J. Rogers, SE Ohio, 15 Jan 2025 1300
Abstract:
This paper introduces a novel interpretation of gravitational wavelength as the geometric representation of spacetime curvature between two masses. By considering the rest mass of all atoms contributing to the spacetime geometry, we propose that the gravitational wavelength encapsulates the total curvature arising from their interaction. This framework unifies microscopic and macroscopic properties, presenting spacetime curvature as an emergent property of rest mass distributed across a region. The implications extend to understanding forces, motion, and the fundamental nature of curved spacetime.
1. Introduction
In classical and relativistic physics, gravity is described as the curvature of spacetime induced by mass and energy. However, this curvature is often treated as an abstract concept without a direct physical representation. Here, we argue that the gravitational wavelength between two masses can serve as a tangible descriptor of the spacetime curvature. This approach provides a unified understanding of how rest mass, energy, and geometry interact.
2. Defining Gravitational Wavelength
We define the gravitational wavelength, G_λ, where:
m_P is the Planck mass,
c is the speed of light,
m1 and m2 are the interacting masses,
r is the distance between them.
This wavelength represents the spatial extent of the curvature created by the interaction of and . Importantly, it depends on the combined rest masses and the separation, encapsulating both microscopic and macroscopic contributions.
This is the steps to convert G into G_λ. We are using our framework that factors h and G with common factors to simplify the equation in the physical quantities and unit scaling that is actually being performed.
λ = h / (Gm1m2 / r² ) * 1s
This is the starting point, substituting the gravitational force (times 1s to make units compatible with momentum) into the de Broglie relation.
λ = (s_l³ * m_P) / (1s² * c) / ((s_l³ / (1s² * m_P)) * m1 * m2 / r²) * 1s We've substituted your definitions of G and h in terms of s_l, m_P, and c. the 1s is to look at the force with respect to a second of time.
λ = (m_P) / (c) / ((1 / (m_P)) * m1 * m2 / r²) * 1s We've canceled out the s_l³ and 1s² terms.
λ = (m_P² * 1s) / (c * (m1 * m2 / r²)) We've simplified the expression by multiplying the numerators and denominators.
λ = (m_P² * r² * 1s) / (c * m1 * m2 )We move the radius squared to the top of the formula.
3. Rest Mass and Spacetime Curvature
Every atom contributes to spacetime curvature through its rest mass. The cumulative effect of these contributions results in the observed gravitational interaction. The gravitational wavelength, therefore, can be interpreted as the total curvature generated by the aggregated rest masses.
4. Wavelength as Curved Spacetime
The gravitational wavelength can be viewed as the geometric representation of curved spacetime. It bridges:
Microscopic Contributions: Individual rest masses of atoms.
Macroscopic Geometry: The overall curvature between two masses.
This duality offers a new perspective on the nature of gravitational interactions, where rest mass and curvature are inherently linked.
5. Implications
5.1. Emergent Nature of Gravity
The gravitational wavelength provides a framework for viewing gravity as an emergent phenomenon. The curvature of spacetime, as described by G_λ, arises from the aggregated rest mass energy density of all atoms in the system.
5.2. Unified Description of Forces
This model ties gravitational interactions directly to rest mass and spacetime curvature, aligning with efforts to unify the fundamental forces under a geometric framework.
5.3. Applications to Large-Scale Systems
In systems like the Earth-Moon or Sun-Earth, the gravitational wavelength is extremely small, reflecting the immense spacetime curvature generated by these large masses over vast distances. This small wavelength underscores the profound impact of curvature on macroscopic phenomena.
6. Connection to Broader Theories
6.1. Motion as Curved Spacetime
This interpretation aligns with the view that all motion arises from spacetime curvature. The gravitational wavelength quantifies the curvature guiding the motion of masses.
6.2. Mass as Spacetime Geometry
Rest mass itself may be understood as a manifestation of curved spacetime. The gravitational wavelength provides a measurable quantity to explore this relationship further.
7. Conclusion and Future Directions
The gravitational wavelength offers a novel way to conceptualize spacetime curvature as a measurable and geometric property of gravitational interactions. By tying it to the rest mass of all atoms contributing to a system, this framework provides insights into the emergent nature of gravity and its connection to fundamental properties of matter and energy.
Future work will explore applications to:
Black holes, where curvature and wavelength reach extreme conditions.
Cosmology, to understand the role of rest mass in large-scale spacetime dynamics.
Quantum gravity, to unify this framework with quantum-scale interactions.
References:
Einstein, A. (1916). The Foundation of the General Theory of Relativity. Annalen der Physik.
