>[!warning] >This content has not been peer reviewed. # RST PPN Constraints — Solar System Tests ## Overview The Parameterized Post-Newtonian (PPN) formalism tests theories of gravity against precision Solar System measurements. RST must satisfy these constraints to be viable. However, RST's relational nature requires the adjustments specified in the [[Relational Substrate Theory (RST)|Relational Adjustment Protocol]] (Part VII-B) before standard PPN bounds can be applied. --- ## The PPN Parameters The PPN formalism expands the spacetime metric around a Newtonian source to first post-Newtonian order: $g_{00} = -1 + 2U - 2\beta U^2 + \cdots$ $g_{ij} = (1 + 2\gamma U)\delta_{ij} + \cdots$ where $U = GM/(rc^2)$ is the Newtonian potential. The key parameters are: | Parameter | GR Value | Best Constraint | Source | |:---|:---|:---|:---| | $\gamma$ | 1 | $|\gamma - 1| < 2.3 \times 10^{-5}$ | Cassini (Bertotti, Iess & Tortora 2003, Nature 425, 374) | | $\beta$ | 1 | $|\beta - 1| < 8 \times 10^{-5}$ | Perihelion precession (Mercury + LLR) | | $\alpha_1$ | 0 | $|\alpha_1| < 10^{-4}$ | Lunar Laser Ranging | | $\alpha_2$ | 0 | $|\alpha_2| < 10^{-7}$ | Solar alignment with ecliptic | --- ## RST's Two Sources of PPN Corrections RST has two fields beyond the metric: the workload scalar $q$ and the aether vector $A^\mu$. Each contributes independently to the PPN parameters. ### Source 1: The Workload Scalar $q$ (via PCG Conformal Coupling) With the physical metric $\tilde{g}_{\mu\nu} = e^{2q}g_{\mu\nu}$, the scalar field sources from baryonic matter and, in the deep Newtonian regime ($g \gg a_0$), locks to the Newtonian potential: $q \approx \Phi_N/c^2$. The conformal factor contributes to the metric as: $\tilde{g}_{00} \approx -(1 - 2\Phi_N/c^2 + 2\Phi_N/c^2) = -(1 - 2\Phi_N/c^2)(1 + 2q)$ In the weak-field, this means the scalar field effectively doubles the gravitational potential in the time-time component — which is precisely what GR already does. The scalar contribution is *degenerate* with the definition of $G_\text{measured}$. **Key point:** The scalar field's PPN contribution is absorbed into the renormalization of $G$ (Relational Adjustment Protocol, Adjustment 1). The effective Newton's constant measured in the Solar System is: $G_\text{measured} = G_\text{bare}(1 + \alpha_q)$ where $\alpha_q$ is the scalar field's contribution. This means the scalar field does not produce an anomalous $\gamma$ — its effect is indistinguishable from a redefinition of $G$. **Residual correction:** The MOND-regime tail of the interpolation function produces a tiny additional acceleration $\delta a \sim a_0^2/g_N \sim 10^{-17}$ m/s$^2$ at 1 AU. This is $\sim$14 orders of magnitude below current measurement precision. ### Source 2: The Aether Vector $A^\mu$ The aether field produces preferred-frame effects that are well-studied in the Einstein-Aether literature. The PPN parameters for the aether sector are known functions of the coupling constants $c_1, c_2, c_3, c_4$. #### The $\gamma$ Parameter In Einstein-Aether theory (with $c_{13} = c_1 + c_3$, $c_{14} = c_1 + c_4$): $\gamma - 1 = -\frac{c_{14}(2 - c_{14})}{2(1 - c_{14})(2 + c_{14} + 