The Spectral Gravity Sub-Programme

Overview

The emergent geometry sub-programme (Presentation Note 2) reconstructs the effective Lorentzian metric $g^{\mu\nu} = 2\eta^{\mu\nu}$ as a forced output of admissibility. What dynamics does this metric obey, and why does it take the Einstein form?

The spectral gravity sub-programme answers this in four stages built on a single functional, the projective spectral entropy $\mathcal{S}_\Pi[g] = \tfrac{1}{2}\log\det' A_g$ associated with the minimal admissible Laplace-type operator $A_g = -\nabla_g^2$. The Einstein tensor arises as the infrared $a_2$ response of the renormalized metric variation; the field equations $G_{\mu\nu} + \Lambda_{\mathrm{ren}}g_{\mu\nu} = 8\pi G_N T_{\mu\nu}^{(\Pi)}$ are recovered as the Euler–Lagrange condition of a conditional variational principle, given supplied renormalized coefficients; the construction admits an Eddington-inspired Born–Infeld action as an admissible (not uniquely selected) ultraviolet completion, while its Lorentzian counterpart remains an open problem. This note covers the gravitational sector in the absence of matter; the joint gauge–gravity stratification belongs to Presentation Note 8.

Four stages, one functional. $\mathcal{S}_\Pi[g]$ is the only input. The $a_2$ sector gives the Einstein tensor, but not the Newton constant: the proper-time cutoff is separated from zeta regularization, and the finite coefficient $c_{\mathrm{EH}}^{\mathrm{ren}}$, its sign, and hence $G_N > 0$ are matching data, not predictions. The heat-kernel quantity $u \sim L^{-2}$ is an eigenvalue scale, not a temperature, so no local first law follows from the heat kernel alone. What the spectral stationarity establishes is a conditional variational coupling (no horizon, no Rindler wedge, no Raychaudhuri theorem required); reading it as equilibrium is an interpretation, not a result. In Lorentzian signature the transverse-traceless kernel is posited, not derived — a Lorentzian counterpart of $\mathcal{S}_\Pi$ requires a closed-time-path in-in construction, which is open — and under that posit the massless branch is undeformed. The Eddington-inspired Born–Infeld action is an admissible UV completion, conditional on Hypothesis [H-ext] and on a supplied Einstein term; even under [H-ext] it is not uniquely selected — the admissible completers form an explicitly parametrised family (Gravity Theorem 1).

Quick facts

Author

Jérôme Beau

Work type

Synthesis (presentation note, working paper)

License

CC BY 4.0

Scope

4 papers (Gravity, Thermodynamics, Lorentz, BornInfeld); $a_2$ Einstein response in the matter-free IR; Lorentzian truncation; Born–Infeld UV completion

Status

Matter-free IR sector conditional on supplied renormalized coefficients; BI tensorial result conditional on [H-ext] and a supplied Einstein term; open: $c_{\mathrm{EH}}^{\mathrm{ren}}$ sign and magnitude, the thermodynamic bridge, the Lorentzian in-in construction, coupled $G_{\mu\nu} = 8\pi G_N T_{\mu\nu}$ with explicit matter

The structural chain

$\mathcal{S}_\Pi[g] = \tfrac{1}{2}\log\det' A_g \;\Longrightarrow\; \delta_g\mathcal{S}_\Pi \ni c_{\mathrm{EH}}^{\mathrm{ren}}\, G_{\mu\nu} \;\Longrightarrow\; G_{\mu\nu} + \Lambda_{\mathrm{ren}}\,g_{\mu\nu} = 8\pi G_N T_{\mu\nu}^{(\Pi)} \;\Longrightarrow\; \sqrt{-\det(g + \ell_{\mathrm{sp}}^2 R)}.$

The stages arise from distinct constituent papers: the IR hierarchy from the heat-kernel $a_2$ coefficient of $\mathcal{S}_\Pi$, controlled by the ratio $L_4^2 = |\beta^{\mathrm{ren}}/c_{\mathrm{EH}}^{\mathrm{ren}}|$ of renormalized coefficients (Gravity); the Einstein–matter coupling as the Euler–Lagrange condition of the stationarity of $\Gamma_\Pi = \mathcal{S}_\Pi^{\mathrm{ren}} - W_\Pi$, given supplied renormalized coefficients (Thermodynamics); and the Eddington-inspired Born–Infeld action as an admissible — not uniquely selected — tensorial UV completion under conditions (C1)–(C5), conditional on Hypothesis [H-ext] and on a supplied Einstein term (Gravity Theorem 1, BornInfeld).

