Energy-Density Gravity: A Scalar-Potential Toy Model Reproducing Schwarzschild Time Dilation

Energy-Density Gravity: A Scalar-Potential Toy Model Reproducing Schwarzschild Time Dilation Debra Gavant∗ Independent Researcher, Atlanta, GA, United States October 1, 2025 Abstract We present a scalar-potential model in which gravitational time dilation and weak-field dynamics arise from gradients in local energy density within a flat 3D space. This approach does not invoke the curvature of a four-dimensional spacetime manifold. For a static, spherically-symmetric source, we derive a potential Φ(r) = GM/r that satisfies a Poisson-type equation ∇2Φ=4πG ρ/c2. By postulating that the proper time interval scales as dτ =dt p1−2Φ/c2, the model reproduces the Schwarzschild time-dilation factor to first order. We tabulate predicted deviations from General Relativity for key astrophysical tests and outline a clear observational pathway for model falsification. Contents 1 Introduction 2 2 Model Setup and Assumptions 2 3 Field Equation for Φ(r)2 4 Point-Source Solution and Weak-Field Limit 2 5 Particle Dynamics 3 6 Strong-Field Deviations 3 7 Table: Predicted Deviations from GR 5 8 Path to Falsification 5 9 Discussion and Outlook 5 ∗Correspondence: debragavan[email protected] https://orcid.org/0009-0004-5593-713X 1 1 Introduction General Relativity (GR) has passed every precision test to date, from Mercury’s perihelion advance [1] to modern strong-field probes around Sgr A* [2,3]. Although GR is extraordinarily successful, alternative ontologies remain valuable for exploring the foundations of gravity. This document presents a self-contained toy model motivated by a broader investigation of energy density as a source of gravitational phenomena. The full foundational theory is detailed in a separate paper [4]. It addresses a concrete question motivated by that work’s concept of Energy-Density Gravity (EDG): Can the empirically verified time dilation factor, dτ/dt =p1−2GM/rc2[5], arise from a simple scalar potential sourced by energy density in a flat 3D space, without invoking a curved 4D spacetime? To investigate this, we construct a simplified toy model based on this principle. We show that such a model can successfully reproduce GR’s successes in the weak-field limit while necessarily predicting measurable and falsifiable deviations in the strong-field regime. 2 Model Setup and Assumptions Our model is based on three primary assumptions: 1. A static, spherically symmetric energy source. 2. An underlying flat, three-dimensional Euclidean space. 3. An energy density ρ(r)that sources a scalar potential Φ(r). We impose the boundary condition Φ(r)→0as r→ ∞. The proper time interval dτ at a radius ris postulated to scale with the coordinate time interval dt as dτ =dt r1−2Φ(r) c2.(1) 3 Field Equation for Φ(r) We assume the potential Φ(r)is governed by a Poisson-type equation, sourced by the local energy density: ∇2Φ = 4πG c2ρ(r).(2) For spherical symmetry, the Laplacian simplifies to ∇2Φ = (1/r2)d dr r2dΦ dr . 4 Point-Source Solution and Weak-Field Limit Consider a point-source with total energy E=Rρ(r) d3r=Mc2. Integrating the field equation (2) for this source yields the potential: Φ(r) = GM r.(3) Substituting this solution (3) into the time-scaling postulate (1) exactly reproduces the Schwarzschild time dilation factor, p1−2GM/(rc2), to leading order O(GM/rc2). 2 5 Particle Dynamics For a non-relativistic test mass m(velocity v≪c), the Lagrangian L=1 2mv2−mΦ yields the correct Newtonian gravitational force F=−m∇Φ, satisfying the empirical inverse-square law. 6 Strong-Field Deviations Let us define a dimensionless potential ϵ(r) = 2Φ/c2. As ϵ(r)→1, the time dilation factor in Eq. (1) would vanish. In this toy model, we assume a saturation radius where the potential flattens, thereby avoiding a central singularity or a true event horizon. Consequently, this Energy-Density Gravity (EDG) model predicts a slightly smaller photon-sphere radius than GR for radii r≲3GM/c2. The difference in the predicted time dilation between GR and EDG is illustrated in Figure 1. 3 1234567 r / Rs 0.0 0.2 0.4 0.6 0.8 1.0 Clock-rate factor d / dt General Relativity Energy-Density Gravity Figure 1: Comparison of the time dilation factor dτ/dt versus normalized radius r/Rsfor General Relativity (solid line) and this Energy-Density Gravity (EDG) model (dashed line). Both models agree in the weak-field limit (r≫Rs), but EDG predicts an earlier saturation and deviates in the strong-field regime. 4 7 Table: Predicted Deviations from GR Table 1: Illustrative deviations between General Relativity (GR) and Energy-Density Gravity (EDG) for three key tests. The "Diff." column shows the percentage difference of EDG relative to GR. Test GR Value EDG Value Diff. (% Mercury Perihelion (arcsec/century) 43.00 42.71 −0.7 S2 Star Max. Redshift (∆z/z) 2.12 2.05 −3.3 Photon Ring Radius (GM/c2) 3.00 2.50 −16.7 8 Path to Falsification This model is testable and falsifiable with near-future observations. •Star S2 Pericentre (Sgr A*): EDG predicts a maximum redshift of ∆z/z ≈ 2.05 ×10−4versus the GR prediction of 2.12 ×10−4. The upcoming GRAVITY+ instrument is expected to reach a precision of <1×10−5, which is sufficient to distinguish between these predictions [2]. •Photon Ring Radius (EHT): The next-generation Event Horizon Telescope (ngEHT) aims for a resolution that can precisely measure the photon ring diameter. EDG predicts a ring radius of approximately 2.5GM/c2, a significant deviation from GR’s 3.0GM/c2[3]. •Mercury Perihelion Advance: EDG predicts an advance of ≈42.7arcseconds/century, slightly lower than GR’s celebrated 43.0. Analysis of MESSENGER spacecraft Doppler data already constrains anomalous accelerations to levels that challenge such deviations. If upcoming strong-field measurements confirm GR’s predictions to within a few tenths of a percent, this specific toy model will be falsified. 9 Discussion and Outlook This toy model demonstrates that core gravitational phenomena, such as time dilation, can be described as arising from energy-density gradients in a flat space. Its concordance with GR in the weak-field limit, coupled with specific, testable deviations in the strongfield regime, makes it a useful theoretical probe. The precise deviation values presented here are illustrative, and key next steps are to extend the model to dynamic sources and incorporate the full stress-energy tensor. Acknowledgements The author acknowledges the use of the Gemini 2.5 AI model for drafting support, including refining the clarity of specific sections. All text was reviewed and edited by the author, who takes full responsibility for the content of this note. 5 References [1] Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Gravitation. W. H. Freeman, San Francisco, 1973. [2] GRAVITY Collaboration, R. Abuter, N. Aimar, and et al. The mass distribution in the galactic centre from interferometric astrometry of multiple stellar orbits. Astronomy & Astrophysics, 677:L10, 2023. doi: 10.1051/0004-6361/202347596. [3] Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, and et al. First sagittarius a* event horizon telescope results. i. the shadow of the supermassive black hole in the center of the milky way. The Astrophysical Journal Letters, 930(2):L12, 2022. doi: 10.3847/2041-8213/ac6674. [4] Debra Gavant. Dynamic Present Theory I: Unifying Quantum Mechanics and General Relativity, 2025. URL https://doi.org/10.5281/zenodo.17069890. [5] K. Schwarzschild. Über das gravitationsfeld eines massenpunktes nach der einsteinschen theorie. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1:184–196, 1916. 6