🌌 Quantum Gravity in Marias Theory

Explanation of the formula m = (A × h × f) / c_local² in the Marias theory context

The formula relates mass m to fundamental constants and variables:

• h — Planck’s constant, representing the quantum of action
• f — frequency of the photon or oscillation considered
• c_local — the local speed of light, which can vary depending on the physical environment
• A — a dimensionless or context-dependent proportionality constant specific to the theory

1. Starting point: Energy of a photon

From quantum mechanics, the energy E of a photon is given by the Planck-Einstein relation:

\( E = h f \)

2. Energy-mass equivalence

Einstein’s mass-energy equivalence states:

\( E = m c_{local}^2 \)

Rearranged, the mass equivalent of energy is:

\( m = \frac{E}{c_{local}^2} \)

3. Combining the two

Substituting E = h f into the mass-energy equivalence:

\( m = \frac{h f}{c_{local}^2} \)

This gives a basic estimate of the “mass equivalent” of a photon with frequency f using the local speed of light.

4. Introducing the constant A

In the Marias theory, the factor A is introduced to refine or generalize this relation, possibly accounting for:

• Interaction effects,
• The photon’s behavior under specific physical conditions,
• Geometrical or holographic considerations,
• Or the new unified force framework proposed.

Thus, the formula becomes:

\( m = \frac{A \times h \times f}{c_{local}^2} \)

5. Possible derivation or justification for A

A could be derived from the specifics of light’s interaction with gravitational fields, such as in the 1919 solar eclipse experiment.

It may incorporate correction terms for photon mass or effective mass arising from the proposed unified force model.

Alternatively, A might encapsulate dimensional constants or scaling factors derived from observational data.

6. Extended example with force analogy

Given that force F might be related to energy gradients, one can hypothesize:

\( F = \frac{dE}{dx} \approx \frac{d}{dx}(h f) = h \frac{df}{dx} \)

where spatial variation in frequency or energy relates to force fields in the theory.

Connecting this back to mass via acceleration a:

\( F = m a \implies m = \frac{F}{a} \)

With suitable theoretical substitutions, this could tie back into the formula for m.

Summary

The formula

\( m = \frac{A \times h \times f}{c_{local}^2} \)

extends the classic photon mass-energy relation by adding a theory-specific constant A and by recognizing that the speed of light c is a local constant c_local. This local speed of light can vary depending on physical conditions and environment, providing a more nuanced and potentially accurate description of mass-energy relations in different regions of the universe.

Let's provide a more mathematical demonstration of the formula:

\[ F(t) = \frac{B_1 \cdot B_2 \cdot \cos(\omega_1 t + \phi_1) \cdot \cos(\omega_2 t + \phi_2)}{d^2} \]

Step 1: Use trigonometric product-to-sum identity:

\[ \cos A \cdot \cos B = \frac{1}{2} \left[\cos(A + B) + \cos(A - B)\right] \]

Applying this to our expression:

\[ F(t) = \frac{B_1 B_2}{2 d^2} \left[ \cos\big((\omega_1 + \omega_2) t + \phi_1 + \phi_2\big) + \cos\big((\omega_1 - \omega_2) t + \phi_1 - \phi_2\big) \right] \]

Step 2: Interpret \( F(t) \) as a force arising from the interaction of two oscillatory fields.

We consider \( F(t) \) to be the resultant force at time \( t \) due to two sources producing oscillations with angular frequencies \( \omega_1 \) and \( \omega_2 \).

Step 3: Demonstrate time-averaged force over one period \( T \)

The period corresponding to the difference frequency is:

\[ T = \frac{2\pi}{|\omega_1 - \omega_2|} \]

The time-averaged force \( \langle F \rangle \) over one period \( T \) is:

\[ \langle F \rangle = \frac{1}{T} \int_0^T F(t) \, dt \]

Substituting the expanded form of \( F(t) \):

\[ \langle F \rangle = \frac{B_1 B_2}{2 d^2 T} \int_0^T \left[ \cos\big((\omega_1 + \omega_2) t + \phi_1 + \phi_2\big) + \cos\big((\omega_1 - \omega_2) t + \phi_1 - \phi_2\big) \right] dt \]

Because the integral of cosine over its full period is zero:

\[ \int_0^T \cos(k t + \theta) dt = 0 \quad \text{for any integer } k \neq 0 \]

Thus, the first term with frequency \( \omega_1 + \omega_2 \) averages to zero over \( T \). For the second term:

\[ \int_0^T \cos\big((\omega_1 - \omega_2) t + \Delta \phi\big) dt = \left[ \frac{\sin\big((\omega_1 - \omega_2) t + \Delta \phi\big)}{\omega_1 - \omega_2} \right]_0^T = 0 \]

because \( \sin(\theta + 2\pi) = \sin(\theta) \).

Step 4: Implication

The strict time average of \( F(t) \) is zero, which means the force oscillates around zero. However, the instantaneous force \( F(t) \) fluctuates with beat frequencies and can produce effective forces when interacting with nonlinear systems or when integrated over shorter time scales.

Step 5: Relation to physical scenarios

Such formulas are used to describe interference forces, radiation pressure modulation, or forces mediated by coupled oscillatory fields. The \( 1/d^2 \) factor aligns with forces following inverse-square laws, like gravitational or electrostatic forces.