>[!warning] >This content has not been peer reviewed. # RST Matter Coupling — The PCG Fix ## The Problem In RST 1.5, the matter action is written as minimally coupled to the Einstein metric: $S_m = S_m[g_{\mu\nu},\, \psi_m]$ If matter couples only to $g_{\mu\nu}$, then varying the total action with respect to the workload field $q$ yields: $\nabla \cdot \left[\mu\,\nabla q\right] = 0$ There is no baryonic source term ($4\pi G \rho_b$) on the right side. The workload field has no information about where the mass is. It satisfies a homogeneous equation and produces no gravitational enhancement. This is a structural incompleteness in the RST 1.5 action. The field equation we have been *using* (and which successfully fits 171 SPARC galaxies; Lelli et al. 2016) is: $\nabla \cdot \left[\mu\,\nabla q\right] = 4\pi G \rho_b$ For this equation to emerge from the action, $q$ must couple to matter. --- ## The Fix: Bekenstein's Phase Coupling (PCG) The solution is imported from Bekenstein's Phase Coupling Gravity (Bekenstein 1988, Phys. Lett. B 202, 497). Matter does not couple to the bare substrate metric $g_{\mu\nu}$. It couples to a **physical metric** $\tilde{g}_{\mu\nu}$ that is conformally related to $g_{\mu\nu}$ through the workload field: $\tilde{g}_{\mu\nu} = e^{2q}\,g_{\mu\nu}$ The matter action becomes: $S_m = S_m[\tilde{g}_{\mu\nu},\, \psi_m]$ ### Why This Works When varying the total action with respect to $q$, the chain rule applied to $S_m[\tilde{g}]$ produces: $\frac{\delta S_m}{\delta q} = \frac{\delta S_m}{\delta \tilde{g}_{\mu\nu}} \cdot \frac{\delta \tilde{g}_{\mu\nu}}{\delta q} = \tilde{T}^{\mu\nu} \cdot 2\tilde{g}_{\mu\nu} \cdot \sqrt{-\tilde{g}} = 2\tilde{T}\sqrt{-\tilde{g}}$ where $\tilde{T} = \tilde{g}^{\mu\nu}\tilde{T}_{\mu\nu}$ is the trace of the matter energy-momentum tensor. For non-relativistic matter, $\tilde{T} \approx -\rho_b c^2$. In the weak-field limit ($q \ll 1$, so $e^{2q} \approx 1$), this gives precisely: $\nabla \cdot \left[\mu\,\nabla q\right] = 4\pi G \rho_b$ The conformal coupling sources the workload field with baryonic matter, producing the MOND field equation that RST has been using successfully. --- ## The Updated RST 1.6 Action $S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi G} - \frac{M^2}{2}\mathcal{K}(A,\nabla A) - \frac{a_0(\theta)^2}{8\pi G}\mathcal{F}(y,\,n) - \varepsilon_{\min} + \lambda(A^\mu A_\mu + 1) \right] + S_m[\tilde{g}_{\mu\nu},\,\psi_m]$ where: $\tilde{g}_{\mu\nu} = e^{2q}\,g_{\mu\nu}$ The only change from RST 1.5 is the matter coupling: $S_m[g_{\mu\nu}] \to S_m[\tilde{g}_{\mu\nu}]$. All other terms are unchanged. The field equations for the metric, aether, and workload field all receive corrections from this coupling, but in the weak-field, quasi-static limit, the galactic-scale phenomenology is identical to RST 1.5. --- ## Physical Interpretation In RST's language, the physical metric $\tilde{g}_{\mu\nu}$ is the metric that matter "experiences." The bare metric $g_{\mu\nu}$ is the substrate's geometric structure. The conformal factor $e^{2q}$ encodes the workload field's influence on how matter moves through the substrate. This is consistent with the relational ontology: matter does not interact with "raw spacetime" — it interacts with the substrate as modified by the workload of maintaining gravitational structure. The workload field $q$ acts as a mediator between the substrate geometry and the