Photon Energy, Effective Mass, and Gravitational Interaction in ECM: A Unified Approach

Soumendra Nath Thakur

29-01-2025

Abstract

This study explores the fundamental concepts of photon energy, effective mass, and gravitational interaction within the framework of Extended Classical Mechanics (ECM). By incorporating the role of apparent mass (Mᵃᵖᵖ)—a negative mass component—this approach provides a refined perspective on photon behaviour in gravitational fields, gravitational redshift, and interactions between massive bodies. The total energy of a photon is analysed as the sum of its inherent energy (E) and gravitational interaction energy (Eg), leading to a deeper understanding of gravitational frequency shifts and momentum variations. Additionally, the study extends ECM principles to massive bodies, redefining gravitational potential energy and effective mass to account for apparent mass effects. The force equations for both photons and massive bodies are formulated, revealing the impact of apparent mass on motion, gravitational attraction, and antigravitational effects—such as those influenced by dark energy. These findings present a unified framework that bridges classical mechanics, quantum principles, and cosmological implications, offering novel insights into gravitational dynamics and photon interactions.

Keywords: Photon Energy, Effective Mass, Apparent Mass, Gravitational Interaction, Gravitational Redshift, Extended Classical Mechanics (ECM), Dark Energy, Antigravitational Effects, Gravitational Potential, Photon Momentum

1. Photon energy, described by Planck's equation E = hf, consists of two components: the inherent energy (E) of the photon emitted from a gravitational source, and the additional gravitational energy (Eg) acquired due to interactions with the source's gravitational field. As photons escape the source's gravitational field, they retain their inherent energy (E) while gradually expending their gravitational energy (Eg), completely losing it once they exit the gravitational influence.

Photon Energy and Gravitational Redshift

• Total Energy of a Photon: Eₜₒₜₐₗ = E + Eg = h (f + Δf)

• Photon’s Inherent and Interaction Energy: E = hf, Eg = hΔf = ΔE

• Photon Energy Shift in a Gravitational Field (Gravitational Redshift): hΔf = hf/c²·g·∆H. Where ΔH is the height in the gravitational field and g is the gravitational field strength.

Photon Energy, Momentum, and Interaction in a Gravitational Field:

Photon Momentum and Effective Mass

• Photon’s Momentum: ρ = h/λ

• Effective Mass of a Photon: Mᵉᶠᶠ = hf/c² = E/c² 

Photon Interaction in a Gravitational Field

• Gravitational Interaction Energy (Additional Energy): Eg = E + Δρ

• Photon’s Apparent Kinetic Energy: KEₚₕₒₜₒₙ = (1/2) −Mᵃᵖᵖ·c². Where: Mᴍ₁c² =−Mᵃᵖᵖ·c², Mᴍ = 0,

Force and Gravitational Mass of a Photon

• Force of a Photon in a Gravitational Field: −Mᵃᵖᵖ⋅aᵉᶠᶠ

• Gravitational Mass Relation: Mɢ = Mᴍ + (−Mᵃᵖᵖ) and Mᵉᶠᶠ = Mɢ = Mᴍ + (−Mᵃᵖᵖ)

2. Effective Mass, Apparent Mass, and Gravitational Interaction for Photons and Massive Bodies in Extended Classical Mechanics (ECM)

Photon’s Effective Mass and Gravitational Interaction

• Photon’s Inherent Effective Mass and Relationship with Apparent Mass: In Extended Classical Mechanics (ECM), the inherent effective mass of a photon is derived from its energy, given by:

Mᵉᶠᶠ = hf/c² = E/c²

 

where h is Planck’s constant, f is the photon’s frequency, and c is the speed of light. While photons have no rest mass (Mᴍ = 0), they exhibit an effective mass due to their energy, which influences their behaviour in gravitational fields.

• Gravitational Interaction Energy and Apparent Mass for a Photon: When a photon travels through a gravitational field, it experiences a shift in frequency due to gravitational redshift or blueshift. This frequency shift alters the photon’s energy and consequently, its effective mass. The energy change due to this shift is represented by Eg = hΔf, and this contributes to the photon’s total effective mass in the gravitational field. The total effective mass of a photon is given by:

Mᵉᶠᶠ,ₜₒₜₐₗ = Mᵉᶠᶠ + Eg/c² = hf/c² + hΔf/c²

This equation accounts for both the photon’s inherent energy and the additional energy due to gravitational interaction, which modifies its effective mass.