3c_2)}$ **RST constraint:** We already require $c_{13} = c_1 + c_3 = 0$ from GW170817. To achieve $\gamma = 1$, we additionally need $c_{14} = 0$ (i.e., $c_1 + c_4 = 0$, or equivalently $c_4 = -c_1$). With $c_3 = -c_1$ and $c_4 = -c_1$: $\gamma - 1 = 0 \quad \checkmark$ #### The Preferred-Frame Parameters $\alpha_1, \alpha_2$ In Einstein-Aether theory: $\alpha_1 = -\frac{8(c_3^2 + c_1 c_4)}{2c_1 - c_1^2 + c_3^2}$ $\alpha_2 = \frac{\alpha_1}{2} - \frac{(c_1 + 2c_2 - c_3)(2c_1 + 3c_2 + c_3 + c_4)}{c_{123}(2 - c_{14})}$ where $c_{123} = c_1 + c_2 + c_3$ and $c_{14} = c_1 + c_4$. With $c_3 = -c_1$ and $c_4 = -c_1$: $\alpha_1 = -\frac{8(c_1^2 - c_1^2)}{2c_1 - c_1^2 + c_1^2} = -\frac{8 \cdot 0}{2c_1} = 0 \quad \checkmark$ $\alpha_2 = 0 - \frac{(c_1 + 2c_2 + c_1)(2c_1 + 3c_2 - c_1 - c_1)}{c_2 \cdot (2)} = -\frac{(2c_1 + 2c_2)(3c_2)}{2c_2}$ This requires further constraint on $c_2$ relative to $c_1$ to ensure $\alpha_2 = 0$. The condition is: $2c_1 + 2c_2 = 0 \implies c_2 = -c_1$ --- ## The Constraint Solution Combining all constraints: | Constraint | Source | Condition | |:---|:---|:---| | GW speed: $c_T = c$ | GW170817 | $c_3 = -c_1$ | | Eddington: $\gamma = 1$ | Cassini | $c_4 = -c_1$ | | Preferred frame: $\alpha_1 = 0$ | Lunar Laser Ranging | Satisfied by $c_3 = -c_1$, $c_4 = -c_1$ | | Preferred frame: $\alpha_2 = 0$ | Solar alignment | $c_2 = -c_1$ | **Result:** All four constraints are satisfied by a one-parameter family: $\boxed{c_2 = c_3 = c_4 = -c_1}$ The entire aether sector is controlled by a single free parameter $c_1$. This parameter sets the overall strength of the aether's kinetic energy and is bounded by additional constraints (Cherenkov radiation, Big Bang nucleosynthesis, black hole stability). --- ## Remaining Constraints on $c_1$ With $c_2 = c_3 = c_4 = -c_1$, the aether sector reduces to: $\mathcal{K} = c_1\left[(\nabla_\mu A_\nu)(\nabla^\mu A^\nu) - (\nabla_\mu A^\mu)^2 - (\nabla_\mu A_\nu)(\nabla^\nu A^\mu) - (A^\mu\nabla_\mu A_\nu)(A^\rho\nabla_\rho A^\nu)\right]$ ### Spin-0 Mode Speed The squared speed of the spin-0 aether mode is: $c_0^2 = \frac{c_{123}(2 - c_{14})}{c_{14}(1 - c_{13})(2 + c_{14} + 3c_2)}$ With $c_2 = c_3 = c_4 = -c_1$: - $c_{14} = c_1 + c_4 = 0$ - $c_{123} = c_1 + c_2 + c_3 = -c_1 \neq 0$ The numerator is $c_{123}(2 - c_{14}) = -2c_1 \neq 0$ while the denominator contains a factor of $c_{14} = 0$, so the formula has a **pole** (divergence), not an indeterminate form. This is a known feature of the $c_{14} = 0$ subspace in Einstein-Aether theory: the spin-0 aether-gravity coupling vanishes and the mode **decouples** from the graviton sector (Jacobson 2007). >[!note] >The $c_{14} = 0$ condition removes the spin-0 aether-gravity coupling. This is physically sensible: it means the aether's "breathing mode" does not radiate gravitationally. The workload scalar $q$ handles the scalar gravitational degree of freedom instead. ### Spin-1 Mode Speed $c_1^{\text{mode}} = \frac{c_1 - c_3}{2(1 - c_{13})} = \frac{c_1 + c_1}{2(1 - 0)} = c_1$ For stability, we need $c_1 > 0$. For subluminal propagation (to avoid vacuum Cherenkov radiation), we need $c_1 \leq 1$. $\boxed{0 < c_1 \leq 1}$ ### Big