Looking at extremal black holes from very far away

TL;DR

The paper addresses how quantum corrections to near-extremal black holes can be captured from the full higher-dimensional geometry, rather than solely from the near-horizon throat. It identifies nearly-gapless light modes (Schwarzian, gauge, and rotational) in the full geometry, whose eigenvalues scale linearly with temperature and which localize near the horizon as extremality is approached. A contour prescription for the Euclidean Einstein-Maxwell action is developed to regulate sign-indefinite contributions, and the authors provide analytic results for BTZ and numerical results for hyperbolic and RN-AdS$_4$ cases, showing the universal $Z \sim e^{S_0}\, T^{3/2}$ behavior from Schwarzian modes. The work clarifies when rotational and gauge modes extend to the full geometry and discusses the flat-space limit, with implications for the interpretation of low-temperature thermodynamics and for connecting throat analyses to higher-dimensional gravity.

Abstract

Near-extremal black holes are subject to large quantum effects, which modify their low-temperature thermodynamic behavior. Hitherto, these quantum effects were analyzed by separating the geometry into the near-horizon region and its exterior. It is desirable to understand and reproduce such corrections from the full higher-dimensional asymptotically flat or AdS geometry's perspective. We address this question in this article and fill this gap. Specifically, we find off-shell eigenmodes of the quadratic fluctuation operator of the Euclidean gravitational dynamics, with eigenvalues that vanish linearly with temperature. We illustrate this for BTZ and neutral black holes with hyperbolic horizons in AdS in Einstein-Hilbert theory, and for the charged black holes in Einstein-Maxwell theory. The linear scaling with Matsubara frequency, which is a distinctive feature of the modes, together with the fact that their wavefunctions localize close to the horizon as we approach extremality, identifies them as responsible for the aforementioned quantum effects. We provide a contour prescription to deal with the sign indefiniteness of the Euclidean Einstein-Maxwell action, which we derive to aid our analysis. We also resolve a technical puzzle regarding modes associated with rotational isometries in stationary black hole spacetimes.

Figures (12)

• Figure 2: The uplift of Schwarzian modes to the full hyperbolic 4 black hole geometry. We show the data for very low temperatures in the top panel, \ref{['fig:hypschwzoom']}, demonstrating linear behavior with temperature. In the bottom left panel, \ref{['fig:hypschwfull']}, we show the behavior over a greater range of temperatures, making the departures from linear growth manifest. Finally, in the bottom right panel \ref{['fig:hypschwcollapse']}, we plot the eigenvalues rescaled by the overtone number $\frac{2}{n}\,\lambda_n$ to demonstrate that $\lambda_n \sim n\,T$ (notice the three curves are overlapping, which was achieved by working with extended precision numerics).
• Figure 3: The uplift of probe Maxwell field zero modes to the full hyperbolic 4 black hole geometry. We show the data for very low temperatures in the top panel \ref{['fig:hypu1zoom']}, demonstrating linear behavior with temperature. In the bottom left panel \ref{['fig:hypsu1full']}, we show the behavior over a greater range of temperatures, making the departures from linear growth manifest. Finally, in the bottom right panel \ref{['fig:hypu1collapse']}, we plot the eigenvalues rescaled by the overtone number $\frac{1}{n}\,\lambda_n$ to demonstrate that $\lambda_n \sim n\,T$.
• Figure 4: The uplift of the (formal) rotational field zero modes to the full hyperbolic 4 black hole geometry. We show the data for very low temperatures in the top panel, \ref{['fig:hyprotzoom']}, demonstrating linear behavior with temperature. In the bottom left panel, \ref{['fig:hyprotfull']}, we show the behavior over a greater range of temperatures, making the departures from linear growth manifest. Finally, in the bottom right panel, \ref{['fig:hyprotcollapse']}, we plot the eigenvalues rescaled by the overtone number $\frac{1}{n}\,\lambda_n$ to demonstrate that $\lambda_n \sim n\,T$. While the plot very closely resembles that of the probe Maxwell field \ref{['fig:hyperbolic_gauge']}, there are subtle differences in the numerical data.
• Figure 5: The uplift of Schwarzian modes to the full Reissner-Nordström-4 black hole geometry with $\mu=10$. We show the data for very low temperatures in the top panel, \ref{['fig:rnschwzoom']}, demonstrating linear behavior with temperature. In the bottom left panel, \ref{['fig:rnschwfull']}, we show the behavior over a greater range of temperatures, making the (mild) departures from linear growth manifest. Finally, in the bottom right panel \ref{['fig:rnschwcollapse']}, we plot the eigenvalues rescaled by the overtone number $\frac{2}{n}\,\lambda_n$ to demonstrate that $\lambda_n \sim n\,T$ (notice the three curves are overlapping, which was achieved by working with extended precision numerics).
• Figure 6: The behavior of the norm of the perturbations, with $n=2$, as a function of proper distance $\rho \sim \lads \, \log(r/r_+)$ from the horizon. As with earlier results, this is for a Reissner-Nordström-4 black hole geometry with $\mu=10$. The norm $\norm{X}^2$ defined as the integrand of \ref{['eq:schnorm']}, viz., the specific combination of the functions $f_i$, $a_t$ and $a_r$. The data is shown for $\kappa \lads \in \{0.01, 0.005,0.002,0.001,0.0005,0.0002, 0.0001\}$. On the left panel \ref{['fig:rnschhnorm']} we show the bare norm which increases in amplitude as we go to lower temperature, similar to the feature seen in the BTZ case \ref{['fig:h_squared_btz']}. However, the norm density, which is normalized to integrate to unity, displays data collapse as depicted in \ref{['fig:rnschhnormdensity']}.