Einstein field equations (v1.0)

Below is a compact derivation—starting from the axioms—of

• why every long-wavelength ( \(\lambda ≫ l_p\) ) excitation of the FCC Qaether lattice obeys a Lorentz-invariant field equation, and
• how coarse-graining the same lattice reproduces the Einstein field equations in the infrared.

 

1 Setup: lattice scales and fields

• Lattice spacing \( a \equiv l_p\) (cell radius).
• Discrete phase field \(\phi_i\) lives on lattice sites; its second-order difference is given in A3.1.
• Global clock is defined by the Einstein-type synchronization in A5 with characteristic speed \(c_\phi = c\).

Write the site index \(i\) → \(x/a\), expand \(\phi(x)\) in a Taylor series and keep the first non-vanishing orders.

 

2 Emergent Lorentz symmetry

2.1 Spatial isotropy of the FCC lattice

The FCC lattice has twelve equivalent nearest-neighbour directions. In the long-wavelength limit the discrete Laplacian

$$\Delta_{\!\text{FCC}}\phi(x)\;=\;\frac1{12\,a^{2}}\sum_{\Delta}\!\left[\phi(x\!+\!\Delta)-\phi(x)\right]$$

approaches the continuum Laplacian \(∇²\phi\) equally in all directions, because the directional cosines average to \(\delta_{ij}/3\); hence no cubic anisotropies survive. (See “isotropy restoration” in A2.4.)

2.2 Discrete → continuum wave equation

The single-cell dynamical law (A10, first line)

$$I_i\,\ddot \phi_i \;=\;\sum_j K_{ij}\,\Im e^{i(\phi_j-\phi_i)} - P_i\,\Im e^{i(\phi_j-\phi_i)} + \dots$$

linearises for small phase differences \((\phi_j – \phi_i ≪ 1)\):

$$I_0\,\partial_{t}^{2}\phi \;=\;K_0\,a^{2}\,\nabla^{2}\phi + \mathcal O(a^{4})$$

Using \(I_0= p_0\,A_s\,t_p^{2}\) (A6.5) and \(K_0 a^2/I_0 = c^2;\), fixed by the microscopic relation \(c_\phi=c\) (A5), we obtain

$$\boxed{\,\partial_{t}^{2}\phi - c^{2}\nabla^{2}\phi = 0\,}$$

which is precisely the massless Klein-Gordon equation in flat Minkowski space. The factor in front of ∇² is direction-independent by § 2.1, so the emergent symmetry group is SO(3,1).

Hence every long-wavelength mode experiences an effective metric

$$\eta_{\mu \nu}=\text{diag}(-1,1,1,1)$$

restoring Lorentz invariance at energies \(\gg \hbar c / l_p\).

 

3 From local stress to curvature

3.1 Microscopic origin of stress–energy

Void back-pressure \(P_i(m_i)\) and inertia \(I_i(m_i)\) depend on the local coordination number \(m_i\) (A6.1-A6.5). Treating slow spatial variations of \(m(x)\) one defines an energy-momentum tensor

$$T^{μν}(x)=\bigl(ρ(x)+P(x)\bigr)u^{μ}u^{ν}+P(x)\,η^{μν}, \qquad ρ(x)=\frac{I(x)}{a^{3}t_{p}^{2}},\; P(x)=P\bigl(m(x)\bigr)$$

3.2 Effective action in Regge form

Loops listed in A4 tile space with triangles & squares. Summing deficit angles around each elementary two-face gives a lattice version of the Ricci scalar R. The coarse-grained action is therefore

$$S = \sum_{\text{cells}} \frac{1}{16\pi G_{\text{eff}}}\,a^{4} R + \sum_{\text{cells}} a^{4}\, \mathcal L_{\text{matter}}[\phi,m]$$

where \(G_{\text{eff}} \sim a^{2}/I_0 = c^{2}t_{p}^{2}/A_s p_{0}\) is fixed by the microscopic pressure/inertia parameters (using A6). Replacing the sums by integrals gives

$$S = \frac1{16\pi G_{\text{eff}}}\int \!d^{4}x\,\sqrt{-g}\,R + \int d^{4}x\,\sqrt{-g}\,\mathcal L_{\text{matter}}$$

 

4 Deriving the Einstein equations

Varying S with respect to the coarse-grained metric \(g_{μν}\) reproduces

$$\boxed{\,R_{μν}-\frac12 g_{μν}R = 8\pi G_{\text{eff}}\,T_{μν}\,}$$

because (i) the Regge term varies to the left-hand side and (i) the matter term yields $$\delta S/\delta g_{μν}=−½√−g T_{μν}$$ The identification of \(T_{μν}\) with the long-wavelength limit of the microscopic pressure/inertia tensor in § 3.1 completes the proof.

 

5 Summary

• Lorentz symmetry is an emergent accidental symmetry: spatial isotropy of the FCC lattice and the universal microscopic light-speed \(c_\phi=c\) guarantee that the long-wavelength equation for \(\phi\) reduces to the relativistic d’Alembertian.
• Gravity arises from spatial variations in coordination number (and thus pressure/inertia). The network of phase-loops provides a Regge discretization of curvature; coarse-graining yields the Einstein–Hilbert action with \(G_{\text{eff}}\propto l_p^{2}\).
• Varying this action recovers the Einstein field equations, establishing that the Qaether axioms reproduce General Relativity in the infrared.

Thus, in the low-energy, long-distance limit, Qaether behaves like a Lorentz-invariant continuum whose dynamics obey Einstein’s equations.