• Force Equation for Photons in Gravitational Fields: The force acting on a photon in a gravitational field is described by the equation:

F = (Mᴍ − Mᵃᵖᵖ)·aᵉᶠᶠ = Mᵉᶠᶠ·aᵉᶠᶠ 

Since photons have no rest mass (Mᴍ = 0), this simplifies to:

F = −Mᵃᵖᵖ·aᵉᶠᶠ = Mᵉᶠᶠ·aᵉᶠᶠ 

where Mᵉᶠᶠ = hf/c² represents the photon’s effective mass, and the force acting on the photon depends on its gravitational interaction.

3. Effective Mass of Massive Bodies and Gravitational Interaction in ECM

In ECM, the gravitational force equation is given by:

F𝑔 = G·Mɢ·mₘ/r², 

where Mɢ is the gravitational mass of the larger body (e.g., a planet or star), mₘ is the inertial mass of the smaller body (e.g., a satellite or particle), and r is the distance between the two bodies.

In ECM, the gravitating mass Mɢ is related to the effective mass Mᵉᶠᶠ, which accounts for the influence of both the inertial matter mass mₘ and the apparent mass Mᵃᵖᵖ in a gravitational field. The relationship is expressed as:

Mɢ = Mᵉᶠᶠ = mₘ + (−Mᵃᵖᵖ)

This equation shows how the gravitational mass Mɢ of a body is influenced by the apparent mass in a gravitational field.

• Gravitational Potential Energy and Effective Mass of Massive Bodies: The gravitational potential energy Eɢᵣₐᵥ of a massive body mₘ in the gravitational field of a larger body Mᴍ at distance r is given by:

Eɢᵣₐᵥ =− G·Mᴍ·mₘ/r 

where G is the gravitational constant. The gravitational potential energy modifies the total effective mass of the body, which can be expressed as:

Mᵉᶠᶠ = mₘ + Eɢᵣₐᵥ/c²

 

This shows that the gravitational potential energy increases the effective mass of a massive body, which affects its interaction with other bodies in a gravitational field.

• Force Equation for Massive Bodies in Gravitational Fields: The force acting on a massive body in a gravitational field can be described using the following equation:

F = (mₘ − Mᵃᵖᵖ)·aᵉᶠᶠ = Mᵉᶠᶠ·aᵉᶠᶠ 

where Mᵉᶠᶠ is the total effective mass, which includes both the inertial matter mass mₘ and the apparent mass Mᵃᵖᵖ. This equation explains how the motion of a massive body is influenced by the combined effect of its gravitational and apparent masses.

Apparent Mass in Gravitational, Antigravitational, and Everyday Contexts

The concept of apparent mass is crucial not only for gravitational interactions but also for understanding motion, gravitational, and antigravitational effects.

• Apparent Mass in Motion and Gravitational Effects: When an object is in motion, its apparent mass can change as it accelerates or decelerates. For example, a bicycle in motion experiences a decrease or increase in effective mass depending on whether it is speeding up or slowing down. The inertial matter mass mₘ interacts with the negative apparent mass −Mᵃᵖᵖ, modifying the forces acting on the object. As the bicycle accelerates or decelerates, the change in effective mass influences how it responds to gravity and other external forces. This dynamic is a key feature of ECM, where the interaction of mass and motion influences the forces acting on objects.

• Gravitational and Antigravitational Effects: Objects exhibiting negative apparent mass can behave in unusual ways, such as experiencing repulsion from larger masses. This antigravitational effect is associated with dark energy and its influence on the apparent mass of objects. For example, in the presence of dark energy, negative apparent mass can cause objects to move away from larger bodies instead of towards them, displaying an antigravitational effect. This is an important concept in cosmology and the study of large-scale gravitational dynamics, where dark energy and apparent mass contribute to the overall behaviour of the universe.

Conclusion:

The concepts of effective mass and apparent mass in Extended Classical Mechanics (ECM) provide a comprehensive understanding of gravitational interactions for both photons and massive bodies. For photons, their effective mass, derived from their energy, governs their behaviour in gravitational fields. For massive bodies, gravitational potential energy contributes to the effective mass, altering their gravitational interactions. The concept of apparent mass, especially when negative, plays a crucial role in understanding the effects of motion, gravitational, and antigravitational forces. These principles offer a unified framework for understanding how mass and energy interact in gravitational contexts, ranging from everyday experiences to extreme conditions influenced by dark energy.

4. Key Definitions in Extended Classical Mechanics (ECM):

Below is an alphabetically ordered list of key terms and their definitions:

A

Apparent Mass (Mᵃᵖᵖ) – A negative mass component that arises in motion, gravitational interactions, and antigravitational effects. It plays a role in modifying an object's effective mass and influences phenomena such as gravitational redshift and dark energy effects.

