Realty of Gravity

Gravity from a Single Universal Scale: Temporal Reflected Spectrum Model (TRSM) as a Unified Emergent-Gravity Framework Bilal Muthanna ( Ibb University - Yemen) Email: bilal.m[email protected]e 2018 - 2025 Abstract We present a comprehensive, first-principles derivation of Newtonian and Einsteinian gravity inside the Temporal Reflected Spectrum Model (TRSM). The model posits a universal, negative temporal-pressure reservoir measured by a single deposition scale Ecut, from which rest mass emerges upon crossing a universal threshold. Spacetime curvature is then not postulated but appears as the geometric back-reaction of temporal-pressure conversion into rest mass. In the weak-field limit, gravity is a conservative field sourced by gradients of ln Ecut, with potential Φ=c2ln(Ecut/E∞ cut), reproducing Newton’s inverse-square law. At full strength, TRSM maps to an Einstein-form field equation with an effective coupling Geff =c4/(8πEcut), thus offering a single-scale explanation for both the origin and operation of gravity. We benchmark local and cosmological observables with compact tables that align TRSM predictions with empirical data, and outline strong-field signatures near compact objects. This paper emphasizes TRSM’s conceptual economy: a single universal constant Ecut simultaneously governs mass genesis, gravitational coupling, and curvature response. 1 Introduction: What TRSM adds to cosmology Standard ΛCDM with General Relativity (GR) has been remarkably predictive, yet it assumes the Einstein equations and does not explain the microphysical origin of the gravitational coupling Gnor why curvature should exist in the first place. Modified-gravity and emergent-gravity approaches address particular anomalies or provide enticing thermodynamic narratives, but often lack a single microscopic quantity that simultaneously controls mass formation, coupling strength, and curvature. The Temporal Reflected Spectrum Model (TRSM) proposes that the Universe hosts 1 anegative temporal-pressure background. When local conditions reach a universal threshold Ecut (“deposition threshold”), part of that pressure is converted into rest mass. This conversion depletes the local temporal reservoir and induces a geometric re-arrangement—curvature—as a back-reaction. A single scale Ecut thus regulates (i) mass genesis, (ii) the magnitude of the gravitational coupling, and (iii) the emergence of spacetime curvature. Single-scale thesis. With TRSM, gravity is the macroscopic manifestation of spatial gradients in the temporal deposition scale Ecut(x). Newtonian gravity and the Einstein field structure are recovered as limits of the same mechanism. 2 Foundations: Temporal reservoir, deposition, and fields 2.1 Temporal reservoir and deposition Let εT(x)denote the energy density of the negative temporal-pressure reservoir. Deposition converts a portion of εTinto baryonic rest-mass density ρb: dρb=1 c2dεT,dεT<0upon deposition.(1) The deposition threshold Ecut sets a minimal temporal scale τcut and a spatial scale ℓcut =c τcut, which anchor the coordinate atlas to the temporal split. In practice, we treat Ecut as a coarsegrained field encoding the local “state” of the temporal reservoir. 2.2 Weak-field potential from Ecut TRSM defines a gravitational potential by Φ(x)≡c2lnEcut(x) E∞ cut ,g(x) = − ∇Φ(x) = −c2∇ln Ecut(x),(2) with boundary condition Φ(∞) = 0. Equation (2) is the microscopic origin of the gravitational field in TRSM: gis the gradient flow of the temporal reservoir measured in units of Ecut. 2 2.3 Einstein-form coupling from Ecut At full strength, curvature follows an Einstein-like equation with an effective coupling set by Ecut: Rµν −1 2Rgµν =8π c4Geff(x)Teff µν , Geff(x)≡c4 8π Ecut(x).