The chain is Riemannian throughout. A Lorentzian counterpart of $\mathcal{S}_\Pi$ is not available: obtaining one requires an in-in construction on a closed time path, which is open. Lorentz analyses a posited local four-derivative kernel and returns a negative result about it (see below), so the Lorentzian link in this chain is an open problem rather than a delivered stage.

Papers of the sub-programme

Conditions for an IR Einstein sector.

Gravity — Conditions for an Infrared Einstein Sector from Spectral Geometry. Establishes $\mathcal{S}_\Pi[g] = \tfrac{1}{2}\log\det' A_g$ on a four-dimensional Riemannian manifold and derives the renormalized metric variation, strictly separating the proper-time cutoff $\Lambda = \ell_{\mathrm{sp}}^{-1}$ (which carries the $a_0\Lambda^4$ and $a_2\Lambda^2$ divergences) from zeta regularization (which carries the finite determinant and the logarithmic dependence, governed in $d = 4$ by $a_4 = \zeta_A(0)$). Results: a single minimal scalar contributes $\Delta c_{\mathrm{EH}}^\Lambda = -\Lambda^2/[12(4\pi)^2]$, which is negative; the finite renormalized coefficient $c_{\mathrm{EH}}^{\mathrm{ren}}$, its sign, and hence $G_N > 0$ are matching data — fixed by a renormalization condition and by the complete operator content, not predicted by the determinant; the IR hierarchy is set by the ratio of renormalized coefficients, $R\,L_4^2 \ll 1$ with $L_4^2 = |\beta^{\mathrm{ren}}/c_{\mathrm{EH}}^{\mathrm{ren}}|$; a conditional regime $G_N = O(\ell_{\mathrm{sp}}^2)$ survives only if the matched coefficient is cutoff-dominated and positive; BI tensorial theorem under (C1)–(C5), conditional on [H-ext] and on a supplied Einstein term; structural necessity of non-injective projection for $\mathcal{S}_\Pi > 0$. The Born–Infeld completion is not uniquely selected even under [H-ext] — the admissible completers form an explicitly parametrised family — and no algebraic relation fixes $G_N$ in terms of $\ell_{\mathrm{sp}}$ alone.

Einstein–matter coupling from spectral stationarity.

Thermodynamics — Einstein–Matter Coupling from Spectral Stationarity: A Conditional Variational Result and the Open Thermodynamic Bridge. Results: given supplied renormalized coefficients, stationarity of $\Gamma_\Pi = \mathcal{S}_\Pi^{\mathrm{ren}} - W_\Pi$ yields $G_{\mu\nu} + \Lambda_{\mathrm{ren}}g_{\mu\nu} = 8\pi G_N T_{\mu\nu}^{(\Pi)}$ — including the cosmological term $\Lambda_{\mathrm{ren}}$ — with no horizon, Rindler wedge, or Raychaudhuri theorem invoked. The heat-kernel quantity $u = -\partial_t\log K \sim L^{-2}$ is an eigenvalue scale, not a temperature (a temperature is $L^{-1}$ in natural units), so no local first law follows from the heat kernel alone. Reading the stationarity as equilibrium is a programme-level interpretation, not a result. The bridge to thermodynamics is open: it requires a heat one-form, an energy notion, and a Carathéodory–Jauch–Roberts integrability theorem.

The Lorentzian truncation: a negative result.

Lorentz — No Quartic Graviton Dispersion from the Local Covariant Four-Derivative Truncation. The local Lorentzian transverse-traceless kernel $\mathcal{O}_{\mathrm{TT}} = \Box(c_{\mathrm{EH}} + \beta\Box)$ is posited, not derived: a Lorentzian counterpart of $\mathcal{S}_\Pi$ requires an in-in construction on a closed time path, which is open, so every statement below is conditional on that posit. Results: for $c_{\mathrm{EH}} \neq 0$ the kernel is the direct sum $\ker\Box \oplus \ker(c_{\mathrm{EH}} + \beta\Box)$, and the massless summand obeys $\omega^2 = |\vec k|^2$ exactly, with no correction at any order in $|\vec k|$. Hence $A_4 = 0$ on the massless branch, there is no mapping $A_4 \leftrightarrow \ell_{\mathrm{sp}}$, and no bound on $\ell_{\mathrm{sp}}$ rests on that identification. For $\beta \neq 0$ a helicity-$2$ pole with opposite residue — a ghost, independently of the sign of $c_{\mathrm{EH}}/\beta$ — sits at signed mass squared $m_2^2 = -c_{\mathrm{EH}}^{\mathrm{ren}}/\beta^{\mathrm{ren}}$, a matching datum not tied to $\ell_{\mathrm{sp}}$. The resemblance to quadratic gravity is a comparison: the five-polarization massive spin-$2$ multiplet is not established, and the $R^2$ sector leaves a massive scalar unexcluded. Status: open.

