By Soumendra Nath Thakur, ORCiD: 0000-0003-1871-7803
12-11-2024
The study distinguishes between the intrinsic photon energy (E) and the gravitational-interaction energy (Eg), which is treated as separate but interrelated components when photons interact with gravitational fields. Using key quantum mechanical principles, including Planck's energy-frequency relation E=hf and de Broglie's photon momentum-wavelength relation ρ=h/λ, we establish a mathematical framework for understanding these interactions. Additionally, Planck scale parameters are incorporated to define observational limits within quantum-gravitational contexts, ensuring that the formulation aligns with established measurement constraints.
Photon Interactions in Gravitational Fields
This section refines the framework by exploring the distinct types of photon energy interactions under various gravitational conditions. Building on earlier discussions about symmetry in energy and momentum exchange, we now recognize that intrinsic photon energy (E) and gravitational-interaction energy (Eg) are distinct yet symmetrically gained and lost during photon interactions with gravitational fields.
1. Photon Emission and Energy Composition: At the moment of emission, the photon carries its intrinsic energy, E=hf, along with an additional gravitational interaction energy, Eg=hΔf, due to the influence of the gravitational field. The photon’s total energy at emission is therefore E+Eg = h(f+Δf), where Δf represents the frequency shift induced by the gravitational field.
2. Energy Expenditure during Ascent from the Gravitational Well: As the photon ascends from the gravitational well, it expends energy from the gravitational interaction component (Eg), rather than its intrinsic energy (E). This expenditure is reflected by a gradual reduction in Δf, corresponding to the observed gravitational redshift. As the photon escapes the gravitational influence, Eg diminishes, leaving only the photon’s intrinsic energy, E=hf, intact in regions of negligible gravitational potential.
3. Distinct Energy Types: The photon’s inherent energy (E) is fundamentally distinct from the interactional energy (Eg). While E is intrinsic to the photon and constant across gravitational fields, Eg arises from the photon’s interaction with the gravitational field, being temporary and dependent on the photon’s position within that field.
4. Symmetry of Energy and Momentum Exchange: The interactional energy (Eg) is symmetrically gained when the photon enters a gravitational field and symmetrically lost as it exits. This symmetry reflects the reversible nature of gravitational influence on the photon’s total energy. The inherent energy (E), however, remains unaffected by the gravitational field and represents a constant property of the photon, independent of gravitational influence.
5. Gravitational Redshift and Blueshift: As the photon moves away from the gravitational source, it experiences a redshift due to the progressive loss of Eg, with the photon’s frequency shifting from f+Δf to its inherent frequency f as the gravitational interaction energy Eg is expended. Conversely, as the photon moves into a stronger gravitational field, it would experience a blueshift, with an increase in Δf as Eg is symmetrically gained.
Mathematical Presentation:
The photon’s energy state at emission is represented by the sum of its intrinsic energy (E) and its gravitational-interaction energy (Eg), with the total energy given by:
E + Eg = h(f+Δf)
As the photon moves away from the gravitational source:
1. The expenditure of Eg: The photon loses Eg gradually due to gravitational redshift, with the frequency shift Δf diminishing as the photon climbs out of the gravitational well.
2. The constant inherent energy: The intrinsic energy E=hf remains constant throughout the photon’s journey, unaffected by gravitational influence.
Once the photon has moved beyond the gravitational field’s influence, Eg is fully expended, leaving only the inherent energy E=hf.
Expansion on Photon Energy Interactions in Gravitational Fields:
1. As the photon moves away from the source, it loses Eg due to the gravitational redshift, eventually stabilizing to its intrinsic E=hf when it reaches a region with negligible gravitational potential. This perspective frames the gravitational interaction energy as a component that modifies the photon’s total energy specifically due to its position within the gravitational field, influencing its energy state but diminishing as it escapes the well.
2. Inherent Photon Energy (E): This is given by E=hf, where h is Planck’s constant, and f is the intrinsic frequency of the photon as it is emitted. This energy represents the photon's baseline or inherent energy.
3. Gravitational Interaction Energy (Eg): This additional energy, represented as Eg=hΔf, accounts for the photon's interaction with the gravitational field. Here, Δf represents the frequency shift induced by the gravitational potential at the point of emission.
4. Total Initial Energy at Emission (E+Eg): Combining these, the photon’s energy state at emission is indeed E+Eg, the sum of its inherent energy and the gravitational interactional energy. This total is the photon's highest energy point.
