Proper Time as the Foundation of Motion and Its Propagation as Spacetime Curvature: A Speculative Framework

 Abstract:

This paper explores a radical reinterpretation of motion and gravity, proposing that fundamental motion is intrinsically linked to changes in proper time experienced by particles. We hypothesize that the proper time of a particle is not merely a measure along its worldline, but a dynamic entity that propagates outwards, and that spacetime curvature arises as a cumulative effect of this propagating proper time from all matter. This framework aims to provide a potential mechanism for the profound interdependence captured by Wheeler's famous dictum: "Spacetime tells matter how to move; matter tells spacetime how to curve." Within this perspective, motion is redefined as a manifestation of differential proper time, and gravity emerges not as a force, but as the geometric consequence of proper time propagation, offering a potentially unified perspective on mass, gravity, motion, and the nature of time itself.

1. Introduction: Re-examining the Nature of Motion and Wheeler's Dictum

The standard framework of physics, while remarkably successful in describing and predicting motion, may lack a deeply satisfying explanation of what motion fundamentally is. Classical mechanics, special relativity, and general relativity provide precise mathematical tools to calculate trajectories and understand the influence of forces and gravity. However, the underlying essence of motion itself, particularly in the context of spacetime, remains a subject of ongoing inquiry. John Archibald Wheeler famously summarized the intimate relationship between spacetime and matter with the phrase: "Spacetime tells matter how to move; matter tells spacetime how to curve."

This paper proposes a speculative framework that challenges conventional understandings by suggesting that motion is fundamentally a change in proper time. We hypothesize that this change in proper time is not merely a consequence of motion but is, in fact, the very definition of motion in a relativistic spacetime. Furthermore, we explore the radical idea that the proper time of a particle is not a static property but a dynamic entity that propagates outwards, and that the cumulative effect of this propagating proper time is what we perceive as mass, spacetime curvature, and gravity. We aim to explore if the concept of "propagating proper time" can provide a deeper, more fundamental basis for understanding Wheeler's profound interdependence.

2. Motion as Change in Proper Time: A Redefinition

We propose a redefinition of motion, moving beyond the classical notion of spatial displacement over time. Instead, we suggest that motion, at its most fundamental level, is a manifestation of a change in proper time experienced by a particle relative to a chosen frame of reference.

Consider setting an object in motion. In our framework, this action is interpreted as fundamentally altering the object's proper time. The application of a force, such as when firing an arrow from a bow, is not merely imparting momentum or kinetic energy in a spatial sense, but is primarily changing the arrow's worldline in spacetime in such a way that its proper time, as measured by a stationary observer, is altered. The observed "motion" – the spatial displacement – is a consequence of this fundamental change in proper time.

This perspective implies a 1:1 equivalence: the degree of motion is directly reflected in the magnitude of the change in proper time. Inertia, then, becomes the tendency of a particle to maintain its current state of proper time flow until acted upon by an external influence that alters this state.

3. Propagating Proper Time and Spacetime Curvature: Gravity Emerges

We extend this concept to gravity by proposing that the proper time of a particle is not confined to its worldline, but actively propagates outwards into spacetime. We hypothesize that mass-energy acts as a source for this proper time propagation.

Imagine each particle as a source emitting or "propagating" its intrinsic time experience into the surrounding spacetime. As these "propagating proper times" from all matter in the universe interact and superpose, they create a cumulative effect. We propose that spacetime curvature is precisely this cumulative manifestation of propagating proper time.

In this view, curved spacetime is not merely an abstract geometric arena, but a tangible consequence of the dynamic propagation of time experience from all particles. Gravity, therefore, emerges not as a separate force, but as the geometric effect of this spacetime curvature, guiding the motion of particles along geodesics that are shaped by the collective "stacking" of these propagating proper times.

4. A Unified View of Mass, Gravity, Motion, and Time: Realizing Wheeler's Vision

This speculative framework offers a potentially unified view of several fundamental concepts, providing a possible realization of Wheeler's dictum:

• Mass as Source of Proper Time Propagation: "Matter tells spacetime how to curve." Mass-energy becomes the fundamental source of proper time propagation, linking mass directly to the dynamic nature of time. Mass, in this view, is not just an inert quantity, but an active source that shapes spacetime through its influence on proper time.

• Gravity as Spacetime Curvature Arising from Proper Time Propagation: "Spacetime tells matter how to move." Gravity is reinterpreted as the geometric consequence of the cumulative effect of propagating proper time, eliminating the need for a separate gravitational force. Spacetime curvature, in turn, dictates the geodesics along which particles move, thus "telling matter how to move."