Born–Infeld UV completion.

BornInfeld — states the capacity axiom [A-cap], bounding the per-node relaxation rate by $c_{\mathrm{BI}}$, and presents the Born–Infeld action as the saturation candidate compatible with it; the selection criterion established in the literature (exceptional propagation) is not derived from projection structure, which is stated as an open problem. The parity of the candidate action supplies the minimal fibre condition $\Pi^{-1}(y) \supseteq \{\chi, -\chi\}$, implemented in the Weil setting by O18.

Inputs and outputs

Upstream inputs. Non-injectivity as a structural necessity for $\mathcal{S}_\Pi > 0$ (ENI); the effective four-dimensional Lorentzian manifold $(M, g^{\mu\nu} = 2\eta^{\mu\nu})$ with minimal admissible Laplace-type operator $A_g = -\nabla_g^2$ from the emergent geometry sub-programme (Presentation Note 2, Q5b–Q11); the spectral length scale $\ell_{\mathrm{sp}}$ from Branch I and the Born–Infeld saturation constant $c_{\mathrm{BI}}$ of the projection dynamics; the BI parity involution and minimal fibre condition from BornInfeld and O18.

Outputs. The Einstein equations $G_{\mu\nu} + \Lambda_{\mathrm{ren}}g_{\mu\nu} = 8\pi G_N T_{\mu\nu}^{(\Pi)}$ in the matter-free IR, conditional on supplied renormalized coefficients; the conditional regime $G_N = O(\ell_{\mathrm{sp}}^2)$, holding only if the matched $c_{\mathrm{EH}}^{\mathrm{ren}}$ is cutoff-dominated and positive; the Eddington-inspired Born–Infeld action conditional on [H-ext] and on a supplied Einstein term; the $a_2 \to$ gravity, $a_4 \to$ gauge spectral stratification — consumed by Q12, Q13, the gauge–gravity stratification sub-programme (Note 8), and cosmological/phenomenological applications.

Status

Gravity in the matter-free infrared is conditionally established, not closed: the $a_2$ Einstein term dominates under $R\,L_4^2 \ll 1$, the hierarchy being set by the ratio $L_4^2 = |\beta^{\mathrm{ren}}/c_{\mathrm{EH}}^{\mathrm{ren}}|$ of renormalized coefficients rather than by $\ell_{\mathrm{sp}}$ alone (Gravity); the Einstein–matter coupling is recovered as the Euler–Lagrange condition of the stationarity of $\Gamma_\Pi$, given supplied renormalized coefficients (Thermodynamics). The Lorentzian sector is open: its transverse-traceless kernel is posited rather than derived, and under that posit the massless branch is undeformed, so the sub-programme supplies neither a gravitational-wave prediction nor an observational handle on $\ell_{\mathrm{sp}}$ (Lorentz). Scalar BI uniqueness and the parity involution are proved unconditionally (BornInfeld), as are the spectral carrier identification and the scalar BI reduction (Gravity Lemmas 2–3). The tensorial BI theorem (Gravity Theorem 1) is conditional on Hypothesis [H-ext] (admissible coherence extensivity, Lemma 1 only) and on a supplied Einstein term. Items remain open: the sign and magnitude of $c_{\mathrm{EH}}^{\mathrm{ren}}$, which are matching data rather than predictions of the determinant; the thermodynamic bridge, which requires a heat one-form, an energy notion, and a Carathéodory–Jauch–Roberts integrability theorem; the analytical proof of [H-ext]; the closed-time-path in-in construction that would yield a Lorentzian counterpart of $\mathcal{S}_\Pi$, together with the scalar and vector sectors and the full Lorentzian Einstein stationarity beyond linearisation; and the coupled $G_{\mu\nu} = 8\pi G_N T_{\mu\nu}$ with explicit matter, pending the Fermionic Matter Sub-Programme (Note 6). This note does not address Yang–Mills dynamics — that belongs to the gauge–gravity stratification sub-programme (Presentation Note 8).