5. As the photon ascends from the gravitational well:
6. Expenditure of Gravitational Interaction Energy (Eg): The photon’s apparent energy reduction due to gravitational redshift occurs from the gravitational interaction energy, Eg=hΔf, rather than its inherent energy E=hf. This distinction is crucial, as Eg is specifically associated with the photon’s interaction with the gravitational field and reflects an additional energy component that only exists while the photon is within the gravitational influence of its source.
7. Inherent Energy (E) Remains Constant: The intrinsic energy, E=hf, remains unaffected by the gravitational field as it is a fundamental property of the photon. Thus, as the photon climbs out of the gravitational well, it "sheds" Eg progressively, aligning with the redshift observed. Eventually, Eg is fully expended when the photon reaches a region of negligible gravitational influence, leaving only its inherent energy, E=hf, intact.
This interpretation reinforces the idea that gravitational redshift involves only the additional gravitational interactional energy, allowing the photon’s inherent energy to remain consistent across different gravitational potentials.
8. The energy of the photon at emission within a gravitational well effectively. At the moment of emission, the photon's total energy reflects both its inherent frequency and an additional frequency component due to the gravitational field. Here’s how it unfolds:
9. Inherent Energy and Frequency (E = hf): The photon's inherent energy is represented by E=hf, where f is its intrinsic frequency—an unaltered property of the photon that represents its baseline energy state.
10. Additional Frequency Due to Gravitational Interaction (Δf): When the photon is emitted from within the gravitational field of its source, the gravitational interaction imparts an additional frequency shift, Δf. This results from the gravitational influence exerted on the photon at the point of emission, causing it to emerge with a total frequency of f+Δf due to the local field.
11. Total Energy at Emission (E + Eg): Consequently, the total energy of the photon at emission is E+Eg=h(f+Δf). This value represents the photon's highest energy state, with Eg=hΔf being the extra energy due to the gravitational field's interaction with the photon.
12. Energy Expenditure as Photon Escapes the Gravitational Well: As the photon moves away from its source’s gravitational field, it “loses” Eg, represented by a gradual reduction in Δf due to gravitational redshift. This results in the photon’s frequency gradually decreasing to its inherent frequency f, and thus only E=hf remains in regions of negligible gravitational influence.
This approach clearly distinguishes between the photon's intrinsic properties (frequency f and energy E) and the additional, temporary gravitational effects (Δf and Eg) it experiences due to the source's gravitational well.
13. The additional frequency component, Δf, and its corresponding energy Eg=hΔf, are present only while the photon remains within the gravitational influence of its source. This gravitational interaction effect can be summarized as follows:
14. Gravitational Influence on Frequency: The photon's total frequency at emission, f+Δf, includes both its inherent frequency f and the additional gravitationally induced frequency Δf. This additional frequency represents the photon's gravitational interaction energy Eg within the source’s gravitational well.
15. Persistence of Δf Within the Gravitational Field: As long as the photon remains within the gravitational field, Δf persists as a measurable shift. This implies that the photon’s total energy E+Eg=h(f+Δf) remains higher than its inherent energy E=hf.
16. Redshift and Loss of Δf with Distance: As the photon travels away from the gravitational source, Δf gradually diminishes due to gravitational redshift, which effectively reduces Eg. Once the photon is beyond the gravitational field's influence, Δf becomes negligible, leaving only the inherent frequency f and intrinsic energy E=hf.
In summary, Δf and Eg are directly tied to the photon's position within the gravitational well and disappear as the photon escapes, highlighting the temporary nature of gravitational interaction energy while the photon is within the field.
Symmetry in Momentum Exchange
This phase extends the derived equations to analyse the symmetry of momentum exchange in photon interactions with gravitational fields. When photons undergo wavelength or phase shifts due to gravitational influences, the resulting momentum exchange is symmetrical, preserving both intrinsic energy (E) and gravitational-interaction energy (Eg). The proposed framework suggests that, as photons traverse external gravitational wells, they symmetrically gain and lose Eg in a balanced manner, maintaining conservation of total energy and momentum throughout their trajectories.