matter content. --- ## Implications ### 1. Lensing (Photon Trajectories) Photons are conformally invariant in 4D. A conformal rescaling $\tilde{g}_{\mu\nu} = e^{2q}g_{\mu\nu}$ does not affect null geodesics. Therefore, the scalar field $q$ alone does not bend light — photons follow the geodesics of $g_{\mu\nu}$, not $\tilde{g}_{\mu\nu}$. This means the conformal coupling alone produces **no extra lensing** beyond GR. The gravitational lensing signal must come from the aether field $A^\mu$ via a **disformal** term. **Resolution (RST 1.6.1):** Light is the substrate's bandwidth limit ($c$). Where the substrate is busy maintaining workload $q$, bandwidth is restricted ([[Relational Friction]], A5). The physical metric for photons is: $\tilde{g}_{\mu\nu} = e^{2q}g_{\mu\nu} - \beta(q)\, A_\mu A_\nu$ The $A_\mu A_\nu$ term acts as a refractive index: light is **deflected by information congestion**, not by a dark matter halo. **Mechanism identified.** Full derivation of $\beta(q)$ from the action is programmatic. See [[RST Lensing (Disformal)]] and [[RST Charter (Scientific Status)]]. ### 2. Solar System (PPN) The conformal coupling modifies the post-Newtonian metric. In the deep Newtonian regime ($g \gg a_0$), $q$ tracks the Newtonian potential: $q \approx \Phi_N / c^2$. The conformal factor becomes $e^{2q} \approx 1 + 2\Phi_N/c^2$, which is precisely the standard GR time-time metric perturbation. The PPN parameters ($\gamma$, $\beta$) depend on the interplay between the scalar correction and the aether correction. See [[RST PPN Constraints]]. ### 3. Cosmological Background On the homogeneous FLRW background, $q$ is spatially uniform, so $\tilde{g}_{\mu\nu} = e^{2\bar{q}}g_{\mu\nu}$ is just a constant rescaling of the scale factor: $\tilde{a}(t) = e^{\bar{q}}a(t)$. This is absorbed into the definition of the scale factor and does not change the Friedmann equation's structure. Background cosmology is unaffected. --- ## Relation to Existing Work | Theory | Matter Coupling | Lensing? | RST Compatibility | |:---|:---|:---|:---| | AQUAL (Bekenstein & Milgrom 1984) | Non-relativistic, no metric | N/A | Recovered in weak-field limit | | PCG (Bekenstein 1988) | Conformal: $\tilde{g} = e^{2\phi}g$ | No (photons conformally invariant) | **Adopted** | | TeVeS (Bekenstein 2004) | Disformal: $\tilde{g} = e^{-2\phi}(g + \sinh(2\phi)A_\mu A_\nu)$ | Yes | Disformal term to be adapted | | RST 1.5 (minimal) | Minimal: $S_m[g]$ | N/A (no scalar source) | **Superseded** | | RST 1.6 | Conformal: $\tilde{g} = e^{2q}g$ | Mechanism: disformal $\beta(q)A_\mu A_\nu$ ([[RST Lensing (Disformal)]]) | **Current** | --- ## References - Lelli, F., McGaugh, S. S. & Schombert, J. M. (2016). *SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves.* AJ 152, 157. [arXiv:1606.09251](https://arxiv.org/abs/1606.09251). — empirical calibration; [[SPARC Evaluation Verification]]. - Bekenstein, J. D. & Milgrom, M. (1984). *Does the missing mass problem signal the breakdown of Newtonian gravity?* ApJ 286, 7. — AQUAL. - Bekenstein, J. D. (1988). *Phase coupling gravitation: symmetries and gauge invariance.* Phys. Lett. B 202, 497. — PCG (conformal coupling adopted in RST 1.6). - Bekenstein, J. D. (2004). *Relativistic gravitation theory for the modified Newtonian dynamics paradigm.* Phys. Rev. D 70, 083509. — TeVeS. - Milgrom, M. (1983). *A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis.* ApJ 270, 365.