|
|
Gravitational force and acceleration are written as:
F_g = G M m / r^2 a = F_g / m = G M / r^2
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
M_eff = M_m - M_app
incorporates both the intrinsic matter mass M_m and corrections due to negative apparent mass (M_app) arising from motion, gravitational potential, or field interactions.
Gravitational force:
F_g = G M_eff_source M_eff_test / r^2
Gravitational acceleration of a test particle:
a_eff = F_g / M_eff_test = G M_eff_source / r^2
Explicitly using the ECM decomposition:
a_eff = G (M_m - M_app_source) / r^2
M in dynamic, gravitational, or relativistic expressions is replaced by M_eff = M_m - M_app in ECM to preserve dynamic consistency.2 G M / r c^2 → 2 G M_m / r c^2 (source mass explicit)must be combined with acceleration calculations using effective mass:
a_eff = F_g / M_eff_test
M_m identifies the source; M_eff governs the actual dynamical response.| Classical Concept | Classical Expression | ECM Equivalent | Notes / ECM Insight |
|---|---|---|---|
| Source mass | M | M_eff_source = M_m - M_app_source | Effective source mass incorporates apparent mass; replaces bare mass in all gravitational contexts. |
| Test particle mass | m | M_eff_test = M_m - M_app_test | Effective mass governs inertial response; includes negative apparent mass contributions. |
| Gravitational force | F_g = G M m / r^2 | F_g = G M_eff_source M_eff_test / r^2 | Force depends on both effective masses; fully dynamical. |
| Gravitational acceleration | a = F_g / m = GM / r^2 | a_eff = F_g / M_eff_test = G M_eff_source / r^2 | Acceleration depends on source effective mass; test mass cancels properly in ECM context. |
| Schwarzschild factor | 2GM / r c^2 | 2 G M_eff_source / r c^2 | Relativistic corrections reference effective source mass; test particle dynamics enter separately via a_eff. |
| Potential energy | PE = -GM m / r | PE_eff = -G M_eff_source M_eff_test / r | Reflects full dynamic interaction between effective masses. |
| Escape velocity | v_esc = √(2GM/r) | v_esc_eff = √(2 G M_eff_source / r) | Escape velocity depends solely on effective source mass. |
| Gravitational redshift | Δf/f = GM / r c^2 | Δf/f = G M_eff_source / r c^2 | Redshift depends on effective source mass; energy-momentum exchanges accounted via ECM. |
| Momentum exchange in field | – | Δρ = ± G M_eff_source / r^2 ⋅ Δt | Photon or particle momentum responds symmetrically to the gravitational field; total energy preserved. |
| Classical kinetic energy | KE = ½ m v^2 | KE_eff = ½ M_eff_test v^2 | Effective mass governs kinetic energy in ECM dynamics. |
| Energy conservation (photons) | E = h f | E + E_g = h f + ΔE_g | Distinguishes intrinsic photon energy and gravitational-interaction energy. |
| Classical Term / Expression | ECM Replacement | Notes / ECM Insight |
|---|---|---|
| M (mass, source or generic) | M_eff = M_m - M_app | Effective mass; replaces bare mass in all dynamic or gravitational contexts. |
| m (test particle) | M_eff = M_m - M_app | Test particle’s effective mass includes apparent mass effects. |
| F_g = G M m / r^2 | F_g = G M_eff_source M_eff_test / r^2 | Gravitational force includes effective masses. |
| a = GM / r^2 | a_eff = G M_eff_source / r^2 | Acceleration depends solely on source effective mass. |
| PE = -G M m / r | PE_eff = -G M_eff_source M_eff_test / r | Potential energy incorporates both source and test effective masses. |
| v_esc = √(2GM/r) | v_esc_eff = √(2 G M_eff_source / r) | Escape velocity governed by effective source mass. |
| Δf/f = GM / r c^2 | Δf/f = G M_eff_source / r c^2 | Redshift depends on effective source mass. |
| 2GM / r c^2 (Schwarzschild term) | 2 G M_eff_source / r c^2 | Relativistic correction uses effective mass of the gravitating body. |
| ρ = h / λ (Photon momentum) | ρ_eff = h / λ ± Δρ | Momentum exchange with gravitational field. |
| E = hf (Energy conservation) | E + E_g = hf + ΔE_g | Distinguishes intrinsic and field-interaction energy. |
| KE = ½ m v^2 | KE_eff = ½ M_eff_test v^2 | Kinetic energy governed by effective mass. |