Classical emergence of the quantum-backreacted BTZ black hole from exponential electrodynamics

TL;DR

The paper demonstrates that the logarithmically deformed BTZ black hole, originally found as a quantum-backreacted solution in three-dimensional New Massive Gravity, has a purely classical realization as a static solution of Einstein gravity coupled to exponential nonlinear electrodynamics. By analyzing the on-shell content under staticity and circular symmetry, it proves a dynamical equivalence between the quantum-corrected NMG and Einstein–NLED gravity, identifying a unique shared solution class f(r) = c_0 + c_1 r + c_2 r^2 + c_3 r \ln r and providing an explicit parameter mapping between quantum backreaction and classical charges. The thermodynamic analysis via Iyer–Wald reveals that, while the global BTZ mass is preserved, the backreaction shifts intensive quantities and response functions, with an extended first law and Smarr relation incorporating the NLED couplings. The work further clarifies the horizon structure and shows how the classical limit of the quantum configuration emerges as the Maxwell-like regime of the ENLED theory, suggesting a broader correspondence between higher-curvature quantum corrections and nonlinear classical electrodynamics in 3D. Together, these results offer a concrete route to interpret semiclassical spacetimes within a purely classical framework and motivate exploration of similar dualities beyond the static, purely electric sector.

Abstract

In this work, we revisit a recently reported generalization of the Bañados--Teitelboim--Zanelli black hole arising in New Massive Gravity sourced by the quantum fluctuations of scalar matter, now examined through the lens of a purely classical framework. We show that the same geometry, distinguished by its logarithmic asymptotic structure, emerges as the unique static solution of Einstein gravity coupled to an exponential nonlinear electrodynamics. We trace the origin of this correspondence and prove that this geometry belongs to a unique class of metrics constituting the intersection of the moduli spaces of the static and circularly symmetric sectors of the two theories, thereby revealing a dynamical equivalence between them. An explicit mapping is established between the global charges of the nonlinearly charged black holes and the parameters governing the quantum backreaction in New Massive Gravity, allowing for a natural reinterpretation of the quantum imprints in terms of classical charges. A detailed analysis of the horizon structure of these spacetimes is presented. In addition, the full thermodynamics of the more general configurations is constructed using the Iyer--Wald formalism, from which we derive the first law and the associated Smarr relation. Altogether, our results provide a classical realization of a semiclassical spacetime and point toward a broader correspondence between higher-curvature corrections in quantum gravity and nonlinear effects in self-gravitating electrodynamics in three dimensions.

Figures (2)

• Figure 1: Horizon structure in the three possible classes of critical points. (a) When a single critical point is present, the solution develops only one horizon, and the sign of the Killing norm reverses once this surface is crossed. (b) For configurations admitting two critical points—associated with the two real branches of the Lambert function—the metric can exhibit one, two, or three horizons. In the three-horizon case, the time direction is restored after the first Cauchy horizon, but reverses again upon entering the innermost region. (c) In the degenerate case, where the two critical points degenerate into one, the geometry again has a single horizon, with the same reversal of the time direction beyond it. In all panels we have fixed $c_{2}=1$ to enforce AdS asymptotics and chosen $c_{0}=-5$, corresponding to a positive mass as discussed in Sec. \ref{['sec:Thermo']}.
• Figure 2: Representative example of the allowed parameter region in the $(\beta,\gamma)$ subspace for which all local thermodynamic stability conditions are simultaneously satisfied. The upper branch corresponds to configurations with positive electric charge ($Q>0$), while the lower branch corresponds to negative charge ($Q<0$). In this plot, the pressure $P$, temperature $T$, and entropy $S$ are held fixed; varying these quantities results only in a rescaling of the admissible region and does not alter its qualitative structure.