exponents n i are rational numbers. Most often, the dimensional analysis is applied in the mechanics, aerodynamics, astrophysics and other fields of the modern physics, where there are many problems with a few determinative quantities. The Planck mass as defined by Planck (1906) in terms of three fundamental constants, speed of light in vacuum (c), gravitational constant (G) and reduced Planck constant ℏ , is P c m G = ℏ . Since the constants c, G and ℏ represent three very basic aspects of the universe (i.e. the relativistic, gravitational and quantum phenomena), the Planck mass appears to a certain degree a unification of these phenomena. The Planck mass has many important theoretical ramifications in modern physics. One of them is that the energy equivalent of Planck mass GeV appears to be the unification energy of the fundamental interactions (Georgi et al., 1974). Additionally, the Planck mass can be derived approximately by defining it as a mass whose Compton wavelength and Schwartzchild radius are equal (Bergmann, 1992). The Hubble constant H has been added to the set of constants c, G, ℏ and thus a unique mass dimension quantity has been derived for every triad (c, G, H), (c, ℏ , H) and (G, ℏ , H) by dimensional analysis (Valev, 2013 In the present work we seek a mass dimension quantity that represents as a product of rational exponents of the four constantsc, G, ℏ and H.
General Solution of the Problem for Finding of a Mass Dimension Quantity by Means of Fundamental Constants c, G and H By dimensional analysis, we search for a mass dimension quantity m in the form of product of rational exponents n 1 , n 2 , n 3 and n 4 of the constants c, G, ℏ and H: The exponents n 1 , n 2 , n 3 and n 4 are unknown quantities that can be found by matching dimensions on both sides of equation (1) and k is a dimensionless parameter (coefficient) on the order of unity.
Replacing dimensions of m, c, G, ℏ and H in (1) we find the dimensional equation: From Equation (2) we find system of linear equations for unknown quantities n 1 , n 2 , n 3 and n 4 : The rank of augmented matrix of the system 3 r = is equal to the rank of the coefficient matrix. Thus, in accordance with the Rouche-Capelli theorem the system is consistent and so must have at least one solution. The solution is unique if and only if the rank equals the number of variables. In the system (3) the number of variables 4 3 m r = > = , therefore the solution is not unique, but having infinitely many solutions. However, upon introducing the concept of a free parameter p, where in it is accepted that n 4 = p, system (3) can be transform to: The determinant of system (4) is ∆ = 2 ≠ 0 and the system has a solution that is dependent upon a free parameter p. We find the solution of the system (4) by means of Cramer's rule: where, p is a free parameter. Replacing the solution (5) in Equation (1) we find Equation (6) for the mass m: Obviously, the Equation (6) can be transformed in Equation (7): Therefore, we find the general solution (8) Carvalho, 1995;Valev, 2009) and the minimum measurable mass/energy in the (Sivaram, 1982;Alfonso-Faus, 2012). The exceptionally small mass m 3 seems close to the graviton mass obtained by different methods (Woodward et al., 1975;Gershtein et al., 1998;Valev, 2008;Alves et al., 2011).
According to Ockham's razor principle, all other things being equal, the simplest theory is the most likely to be true (Rodriguez-Fernandez, 1999). In science, this principle is used as a heuristic technique (discovery tool) to guide scientists in the development of theoretical models (Gauch, 2003). Therefore, in the following section, we consider particular solutions where the free parameter |p|≤1 represents as a fraction with a small numerator and denominator, i.e.
1 1 1 1 2 0, 1, , , , , 3 2 4 5 3 p = ± ± ± ± ± ± . We will show that some such solutions result in mass formulas that could be interesting for contemporary particle physics and astrophysics. Particular Solutions where the Free Parameter |p|≤1 Represents as Fractions having Small Numerators and Denominators From the general solution (8) we find the Planck mass m p as a particular solution (9) in case of p = 0: As it has been mentioned in Section 2, from Equation (8) Analogously, from the general solution (8) we find the minimum quantum of mass/energy m G as a particular solution (11) at p = 1: The Equation (12) represents the well known Weinberg mass formula (Weinberg, 1972). The physical meaning of the Weinberg mass was found from Sivaram (1982). He shows the Weinberg mass represents the lightest mass whose self-gravitational energy has measurable value for the time of existence of the universe H −1 ≈ 1.38×10 10 years.
From the general solution (8) Obviously, the mass m 6 , obtained from Equation (8) at 1 2 p = is of the order of the neutrino rest mass (Goobar et al., 2006). From (8) This energy is typical for the energy of protons in Large Hadron Collider (LHC) and possibly is connected with mass of yet unobserved heavy particle or fundamental energetic scale.
This mass is close to one of the seven fundamental equidistant masses found in (Forsythe and Valev, 2014), namely the mass The mass m 12 has been identified in (Forsythe and Valev, 2014) with Eddington mass limit of the most massive stars M E = 6.6×10 32 kg.
The above derived masses, whose free parameters are in the range |p|≤1 and appear in the general solution as fractions having small numerators and denominators, are presented in Table 1.
Probably, the general solution (8) includes additional masses interesting from the physical view point, but indefiniteness of the parameter p doesn't allow unambiguous finding of these masses.
Time dependence of some derived masses is natural and clear. For example mass of the Hubble sphere years (Sivaram, 1982). The Planck mass include constants c, G and ℏ and is time independent, but the rest derived masses depend from the expansion. In result, the microscopic masses m 2 , m 3 , m 4 , m 6 , m 8 and m 11 decrease with cosmological expansion, while the macroscopic masses m 1 , m 5 , m 7 , m 9 , m 10 and m 12 and increase with the expansion.

Conclusion
The Hubble constant H has been added to the three fundamental constants (the speed of light in vacuum, Newtonian gravitational constant and reduced Planck constant) used from Max Planck for derivation of Planck mass by dimensional analysis.
We search by dimensional analysis a mass dimension quantity that represents a product of rational exponents of the four constantsc, G, ℏ and H. In result, a general solution has been found of mass dimension quantity is a small dimensionless quantity and p is an arbitrary parameter in the interval [-1, 1]. According to Ockham's razor principle, all other things being equal, the simplest theory is the most likely to be true. Therefore, we consider particular solutions where the free parameter |p|≤1 represents as a fraction with a small numerator and denominator, i.e.
In result, it has been found that the Planck mass , mass of hypothetical quantum gravity atom M G = 3.8×10 12 kg, Eddington mass limit of stars M E = 6.6×10 32 kg and some more masses potentially important for the physics and astrophysics represent particular solutions for values of p, expressed as fractions with small numerators and denominators. Likely, some of unidentified masses could have heuristic meaning for astrophysics and high energy physics.