Anton Rize

@studiot Thanks for engaging I would like to see it if someone already posted something like this. Id be grateful if you could give the link to this post. To clarify: when I write “Spacetime ≡ Energy”, it is not a dimensional equation or a metaphysical claim. It means that what we call spacetime geometry and what we call energy dynamics are two complementary aspects of a single conserved structure. In standard formulations, geometry provides the arena and energy fills it; here, that distinction is removed. Geometry is the bookkeeping of energy transformations, not their container. The framework uses two closure relations: \[ \beta_X^2 + \beta_Y^2 = 1, \qquad \kappa_X^2 + \kappa_Y^2 = 1, \] describing directional (kinematic) and omnidirectional (gravitational) transformations. From these, the usual SR/GR relations emerge algebraically - for example \(E^2 = p^2 + m^2\) and the gravitational time-dilation factor. So “Spacetime ≡ Energy” is shorthand for removing the last unnecessary separation in physics - between structure and dynamics. If anyone is interested, I can post the short derivation of these closure relations directly here. Thanks for the comment. In standard formulations energy is indeed treated as a property within spacetime - derived from the stress-energy tensor or Hamiltonian of a given system. Here, “Spacetime ≡ Energy” does not equate two things, but removes an unnecessary logical separation: if every geometric property (metric, curvature, causal structure) is inferred only through energy distributions, then treating spacetime as an independent background duplicates ontology. In this FRAMEWORK, energy is defined relationally - as the conserved measure of transformation between possible states. Spacetime geometry then becomes the representation of that conservation, not its container. So it’s not that spacetime “is made of energy,” but that both are two sides of the same relational bookkeeping. Yes, precisely - and I’m glad you mentioned Eddington (though my knowledge of his work is quite shallow). His idea that “a coordinate system is strictly unnecessary” is very close to what I’m doing here. In my approach, the physical and mathematical content remain intact - what changes is what we treat as fundamental. Instead of embedding energy inside geometry, I treat geometry itself as the algebraic closure of relational energy transformations. So the equations of SR/GR still hold - they simply emerge as different projections of one conserved relational structure. Fair point and I completely understand the suspicion. A lot of LLM-written physics have made the word framework radioactive lately. But no, this is just me a very human person trying to simplify how we look at SR/GR algebraically. If it reads a bit too polished, that’s probably because I’ve been refining it for months, not minutes. Still, I appreciate the caution we probably need more of it in modern discussions.

This post is for peer testing and discussion. I'm posting this to spark discussion – has anyone seen similar geometric approaches? Students, if you're learning relativity and frustrated with the math, try these derivations and let me know if they click. Test the code, replicate the algebra for other cases (like Mercury's orbit etc), and share your thoughts/results. Could this simplify teaching GR? All equations are purely algebraic and reproduce known results of Special and General Relativity exactly. Feedback on mathematical structure, reproducibility, and numerical testing is welcome. --- Abstract I’m an independent researcher working on a framework called WILL Relational Geometry, which reproduces the main results of SR and GR using only simple algebraic projections on circles and spheres — no tensors, metrics, or differential equations. The model is based on one principle: Spacetime ≡ Energy — meaning geometry and dynamics are not separate but mutually defined. Paper: https://doi.org/10.5281/zenodo.17115270 (October 2025) Full text (PDF): https://github.com/AntonRize/WILL/blob/46ed9a336bd99607033a811cd3160088a07d3851/documents/WILL_PART_I_SR_GR.pdf --- Core Summary - Everything emerges from normalized projections on S¹ and S²: β → kinematic (v/c) κ → potential (√(Rₛ / r)) - Closure rule: κ² = 2β² — a purely geometric relation similar to the virial condition but derived geometrically. - SR and GR factors appear as projections of the same conserved energy relation. - Spacetime and energy transformations are two faces of one invariant. (Full symbolic table attached as image for clarity.) --- Test 1: Photon Sphere Radius GR: Derived via geodesics → r = 1.5 Rₛ. WILL: Set equilibrium θ₁ = θ₂ ⇒ β² + κ² = 1 ⇒ κ² = 2β² ⇒ κ² = 2/3 ⇒ r = Rₛ / κ² = 1.5 Rₛ. For the Sun (M ≈ 1.989×10³⁰ kg, Rₛ ≈ 2.95 km) → r ≈ 4.425 km. Matches GR exactly. --- Test 2: GPS Time Dilation Offset Known value: net +38.5 μs/day (satellite runs faster). WILL: Unified equation for both effects: τ = √(1 – κ²) × √(1 – β²) where κ² = 2GM/(c²r), β = v/c. Python verification (if python not your thing - down below there's link to Desmos projects ready to go): import numpy as np # Constants G = 6.67430e-11 c = 2.99792458e8 # Earth / GPS parameters (SI) M_earth = 5.972e24 # kg R_earth = 6.37e6 # m (mean radius) r_gps = 2.6571e7 # m (GPS orbital radius) # Orbital speed (circular) v_gps = np.sqrt(G * M_earth / r_gps) # Dimensionless parameters beta = v_gps / c kappa_gps = np.sqrt(2 G M_earth / (c**2 * r_gps)) kappa_earth = np.sqrt(2 G M_earth / (c**2 * R_earth)) # Proper-time factors (WILL unified form) tau_gps = np.sqrt(1 - kappa_gps**2) np.sqrt(1 - beta*2) tau_earth = np.sqrt(1 - kappa_earth**2) # Net daily offset (satellite clock faster => positive) offset_us_per_day = (1 - tau_earth / tau_gps) 86400 1e6 print(f"beta = {beta:.9e}") print(f"kappa_gps = {kappa_gps:.9e}, kappa_earth = {kappa_earth:.9e}") print(f"tau_gps = {tau_gps:.12f}, tau_earth = {tau_earth:.12f}") print(f"Offset ≈ {offset_us_per_day:.2f} μs/day") Result: 38.52 μs/day — perfect match. --- Discussion The same framework reproduces: - Lorentz factor and energy–momentum relation. - Schwarzschild potential without curvature formalism. - ISCO (3 Rₛ) and Kerr limits algebraically. - Natural singularity removal without extra assumptions. Interactive demonstrations with ready desmos projects: https://antonrize.github.io/WILL/relativistic-foundations/ https://antonrize.github.io/WILL/ https://antonrize.github.io/WILL/predictions/ https://antonrize.github.io/WILL/results/ --- Invitation This post invites independent testing of a fully algebraic approach to SR and GR that replaces tensor formalism with geometric projections If you reproduce the above results — or find where it fails — please share your numbers. Constructive criticism and falsification proposals are very welcome. --- License: CC BY-NC 4.0 — free for scientific use and replication. © 2025 Anton Rize