RTG Gravity I: Emergent Metric → Einstein–Hilbert (tree‑level)

Revision: 1.3 (clarity & units update)  |  Date: 14 Aug 2025
Authors: Mustafa Aksu · Grok · ChatGPT

0 · Scope & what this page covers

This is the canonical tree‑level gravity page for RTG. It defines the emergent metric from the microscopic resonance kernel and derives the Einstein–Hilbert (EH) action plus minimal coupling to effective matter and \( \mathrm{U}(1) \) gauge fields obtained after coarse‑graining. All loop effects, running couplings, and anomaly conditions are deferred to RTG Gravity II: Quantum Corrections, Running & Anomalies.

1 · Units & symbols (standardised)

• All angular frequencies \( \omega,\Delta\omega,\delta\omega \) are in rad·s\(^{-1}\); convert to Hz by \( f=\omega/(2\pi) \). Unless stated, \( \Delta\omega^\ast=(1.45\pm0.08)\times10^{23}\,\mathrm{s}^{-1} \).
• Kernel vs gauge (lattice): \( A_{ij} \) = resonance amplitude (kernel). Use a dimensionless gauge phase \( B_{ij}\equiv a\,\mathcal A_{ij} \) and the link \( U_{ij}=e^{\,i(\phi_i-\phi_j-B_{ij})} \). This avoids any clash with the kernel amplitude \( A_{ij} \).
• \( \sigma_{\mathrm{exch}} \) = UV width for the exchange term (model/MD regulator); \( \sigma_{\mathrm{noise}} \) = CHSH noise parameter; \( \sigma \) (unsuffixed) is reserved for RG smooth cutoffs in Gravity II. Independence: \( \sigma_{\mathrm{exch}} \), \( \sigma_{\mathrm{noise}} \), the RG cutoff \( \sigma \), and \( \Delta\omega^\ast \) are mutually independent.
• Natural units are used in derivations (\( \hbar=c=1 \)) unless explicitly restored; numerical mapping sections keep \( c \) visible where helpful.

2 · RTG kinematics & fields (recap)

Microscopic nodes carry a compact phase \( \phi \), angular frequency \( \omega \), and a binary spin gate \( s=\pm i \) (code spins \( \sigma_i=\pm1 \), with \( s_i=i\,\sigma_i \)). The split resonance kernel factorises as \( \mathcal R_{ij}\equiv A_{ij}(1+s_is_j) \), with \( A_{ij}=\tfrac{3}{4}[1+\cos(\phi_i-\phi_j)]\exp[-(\Delta\omega_{ij}/\Delta\omega^\ast)^2] \). The open/closed gate factor \( 1+s_is_j\in\{0,2\} \) controls whether a link contributes.

Windows. For cross‑document consistency (Gauge/Gravity II/EFT), define \( x\equiv|\Delta\omega|/\Delta\omega^\ast \). The commonly used windows are: \( \mathrm{U}(1): 0 \le x < 0.28 \); soft‑spin SU(2): \( 0.28 \le x \le 0.70 \); \( \mathrm{U}(1)^2: 1.55 \le x \le 1.70 \) (narrow; avoid \( x \ge 1.70 \) where 5‑D anomalies appear). In EFT/MC contexts a Litim window may replace the Gaussian factor when measuring geometry constants.

After coarse‑graining, the long‑wavelength sector admits continuum fields: a compact phase \( \phi(x) \), a coarse \( \mathbb Z_2 \)‑gate field \( G(x)\in[0,1] \), and (when present) a \( \mathrm{U}(1) \) gauge potential \( A_\mu(x) \) associated with lattice phases \( B_{ij} \) and links \( U_{ij} \).

3 · Constructing the emergent metric

3.1 · Local structure tensor from the kernel

Let \( x \) be a point in the coarse lattice/continuum and \( \mathcal N_x \) a neighbourhood of links \( \langle ij\rangle \) whose midpoint lies near \( x \). Define a symmetric “structure tensor”

\( \mathbf{S}_{\mu\nu}(x)=\frac{1}{\mathcal N}\sum_{\langle ij\rangle\in \mathcal N_x} w_{ij}\,\Delta x_{ij,\mu}\,\Delta x_{ij,\nu},\quad \Delta x_{ij}=x_i-x_j,\quad \mathcal N=\sum_{\langle ij\rangle\in \mathcal N_x} w_{ij}. \)

Choice of weights. Use \( w_{ij}=\mathcal R_{ij} \) for gated geometry (preferred in mixed‑gate regions, as it encodes the binary open/closed channel). Use \( w_{ij}=A_{ij} \) only in open‑gate‑dominant neighbourhoods when you explicitly want to isolate phase interference without gate masking. This “weighted Laplacian” viewpoint evokes a local Sturm–Liouville/diffusion geometry where the weights define the measure.

