Einstein Probabilistic Units/Convergence

CONVERGENCE

$Classical\;\;\between \;\;Quantum$

In this essay Einstein’s General Theory of Relativity and the Quantum Theory of Radiation are converged providing new models and insights into the nature of gravity.

As the Cosmic Microwave Background spectrum is strongly correlated to Planck’s law for black body radiation this is a natural point of convergence for general relativity and quantum radiation.

With these new models this essay will examine the cosmological constant problem and gravitational hierarchy problem. Though not intended to be a theory of quantum gravity, next steps toward quantization of gravity will be presented.

During the years 1916-1917 Albert Einstein published his seminal papers on the General Theory of Relativity and the Quantum Theory of Radiation [1, 2, 3, 4]. The general theory of relativity describes how matter behaves in space-time while the quantum theory of radiation describes how matter behaves in a radiation field. In this essay these two theories are joined providing new insights into the nature and phenomenon of gravity.

In the quantum theory of radiation Einstein developed an alternative approach to Planck’s law of black body radiation. Einstein’s quantum theory of radiation introduced two coefficients that are a measure of the probability of absorption or emission of radiation. The A coefficient is the rate of spontaneous emission, and the B coefficient is the rate of stimulated absorption or emission. In order to generalize the principals of quantum radiation these coefficients can be arranged into a system of probabilistic units.

For example using these units; force is represented by c2/B, energy is represented by c3/AB, and energy density is represented by A2/B.

General relativity and quantum radiation relationship

The energy density of the cosmic microwave background is strongly correlated to Planck’s law for black body radiation. The cosmic microwave background radiation is expressed below where u is energy density, ħ is Planck’s reduced constant, c is the speed of light, T is temperature, k is Boltzmann constant and υ is frequency.

Planck’s formulations,

$u_{\nu }(\nu )\;d\nu =16\;\pi ^{2}{\frac {\hbar \;\nu ^{3}}{c^{3}}}{\frac {1}{e^{(\hbar \nu /kT)}-1}}d\nu$

Einstein’s A B coefficient formulation,

$u_{\nu }(\nu )\;d\nu =8\;\pi \;{\frac {A}{B}}{\frac {1}{e^{(\hbar \nu /kT)}-1}}d\nu$

The above cosmic microwave background relationships suggests energy density is a natural point of convergence for general relativity and quantum radiation theories.

In general relativity, a form of energy density is referred to as the vacuum energy and is derived using two constants Lambda Λ (Einstein’s cosmological constant) and Kappa k (Einstein’s gravitational constant). In quantum radiation the energy density of a quantum system can be derived using the two Einstein’s coefficients A and B. The vacuum energy density uΛ relationship is.

$u_{\Lambda }={\frac {\Lambda }{\kappa _{0}}}={\frac {A_{\Lambda }^{2}}{B_{P}}}$

Einstein’s Gravitational Constant and Einstein’s B coefficient

At the Planck scales Kappa and Einstein’s B coefficient are equal and equivalent to the inverse Planck linear mass density lP/mP. These relationships are expressed below.

$\kappa _{0}=B_{P}={\frac {l_{P}}{m_{P}}}={\frac {G}{c^{2}}}$

The Planck linear mass density relationship holds from the Planck scale to the scale of the universe. This is evident in Einstein’s “Cosmological consideration on the general theory of relativity” [4]. For any region of the universe the following holds.

${\frac {1}{\kappa _{0}}}={\frac {m_{P}}{l_{P}}}={\frac {1}{4\;\pi ^{2}}}{\frac {(M)\;mass\;universe}{(R)\;radius\;universe}}$

Scaling Ratios

Scaling ratios play critical roles in both general relativity and quantum theories.

The ratio of Kappa and Einstein’s B coefficient ΘZ is a dimensionless ratio that plays a significant role in various aspects of general relativity (Schwarzschild radius, gravitational lensing, gravitational waves, and cosmic scale factor …). Similarly in theory of radiation scaling ratio ΘZ also plays a critical role.

$SR_{Z}={\frac {\kappa _{0}}{B_{Z}}}={\frac {G}{c^{2}}}\times {\frac {M_{Z}}{R_{Z}}}={\frac {l_{p}}{m_{P}}}\times {\frac {M_{Z}}{R_{Z}}}={\frac {\nu _{Z}^{2}}{\nu _{P}^{2}}}$

A useful form of the above relationship is.

$B_{Z}=\kappa \;{\frac {\nu _{P}^{2}}{\nu _{Z}^{2}}}$

Cosmological Constant Lambda and Einstein’s A Coefficient Using Einstein’s “Cosmological consideration on the general theory of relativity” [4] we can equate Lambda and Einstein’s A Coefficient.

$\Lambda ={\frac {A_{\Lambda }^{2}}{c^{2}}}={\frac {1}{((R)\;radius\;universe)^{2}}}$

Lambda and Einstein’s A Coefficient can also be expressed in units of time and frequency. In terms of Einstein’s A Coefficient this would be the effective energy density frequency required to flatten space-time. This frequency would also be directly related to the quantum space-time field created by matter and energy in the universe.

Insights into the cosmological problem

In physics the cosmological constant problem is the disagreement between general relativities’ vacuum energy density and the quantum zero point energy density by some 120 orders of magnitude.

The vacuum density uΛ as defined by Einstein [4]

$u_{\Lambda }={\frac {\Lambda }{\kappa _{0}}}={\frac {A_{\Lambda }^{2}}{B_{P}}}$

The Schwarzschild density represents the limiting energy density of universe and is equivalent to the vacuum energy density.

$u_{s}={\frac {3\;c^{2}}{32\;\pi \;G^{3}\;M_{S}^{2}}}={\frac {3\;c^{2}}{8\;\pi \;G\;r_{S}^{2}}}={\frac {A_{\Lambda }^{2}}{B_{P}}}$

There are many derivations for the zero-point energy density u0, of which many are representative of Planck energy density uP. Unlike the vacuum energy density the zero point energy density needs to be virtually free of curvature and any spontaneous emission so as not to introduce curvature. Such a formulation is given below.

$u_{0}=u_{P}={\frac {c^{7}}{\hbar \;G^{2}}}={\frac {c^{3}/\hbar }{B_{P}^{2}}}$

Account for the universes accelerated expansion.

$u_{0}={\frac {Acceleration\;per\;per\;unit\;mass}{\kappa _{0}^{2}}}$

Some insight into the cosmological problem;

From the above it is evident that the zero point energy density and the vacuum energy density are not one and the same. The role of the vacuum energy density is to flatten space-time curvature due to “ponderable matter” and is associated with Einstein’s A Coefficient The zero point energy density is a specific form of the Planck energy density and is a quantum field associated with Einstein’s B Coefficient

The vacuum energy density is coupled to the zero point energy density via k0 or BP constants at the Planck scale. As Kappa, Lambda and Einstein’s B coefficient are constants the difference in AP and AΛ accounts for scale difference in these energy density’s.