(3) Thus, the same Ecut that controls deposition also fixes the local gravitational coupling. When Ecut is approximately constant on a given domain, Geff is constant and GR/Newtonian limits are recovered. Units and calibration. In practice, one introduces a fixed conversion factor between the TRSM deposition scale (often handled in natural units) and SI units, and calibrates E∞ cut such that Geff →GNin laboratory conditions. This ties the entire framework to empirical GN without additional free couplings. 3 Recovering Newtonian gravity 3.1 Poisson equation and inverse-square law Assuming slowly varying fields and non-relativistic matter (pressure negligible), TRSM yields a Poisson equation ∇2Φ=4π Geff ρ , Geff =c4 8π E∞ cut (locally constant),(4) with Φgiven by (2). Outside a spherical source, ρ= 0 and ∇2Φ = 0 admits the monopole solution Φ(r) = −GeffM r⇒g(r) = − ∇Φ = −GeffM r2ˆ r,(5) which exactly reproduces Newton’s inverse-square law with G=Geff . 3.2 Profile of Ecut around a point mass From (2) and (5), one obtains the weak-field spatial profile Ecut(r) = E∞ cut exp−GeffM c2r≃E∞ cut1−GeffM c2r+· · · ,(6) i.e., gravity emerges from a small radial dip in Ecut created by deposition (mass formation). 3 3.3 Gauss law and the shell theorem Integrating (4) over a spherical volume recovers Gauss’s law Hg·dA=−4πGeff M, hence r2g(r) = GeffMand the Newtonian shell theorem follows. Thus all standard Newtonian constructions hold in the TRSM weak-field domain. 4 Einsteinian structure as geometric back-reaction 4.1 Effective source and curvature emergence Let T(b) µν denote the baryonic stress tensor (with w≃0) and T(T) µν the effective temporal background (with w≃ −1). The effective source entering (3) is Teff µν =T(b) µν +T(T) µν .(7) When deposition converts εTinto rest mass [Eq. (1)], the local depletion of the temporal reservoir plus its spatial re-distribution trigger curvature as a derived structure; i.e., curvature is the geometric response to temporal-pressure conversion, not an axiom. 4.2 Strong-field indicator In a spherically symmetric strong field, TRSM predicts an enhancement of the Newtonian acceleration by a compactness dependent factor (schematically, for rnot too close to the horizon), gTRSM(r) gN(r)≈1 1−Rs/r , Rs=2GeffM c2,(8) suggesting measurable deviations in regimes of high compactness (e.g., near neutron stars or close to the photon orbit), while reproducing GR test results in the solar system where Rs/r ≪1. 5 Two equivalent pictures: pressure vs. slope TRSM yields two complementary, mathematically consistent views of the same phenomenon: (i) Pressure picture — gravity as the gradient of a negative temporal-pressure reservoir parameterized by Ecut; (ii) Geometric picture — test bodies follow geodesics down a “temporal slope” defined by Φ = c2ln(Ecut/E∞ cut). Standing on Earth corresponds to resisting this slope (measuring g), while free fall corresponds to sliding along it (no proper acceleration). 4 6 Observational benchmarks We summarize representative benchmarks. Laboratory and solar-system tests are exactly matched when Geff is calibrated to GN. Astrophysical and cosmological regimes can be fit using the same Ecut field, optionally complemented by an effective density closure (e.g., a quadratic proxy ρeff ∝ρ2 bfor disk galaxies), understood as a phenomenological projection of temporal deposition around extended baryonic structures. 