The Spectral Gravity Sub-Programme

Overview

This sub-programme works on a supplied Riemannian metric. Its reading as the emergent metric of the emergent geometry sub-programme (Presentation Note 2) is conditional on the unestablished spatial-limit hypothesis [H-L], the open lifting hypothesis [H-lift], a supplied rank-four extension and a continuation from Riemannian signature that is not performed, and no metric coefficient value is established. Under which conditions does the dynamics of such a metric 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 $A_g = -\nabla_g^2$ on that metric. The Einstein tensor locally dominates the renormalized metric variation when the matched Einstein coefficient is non-zero and the curvature is small, its cutoff contribution coming from $a_2$; 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 and at leading local infrared order; 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. $\mathcal{S}_\Pi[g]$, on a supplied metric, is the only input. The $a_2$ cutoff sector carries the Einstein–Hilbert term, 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), version 2.4

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

On a supplied Riemannian metric, matter-free IR sector conditional on supplied renormalized coefficients; emergent reading of the metric conditional on [H-L], [H-lift], a supplied rank-four extension and a continuation; 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; non-injectivity as a structural necessity for genuinely emergent descriptions (ENI), the identification of $\mathcal{S}_\Pi$ with the projection entropy being open. 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 and at leading local infrared order, 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 structure of the candidate action is a response-family covariance established in the Weil setting by O18; by O18's no-go, it does not force the fibre condition $\Pi^{-1}(y) \supseteq \{\chi, -\chi\}$, which is carried as a structural hypothesis (the Weil-level conjugation is an O17 theorem, the physical-fibre identification an open bridge).

Inputs and outputs

Upstream inputs. Non-injectivity as a structural necessity for genuine emergence, with strictly positive projection entropy (ENI; the spectral functional $\mathcal{S}_\Pi[g]$ is a sibling realisation, its identification with the projection entropy being open); a supplied four-dimensional Riemannian metric, on which the minimal Laplace-type operator $A_g = -\nabla_g^2$ is elliptic, whose reading as the emergent metric of the emergent geometry sub-programme (Presentation Note 2) is conditional on the unestablished spatial-limit hypothesis [H-L], the open lifting hypothesis [H-lift], a supplied rank-four extension and a continuation from Riemannian signature that is not performed, no metric coefficient value being established; the spectral length scale $\ell_{\mathrm{sp}}$ from Branch I and the Born–Infeld saturation constant $c_{\mathrm{BI}}$ of the projection dynamics; the capacity axiom [A-cap] from BornInfeld; the parity structure of the Born–Infeld candidate action as a response-family covariance from O18, with the fibre involution carried as a structural hypothesis (not forced by it, per O18's no-go).

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 and at leading local infrared order; 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 organising reading $a_2 \to$ gravity, $a_4 \to$ gauge — consumed by Q12, Q13, the gauge–gravity stratification sub-programme (Note 8), and cosmological/phenomenological applications.

Status

On a supplied Riemannian metric, gravity in the matter-free infrared is conditionally established, not closed: the Einstein term, a matching datum whose cutoff contribution comes from $a_2$, dominates locally, by power counting, when it is non-zero and $R\,L_4^2 \ll 1$, 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, at leading local infrared order (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). The constraints on the sector function are proved without [H-ext] (Gravity Lemma 2) and show that the Born–Infeld member is not selected; among carriers linear in the curvature, of two-derivative order and with a spectrum that is not a multiple of the identity, the spectral carrier is classified up to a trace parameter (Gravity Lemma 3), and the saturation condition applied to the Ricci endomorphism selects the mixed Ricci carrier (Gravity Theorem 1); [A-cap] is an axiom and the Born–Infeld action a saturation candidate, not a derived or unique completion (BornInfeld); the parity structure of the candidate action is a response-family covariance established in the Weil setting by O18, and the fibre involution itself is a structural hypothesis, not forced by it (O18 no-go). 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).