Cosmology Without Expansion: Relational Conformal Projection / Emergent Acceleration from Informational Rapidity Fields

Abstract & Full Description This manuscript introduces the Relational Conformal Projection (RCP) framework, a comprehensive theoretical model that fundamentally reinterprets cosmological observables. By departing from the standard Friedmann-Lemaître-Robertson-Walker (FLRW) paradigm of metric expansion, this work demonstrates that the empirical phenomena defining modern cosmology—such as cosmological redshift, the time dilation of Type Ia Supernovae, and apparent accelerated expansion—can be rigorously derived as geometric consequences of an informational conformal projection within a static 4D Lorentzian manifold $(\mathcal{M}, g_{\mu\nu})$. The RCP framework resolves the historical vulnerabilities of static universe models (e.g., the Tolman surface brightness paradox and the Shapiro delay modulation) by replacing ad-hoc kinematic assumptions with a complete variational field theory. 1. Core Mathematical Ontology Instead of a time-varying scale factor $a(t)$, the RCP model postulates that the observational differences between an emission event $B$ and an observation event $A$ are governed by a scalar informational rapidity field $\Phi$. The metric perceived by the observer is conformally related to the static background metric: $$\tilde{g}_{\mu\nu} = e^{-2\Phi} g_{\mu\nu}$$ Because conformal transformations preserve the causal structure (null geodesics), the geometric paths of light rays remain invariant, but the affine parametrization and transverse lengths are rescaled. This yields the fundamental observable mappings: Redshift: $1+z = e^\Phi$ Time Dilation: $dt_{obs} = e^\Phi dt_{emit}$ Transverse Area: $A_{obs} = e^{-2\Phi} A_{emit}$ 2. Dynamical Foundations and the Action Principle The field $\Phi$ is not introduced as a phenomenological fit, but is governed by a principle of informational coherence. The assignment of $\Phi$ minimizes a 4D action, which, when integrated along null geodesics parametrized by an affine parameter $s$, reduces to an effective 1D Lagrangian: $$\mathcal{L}(\Phi, \partial_s \Phi; s) = \frac{1}{2} (\partial_s \Phi)^2 - S(\Phi; s)$$ Applying the Euler-Lagrange equations yields the deterministic coherence equation for the relational field: $$\frac{d^2\Phi}{ds^2} = \partial_\Phi S(\Phi)$$ 3. The Origin of Dark Energy and the Coupling Constant A major result of this paper is the elimination of Dark Energy as a physical fluid or vacuum energy. By postulating a minimal exponential potential for the informational field, $S(\Phi) = -\lambda^2 e^{\alpha \Phi}$ (where $\lambda \approx H_0/c$), the model naturally produces an accelerating apparent expansion. Expanding $\Phi(s)$ locally and matching it with the kinematic deceleration parameter $q_0$ of the local Hubble law, we analytically derive the exact coupling constant $\alpha$: $$\alpha = q_0 - 1$$ Given the observational consensus of $q_0 \approx -0.55$, the field potential is strictly constrained to $\alpha \approx -1.55$. This proves that cosmic acceleration is not a dynamical force pulling galaxies apart, but an inherent non-linearity in the relational projection of spacetime. 4. Resolution of the Tolman Paradox Historically, static models failed the Tolman surface brightness test. In the RCP framework, the conformal rescaling of the metric dictates that the observed surface brightness $B_{obs}$ scales exactly as predicted by expanding universe models. Factoring in the energy reduction of photons ($e^{-\Phi}$), the time dilation ($e^{-\Phi}$), and crucially, the conformal transverse area contraction ($A \to e^{-2\Phi}A$), the model inherently yields: $$B_{obs} \propto \frac{e^{-\Phi}}{e^{-2\Phi} e^{\Phi}} = e^{-4\Phi} = (1+z)^{-4}$$ This provides an exact geometric match to observations without requiring physical space to stretch. 5. Implications for CMB and Primordial Nucleosynthesis (BBN) The manuscript extends the RCP formalism to the early universe. It outlines a novel operational program where: BBN: The standard Boltzmann equations for nuclear reaction rates $\Gamma$ are rewritten using relational time gradients, yielding an effective interaction rate $\Gamma_{eff}(s) = \Gamma(T) e^{-\Phi(s)}$. This successfully mimics the "freeze-out" mechanics of an expanding universe. CMB: The acoustic peaks of the Cosmic Microwave Background are reinterpreted not as physical sound waves in a dilating plasma, but as conditions of constructive interference of the informational phase $\varphi_k = \int \kappa_k \Phi(s) ds$ integrated across adjacent lines of sight. Conclusion This paper provides the mathematical scaffolding for a new cosmological paradigm. By demonstrating that a static geometry can produce the exact kinematic signatures of the $\Lambda$CDM model through conformal relationality, it invites a profound ontological shift in theoretical physics. The framework is strictly falsifiable and provides a direct computational pipeline for confronting MCMC parameter estimations with Type Ia Supernovae, Strong Lensing time delays, and CMB angular power spectra. This manuscript is current in Official Peer Review. Not final version.Copyright©2026 Alex De Giuseppe.All rights reserved. This work is protected by copyright. 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