• Motion as Change in Proper Time: Motion is fundamentally redefined as a change in proper time, establishing a direct and intrinsic link between motion and time itself, consistent with the relativistic understanding of spacetime.

• Time as Active and Foundational: Time, in the form of propagating proper time, becomes not just a passive dimension, but an active and foundational element of reality, from which spacetime and gravity emerge, providing a deeper basis for the interdependence described by Wheeler.

5. Challenges and Speculations

This framework, being highly speculative, faces numerous challenges and raises many open questions:

• Lack of Mathematical Formalism: A major challenge is the absence of a rigorous mathematical formulation for "propagating proper time" and its relation to spacetime curvature. Developing such a formalism would be a monumental task.

• Departure from Established Physics: This idea departs significantly from established paradigms in physics, requiring a fundamental rethinking of core concepts.

• Empirical Verification: Direct empirical verification of these ideas in their current conceptual form is likely not immediately feasible. The framework would need to make testable predictions to be considered scientifically viable in the long term.

• Nature of "Propagation": The mechanism and nature of "proper time propagation" need to be clarified. What is the medium of propagation? How does it interact and superpose?

6. Conclusion

This paper has presented a speculative framework that reimagines motion as fundamentally linked to changes in proper time, and proposes that spacetime curvature and gravity emerge from the dynamic propagation of proper time from all matter. This framework attempts to offer a potential underlying mechanism for the profound interdependence between spacetime and matter articulated by Wheeler's quote. While highly conceptual and facing significant challenges, this perspective offers a potentially unified and novel way to understand the deep interconnectedness of mass, gravity, motion, and the very nature of time. Further exploration of these radical ideas, even if only at a conceptual level, may offer new avenues for thinking about the fundamental structure of reality and the ongoing quest for a deeper understanding of the universe.

Unifying Mass Equivalence: Revealing the Hidden Conversions in Fundamental Constants

 J. Rogers, SE Ohio, 27 Feb 2025, 1200

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Abstract

In the standard framework of physics, fundamental constants such as Planck’s constant (h), Boltzmann’s constant (k), and the vacuum permittivity (ε₀) are presented as empirically determined numbers with units that facilitate dimensional consistency. However, this treatment offers little insight into the underlying reasons for their specific values or their interconnections. In this paper, we propose a framework that reveals a hidden structure within these constants by reinterpreting them as conversion factors directly linking energy to mass via Einstein’s mass–energy equivalence (E = mc²). We demonstrate how frequency, temperature, and charge—properties traditionally viewed as distinct—each possess an inherent mass equivalence when their associated constants are decomposed to expose a c² factor. This unification not only demystifies the origin and value of these constants but also provides a deeper, mechanistic understanding of their role in the fabric of spacetime.

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1. Introduction

Fundamental constants form the bedrock of our physical theories, yet their treatment in standard physics is often limited to “plug-and-chug” parameters derived from experimental measurement. For example, Planck’s constant (h) is introduced to relate the energy of a photon to its frequency, Boltzmann’s constant (k) converts temperature to energy, and vacuum permittivity (ε₀) appears in Coulomb’s law and electromagnetic energy density. In each case, the constants are presented as axiomatic givens with little attention paid to the “why” behind their values.

This paper challenges the traditional view by proposing that these constants are not arbitrary numbers but rather encode a deeper equivalence between mass and various physical properties. Through the lens of E = mc², we show that:

• Frequency is intrinsically tied to mass via Planck’s constant.
• Temperature carries an inherent mass equivalence mediated by Boltzmann’s constant.
• Charge and electric field energy reveal a mass density equivalent through vacuum permittivity.

By reinterpreting these constants as conversion factors, we begin to answer longstanding questions about their origin and demonstrate a hidden unity among seemingly disparate areas of physics.

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2. Mass Equivalence of Frequency: The Case of Planck’s Constant

2.1 Standard Formulation

Traditionally, Planck’s constant is introduced through the relation

$$E = hf,$$

where $E$ is the energy of a photon and $f$ is its frequency. The experimental value of $h \approx 6.626 \times 10^{-34} \, \text{J·s}$ is taken as a given.

