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Gravitational force and acceleration are written as:
Fg = G M m / r2 a = Fg / m = G M / r2
where M is the static source mass and m is the test particle mass.
The test particle’s mass cancels in acceleration, giving
universal acceleration independent of the test mass.
Mass is dynamic and context-dependent. The effective mass
Meff = MM − Mapp
incorporates both the intrinsic matter mass MM and corrections due to
negative apparent mass (Mapp)
arising from motion, gravitational potential, or field interactions.
Gravitational force:
Fg = G Meffsource Mefftest / r2
Gravitational acceleration of a test particle:
aeff = Fg / Mefftest
= G Meffsource / r2
Explicitly using the ECM decomposition:
aeff = G ( MM − Mappsource ) / r2
M in dynamic, gravitational, or relativistic expressions is replaced by
Meff = MM − Mapp in ECM.
2 G M / r c2 → 2 G MM / r c2must be combined with acceleration calculations using effective mass:
aeff = Fg / Mefftest
MM identifies the source;
Meff governs the actual dynamical response.
| Classical Concept | Classical Expression | ECM Equivalent | Notes / ECM Insight |
|---|---|---|---|
| Source mass | M | Meffsource = MM − Mappsource | Effective source mass incorporates apparent mass; replaces bare mass in all gravitational contexts. |
| Test particle mass | m | Mefftest = MM − Mapptest | Effective mass governs inertial response; includes negative apparent mass contributions. |
| Gravitational force | Fg = G M m / r2 | Fg = G Meffsource Mefftest / r2 | Force depends on both effective masses; fully dynamical. |
| Gravitational acceleration | a = Fg / m = G M / r2 | aeff = Fg / Mefftest = G Meffsource / r2 | Acceleration depends on source effective mass; test mass cancels properly in ECM context. |
| Schwarzschild factor | 2 G M / r c2 | 2 G Meffsource / r c2 | Relativistic corrections reference effective source mass; test particle dynamics enter separately via aeff. |
| Potential energy | PE = −G M m / r | PEeff = −G Meffsource Mefftest / r | Reflects full dynamic interaction between effective masses. |
| Escape velocity | vesc = √(2 G M / r) | vesceff = √(2 G Meffsource / r) | Escape velocity depends solely on effective source mass. |
| Gravitational redshift | Δf / f = G M / r c2 | Δf / f = G Meffsource / r c2 | Redshift depends on effective source mass; energy–momentum exchanges accounted via ECM. |
| Momentum exchange in field | – | Δρ = ± G Meffsource / r2 · Δt | Photon or particle momentum responds symmetrically to the gravitational field; total energy preserved. |
| Classical kinetic energy | KE = ½ m v2 | KEeff = ½ Mefftest v2 | Effective mass governs kinetic energy in ECM dynamics. |
| Energy conservation (photons) | E = h f | E + Eg = h f + ΔEg | Distinguishes intrinsic photon energy and gravitational-interaction energy. |
| Classical Term / Expression | ECM Replacement | Notes / ECM Insight |
|---|---|---|
| M (mass, source or generic) | Meff = MM − Mapp | Effective mass; replaces bare mass in all dynamic or gravitational contexts. |
| m (test particle) | Meff = MM − Mapp | Test particle’s effective mass includes apparent mass effects. |
| Fg = G M m / r2 | Fg = G Meffsource Mefftest / r2 | Gravitational force includes effective masses. |
| a = G M / r2 | aeff = G Meffsource / r2 | Acceleration depends solely on source effective mass. |
| PE = −G M m / r | PEeff = −G Meffsource Mefftest / r | Potential energy incorporates both source and test effective masses. |
| vesc = √(2 G M / r) | vesceff = √(2 G Meffsource / r) | Escape velocity governed by effective source mass. |
| Δf / f = G M / r c2 | Δf / f = G Meffsource / r c2 | Redshift depends on effective source mass. |
| 2 G M / r c2 (Schwarzschild term) | 2 G Meffsource / r c2 | Relativistic correction uses effective mass of the gravitating body. |
| ρ = h / λ (Photon momentum) | ρeff = h / λ ± Δρ | Momentum exchange with gravitational field. |
| E = h f (Energy conservation) | E + Eg = h f + ΔEg | Distinguishes intrinsic and field-interaction energy. |
| KE = ½ m v2 | KEeff = ½ Mefftest v2 | Kinetic energy governed by effective mass. |