Nonstandard physics/Planck mass

In physics, the Planck mass (mP) is the unit of mass in the system of natural units known as Planck units. It is defined so that

$m_P = \sqrt{\frac{\hbar c}{G}} = 1.2209\cdot 10^{19}GeV/c^2 = 2.17644\cdot 10^{-8}kg \$

where c is the speed of light in a vacuum, G is the gravitational constant, and h is the reduced Planck constant.

Particle physicists and cosmologists often use the reduced Planck mass, which is

$\sqrt\frac{\hbar{}c}{8\pi G}$ = 4.340 µg = 2.43 * 1018 GeV/c2.

The added factor of $1/\sqrt{8\pi}$ simplifies a number of equations in general relativity.

The name honors Max Planck, who was the first to propose it.

Derivation

Units dimension approach

Standard units dimension approach is based on the dimensions of the three fundamental constants:

$[c] = LT^{-1} \$
$[G] = M^{-1}L^3T^{-2} \$
$[\hbar] = M^1L^2T^{-1} \$.

So, the Planck mass could be derived from the following dimension relationship:

$[m_P]^1 = [c]^{n1}\cdot [G]^{n2}\cdot [\hbar]^{n3} \$

or in the detailed form:

$M^1 = (LT^{-1})^{n1}\cdot (M^{-1}L^3T^{-2})^{n2}\cdot (M^1L^2T^{-1})^{n3}, \$

which is equivalent to the three algebraic equations:

$-n_2 + n_3 = 1 \$
$n_1 + 3n_2 + 2n_3 = 0 \$
$n_1 + 2n_2 + n_3 = 0 \$.

The solution of this system is:

$n_1 = 1/2, n_2 = -1/2, n_3 = 1/2. \$

Thus, the resulting Planck mass will be:

$m_P = c^{1/2}G^{-1/2}\hbar^{1/2} = \sqrt{\frac{c\hbar}{G}}.$

Natural force approach

Coulomb law (or electric charge force) is:

$F_E = \frac{1}{4\pi \epsilon_E}\cdot \frac{e^2}{r^2} = \alpha_E\cdot \frac{\hbar c}{r^2}, \$

where $\alpha_E = \frac{e^2}{2\epsilon_E hc}$ is the electric fine structure constant for electric charge quantum - $e \$(electron charge), and $\epsilon_E \$ is the dielectric constant. Newton law (or gravitatational mass/charge force) is:

$F_G = \frac{1}{4\pi \epsilon_G}\cdot \frac{m_{\alpha}^2}{r^2} = \alpha_G\cdot \frac{\hbar c}{r^2}, \$

where $\alpha_G = \frac{m_{\alpha}^2}{2\epsilon_G hc}$ is the gravitational fine structure constant for the gravitational mass quantum $m_{\alpha} \$(will be defined later), and $\epsilon_G = \frac{1}{4\pi G} \$ is the gravitational "dielectric" constant [1] [2]. In the case of equality of the above forces, we shall get the equality of the fine structure constants for fields with gradient potentials:

$\alpha_E = \alpha_G, \$

from which the gravitational mass quantum could be derived:

$m_{\alpha} = e\sqrt{\frac{\epsilon_G}{\epsilon_E}} = \frac{e}{\sqrt{4\pi G\epsilon_E}} = \sqrt{\alpha_E}\cdot m_P, \$

where $m_P \$ is the Planck mass. George Johnstone Stoney (1881) first proposed this mass scale, now called the Stoney scale, before quantum theory was created.

Rotor potential fields.

Coulomb law for magnetic charges (or magnetic flux) is:

$F_M = \frac{1}{4\pi \mu_E}\cdot \frac{\phi_E^2}{r^2} = \beta_E\cdot \frac{\hbar c}{r^2}, \$

where $\beta_E = \frac{\phi_E^2}{2\mu_E hc}$ is the magnetic fine structure constant (first proposed by Yakymakha in 1989 [3]) for magnetic charge quantum - $\phi_E = \frac{h}{e} \$ (magnetic flux quantum), and $\mu_E \$ is the magnetic constant. Newton law for "magnetic like" gravitational mass/charge is:

$F_{GM} = \frac{1}{4\pi \mu_G}\cdot \frac{\phi_{G\alpha}^2}{r^2} = \beta_G\cdot \frac{\hbar c}{r^2}, \$

where $\beta_G = \frac{\phi_{G\alpha}^2}{2\mu_G hc}$ is the gravitational fine structure constant for the "magnetic like" gravitational mass/charge quantum $\phi_{G\alpha} \$(will be defined later), and $\mu_G = \frac{4\pi G}{c^2} \$ is the gravitational "magnetic-like" constant [1]. In the case of equality of the above forces, we shall get the equality of the fine structure constants for fields with rotor potentials:

