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

TL;DR

This work builds a direct, first-principles correspondence between thermodynamic criticality and geometric observables of black holes by linearizing Einstein equations about a vacuum background and sourcing conserved, compact perturbations of the stress-energy tensor. It constructs $L^{1}$-bounded kernel maps that relate perturbations to metric responses, and then to optical observables such as the shadow radius $R_{sh}$ and the photon-sphere frequency $Ω_{ph}$, with a clear local-plus-tail structure separating near-field and far-field contributions. Under mild near-critical assumptions, dominated convergence transfers the thermodynamic critical exponent to the geometric susceptibility with $oxed{γ_{sh}=γ_{th}}$ and analytic corrections, while AdS tails furnish explicit outside-support bounds. The authors validate the approach with a reproducible numerical pipeline, demonstrate a universal scaling collapse across models, and derive an observability criterion showing horizon-scale imaging could detect criticality at the few-percent level in next-generation observations.

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, $γ_{\rm sh}=γ_{\rm th}$, 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.

Figures (6)

• Figure 1: Equation-of-state–driven source triplets $\{\Delta\rho,\Delta p_r,\Delta p_t\}$ for several $T/T_c$ values. The compact support and edge regularity are evident across all families.
• Figure 2: Metric perturbation $\delta f(r)$ and cumulative mass $\mathcal{M}_{\rm eff}(r)$. All cases saturate by $r\gtrsim50M$, with near-field slopes set by source anisotropy.
• Figure 3: Asymptotic consistency of the metric perturbation.
• Figure 4: Shadow radius and susceptibility scaling.
• Figure 5: Grid refinement (a) and domain truncation (b) diagnostics for numerical convergence.