3.2 · Orthonormal frame and metric normalization

Diagonalise \( \mathbf S \) to get eigenpairs \( (\lambda_a,\mathbf v^{(a)}) \), \( a=0,1,2,3 \). Define a tetrad \( e^a{}_\mu(x) \) by normalising these eigenvectors and choose a conformal factor \( \Omega(x) \) such that

\( g_{\mu\nu}(x)=\Omega^2(x)\,\eta_{ab}\,e^a{}_\mu(x)\,e^b{}_\nu(x),\qquad \eta_{ab}=\mathrm{diag}(-1,+1,+1,+1). \)

3.3 · Signature, time orientation, and the speed scale

• Signature selection. The direction aligned with the local phase flow (principal axis of \( \mathbf S \) correlated with \( \nabla\phi \)) is designated timelike; this fixes the sign in \( \eta_{ab} \).
• Conformal factor. Fix \( \Omega(x) \) by small‑\( q \) phase‑mode dispersion or graph‑Laplacian matching to \( \nabla_\mu(\sqrt{-g}g^{\mu\nu}\nabla_\nu) \), so that phase waves propagate at speed \( c \). Practical fit: estimate dispersion from the Fourier spectrum of phase two‑point functions, or match propagation delay/spectral gaps to the continuum operator.
• Units. If \( c\neq 1 \) is explicit, \( \Omega \) carries the necessary power of \( c \) so that the EH action has energy dimension.

4 · Continuum action at tree level

4.1 · Gravitational sector (Einstein–Hilbert + Λ)

\( S_{\rm grav}[g]=\frac{1}{16\pi G_0}\int d^4x\,\sqrt{-g}\,(R-2\Lambda_0), \quad \Rightarrow \quad G_{\mu\nu}+\Lambda_0 g_{\mu\nu}=8\pi G_0\,T_{\mu\nu}. \)

Here \( G_0,\Lambda_0 \) are coarse‑grained constants (no running on this page); \( T_{\mu\nu} \) is supplied by the effective matter/gauge sectors below.

4.2 · Matter sector from RTG (compact phase + Z₂ gate)

\( S_{\phi,G}=\int d^4x\,\sqrt{-g}\left[ \frac{\rho_s}{2}\,G(x)\,g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi + \kappa\,(1-G(x)) \right], \qquad \rho_{s,\mathrm{eff}}=\langle G\rangle\,\rho_s. \)

4.3 · U(1) gauge sector and minimal coupling

The lattice gauge phase \( B_{ij}\equiv a\,\mathcal A_{ij} \) (dimensionless) induces the link \( U_{ij}=e^{\,i(\phi_i-\phi_j-B_{ij})} \). In the continuum this maps to a \( \mathrm{U}(1) \) connection \( A_\mu \) with \( F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu \) and Maxwell term \( S_{\rm U(1)}=-\tfrac14\int d^4x\,\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu} \).

Coupling the compact phase field. The \(\phi\)‑sector’s \( \mathrm{U}(1) \) is an internal phase symmetry (global by default). If promoted locally, replace \( \partial_\mu\phi \) by the covariant derivative \( D_\mu\phi=\partial_\mu\phi – i q A_\mu \) throughout the kinetic term so that the gate multiplies the gauged kinetic energy:
\( S_{\phi,G}^{\rm gauged}=\int d^4x\,\sqrt{-g}\left[ \frac{\rho_s}{2}\,G(x)\,g^{\mu\nu}D_\mu\phi\,D_\nu\phi + \kappa(1-G) \right] \).
(Alternatively, on the lattice use the covariant difference \( \Delta\phi_{ij}-B_{ij} \).)

Exchange‑channel convention. Use either \( +J_{\rm ex}\sin(\Delta\phi-B_{ij}) \) or the gauge‑equivalent XY form \( -J_{\rm ex}\cos(\Delta\phi-B_{ij}) \); choose one, not both.

4.4 · Variations and field equations

• \( T_{\mu\nu}^{(\phi,G)}=\rho_s G\,(\partial_\mu\phi\,\partial_\nu\phi-\tfrac12 g_{\mu\nu}(\partial\phi)^2)-\kappa(1-G)\,g_{\mu\nu} \),   \( T_{\mu\nu}^{\rm (EM)}=F_{\mu\alpha}F_\nu{}^\alpha-\tfrac14 g_{\mu\nu}F^2 \).
• \( \nabla_\mu(G\,\rho_s\,\nabla^\mu\phi)=0 \) (or \( \nabla_\mu(G\,\rho_s\,D^\mu\phi)=0 \) in the gauged case).
• \( \nabla_\mu F^{\mu\nu}=J^\nu \) with the usual conserved current \( J^\nu \) from minimally coupled matter.

5 · Lattice → continuum parameter map (tree‑level)

\( \rho_s=\frac{3}{2}\,J\,a_{\rm lat}^{\,2-d} \quad \Rightarrow \quad (d=3):\ \rho_s=\tfrac{3}{2}J\,a_{\rm lat}^{-1}. \)

\( K’_{\rm TV}=K_{\rm lat}\,a_{\rm lat}^{\,1-d} \), appearing in \( S[\omega]\sim\int K’_{\rm TV}\,\|\nabla\omega(x)\|_1\,\Delta\omega^{\ast\, -1}\,d^d x \). Units: in \( d=3 \), \( K’_{\rm TV} \) has units MeV·fm\(^{-2}\) and sets the scale for frequency‑gradient suppression in the coarse‑grained continuum.

