Abstract

1. High-level explanation (non-mathematical)

The ECM viewpoint differs from the geometric interpretation of General Relativity. ECM asserts that photon energy is not an immutable local invariant but is continuously modulated by the gravitational proximity: a photon at radius \(r\) from a mass has instantaneous energy \(E(r)\) determined by a mass-displacement bookkeeping (\(\Delta M_m\) displaced from binding energy) and by interactional displacement \(\Delta M_g(r)\) induced by the field. Because the photon speed is constant (\(c=f\lambda\)), the wavelength \(\lambda\) and therefore momentum \(p=h/\lambda\) change with \(r\). This changing momentum produces a phase gradient across the wavefront and a transverse impulse that bends the ray. The bending is symmetric during approach and recede because gravitational fields are radial and symmetric; beyond a characteristic distance \(r_{\max}\) the field influence ceases and the photon maintains fixed energy.

Key ECM concepts:

• Mass displacement: Emission corresponds to displacement of bound mass \(\Delta M_m\); apparent mass is \(M^{app}=-\Delta M_m\).
• Negative apparent mass: \(M^{app}<0\) encodes repulsive / non-attractive contributions in gravitational bookkeeping and plays a role in cosmological phenomena in ECM.
• Photon dynamics: \(E(r)=\Delta M_m c^2[1+\chi(r)]\) and \(p(r)=E(r)/c\). The proximity function \(\chi(r)\) models field-induced interactional mass-displacement.
• rₘₐₓ: the radius beyond which \(dp/dr\to 0\) and further gravitational energy exchange ceases; the photon becomes dynamically free.

2. Mathematical presentation - action / phase derivation (ECM)

We derive the ray equation using the eikonal/variational principle: the ray extremizes the action \(S=\int p(r)\,ds\). Use polar coordinates \((r,\theta)\) and arc-length parametrization. With the photon momentum depending on radius, \[ p(r)=p_\infty\,[1+\chi(r)],\qquad p_\infty\equiv \Delta M_m c. \]

Conservation of angular momentum (cyclic \(\theta\)) yields \[ J \;=\; p(r)\, r^2\,\dot\theta \quad\text{(arc-length parametrization)}, \] and parametrizing by arc-length (\(\sqrt{\dot r^2+r^2\dot\theta^2}=1\)) gives the exact ray equation \[ \boxed{\frac{d\theta}{dr} \;=\; \frac{J}{p(r)\, r^2}\,\frac{1}{\sqrt{1 - \dfrac{J^2}{p(r)^2 r^2}}},\qquad J=p_\infty b.} \]

Weak-field linearization and bending integral

For \(\chi(r)\ll1\) expand to first order. After algebra (change of variable \(r=b/\sin\psi\)) the bending angle simplifies to the compact form \[ \boxed{\theta_{\rm bend} \;=\; 2\int_{0}^{\pi/2}\chi\!\bigl(r(\psi)\bigr)\,d\psi.} \] This expresses the net deflection as an angular average of the proximity-coupling \(\chi(r)\) along the unperturbed straight-line path.

Simple model \(\chi(r)=\alpha GM/(r c^2)\)

If the field-driven coupling is proportional to the Newtonian potential, \(\chi(r)=\alpha GM/(r c^2)\), then \[ \theta_{\rm bend} \;=\; 2\frac{\alpha GM}{b c^2}. \] Setting \(\alpha=1\) recovers the Newtonian impulse result \(2GM/(b c^2)\); choosing \(\alpha=2\) reproduces the GR value \(\theta=4GM/(b c^2)\). In ECM the factor \(\alpha=2\) is interpreted as the sum of two physical contributions (inherent + interactional displacement) rather than as separate geometric time/space curvature effects.

Phase-accumulation (eikonal) equivalence

The same result follows from the stationary-phase condition. The accumulated phase along a path \(C\) is \[ \Phi=\int_C k(r)\,ds=\frac{1}{\hbar}\int_C p(r)\,ds, \] and extremizing \(\Phi\) yields the ray equation above. The first-order bending integral is therefore equivalently interpreted as arising from an integrated phase gradient along the path due to \(\chi(r)\).

Worked numeric example - representative stellar photon

Representative photon frequency \(f=6.0368\times 10^{14}\,\mathrm{Hz}\) (green light) gives energy \[ E=hf\approx 4.00\times 10^{-19}\ \mathrm{J}. \] Mass-equivalent displaced: \[ \Delta M_m=\frac{E}{c^2}\approx 4.45\times 10^{-36}\ \mathrm{kg}; \qquad M^{app}=-\Delta M_m. \] Solar-constant based fluxes show aggregated mass-equivalent transfers large on astrophysical scales; ECM uses such aggregation to explain cumulative cosmological apparent-mass effects.

3. r_max and its operational meaning

In ECM, r_max is the operational boundary where field-induced interactional mass displacement vanishes ( \(\chi(r)\to0\) ), so \(dp/dr\to0\). It is not only the escape radius for the source well: it is the radius beyond which the photon is free from any further gravitational energy exchange. Because gravitational effects are radial and symmetric, momentum exchange-and therefore path curvature - is symmetric about the point of closest approach.

4. ECM vs GR - comparison and experimental discriminants

Both GR and ECM predict the same classical bending when parameters are set appropriately. The interpretations differ:

• GR: purely geometric-light follows null geodesics of the curved metric; photon has no local energy exchange due to the field.
• ECM: physically mechanistic-light's wavelength and momentum change with proximity, producing phase gradients and transverse impulses. The factor that yields GR's number corresponds in ECM to specific physical contributions and coupling constants that must be computed from microphysics and tested experimentally.

Discriminating experiments would seek signatures of local energy exchange (e.g., measurable phase/energy bookkeeping in controlled gravitational-like potentials, laboratory phase-shift tests, thermionic/photonic mass-balance experiments). Appendix 40 (thermionic emission in CRT systems) and piezoelectric rotational phase experiments are candidate laboratory contexts for testing ECM predictions.

5. Conclusion

ECM provides a coherent, experimentally-minded alternative foundation for photon–gravity interactions: it attributes light bending to proximity-dependent energy and momentum changes (phase gradients) rather than to abstract spacetime curvature. It reproduces classical results (including the GR bending formula) when the interactional coupling is chosen to match observations, but crucially it gives a route to calculate that coupling from microphysics and therefore to propose decisive laboratory tests. Further development requires (a) first-principles derivations of the coupling function \(\chi(r)\), (b) quantitative laboratory experiments such as precise thermionic calorimetry and phase-shift measurements, and (c) cosmological bookkeeping for large-scale negative apparent mass effects.

References