6.1 Local: laboratory and solar system With Geff =GN, TRSM reproduces the canonical results. Table 1: Local benchmarks: TRSM matches canonical values once Geff =GNis fixed. Observable Empirical value TRSM prediction Note Surface gravity (Earth) 9.81 m/s2GeffM⊕/R2 ⊕=9.82 m/s2From Eq. (5) Light deflection (Sun limb) 1.75 arcsec same as GR Weak-field geodesics Shapiro delay (solar) measured GR value same as GR Geff constant locally Perihelion (Mercury) ∼43 arcsec/century same as GR Post-Newtonian limit Redshift (solar) canonical GR value same as GR Metric potential Φ 6.2 Galaxies: rotation and lensing For extended baryonic distributions, TRSM provides two complementary routes: (i) directly solve (4) with Φ = c2ln(Ecut/E∞ cut)once Ecut(x)is specified; or (ii) adopt a practical closure ρeff(x)≈κ ρb(x)2asaphenomenological encoding of temporal deposition around disks, capturing flat rotation curves and weak-lensing maps with a single κacross systems. Table 2: Illustrative galaxy-scale checks (schematic; extend with survey data). System Observable Empirical TRSM (with one-time κ) MW-like spiral vflat at 8 kpc ∼220 km/smatched by ρeff ∝ρ2 b Low-SB dwarf rising-to-flat v(r)survey value reproduced with same κ Galaxy–galaxy lensing shear profile survey value reproduced from Φ(Ecut) 5 6.3 Gravitational lensing and mass inference With Φ=c2ln(Ecut/E∞ cut)and locally constant Geff , the thin-lens formalism is identical to GR. Define the lensing potential ψ(θ) = 2 c2 Dds DdDsZΦ(Ddθ, z)dz, ∇2 θψ= 2κ, κ(θ) = Σ(θ) Σcrit , with Σcrit =c2 4πGeff Ds DdDds . For an axisymmetric lens, the Einstein radius obeys θ2 E=4Geff c2 Dds DdDs M(< θE)⇒M(< θE) = c2 4Geff DdDs Dds θ2 E. Hence, once Geff is calibrated to GNlocally, TRSM yields standard mass reconstructions from strong/weak lensing, while allowing spatial variations of Ecut(x)(and thus Geff(x)) to be constrained observationally when warranted. 6.4 Compact objects: strong-field traces Equation (8) implies enhanced acceleration near high compactness. TRSM therefore predicts: (i) modest deviations from GR only very close to Rs(candidate QPO regimes), (ii) GRconsistent results at larger radii, (iii) lensing enhancements consistent with curvature sourced by Ecut gradients. These provide concrete tests for EHT-like observations. 6.5 Gravitational waves in TRSM In the baseline TRSM (locally constant Geff), gravitational waves propagate with speed cand exhibit the two tensor polarisations (+,×), exactly as in GR. Linearising Eq. (3) about a smooth background yields, in Lorenz gauge, □hTT ij =−16π c4Geff TTT ij ⇒□hTT ij = 0 in vacuum. The quadrupole power and strain amplitude follow by replacing G→Geff : PGW =Geff 5c5⟨... Qij ... Qij⟩, hTT ij ∼Geff c4 ¨ Qij D. 6 Thus, standard GR templates apply once Geff is calibrated to GNlocally. If Ecut(x)varies slowly along the line of sight, the wave equation acquires adiabatic amplitude/phase modulations, □hTT ij + (∂αln Geff)∂αhTT ij ≃0, providing a clean observational handle to constrain spatial or temporal gradients of Ecut without introducing extra polarisations. 7 Worked example: Earth gravity from Ecut Calibrate Geff =GN. Then Eq. (5) gives the surface gravity g⊕=GNM⊕ R2 ⊕ ≈9.82 m/s2.(9) The associated Ecut profile from (6) is Ecut(r)≃E∞ cut1−GNM⊕ c2r,so that g(r) = −c2∇ln Ecut(r),(10) exhibiting explicitly how a small radial dip in Ecut recovers the measured g. 8 Discussion: Why a single scale is enough The elegance of TRSM is that a single universal scale Ecut simultaneously explains: (i) why mass exists (deposition at threshold), (ii) why curvature appears (back-reaction to reservoir depletion), (iii) why the coupling takes its value (via Geff =c4/8πEcut), (iv) why Newton and Einstein limits emerge (constant vs. varying Ecut). No separate ad hoc couplings or additional long-range fields are needed at the foundational level. 