Comparative Analysis with Classical and Relativistic Perspectives
In the final phase, this framework is compared with both classical and relativistic models of photon behaviour in gravitational fields. The comparison emphasizes the distinct nature of gravitational-interaction energy (Eg) relative to intrinsic photon energy (E), highlighting the model's adherence to the principles of energy conservation while suggesting a departure from interpretations that conflate gravitational effects with spacetime curvature. The analysis presents a fresh perspective on gravitational lensing and dark energy, proposing new interpretations in light of photon-graviton interactions.
Spacetime Curvature vs. Gravitational Field Lensing
1. Background and Title:
The image displays the title "Spacetime Curvature vs. Gravitational Field Lensing" in bold black text. This sets the focus on differentiating between gravitational lensing interpretations based on General Relativity's spacetime curvature and external gravitational fields.
2. Source of Light (Top Right):
Positioned in the top right corner, a small sphere labelled "Source of Light" represents a distant luminous object. This body is drawn small to convey distance, emphasizing that the light travels a vast distance before interacting with gravitational influences.
3. Rays of Light (Extending from Source):
The lines radiate outward from the source of light, symbolizing photon trajectories or light rays moving omni directionally. Several lines are directed toward the bottom left, where they approach the observer, showing how light travels through and interacts with gravitational fields.
4. Observation Point (Bottom Left):
In the bottom left, a larger sphere labelled "Observation of Light" represents the observing body (e.g., Earth). Its larger size suggests proximity, emphasizing that it is the endpoint for analysing the path of light under gravitational influences.
5. Celestial Body (M) as the Moon:
Near the Observation of Light, a smaller sphere labelled "M" represents the Moon, which orbits around the observer (Observation Point). During phenomena like a solar eclipse, M aligns with the observer and the massive body (e.g., Sun), which is crucial for the gravitational lensing demonstration.
6. Massive Body/Sun (Centre):
Cantered between the Source of Light and the Observation of Light, a large sphere labelled "Massive Body/Sun" represents a nearby gravitationally influential object (e.g., the Sun). This body is illustrated as the largest sphere, signifying its strong gravitational influence over light rays passing through its vicinity.
7. Gravitational Fields (Around Massive Body):
The curved lines surround the Massive Body/Sun, representing its gravitational field. This field is extended to visually differentiate between gravitational influences arising from the mass itself rather than from spacetime curvature.
8. Curved Spacetime (Below Massive Body):
Below the Massive Body/Sun, a curvature represents spacetime distortion. This depiction aligns with General Relativity's view of mass-induced spacetime warping, but in this illustration, it is shown as insufficient for redirecting light in a lensing effect, suggesting limitations in the curvature alone.
9. Concept Visualization (Photon Pathways and Interactions):
The visualization emphasizes two distinct photon pathways interacting differently with the massive body, depending on the surrounding fields:
• Lower Ray Path (Interaction with Spacetime Curvature):
Photons traveling along the lower ray pathway encounter the curved spacetime around the Massive Body/Sun. This path is obstructed by the mass of the Massive Body, unable to continue toward the Observation Point. This visualization implies that gravitational lensing is not solely due to the spacetime curvature predicted by General Relativity, as these rays cannot bypass the mass.
• Upper Ray Path (Interaction with Gravitational Fields):
Photons on the upper path bypass the curved spacetime and instead follow the gravitational field lines around the Massive Body/Sun. In this pathway, the photons are redirected by the gravitational field rather than by spacetime curvature. This interaction with the gravitational field allows them to proceed unobstructed toward the Observation Point, proposing that gravitational lensing is actually facilitated by these external gravitational fields.
Observational Alignment during a Solar Eclipse:
It is essential to understand that gravitational lensing is often observed during a solar eclipse, where M (the Moon) aligns between the Earth (Observation Point) and the Sun (Massive Body), casting a shadow on Earth. During this alignment, the Source of Light, Massive Body/Sun, M, and Observation Point are all positioned in a straight line. This alignment reinforces the need for the massive body’s external gravitational field to guide photons to the observation point, rather than the curvature of spacetime alone.
Summary
This image visually argues that gravitational lensing arises from photon interactions within the external gravitational fields surrounding massive bodies rather than the spacetime curvature framework alone, as proposed by General Relativity. By emphasizing the photon energy pathways, this illustration suggests that the gravitational field of a massive body actively guides light toward the observer, demonstrating gravitational lensing without requiring spacetime distortion. This approach aligns with quantum mechanical interpretations, highlighting how external gravitational fields interact with photon energy to produce the lensing effect.
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