Soumendra Nath Thakur
ORCID: 0000-0003-1871-7803
As the photon escapes the source gravitational potential well, the interactional component of its energetic state progressively diminishes, with the resulting decrease in photon frequency and increase in wavelength manifesting as gravitational redshift. Within the ECM mass-accounting framework, this progression may be represented as
ΔMᵃᵖᵖ ← ΔMᴍ ← Δλₚₕₐₛₑ
The reduction of the interactional mass contribution may therefore be considered together with the corresponding frequency displacement and phase-wavelength displacement along the radial propagation path. If the initial interactional quantity is represented by 2ΔMᴍ, and the remaining interactional quantity at a radial position r is represented by ΔMᴍ(r), then the interactional quantity expended between the emission state and that position may be expressed as
ΔMₑₓₚ(r) = 2ΔMᴍ,₀ − ΔMᴍ(r).
When the remaining quantity reaches the corresponding inherent-energy representation,
2ΔMᴍ − ΔMᴍ = ΔMᴍ,
the interactional component has been reduced to the remaining ΔMᴍ state at that position. The associated frequency representation is
ΔMᴍ(r) = hΔf(r)/c².
Accordingly, the cumulative redshift loss from the emission state to the position r is more appropriately represented by the corresponding frequency difference,
Δfɢ(r) = f₀ − f(r),
with the associated interactional energy displacement
ΔEɢ(r) = h[f₀ − f(r)]
and corresponding mass representation
ΔMɢ(r) = h[f₀ − f(r)]/c².
Thus, r f₍ₓ°₎ should not itself be identified as the total redshift energy loss. The quantity r f₍ₓ°₎ belongs to the accumulated phase progression associated with propagation through the radial distance, whereas the redshift loss is represented by the change in the frequency-dependent interactional energy between the initial and subsequent states.
For the phase representation, the accumulated phase-coordinate displacement may be written generally as
x°(r) = (360°/c) ∫₀ʳ f(r′)dr′.
where the use of the integral permits the photon frequency to vary continuously along the radial path. In the special case of a locally specified frequency state, the corresponding phase relation reduces to
x°(r) = 360° f(r) Δt,
with
Δt = r/c.
Hence, the accumulated phase and the cumulative redshift loss describe two related but distinct aspects of the same propagation process: x° describes phase progression through the field, while Δfɢ and ΔEɢ describe the source-dependent frequency and energy displacement.
The effective-force relation must likewise be interpreted dynamically:
−Fᴇᴄᴍ,ᴘʜᴏᴛᴏɴ = 2ΔMᴍ aᵉᶠᶠ.
As 2ΔMᴍ ↓ with increasing radial distance, the corresponding effective acceleration required by the weakening interaction also decreases,
aᵉᶠᶠ ↓,
and the associated effective repulsive force correspondingly decreases:
−Fᴇᴄᴍ,ᴘʜᴏᴛᴏɴ ↓.
The reduction therefore represents a progressively weakening source-dependent interaction rather than an acceleration maintained at a fixed magnitude. Throughout this process, the photon retains its manifested propagation condition
v = Δλ/Δt = ℓᴘ/tᴘ = c.
The gravitational interaction is consequently manifested through the changing frequency, wavelength, phase state, interactional mass representation, effective force, and effective acceleration, while the propagation velocity remains c.
At the limiting radial position where the source-dependent interactional contribution reaches zero within the ECM representation, the corresponding gravitational redshift interaction has been exhausted. The photon then retains its intrinsic energy-frequency state, while the source-dependent interactional component no longer contributes to the continuing energy exchange.