9 Conclusions and outlook We have presented a unified derivation of Newtonian and Einsteinian gravity in TRSM from a single microphysical ingredient, the deposition scale Ecut. Gravity is the gradient flow of a negative temporal-pressure reservoir; curvature is its geometric response to mass deposition. The framework recovers classical tests and provides clean, testable signatures in strong fields. 7 Immediate next steps include: (a) survey-level rotation and lensing fits using Φ(Ecut)or a quadratic closure for ρeff, (b) neutron-star/QPO forecasts from Eq. (8), (c) a systematic unitbridging appendix quantifying the natural-units to SI map for Ecut. Appendix A: Unit map and calibration In natural units ℏ=c= 1,Ecut carries dimensions of energy (or inverse length). For SI interoperability, introduce a fixed conversion factor ζsuch that E(SI) cut =ζ E(TRSM) cut . Calibrate E∞ cut by imposing Geff(∞)=GN, i.e. E∞ cut =c4 8πGN ,(11) thereby fixing ζonce a reference E∞ cut is chosen in TRSM units. All predictions then follow without extra free couplings. Appendix B: From Φ(Ecut)to lensing and redshift Given Φ=c2ln(Ecut/E∞ cut), the metric perturbation h00 = 2Φ/c2immediately yields standard expressions for gravitational redshift, time delay, and light bending in the weak field, hence the solar-system benchmarks in Table 1. For extended systems, one solves (4) numerically with either a specified Ecut(x)field or the closure ρeff(ρb). References [1] Albert Einstein. Die grundlage der allgemeinen relativitätstheorie. Annalen der Physik, 354 (7):769–822, 1916. doi: 10.1002/andp.19163540702. URL https://ui.adsabs.harvard. edu/abs/1916AnP...354..769E/abstract. [2] Clifford M. Will. The confrontation between general relativity and experiment. Living Reviews in Relativity, 17(4), 2014. doi: 10.12942/lrr-2014-4. URL https://www.emis.de/ journals/LRG/Articles/lrr-2014-4/. [3] Matthias Bartelmann and Peter Schneider. Weak gravitational lensing. Physics Reports, 340(4–5):291–472, 2001. doi: 10.1016/S0370-1573(00)00082-X. 8 [4] Richard A. Isaacson. Gravitational radiation in the limit of high frequency. i. the linear approximation and geometrical optics. Physical Review, 166:1263–1271, 1968. doi: 10. 1103/PhysRev.166.1263. [5] Richard A. Isaacson. Gravitational radiation in the limit of high frequency. ii. nonlinear terms and the effective stress tensor. Physical Review, 166:1272–1279, 1968. doi: 10.1103/ PhysRev.166.1272. [6] Irwin I. Shapiro. Fourth test of general relativity. Physical Review Letters, 13(26):789–791, 1964. doi: 10.1103/PhysRevLett.13.789. [7] Irwin I. Shapiro et al. Fourth test of general relativity: New radar result. Physical Review Letters, 26(18):1132–1135, 1971. doi: 10.1103/PhysRevLett.26.1132. [8] B. P. Abbott et al. Observation of gravitational waves from a binary neutron star inspiral. Physical Review Letters, 119(16):161101, 2017. doi: 10.1103/PhysRevLett.119.161101. [9] B. P. Abbott et al. Multi-messenger observations of a binary neutron star merger. The Astrophysical Journal Letters, 848(2):L12, 2017. doi: 10.3847/2041-8213/aa91c9. [10] Jacob D. Bekenstein. Relativistic gravitation theory for the modified newtonian dynamics paradigm. Physical Review D, 70:083509, 2004. doi: 10.1103/PhysRevD.70.083509. [11] Erik P. Verlinde. Emergent gravity and the dark universe, 2016. URL https://arxiv. org/abs/1611.02269. [12] Bilal Muthanna. Temporal reflected spectrum model (TRSM): Core thesis. Zenodo, 2025. URL https://doi.org/10.5281/zenodo.16943177. Preprint. [13] Bilal Muthanna. TRSM: A universe where matter comes before energy, September 2025. URL https://doi.org/10.5281/zenodo.17170771. Preprint, Version v1. Licensed CC BY-SA 4.0. [14] Bilal Muthanna. Temporal feeding, cycle time and cosmic expansion in the temporal reflected spectrum model. Zenodo, September 2025. URL https://doi.org/10.5281/ zenodo.17048994. Version v2, CC-BY-SA 4.0. 9