Antigravitational Effect – A repulsive force experienced by objects with negative apparent mass, often associated with dark energy. It can cause objects to move away from larger gravitational bodies instead of being attracted to them.

E

Effective Acceleration (aᵉᶠᶠ) – The acceleration experienced by an object when accounting for both its inertial mass and apparent mass contributions.

Effective Mass (Mᵉᶠᶠ) – The total mass influencing an object’s motion or interaction in a gravitational field. It is given by the sum of the inertial mass and apparent mass:

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)

For photons, the effective mass is derived from their energy:

Mᵉᶠᶠ =hf/c² 

Energy of Gravitational Interaction (Eg) – The energy change a photon undergoes when moving through a gravitational field, responsible for gravitational redshift or blueshift. Defined as:

Eg = hΔf

Energy of Gravitational Potential (Eɢᵣₐᵥ) – The potential energy of a massive body mₘ in the gravitational field of a larger body Mᴍ, given by:

Eɢᵣₐᵥ =− G·Mᴍ·mₘ/r 

This energy modifies the effective mass of a body in a gravitational field.

F

Force on a Photon in a Gravitational Field (F) – The force acting on a photon due to gravitational interactions, given by:

F = −Mᵃᵖᵖ·aᵉᶠᶠ

Since photons have no rest mass, their motion is determined by their effective mass.

Force on a Massive Body in a Gravitational Field (Fg) – The force acting on a massive body due to gravitational attraction, expressed in ECM as:

F𝑔 = G·Mɢ·mₘ/r²

where Mɢ is the gravitational mass of the larger body, and mₘ is the inertial mass of the smaller body.

G

Gravitational Mass (Mɢ) – The mass responsible for generating a gravitational field, given by:

Mɢ = Mᵉᶠᶠ = mₘ + (−Mᵃᵖᵖ)

It accounts for the influence of apparent mass in a gravitational system.

Gravitational Redshift – The shift in a photon’s frequency when moving through a gravitational field, causing it to lose energy as it moves away from a massive object.

I

Inertial Matter Mass (mₘ) – The mass of an object that determines its resistance to acceleration under an applied force. In ECM, it contributes to the total effective mass.

M

Mass of a Larger Body (Mᴍ) – The mass of a larger gravitational body, such as a planet or star, that influences smaller objects in its gravitational field.

Massless Photon Assumption – The assumption that photons have no rest mass (mₘ = 0), but still possess effective mass due to their energy.

P

Photon’s Effective Mass – The mass associated with a photon due to its energy, given by:

Mᵉᶠᶠ =hf/c² 

Photon Momentum (ρ) – The momentum of a photon is given by:

ρ = h/λ

where λ is the photon’s wavelength

Conclusion

The framework of Extended Classical Mechanics (ECM) provides a unified approach to understanding photon energy, effective mass, and gravitational interaction for both massless and massive bodies. By incorporating the concept of apparent mass (Mᵃᵖᵖ), this study redefines gravitational dynamics, offering a deeper insight into photon behaviour in gravitational fields, gravitational redshift, and momentum variation. The formulation of effective mass as Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ) reconciles traditional gravitational theories with observed phenomena such as frequency shifts in photons, energy conservation, and gravitational potential influences on massive bodies.

For photons, ECM illustrates how their energy consists of both inherent energy (E = hf) and gravitational interaction energy (Eg = hΔf), affecting their motion and force interaction within gravitational fields. The force equation F = −Mᵃᵖᵖ·aᵉᶠᶠ explains the gravitational influence on massless particles. Meanwhile, for massive bodies, ECM accounts for apparent mass effects, leading to a refined expression for gravitational potential energy (Eɢᵣₐᵥ = − G·Mᴍ·mₘ/r) and force interactions.

The concept of negative apparent mass introduces significant implications for antigravitational effects, potentially explaining dark energy and the observed expansion of the universe. By integrating apparent mass into gravitational equations, ECM suggests a possible mechanism for repulsive gravitational behaviour, which could be relevant in cosmology, astrophysics, and high-energy physics.

Ultimately, ECM extends classical mechanics by bridging classical gravity, quantum mechanics, and cosmological effects, providing a more comprehensive framework for analysing both local gravitational interactions and large-scale cosmic phenomena. This refined perspective has the potential to deepen our understanding of gravity, mass-energy interactions, and the role of dark energy in shaping the universe. Photon Energy, Effective Mass, and Gravitational Interaction in ECM: A Unified Approach