$\beta_E = \beta_G = \frac{1}{4\alpha}, \$

from which the gravitational rotor mass quantum could be derived:

$\phi_{G\alpha} = \phi_E\cdot \frac{\mu_G}{\mu_E} = \frac{h}{m_{\alpha}}\cdot \sqrt{4\beta_{E}\alpha_E} = \frac{h}{m_{\alpha}}. \$

Quantum mechanical derivation

Gravitational inductance quantum mechanical operators (nonrelativistic case):

$\hat p_{GL} = -i\hbar \frac{d}{dq_G}, \quad \quad \quad \quad \quad \hat p_{GL}^* = i\hbar \frac{d}{dq_G},\quad \quad \quad \quad \quad (1) \$

Hamilton operator for gravitational LC circuit:

$\hat H_{GLq} = -\frac{\hbar^2}{2L_G}\cdot \frac{d^2}{dq_G^2} + \frac{L_G\omega_0^2}{2}\hat q_G^2 \quad \quad \quad \quad \quad (2) \$

where $L_G \$ and $C_G \$ are the quantum gravitational inductance and capacitance respectively.

$-\frac{\hbar^2}{2L_G}\frac{d^2 \Psi}{dq_G^2} + \frac{L_G\omega_0^2}{2}q_G^2\Psi = W\Psi \quad \quad \quad \quad \quad (3) \$

To solve this equation we need to devine the following dimensionless variables and scales:

$\xi_q = \frac{q_G}{q_{G0}}; \quad \quad q_{G0} = \sqrt{\frac{\hbar}{L_G\omega_0}}; \quad \quad \lambda_q = \frac{2W}{\hbar\omega_0} \quad \quad (4) \$

Then we could rewrite equation (3) in the form of the differential equation of Chebyshev-Ermidt:

$(\frac{d^2}{d\xi^2} + \lambda - \xi^2)\Psi = 0. \quad \quad (5) \$

The eigen values of the Hamiltonian will be:

$W_n = \hbar \omega_0(n + 1/2), \quad \quad n = 0,1,2,.. \$

where $\omega_0 = \frac{1}{\sqrt{L_GC_G}} \$ is resonance frequency. At $n = 0 \$ we shall have zero oscillation:

$W_0 = \hbar \omega_0/2. \$

We can consider the scaling gravitational charge in the detailed form:

$q_{G0} = \sqrt{\frac{\hbar}{L_G\omega_0}} = \sqrt{\frac{\hbar}{\rho_G}} \quad \quad (6) \$

However, the gravitational characteristic impedance of free space has the following value:

$\rho_G = \sqrt{\frac{L_G}{C_G}} = \sqrt{\frac{\mu_G}{\epsilon_G}} = 2\alpha \cdot \frac{h}{m_{\alpha}} \$

Therefore, the induced "gravitational charge" will be:

$q_{G0} = \frac{m_{\alpha}}{\sqrt{4\pi \alpha }} = \frac{m_P}{\sqrt{4\pi}} \quad \quad (7) \$

So, the Planck mass is naturally derived from the Shrodinger equation and correlated with Stoney scale.

Applications

Gravitational characteristic impedance

Stratton (1941) [4] first proposed the electrodynamic characteristic impedance of vacuum in the standard form:

$\rho_{E0} = \sqrt{\frac{\mu_E}{\epsilon_E}} = 2\alpha_ER_{EH}, \$

where $R_{EH} = \frac{h}{e^2} \$ is the electrodynamics von Klitzing constant. By analogy, the gravidynamics characteristic impedance could be proposed (see [2]):

$\rho_{G0} = \sqrt{\frac{\mu_G}{\epsilon_G}} = 2\alpha_GR_{GH} = 2.7967\cdot 10^{-18} m^2s^{-1}kg^{-1}, \$

where $R_{GH} = \frac{h}{m_{\alpha}^2} \$ is the gravidynamics von Klitzing constant. For the sake of comparison some realistic exaples may be presented. Actually, spherical megascopic bodies have the folloving characteristic impedance:

$\rho_{GS} = \frac{v_cr_s}{3m_s}, \$

where $v_c \$ is the equator velocity, $r_s \$ spherical body radius and $m_s \$ body mass.

Venus has the following parameters: $v_c = 1.807 m/s \$ ; $r_s = 6.07\cdot 10^6 m \$ ; $m_s = 4.87\cdot 10^{24} kg \$. So, its characteristic gravitational impedance is $\rho_{GVenus} = 7.508\cdot 10^{-19} m^2s^{-1}kg^{-1} \$, which is about 0.268 of the vacuum impedance value.