Planck, M. (1901). On the Law of Distribution of Energy in the Normal Spectrum. Annalen der Physik.
de Broglie, L. (1924). Recherches sur la théorie des quanta. Annales de Physique.
Modern Physics Texts and Experimental Data on Gravitational Interactions.
Appendix A: code
import math
# Define your constants based on your framework
def calculate_s_l(h, c, G):
"""Calculates s_l based on your formula."""
return (h * c * G)**(1/6)
def calculate_m_P(h, c, G):
"""Calculates m_P based on your formula."""
return (h * c / G)**(1/2)
# Standard physical constants
h = 6.62607015e-34 # Planck's constant (J.s)
c = 299792458 # Speed of light (m/s) - This is an exact value
G = 6.67430e-11 # Gravitational constant (N.m^2/kg^2)
# Calculate s_l and m_P based on your framework
s_l = calculate_s_l(h, c, G)
m_P = calculate_m_P(h, c, G)
def calculate_wavelength(m1, m2, r):
"""Calculates the 'wavelength of gravity' based on your formula."""
wavelength = (m_P**2/(c * 1)) * (r**2 / ( m1 * m2 )) # 1s is just 1
wavelength = 9.927737965030e-24 * (r**2 / ( m1 * m2 )) # 1s is just 1
return wavelength
def calculate_momentum(wavelength):
"""Calculates the momentum using the de Broglie relation."""
momentum = h / wavelength
return momentum
def calculate_force(m1, m2, r):
"""Calculates the gravitational force."""
force = G * m1 * m2 / (r**2)
return force
# Generate examples with different masses and distances
examples = [
{"m1": 1, "m2": 1, "r": 1}, # Example 1: Unit masses and distance
{"m1": 10, "m2": 20, "r": 5}, # Example 2: Larger masses, larger distance
{"m1": 5.972e24, "m2": 7.348e22, "r": 3.844e8}, # Example 3: Earth-Moon system (approximate)
{"m1": 1.989e30, "m2": 5.972e24, "r": 1.496e11}, # Example 4: Sun-Earth system (approximate)
]
# Create a table to display the results
print("----------------------------------------------------------------------------------------------------")
print("| {:^10} | {:^10} | {:^12} | {:^18} | {:^18} | {:^18} |".format("m1 (kg)", "m2 (kg)", "r (m)", "Wavelength (m)", "Momentum (kg.m/s)", "Force (N)"))
print("----------------------------------------------------------------------------------------------------")
for example in examples:
m1 = example["m1"]
m2 = example["m2"]
r = example["r"]
wavelength = calculate_wavelength(m1, m2, r)
momentum = calculate_momentum(wavelength)
force = calculate_force(m1, m2, r)
print("| {:^10.3e} | {:^10.3e} | {:^12.3e} | {:^18.12e} | {:^18.12e} | {:^18.12e} |".format(m1, m2, r, wavelength, momentum, force))
print("----------------------------------------------------------------------------------------------------")
# Print the calculated values of s_l and m_P
print(f"G_wavelength : {calculate_wavelength(1, 1, 1):.16e} m/kg^2")
print(f"Calculated s_l: {s_l:.12e} m")
print(f"Calculated m_P: {m_P:.12e} kg")
# Verify the relationship h * c / m_P**2 = G
print(f"G: h * c / m_P**2: {h * c / m_P**2:.12e}")
print(f"G: s_l^3 / m_P: {s_l**3/m_P:.12e}")
print(f"G: actual {G:.12e}")
Appendix B: Results
----------------------------------------------------------------------------------------------------
| m1 (kg) | m2 (kg) | r (m) | Wavelength (m) | Momentum (kg.m/s) | Force (N) |
----------------------------------------------------------------------------------------------------
| 1.000e+00 | 1.000e+00 | 1.000e+00 | 9.927737965030e-24 | 6.674300000000e-11 | 6.674300000000e-11 |
| 1.000e+01 | 2.000e+01 | 5.000e+00 | 1.240967245629e-24 | 5.339440000000e-10 | 5.339440000000e-10 |
| 5.972e+24 | 7.348e+22 | 3.844e+08 | 3.342936422668e-54 | 1.982110729079e+20 | 1.982110729079e+20 |
| 1.989e+30 | 5.972e+24 | 1.496e+11 | 1.870505160293e-56 | 3.542396081369e+22 | 3.542396081368e+22 |
----------------------------------------------------------------------------------------------------
G_wavelength : 9.9277379650300003e-24 m/kg^2
Calculated s_l: 1.538439454984e-06 m
Calculated m_P: 5.455511861335e-08 kg
G: h * c / m_P**2: 6.674300000000e-11
G: s_l^3 / m_P: 6.674300000000e-11
G: actual 6.674300000000e-11
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