From Thermodynamic Criticality to Geometric Criticality: A Linear Kernel Map from Matter Susceptibilities to Black-Hole Shadows

TL;DR

This paper develops a linear mapping from matter susceptibilities to black-hole shadow observables, revealing how thermodynamic criticality influences geometric features like the shadow radius and photon-sphere frequency.

Contribution

It introduces an explicit linear kernel map connecting thermodynamic perturbations to black-hole shadow measurements, including bounds and a numerical pipeline.

Findings

Thermodynamic critical exponents transfer to geometric susceptibilities.

Explicit kernels separate near-photon-sphere and far-zone effects.

Numerical pipeline with convergence diagnostics is provided.

Abstract

We construct an explicit linear map from compact, conserved thermodynamic/effective-medium perturbations of the stress-energy tensor to the metric response in static, spherically symmetric spacetimes, and from there to geometric observables of direct relevance to horizon-scale imaging: the shadow radius and photon-sphere frequency. The response is expressed through $L^{1}$-bounded kernels written in a piecewise "local $+$ tail" form, which makes transparent the separation between near-photon-sphere sensitivity and far-zone contributions (including AdS tails). Under mild assumptions on the matter susceptibilities near a critical point, dominated convergence transfers the thermodynamic exponent to the geometric susceptibility, $\gamma_{\rm sh}=\gamma_{\rm th}$, with controlled analytic corrections. We further provide AdS far-zone bounds with explicit outside-support constants depending only on background geometric data at the photon sphere and shell geometry. A reproducible numerical pipeline with convergence diagnostics is presented and benchmarked.