Emergent Gravity

The entangleon, the holographic metric, the composite graviton with its ultraviolet cutoff, and Newton's constant from the Sakharov mechanism.

The emergent chain

HET derives gravitational physics in a chain:

bond oscillators \(\;\to\;\) continuum entangleon \(\phi\;\to\;\) composite graviton \(h_{\mu\nu}\;\to\;\) Newton's constant \(\;\to\;\) Newtonian force law.

Each step is taken explicitly. The bond is the harmonic oscillator of the previous section; the continuum field is its long-wavelength limit; the metric perturbation is its second-derivative composite; Newton's constant follows from the Sakharov mechanism on \(Q_3\times I\); the Newtonian force law is the long-wavelength projection of the linearised Einstein equation coupled to conserved stress-energy.

The continuum limit

In the long-wavelength limit the bond coordinate \(D_{\mathrm{KL}}\) is replaced by a continuum scalar field \(\phi(x)\) — the entangleon. Its action is the Klein–Gordon form

\[ \mathcal{S}[\phi] \;=\; \int\! d^{4}x\;\bigl( \tfrac12\partial_\mu\phi\,\partial^{\mu}\phi - \tfrac12 m_\phi^{2}\phi^{2}\bigr), \qquad m_\phi^{2} \;=\; \kappa_P\,m_P^{2} \;=\; 32\,m_P^{2}, \]

so \(m_\phi = 4\sqrt{2}\,m_P\). The factor \(\kappa_P = 32\) is the per-bond pinning constant set by the curvature of the Kubo–Mori potential at the vacuum on \(Q_3\).

Holographic derivation of the metric

The discrete Ryu–Takayanagi first law on a bond reads \(\delta D_{\mathrm{KL}} = \delta A_{\min}/4\ell_P^{2}\), tying bond entropy variations to extremal-surface areas in Planck units. In the continuum this becomes

\[ h_{\mu\nu}(x) \;=\; -\frac{\ell_P^{2}}{\log\chi}\, \partial_\mu\partial_\nu\langle\phi\rangle(x). \]

The metric perturbation is the second-derivative composite of the entangleon's expectation value. The factor \(1/\log\chi\) carries the Bekenstein–Hawking normalisation \(\log\chi = 1/4\) into the metric.

The graviton

The graviton is the spin-2 mode of \(h_{\mu\nu}\). Two-point function structure follows immediately from the composite definition: the graviton propagator is built from two entangleon propagators with two derivatives at each end,

\[ \langle h_{\mu\nu}(x)\,h_{\rho\sigma}(y)\rangle \;\sim\; \Bigl(\frac{\ell_P^{2}}{\log\chi}\Bigr)^{2} \partial_\mu\partial_\nu\partial_\rho\partial_\sigma\,G_\phi(x-y), \]

with \(G_\phi(k) = i/(k^{2} - m_\phi^{2})\). The four derivatives \(\to k_\mu k_\nu k_\rho k_\sigma\) and the entangleon propagator together give the spin-2 two-point function in the long-wavelength regime \(k^{2}\ll m_\phi^{2}\), and a natural ultraviolet cutoff at

\[ k_{\mathrm{UV}} \;=\; m_\phi \;=\; 4\sqrt{2}\,m_P. \]

Above \(k_{\mathrm{UV}}\) the entangleon resolves into the discrete bond degrees of freedom of \(Q_3\) and no continuum graviton exists. The composite-graviton structure, together with the bond discreteness as a physical ultraviolet regulator, is proposed to address the non-renormalisability of perturbative quantum gravity.

The hyperoctahedral \(B_3\) symmetry of \(Q_3\) selects the two-dimensional irreducible representation \(W_2\) on the 24-dimensional hinge space of the prism \(Q_3\times I\). The multiplicity of \(W_2\) is 3, so the spin-2 graviton mode lives on \(\dim W_2 \times m_{W_2} = 2\times 3 = 6\) — the six faces of \(Q_3\).

Newton's constant: the Sakharov derivation

Integrating out the entangleon on the prism \(Q_3\times I\) yields the Einstein–Hilbert action with a finite induced Newton's constant. With the bond oscillator's curvature \(\kappa_P\) and the bond ceiling \(\log\chi = 1/4\), the Sakharov mechanism gives

\[ G_N \;=\; \frac{1}{2\pi\log\chi} \;=\; \tfrac{2}{\pi}\,\ell_P^{2} \quad\text{at } \log\chi=\tfrac14, \]

equivalently \(G_N = \ell_P^{2}\) in Bekenstein–Hawking normalisation. The derivation is closed-form: no continuum loop integral needs to be regulated, since the cube provides its own short-distance structure.

Linearised Einstein equation and Newtonian limit

Coupled to a conserved stress-energy tensor \(T_{\mu\nu}\), the metric \(h_{\mu\nu}\) satisfies the linearised Einstein equation

\[ \Box h_{\mu\nu} \;-\;\partial_\mu\partial^{\alpha}h_{\alpha\nu} \;-\;\partial_\nu\partial^{\alpha}h_{\alpha\mu} \;+\;\partial_\mu\partial_\nu h \;-\;\eta_{\mu\nu}\bigl(\Box h - \partial^{\alpha}\partial^{\beta}h_{\alpha\beta}\bigr) \;=\; -16\pi G_N\,T_{\mu\nu}. \]

In the static, long-wavelength limit with \(T_{00} = \rho(x)\), the 00-component reduces to Poisson's equation \(\nabla^{2}\Phi = 4\pi G_N\rho\), recovering the Newtonian force law

\[ \mathbf{F} \;=\; -\frac{G_N M m}{r^{2}}\,\hat{\mathbf r}. \]

The chain from bond oscillator to Newtonian force is closed without free parameters beyond \(\log\chi = 1/4\), \(d = 3\), and \(\kappa_P = 32\).