Vacuum Gravity from Entropy: Stability, Spectra, and Exact Waves

Abstract

We analyze the vacuum dynamics of Gravity from Entropy, including its algebraically constrained $G$-field formulation. Evaluating the curvature traces over zero-, one-, and two-form sectors, we show that the complete Minkowski Hessian is exactly that of the quadratic-gravity action $AR+B{R}_{\hat{I}¼\hat{I}½}{R}^{\hat{I}¼\hat{I}½}$, with $A=3{\hat{I}}^2/{\ell }_{\mathrm{P}}^4$ and $B=5β^{2}/(2\ell_{\rm P}^{4})$. For diagonalizable curvature blocks, the same action reduces to a sum over eigenvalue logarithms and reproduces these coefficients exactly. A strict diagonal-curvature restriction on the perturbations is instead only a reduced subsector and excludes non-diagonalizable type-N wave curvatures. Linearizing the $G$-field equations and subsequently imposing the algebraic vacuum constraint reproduces the same reduced metric equation and covariant Minkowski Hessian. The spectrum contains the massless graviton, a scalar with $m_0^2=3/(5{\hat{I}}^2)$, and an opposite-residue spin-2 branch with $m_2^2=-6/(5{\hat{I}}^2)=-2m_0^2$. For the foundational choice ${\hat{I}}^2>0$, conventional Einstein normalization therefore implies a tachyonic spin-2 instability. We also show that every four-dimensional Ricci-flat metric solves the local bulk equations through quadratic curvature order, while square-zero Ricci-flat pp-waves are exact local vacuum solutions of the analytic metric-only logarithmic branch. On the isolated massless transverse-traceless eigenspace, the quadratic translation current has the standard general-relativistic normalization.

26 pages