2.2 Revealing the Hidden Mass Conversion

Rewriting the photon energy equation via mass–energy equivalence, we have:

$$E = mc^2.$$

Equating the two expressions for energy yields:

$$hf = mc^2 \quad \Longrightarrow \quad m = \frac{h}{c^2} f.$$

Defining a new conversion constant,

$$Q_m = \frac{h}{c^2},$$

we see that $Q_m$ (with units kg·s) is the mass equivalent per unit frequency. In a natural unit system where $c = 1$, if we define the mass of a 1 Hz photon as 1 kg, then $h$ is normalized to 1. This formulation offers a concrete interpretation: the numerical value of $h$ is determined solely by our definitions of mass and length, revealing an intrinsic 1:1 relation between mass and frequency.

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3. Mass Equivalence of Temperature: Reinterpreting Boltzmann’s Constant

3.1 Conventional Role of k

Boltzmann’s constant appears in the expression for the average kinetic energy of an ideal gas:

$$E_{\text{average}} = \frac{3}{2} kT,$$

linking macroscopic temperature $T$ to microscopic energy.

3.2 Temperature-to-Mass Conversion

Recasting the energy expression via $E = mc^2$ gives:

$$E_{\text{average}} = m_{\text{equivalent}} c^2,$$

which implies a mass equivalent for temperature:

$$m_{\text{equivalent}} = \frac{3}{2} \frac{k}{c^2} T.$$

Introducing

$$k_m = \frac{3}{2} \frac{k}{c^2},$$

the relation becomes:

$$m_{\text{equivalent}} = k_m T.$$

This expression shows that temperature is not just an abstract measure of thermal energy; it also has a direct mass equivalent. As with $h$, the value of $k$ is intimately linked to our unit definitions, and its hidden conversion factor $c^2$ encodes the mass–energy equivalence inherent in thermal phenomena.

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4. Mass Equivalence of Charge: Insights from Vacuum Permittivity

4.1 The Role of ε₀ in Electromagnetism

In electromagnetism, vacuum permittivity appears in the energy density of an electric field:

$$u_E = \frac{1}{2} \varepsilon_0 E^2.$$

Here, $u_E$ represents the energy stored per unit volume in the electric field $E$.

4.2 Converting Electric Field Energy to Mass Density

Expressing the energy density in terms of mass via $E = mc^2$ leads to:

$$\rho_E = \frac{u_E}{c^2} = \frac{1}{2} \frac{\varepsilon_0}{c^2} E^2.$$

We define the conversion factor for charge as:

$$C_m = \frac{1}{2} \frac{\varepsilon_0}{c^2}.$$

Thus, the mass density equivalent becomes:

$$\rho_E = C_m E^2.$$

This reformulation shows that the energy stored in an electric field can be directly translated into a mass density, reinforcing the idea that electromagnetic energy and mass are two sides of the same coin.

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5. Discussion: Implications and the Unification of Constants

5.1 Recasting Constants as Conversion Factors

By decomposing $h$, $k$, and $\varepsilon_0$ to expose a hidden $c^2$ factor, we reinterpret these constants as conversion factors that express mass–energy equivalence. This approach demystifies their roles and suggests that their values are not arbitrary but arise from the very definitions of our units for mass, length, and time.

5.2 Answering the "Why" Questions

Standard physics often states that “these are fundamental constants, and that’s just how the universe is.” In contrast, our approach provides a deeper explanation:

• For $h$: The value is determined by the conversion between frequency and energy (and hence mass), making it an intrinsic measure of the mass equivalent per Hz.
• For $k$: It encapsulates the relationship between temperature and energy, with a hidden $c^2$ factor converting thermal energy to mass.
• For $\varepsilon_0$: It functions as a conversion between electric field energy and mass density, thereby unifying electromagnetism with mass–energy equivalence.

5.3 Unification Through Worldline Equivalence

The reinterpreted constants not only serve as conversion factors but also highlight a broader unity within physical laws. Energy in all its forms—whether it is associated with frequency, temperature, or charge—contributes to the curvature of spacetime along worldlines. This “worldline unification” suggests that mass, energy, and even spacetime geometry are deeply interconnected, with the conversion factors $Q_m$, $k_m$, and $C_m$ serving as bridges between these domains.

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6. Conclusion

We have presented a framework that reinterprets the fundamental constants $h$, $k$, and $\varepsilon_0$ as conversion factors that reveal the intrinsic mass equivalence of frequency, temperature, and charge. This approach not only provides a mechanistic explanation for the values of these constants but also unifies disparate aspects of physics under the umbrella of mass–energy equivalence. By demonstrating that the seemingly arbitrary numerical values of these constants are determined solely by our unit definitions and the inherent conversion factor $c^2$, we offer a fresh perspective that challenges the conventional, descriptive view. In doing so, we open the door to a deeper understanding of how energy, mass, and the geometry of spacetime are fundamentally interconnected—a perspective that may pave the way for future theoretical and experimental breakthroughs in physics.

