Exploring Fundamental Constants Through the Lens of Space-Time Quantization

Abstract

This paper presents an in-depth analysis of how fundamental physical constants, specifically Planck's constant ($h$), the product of $h$ and the speed of light ($hc$), and Newton's gravitational constant ($G$), define a specific volume of space-time. It explores how properties such as energy, momentum, and mass are extracted by dividing this quantized volume by the wavelength and/or powers of the speed of light. Additionally, it examines how this quantized volume scales the radius in the gravitational law formula.

Introduction

Fundamental physical constants play a crucial role in our understanding of the universe. Constants such as $h$, $hc$, and $G$ are not merely abstract values but are intrinsically connected through a specific quantized volume of space-time. This paper aims to elucidate how these constants define this quantized volume and how various properties are extracted through unit scaling.

Defining the Quantized Volume

The specific volume of space-time, denoted as s_l =$(1.538439454984191\times 10^{-6}\text{m}{)}^3(1.538439454984191\; \backslash times\; 10^\{-6\}\; \backslash ,\; \backslash text\{m\})^3$, is approximately$3.641172281610566\times 10^{-18}{\text{m}}^{3}3.641172281610566\; \backslash times\; 10^\{-18\}\; \backslash ,\; \backslash text\{m\}^3$. This volume is determined by the interplay of the constants $hh$, $hchc$, and $GG$. These constants share this volume, which serves as a fundamental quantum of space-time.

$$V_{quantum}\approx (1.538439454984191\times 10^{-6}\text{m}{)}^3\approx 3.641172281610566\times 10^{-18}{\text{m}}^{3}V\_\{quantum\}\; \backslash approx\; (1.538439454984191\; \backslash times\; 10^\{-6\}\; \backslash ,\; \backslash text\{m\})^3\; \backslash approx\; 3.641172281610566\; \backslash times\; 10^\{-18\}\; \backslash ,\; \backslash text\{m\}^3$$

Extraction of Properties

Properties such as energy, momentum, and mass are extracted by dividing this quantized volume by the wavelength and/or powers of the speed of light ($cc$).

1. Energy ($EE$):

   • Energy is scaled by $1\mathrm{/}c1/c$, reflecting how much energy fits into this quantum of space-time.

$$E=\frac{hc}{\lambda }=\frac{(s_l^3\times m_P)\; \times \; f}{1s^2\times c}E\; =\; \backslash frac\{hc\}\{\backslash lambda\}\; =\; \backslash frac\{(s\_l^3\; \backslash times\; m\_P)\}\{1s^2\; \backslash times\; c\}$$

2. Momentum ($pp$):

   • Momentum is scaled by $1\mathrm{/}{c}^{2}1/c^2$, representing the spatial frequency in this framework.

$$p=\frac{E}{c}=\frac{hc}{\lambda c}=\frac{(s_l^3\times m_P)\; \times f}{1s^2\times c^2}p\; =\; \backslash frac\{E\}\{c\}\; =\; \backslash frac\{hc\}\{\backslash lambda\; c\}\; =\; \backslash frac\{(s\_l^3\; \backslash times\; m\_P)\}\{1s^2\; \backslash times\; c^2\}$$

3. Mass ($mm$):

   • Mass is scaled by $1\mathrm{/}{c}^{3}1/c^3$, integrating the volume and temporal aspects.

$$m=\frac{p}{c}=\frac{hc}{\lambda c^2}=\frac{(s_l^3\times m_P)\; \times f}{1s^2\times c^3}m\; =\; \backslash frac\{p\}\{c\}\; =\; \backslash frac\{hc\}\{\backslash lambda\; c^2\}\; =\; \backslash frac\{(s\_l^3\; \backslash times\; m\_P)\}\{1s^2\; \backslash times\; c^3\}$$

These relationships show that energy, momentum, and mass are geometric projections of a progression of powers of $1\mathrm{/}c1/c$, all tied to a specific quantum of space-time.

Scaling the Radius and mass in the Gravity Law Formula

The gravitational law formula can be explored by scaling the radius using the quantized volume:

Gravitational Force Formula

$$F = \frac{G m_1 m_2}{r^2}$$

Gravitational Force with Units for $G$

$$F = \frac{\left( \frac{\text{m}^3}{\text{s}^2 \, \text{kg}} \right) m_1 m_2}{r^2}$$

Rearranged Form

$$F = \left( \frac{\text{m}^3}{r^2} \right) \cdot \left( \frac{m_1 m_2}{\text{s}^2 \, \text{kg}} \right)$$

This indicates that the gravitational force is intrinsically linked to the quantized volume defined by $s_{l}s\_\{va\}$, showing how the radius scales within this framework.

Conclusion

This paper demonstrates how fundamental constants $hh$, $hchc$, and $GG$ define a specific quantized volume of space-time, revealing deep connections between energy, momentum, and mass through unit scaling. By understanding these relationships, we gain new insights into the foundational nature of physical laws and the intrinsic properties of the universe. This framework not only provides a unified approach to understanding fundamental constants but also opens new avenues for exploring the quantization of space-time.