Paper 15: The Scale-Integration Reduction: Effective Four-Dimensional Einstein Equations from the Five-Dimensional Scale-Space Field Theory
Fuente:
Zenodo
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Recurso digital |
| Lingua: | inglese |
| Pubblicazione: |
Zenodo
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866902106448855040 |
|---|---|
| author | Palmer, Donald |
| author_facet | Palmer, Donald |
| contents | <p>Paper 14 of this series derived Newton’s law from the linearised 5D Einstein equations with a point mass source, establishing the kinematic Poisson equation of Paper 1 as a dynamical theorem. It identified a residual open problem: the coupling constant κ5 = 2G/[(L + 2)c^2L] derived in that paper depends on the compactness parameter L = Rc^2/(GM) of the source body, making κ5 body-dependent rather than universal. The present paper performs the s-integration reduction of the 5D field equations, showing how effective 4D Einstein equations emerge and how the body-dependence is resolved.</p> <p>We prove four results. First, the spatial and temporal projections of the 5D equations onto a fixed scale surface s = s0 reduce exactly to the standard 4D Einstein equations in the weak-field limit, recovering GR at a single scale. Second, the matter-weighted s-integration — averaging the 5D equations with the physical volume element W(s) = e^(3s/L) over the support of the matter source — produces effective 4D equations with coupling κeff depending on L. Third, the universal background stress-energy T^(5) bg,tt identified in Paper 11, together with the identity Ls˙ = 2c from Paper 12, provides the normalisation condition that fixes κeff independently of any particular body: κeff = 8πG/c^4 = κ4. Fourth, the effective 4D cosmological constant is Λeff = 3/(c^2L^2), which in the physical limit L → Luniverse ∼ 1060 nats gives Λeff ∼ 10−120 in Planck units — consistent in order of magnitude with the observed value. General relativity is thereby recovered as the effective 4D theory of the scale-space framework, valid when scale structure is averaged over a matter source’s characteristic scale range.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20088927 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Paper 15: The Scale-Integration Reduction: Effective Four-Dimensional Einstein Equations from the Five-Dimensional Scale-Space Field Theory Palmer, Donald scale space dimensional reduction effective 4D Einstein equations Kaluza-Klein Randall-Sundrum cosmological constant GR recovery AdS4 background stress-energy <p>Paper 14 of this series derived Newton’s law from the linearised 5D Einstein equations with a point mass source, establishing the kinematic Poisson equation of Paper 1 as a dynamical theorem. It identified a residual open problem: the coupling constant κ5 = 2G/[(L + 2)c^2L] derived in that paper depends on the compactness parameter L = Rc^2/(GM) of the source body, making κ5 body-dependent rather than universal. The present paper performs the s-integration reduction of the 5D field equations, showing how effective 4D Einstein equations emerge and how the body-dependence is resolved.</p> <p>We prove four results. First, the spatial and temporal projections of the 5D equations onto a fixed scale surface s = s0 reduce exactly to the standard 4D Einstein equations in the weak-field limit, recovering GR at a single scale. Second, the matter-weighted s-integration — averaging the 5D equations with the physical volume element W(s) = e^(3s/L) over the support of the matter source — produces effective 4D equations with coupling κeff depending on L. Third, the universal background stress-energy T^(5) bg,tt identified in Paper 11, together with the identity Ls˙ = 2c from Paper 12, provides the normalisation condition that fixes κeff independently of any particular body: κeff = 8πG/c^4 = κ4. Fourth, the effective 4D cosmological constant is Λeff = 3/(c^2L^2), which in the physical limit L → Luniverse ∼ 1060 nats gives Λeff ∼ 10−120 in Planck units — consistent in order of magnitude with the observed value. General relativity is thereby recovered as the effective 4D theory of the scale-space framework, valid when scale structure is averaged over a matter source’s characteristic scale range.</p> |
| title | Paper 15: The Scale-Integration Reduction: Effective Four-Dimensional Einstein Equations from the Five-Dimensional Scale-Space Field Theory |
| topic | scale space dimensional reduction effective 4D Einstein equations Kaluza-Klein Randall-Sundrum cosmological constant GR recovery AdS4 background stress-energy |
| url | https://doi.org/10.5281/zenodo.20088927 |