> [!warning] > This content has not been peer reviewed. # RST Horndeski Mapping ## Purpose Map RST 1.5's action into Horndeski + Einstein-Aether coordinates to import observational constraints (GW speed, PPN, stability) from the existing literature without re-derivation. --- ## Background: The Horndeski Framework Horndeski gravity (Horndeski 1974, Int. J. Theor. Phys. 10, 363) defines the most general scalar-tensor theory with second-order equations of motion in 4D. The action is parameterized by four functions of the scalar field $\phi$ and its kinetic term $X = -\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$: $S_H = \int d^4x \sqrt{-g} \left[ G_2(\phi, X) + G_3(\phi, X)\Box\phi + G_4(\phi, X)R + G_{4,X}(\phi, X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu\phi)^2\right] + G_5(\phi, X)G_{\mu\nu}\nabla^\mu\nabla^\nu\phi + \cdots \right]$ Because RST contains both a scalar ($q$) and a vector ($A^\mu$), it belongs to the generalized **Scalar-Vector-Tensor (SVT)** class. The two sectors can be analyzed independently for linear perturbation constraints. --- ## The Scalar (Workload) Sector Ignoring the aether terms, the RST action maps to Horndeski as: | Horndeski Function | RST Identification | Properties | | ------------------ | -------------------------------------------------------------------------------------------------------------- | ---------------------------------------- | | $G_2(q, X_q)$ | $-\dfrac{a_0(\theta)^2}{8\pi G}\mathcal{F}\left(\dfrac{X_q}{a_0^2}, n(\theta)\right) - \varepsilon_\text{min}$ | $\theta$-dependent K-essence | | $G_3$ | $0$ | No cubic galileon | | $G_4$ | $\dfrac{1}{16\pi G}$ | Constant — no derivative coupling to $R$ | | $G_5$ | $0$ | No quintic galileon | ### Key property: $G_{4,X} = 0$ Because $G_4$ is a strict constant, its derivative with respect to $X_q$ vanishes identically. This is the critical structural feature. --- ## The Vector (Aether) Sector The aether field $A^\mu$ is governed by the standard Einstein-Aether kinetic terms: $\mathcal{K}(A, \nabla A) = c_1 (\nabla_\mu A_\nu)(\nabla^\mu A^\nu) + c_2 (\nabla_\mu A^\mu)^2 + c_3 (\nabla_\mu A_\nu)(\nabla^\nu A^\mu) + c_4 (A^\mu \nabla_\mu A_\nu)(A^\rho \nabla_\rho A^\nu)$ with coupling constants $c_1, c_2, c_3, c_4$ and the unit-timelike constraint $A^\mu A_\mu = -1$ enforced by a Lagrange multiplier. --- ## Constraint 1: Gravitational Wave Speed (GW170817) ### The Theorem (Ezquiaga & Zumalacárregui 2017; Creminelli & Vernizzi 2017) For a Horndeski theory to satisfy $c_\text{GW} = c$ (required by GW170817 to $|c_\text{GW} - c|/c < 10^{-15}$), two conditions must hold: $G_5 = 0 \quad \text{and} \quad G_{4,X} = 0$ ### RST Status | Condition | RST Value | Satisfied? | | ------------- | ------------------------------ | ---------- | | $G_5 = 0$ | Identically zero | **Yes** | | $G_{4,X} = 0$ | $G_4 = 1/(16\pi G) =$ constant | **Yes** | **Result:** The entire workload/MOND sector trivially passes the GW speed constraint. The scalar field $q$ does not mix with tensor perturbations at linear order. ### The Aether Sector The aether field's tensor perturbations *do* modify GW speed. In standard Einstein-Aether theory, the squared tensor mode speed is: $c_T^2 = \frac{1}{1 - c_{13}}$ where $c_{13} = c_1 + c_3$. **RST requirement:** $c_1 + c_3 = 0$, which gives $c_T^2 = 1$ exactly. ### Combined Result $\boxed{\text{RST is compatible with GW170817.}}$ The scalar sector passes via the Horndeski theorem. The vector sector passes via $c_1 + c_3 = 0$. These are independent conditions on independent sectors. --- ## Constraint 2: Ghost Freedom and Stability For the scalar sector, the no-ghost condition in Horndeski requires: $G_2 + 2XG_{2,X} > 0$ For RST, this translates to a condition on $\mathcal{F}$ and its derivatives. In the AQUAL framework (Bekenstein & Milgrom 1984), ghost freedom is guaranteed when $\mu(x) > 0$ and $d(\mu x)/dx > 0$, both of which hold for the interpolation function $\mu(\eta, n) = \eta/(1 + \eta^n)^{1/n}$ across all physical regimes. --- ## Constraint 3: Post-GW170817 Viability Class After GW170817, the surviving Horndeski theories are precisely those with $G_4 =$ const, $G_5 = 0$. These are called **K-essence + cubic galileon** theories (since only $G_2$ and $G_3$ remain free). RST belongs to the even more restricted subclass: **pure K-essence** ($G_3 = G_5 = 0$, $G_4 =$ const). This is the simplest surviving Horndeski class. It is well-studied, ghost-free, and cosmologically viable. --- ## Implications for PPN The Horndeski mapping alone does not determine the PPN parameters, because PPN depends on the matter coupling. With minimal coupling ($S_m[g_{\mu\nu}, \psi_m]$), a pure K-essence scalar produces no post-Newtonian corrections (because it has no source term). With conformal coupling ($S_m[\tilde{g}*{\mu\nu}, \psi_m]$ where $\tilde{g}*{\mu\nu} = e^{2q}g_{\mu\nu}$), the scalar field acquires a baryonic source and contributes to the metric perturbations. The PPN analysis therefore requires the matter coupling to be specified first. See [[RST Matter Coupling (PCG)]] and [[RST PPN Constraints]]. --- ## Summary | Check | Method | Result | | ------------------------- | ------------------------------------------- | -------------------------------- | | GW speed (scalar sector) | Horndeski theorem: $G_5 = 0$, $G_{4,X} = 0$ | **Pass** | | GW speed (vector sector) | Einstein-Aether: $c_1 + c_3 = 0$ | **Pass (requires $c_3 = -c_1$)** | | Ghost freedom | AQUAL $\mu > 0$, $d(\mu x)/dx > 0$ | **Pass** | | Horndeski viability class | Pure K-essence ($G_2$ only) | **Simplest surviving class** | | PPN parameters | Requires matter coupling specification | See [[RST PPN Constraints]] | --- ## References - Horndeski, G. W. (1974). *Second-order scalar-tensor field equations in a four-dimensional space.* Int. J. Theor. Phys. 10, 363. — most general second-order scalar-tensor theory in 4D; RST workload sector maps to a subclass. - GW170817 / Horndeski constraints: Ezquiaga & Zumalacárregui (2017), Creminelli & Vernizzi (2017); full list [[Relational Substrate Theory (RST)#References]]. - Ezquiaga, J. M. & Zumalacárregui, M. (2017). *Dark Energy After GW170817: Dead Ends and the Road Ahead.* PRL 119, 251304. - Creminelli, P. & Vernizzi, F. (2017). *Dark Energy after GW170817 and GRB170817A.* PRL 119, 251302. - Jacobson, T. & Mattingly, D. (2001). *Gravity with a dynamical preferred frame.* PRD 64, 024028. - Abbott, B. P. et al. (2017). *Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A.* ApJL 848, L13.