Bang Nucleosynthesis (BBN) The aether energy density at BBN modifies the expansion rate, changing the primordial element abundances. The constraint is approximately: $c_1 + 3c_2 + c_3 \lesssim 0.1$ With $c_2 = c_3 = -c_1$: $c_1 - 3c_1 - c_1 = -3c_1$. The absolute value is $3c_1 \lesssim 0.1$, giving: $c_1 \lesssim 0.03$ --- ## Summary: The RST 1.6 Aether Parameter Space | Parameter | Value | Determined by | |:---|:---|:---| | $c_1$ | Free, $0 < c_1 \lesssim 0.03$ | BBN + Cherenkov | | $c_2$ | $-c_1$ | $\alpha_2 = 0$ | | $c_3$ | $-c_1$ | $c_\text{GW} = c$ | | $c_4$ | $-c_1$ | $\gamma = 1$ | RST's aether sector is controlled by a **single free parameter** $c_1$, bounded to a small positive value. All Solar System, gravitational wave, preferred-frame, and nucleosynthesis constraints are simultaneously satisfied. --- ## Theoretical Status | Test | Constraint | RST Status | |:---|:---|:---| | GW speed (GW170817) | $|c_\text{GW}/c - 1| < 10^{-15}$ | **Pass** ($c_3 = -c_1$) | | Eddington parameter (Cassini) | $|\gamma - 1| < 2.3 \times 10^{-5}$ | **Pass** ($c_4 = -c_1$) | | Preferred frame $\alpha_1$ (LLR) | $|\alpha_1| < 10^{-4}$ | **Pass** ($\alpha_1 = 0$ exactly) | | Preferred frame $\alpha_2$ (Solar) | $|\alpha_2| < 10^{-7}$ | **Pass** ($c_2 = -c_1$) | | Cherenkov radiation | Spin-1 mode subluminal | **Pass** ($0 < c_1 \leq 1$) | | BBN | $|c_1 + 3c_2 + c_3| \lesssim 0.1$ | **Pass** ($c_1 \lesssim 0.03$) | | MOND residual in Solar System | $\delta a \sim a_0^2/g \sim 10^{-17}$ m/s$^2$ | **Undetectable** | >[!important] >All constraints in the PPN formalism are applied per the [[Relational Substrate Theory (RST)#Part VII-B The Relational Adjustment Protocol|Relational Adjustment Protocol]]. The scalar field's contribution to PPN is absorbed into $G_\text{measured}$ (Adjustment 1). The aether field's contribution is computed with $\bar{\theta} \neq 0$ at the boundary (Adjustment 2). Deep-Newtonian residuals (Adjustment 3) produce corrections at $\sim 10^{-7}$, below current bounds. --- ## Open Questions 1. **Exact value of $c_1$:** Currently bounded to $(0, 0.03]$. Can it be determined from within the theory, or is it a true free parameter? 2. **Spin-0 mode degeneracy:** The $c_{14} = 0$ condition makes the scalar aether mode degenerate. Is this a feature (the workload field replaces it) or a problem (instability in some regime)? 3. **Black hole solutions:** Einstein-Aether black holes in the $c_2 = c_3 = c_4 = -c_1$ subspace need to be checked for regularity and thermodynamic consistency. --- ## References - Bertotti, B., Iess, L. & Tortora, P. (2003). *A test of general relativity using radio links with the Cassini spacecraft.* Nature 425, 374. - Will, C. M. (2014). *The Confrontation between General Relativity and Experiment.* Living Rev. Relativ. 17, 4. - Foster, B. Z. & Jacobson, T. (2006). *Post-Newtonian parameters and constraints on Einstein-Aether theory.* Phys. Rev. D 73, 064015. - Oost, J., Mukohyama, S. & Wang, A. (2018). *Constraints on Einstein-aether theory after GW170817.* Phys. Rev. D 97, 124023. - Abbott, B. P. et al. (2017). *GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral.* PRL 119, 161101.