Separation of roles. Treat the phase stiffness \( \rho_s \) and the RG coupling \( g=\bar J/K’ \) (with \( \bar J=\tfrac{3}{2}J \)) as separate objects; do not mix them. \( (G_0,\Lambda_0) \) are fixed at a chosen coarse‑graining scale; running belongs to Gravity II.

6 · Practical extraction from simulations

1. Assemble weights. Compute \( \mathcal R_{ij} \) (or \( A_{ij} \) with a gate mask) on links in a neighbourhood \( \mathcal N_x \).
2. Build \( \mathbf S \). Form \( \mathbf S_{\mu\nu}=\sum w_{ij}\,\Delta x_\mu\Delta x_\nu / \sum w_{ij} \).
3. Diagonalise & set tetrad. Take eigenvectors \( \mathbf v^{(a)} \) as an orthonormal frame \( e^a{}_\mu \).
4. Fix \( \Omega \). Calibrate the conformal factor with a long‑wavelength dispersion or Laplace–Beltrami fit so that phase waves propagate at \( c \) (e.g., match the graph Laplacian to \( \nabla_\mu(\sqrt{-g}g^{\mu\nu}\nabla_\nu) \) by measuring phase‑propagation delay or spectral gap).
5. Compute curvature. From \( g_{\mu\nu} \), evaluate \( \Gamma^\lambda{}_{\mu\nu} \), \( R_{\mu\nu} \), and \( R \) on the coarse mesh; validate with geodesic deviation or energy–momentum conservation in test fields.

7 · Consistency checks & limiting cases

• Flat limit. Uniform phases and homogeneous gate statistics (\( G(x)=\)const) give \( \mathbf S\propto \mathbf 1 \) and \( g_{\mu\nu}\to \eta_{\mu\nu} \) up to calibration.
• Gauge independence of spacetime metric. The emergent spacetime metric uses kernel weights and link geometry; internal \( \mathrm{U}(1) \) rephasings do not alter \( g_{\mu\nu} \) at tree level.
• Energy positivity. For \( J>0 \), \( \rho_s>0 \) and the phase sector satisfies the usual positivity bounds for small gradients.
• No loops here. \( G_0,\Lambda_0 \) are constants on this page; scale dependence is treated in Gravity II.

8 · What is not in this page

No two‑loop/renormalization details, no anomaly analyses, no non‑Abelian emergent‑gauge claims. For those, see RTG Gravity II: Quantum Corrections, Running & Anomalies and Gauge Symmetries in RTG.

9 · Related documents

• Enriched Geometric Concepts in RTG — bandwidth thresholds and multi‑D emergence underpinning the window structure.
• From Lattice Hamiltonian to Continuum Action in RTG (workbook) — explicit small‑angle expansion and \( a_{\rm lat} \) mapping for \( \rho_s \) and \( K’_{\rm TV} \).
• Two‑Loop RG Derivation of the Critical Bandwidth — fixes \( \Delta\omega^\ast \), \( g(\mu) \) at two loops, and the scheme table used to set windows.
• Gauge Symmetries in RTG — \( \mathrm{U}(1) \)/soft‑SU(2)/\( \mathrm{U}(1)^2 \) windows; anomaly conditions; Ward‑residual protocol.
• RTG Gravity II: Quantum Corrections, Running & Anomalies — running of \( G,\Lambda \); gauge‑sector running assumptions and checks.
• From RTG to an EFT (minimal, testable) — emergent Maxwell term, \( \kappa_B=C_\kappa J_{\rm ex}^2/K’ \), Litim one‑loop smallness.

Change log

Version	Date (UTC)	Main updates
1.3	2025‑08‑14	Clarified covariant coupling of the phase field: \( D_\mu\phi=\partial_\mu\phi-iqA_\mu \) and gating of the full kinetic term; added units for \( K’_{\rm TV} \) (MeV·fm\(^{-2}\) in 3D) and its role; expanded Laplace–Beltrami fit guidance (delay/spectral gap); fixed U(1) window inequality to \( 0 \le x < 0.28 \) in rendered LaTeX; added guidance on choosing \( w_{ij}=\mathcal R_{ij} \) vs \( A_{ij} \).
1.2	2025‑08‑14	Corrected symmetry windows (narrowed \( \mathrm{U}(1)^2 \) to 1.55–1.70 per RG alignment); unified gauge notation \( B_{ij} \) (phase) and \( U_{ij} \) (link) across §1 and §4.3; clarified local \( \mathrm{U}(1) \) coupling convention (sine vs XY).
1.1	2025‑08‑14	Aligned notation with Gauge/EFT/Gravity II; inserted \( \Delta\omega^\ast \) value and independence statements; added window references; clarified \( B_{ij} \) (dimensionless) → \( A_\mu \) mapping; expanded cross‑links to EFT and Gravity II.
1.0 (canonical)	2025‑08‑12	Original EH + tree‑level gauge‑coupling derivation; standardised units/symbols; adopted lattice→continuum extraction algorithm and parameter maps; moved loops/running/anomalies to Gravity II.