How E=hf works, the simple explanation.

 Imagine you have a special converter. This converter can do two amazing things:

Convert Frequency to Mass: It tells you how much "mass stuff" is packed into a certain "frequency." Let's call this converter "Q_m". It says, "for every unit of frequency, you get this much mass."

Convert Mass to Energy: You already know this one! It's Einstein's famous c². It tells you how much "energy stuff" is inside a certain amount of "mass stuff."

Now, Planck's constant, 'h', is like combining these two converters into one!

Think of it like this:

To find the energy of a photon using its frequency (E=hf), you're actually doing two steps hidden inside 'h':

Step 1: Frequency to Mass (using Q_m): First, 'h' uses its secret ingredient, Q_m, to figure out the "mass equivalent" of the photon's frequency. Higher frequency means more "mass equivalent."

Step 2: Mass to Energy (using c²): Then, 'h' takes that "mass equivalent" and uses c² (just like in E=mc²) to convert it into energy.

So, 'h' is really just a shortcut for doing these two conversions in a row: Frequency → Mass (using Q_m) → Energy (using c²).

This means:

Hidden in 'h' is the idea that frequency and mass are directly related. This is the mass-frequency equivalence. Q_m is the key that unlocks this hidden link inside 'h'.

E=hf and E=mc² are saying the same basic thing: Both are about converting something into energy.

E=mc² says: Convert mass directly into energy using c².

E=hf says: Convert frequency into energy, but it secretly does it by first converting frequency to mass (using Q_m inside 'h'), and then converting that mass to energy (using c² also inside 'h').

In short:

Planck's constant 'h' isn't just a number. It's a package deal containing two fundamental converters: Q_m (frequency to mass) and c² (mass to energy). This "package" reveals that frequency and mass are fundamentally connected, and that both E=hf and E=mc² are just different ways of expressing this same underlying equivalence: everything is fundamentally related to energy, whether we see it as frequency, mass, or something else!

Demystifying Vacuum Permittivity: Unveiling Charge–Mass Equivalence within the E = mc² Paradigm

 

J. Rogers, SE Ohio, 27 Feb 2025, 0200

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Abstract

Vacuum permittivity (ε₀) is traditionally regarded as a measure of free space’s ability to permit electric field lines, appearing in Coulomb’s law and the energy density of electromagnetic fields. In this paper, we propose a novel reinterpretation of ε₀ by revealing its hidden connection to mass–energy equivalence. By recasting the electric field energy density in an E = mc²–like form, we introduce a derived conversion constant that explicitly links the energy stored in an electric field to a corresponding mass density. This approach not only unifies our understanding of electromagnetic energy with relativistic mass but also situates ε₀ within a broader framework alongside constants such as Planck’s constant (h) and Boltzmann’s constant (k), all of which conceal a c² factor fundamental to the fabric of spacetime.

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1. Introduction

Vacuum permittivity, ε₀, is a cornerstone of classical electromagnetism. It appears in Coulomb’s law, which quantifies the force between electric charges, and in the expression for the energy density (u_E) of an electric field:

  u_E = ½ ε₀ E²

While this formulation accurately describes electromagnetic phenomena, it leaves unexplored a deeper interpretation. By invoking Einstein’s mass–energy equivalence (E = mc²), we can reinterpret the energy stored in an electric field as having an equivalent mass density. In doing so, we expose the hidden conversion embedded within ε₀, much like previous reinterpretations of Boltzmann’s constant and Planck’s constant.

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2. Deriving a Charge–Mass Conversion Factor

To bridge the gap between electromagnetic energy and mass, we introduce a conversion factor, Cₘ, defined by extracting the c² factor from the energy density expression:

  u_E = ½ ε₀ E²
      = Cₘ E² c²

where

  Cₘ = (½ ε₀) / c²

Here, Cₘ has units that allow the product Cₘ E² to be interpreted as a mass density (kg/m³). Thus, the equivalent mass density (ρ_E) associated with an electric field becomes

  ρ_E = u_E / c² = Cₘ E²

This reformulation directly mirrors the structure of E = mc², suggesting that the energy stored in the field has a tangible mass equivalent.