Solar has the following parameters: $v_c = 317.4 m/s \$ ; $r_s = 6.96\cdot 10^8 m \$ ; $m_s = 1.989\cdot 10^{30} kg \$. So, its characteristic gravitational impedance is $\rho_{GSolar} = 3.702\cdot 10^{-20} m^2s^{-1}kg^{-1} \$, which is about 0.0132 of the vacuum impedance value.

Mercury has the following parameters: $v_c = 2.99 m/s \$ ; $r_s = 2.425\cdot 10^6 m \$ ; $m_s = 3.311\cdot 10^{23} kg \$. So, its characteristic gravitational impedance is $\rho_{GMercury} = 7.300\cdot 10^{-18} m^2s^{-1}kg^{-1} \$, which is about 2.61 times more than the vacuum impedance value.

Black hole

Gravitational radius of black hole is:

$r_c = \frac{2GM}{c^2}, \$

where $M \$ is the gravitational mass. Considering that rotating velocity is:

$v_c = \frac{c}{\sqrt{2}} \$

we can define the gravitational impedance in the form:

$\rho_{Gc} = \frac{v_cr_c}{M} = \sqrt{2}\frac{G}{c} = \frac{\alpha_G}{\pi \sqrt{2}}R_{GH}. \$

Note that, characteristic impedance relationships

$\frac{\rho_{G0}}{\rho_{Gc}} = 2\pi \sqrt{2} \$

is also fundamental constant.

Gravitational particles

Gravitational bosons

The wavelength of the graviton is:

$\lambda_{Gp} = \frac{h}{cm_P}. \$

$r_{GP} = \frac{\lambda_{Gp}}{2\pi}. \$

The graviton surface:

$S_{GP} = 4\pi r_{GP}^2 = \frac{\lambda_{Gp}^2}{\pi}. \$

The gravitational capacitance:

$C_{GP} = \frac{\epsilon_G}{2\lambda_{GP}}S_{GP}. \$

The gravitational inductance:

$L_{GP} = \frac{\mu_G}{2\lambda_{GP}}S_{GP}. \$

The gravitational characteristic impedance of graviton resonator:

$\rho_{GP} = \sqrt{\frac{L_{GP}}{C_{GP}}} = \frac{4\pi G}{c} = 2\alpha_G R_{G\alpha}.$

The graviton resonator resonance frequency:

$\omega_{GP} = \frac{1}{\sqrt{L_{GP}C_{GP}}} = \frac{m_Pc^2}{\hbar}.$

Note that angular momentum of the particle will be:

$l_{GP} = m_{GP}\omega_{GP}r_{GP}^2 = \hbar \$

i.e - digital, as should be for boson particles.

Gravitational fermions

The wavelength of the gravitational fermion is:

$\lambda_{G\alpha} = \frac{h}{cm_{\alpha}}. \$

$r_{G\alpha} = \frac{\lambda_{G\alpha}}{2\pi \sqrt{2}}. \$

The gravitational fermion surface:

$S_{G\alpha} = 4\pi r_{G\alpha}^2 = \frac{\lambda_{G\alpha}^2}{2\pi}. \$

The gravitational capacitance:

$C_{G\alpha} = \frac{\epsilon_G}{\lambda_{G\alpha}}S_{G\alpha}. \$

The gravitational inductance:

$L_{G\alpha} = \frac{\mu_G}{\lambda_{G\alpha}}S_{G\alpha}. \$

The gravitational characteristic impedance of gravitational fermion resonator:

$\rho_{G\alpha} = \sqrt{\frac{L_{G\alpha}}{C_{G\alpha}}} = \frac{4\pi G}{c} = 2\alpha_G R_{G\alpha}.$

The gravitational fermion resonator resonance frequency:

$\omega_{G\alpha} = \frac{1}{\sqrt{L_{G\alpha}C_{G\alpha}}} = \frac{m_{\alpha}c^2}{\hbar}.$

Note that angular momentum of the particle will be:

$l_{G\alpha} = m_{\alpha}\omega_{G\alpha}r_{G\alpha}^2 = \frac{\hbar}{2} \$

i.e - fractional, as should be for fermion particles.

Significance

Unlike all other Planck base units and most Planck derived units, the Planck mass is a macroscopic amount, having a scale more or less conceivable to humans. For example, the body mass of a flea is roughly 4000 to 5000 mP.

The Planck mass has a Schwarzschild radius equal to the Compton wavelength divided by ?. The Planck mass is also the mass of the Planck particle, a hypothetical minuscule black hole whose Schwarzschild radius equals the Planck length.

The Planck mass is an idealized mass thought to have special significance for quantum gravity when general relativity and the fundamentals of quantum physics become mutually important to describe mechanics.