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3. Recasting Electromagnetic Energy in an E = mc² Format

Expressing the energy density in the form

  u_E = (ρ_E) c²

highlights a clear analogy with the mass–energy equivalence principle. In this framework, the conversion constant Cₘ encapsulates the role of ε₀ as not merely a measure of electric flux but as a bridge between electromagnetic energy and mass. This reveals that:

• Hidden Conversion Factor:
  ε₀, when combined with the factor ½ and divided by c², transforms the squared electric field strength into a mass density.

• Unified Description:
  Just as Boltzmann’s constant (k) and Planck’s constant (h) hide within them a conversion factor (c²) that relates temperature or frequency to mass, ε₀ can similarly be seen as encoding a conversion from electromagnetic energy to mass.

• Physical Interpretation:
  The derived constant Cₘ provides a direct interpretation: for any given electric field strength, one can compute the mass density equivalent of its energy, reinforcing the idea that all energy—even that stored in fields—is fundamentally linked to mass.

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4. Numerical Demonstration

Below is a sample Python code that calculates the mass density equivalent for various electric field strengths using the conversion factor Cₘ:

import numpy as np

# Define constants (SI units)
epsilon_0 = 8.8541878128e-12  # F/m, vacuum permittivity
c = 299792458                 # m/s, speed of light

# Calculate the Charge-to-Mass Conversion Factor (Cₘ)
C_m = (0.5 * epsilon_0) / (c**2)

# Print the value of epsilon_0 and Cₘ
print(f"Vacuum Permittivity (ε₀): {epsilon_0:.2e} F/m")
print(f"Charge-to-Mass Conversion Factor (Cₘ = (0.5 * ε₀)/c²): {C_m:.2e} kg⋅s²/m⁵")

print("\n--- Calculations Table ---")
print(f"{'Electric Field Strength (V/m)':<30} | {'Energy Density (J/m³)':<30} | {'Mass Density Equivalent (kg/m³)':<30}")
print("-" * 90)

# Define example electric field strengths (V/m)
electric_field_strengths = [1e4, 1e6, 1e8, 1e10, 1e12]

for E_field in electric_field_strengths:
    energy_density = 0.5 * epsilon_0 * (E_field**2)        # u_E = 0.5 ε₀ E²
    mass_density = energy_density / (c**2)                   # ρ_E = u_E / c²
    # Alternatively: mass_density = C_m * (E_field**2)
    
    print(f"{E_field:<30.2e} | {energy_density:<30.6e} | {mass_density:<30.6e}")

print("\nNote:")
print(" - 'Energy Density' (J/m³) is calculated as: u_E = 0.5 ε₀ E²")
print(" - 'Mass Density Equivalent' (kg/m³) is obtained by: ρ_E = u_E / c² = Cₘ E²")

This numerical demonstration confirms that the electric field energy can be recast as a mass density, supporting the interpretation that electromagnetic energy, as governed by ε₀, inherently possesses mass.

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5. Discussion and Implications

Reinterpreting vacuum permittivity in this manner yields several significant insights:

• Unified Energy–Mass Picture:
  By expressing the electric field energy density as u_E = (ρ_E) c², we see that electromagnetic energy is not a distinct entity but one that directly contributes to the overall mass–energy content of the universe.

• Revealing Hidden Structures:
  Just as with k and h, decomposing ε₀ to expose the c² factor unveils a hidden layer of physical meaning. The constant Cₘ encapsulates a charge–mass conversion, indicating that every electric field has an associated mass density that influences spacetime curvature.

• Bridging Disciplines:
  This perspective helps bridge the gap between classical electromagnetism and general relativity. If every form of energy—including that stored in electric fields—has a mass equivalent, then electromagnetic interactions play a direct role in shaping the curvature of spacetime.

• Potential Experimental Avenues:
  Although the mass density equivalent for typical electric fields is exceedingly small, recognizing its existence could motivate high-precision experiments in extreme electromagnetic environments, contributing to a deeper understanding of energy–mass interactions.

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6. Conclusion

This paper has presented a novel reinterpretation of vacuum permittivity (ε₀) by demonstrating that it can be decomposed to reveal an implicit conversion factor—analogous to those found in Planck’s constant and Boltzmann’s constant—that directly links electric field energy to mass. By defining the conversion constant

  Cₘ = (½ ε₀) / c²

and recasting the energy density as u_E = (ρ_E) c², we show that the energy stored in an electric field is inherently tied to a mass density. This approach not only demystifies ε₀ but also contributes to a unified view of physical laws in which mass, energy, and the curvature of spacetime are inextricably linked. In doing so, we reaffirm the pervasive nature of E = mc² across all domains of physics.

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