Conformal Boundary Conditions and Higher Curvature Gravity

TL;DR

This work extends conformal boundary conditions to higher-curvature gravity by formulating extended conformal boundary conditions (ECBC) for Einstein-Maxwell-Gauss-Bonnet (EMGB) gravity. It demonstrates that the high-temperature conformal entropy remains extensive and governed by boundary data even with GB corrections, and identifies a universal flat-space limit where leading GB contributions drop out, suggesting a universal flat-space degree-of-freedom count. The Lorentzian analysis shows boundary Weyl-mode dynamics remain unstable for spherical horizons but can exhibit stability for planar and hyperbolic horizons, depending on charge and GB coupling. The results point toward a possible holographic interpretation for finite spacetime regions and introduce a dimensionless boundary scale $\mathcal{P}$ that controls the flat-space limit, with implications for flat-space holography and LR boundary dynamics.

Abstract

We initiate a systematic study of Einstein-Gauss-Bonnet gravity in the presence of boundaries subject to conformal boundary conditions, in which the conformal class of the boundary metric is kept fixed. In Einstein gravity, the trace of the extrinsic curvature is also fixed at the boundary. Here we generalize this boundary condition with the appropriate higher curvature correction. We study the problem both in Euclidean and Lorentzian signature. In Euclidean signature, we show that, similarly to the Einstein gravity case, the entropy at large temperatures exhibits the behavior of a conformal field theory in one lower dimension. We also show that in the flat space limit, the higher curvature corrections do not contribute to the leading behavior at high temperatures. We conjecture that this result is a universal feature of the flat space limit in the presence of conformal boundaries. We test our conjecture by analyzing charged black holes. In Lorentzian signature, we analyze the dynamics of the boundary Weyl factor in black hole backgrounds at the linearized level.

Figures (7)

• Figure 1: The function $\frac{G_N}{\ell_\textrm{\tiny AdS}^{d-1}}\mathcal{N}_{\text{dof}}(\mathcal{P}\ell_\textrm{\tiny AdS})$ for Einstein gravity (GR) \ref{['eq:N_dof(K)_GR_cases']}, the leading Gauss-Bonnet (GB perturbative) correction in perturbation theory \ref{['eq:leading_a_SconfAdS']} and the numerical solution for finite $\tilde{\alpha}=0.01,\,0.1$ (GB non-perturbative). The horizontal dashed semi-lines correspond to the value of each function at the AdS boundary. The numerical GB solution exhibits similar monotonic behavior for any finite value of the coupling $\tilde{\alpha}$ and in any dimension $d$.
• Figure 2: Plot of the function $\frac{G_N}{\ell_\textrm{\tiny dS}^{d-1}}\mathcal{N}_{\text{dof}}(\mathcal{P}\ell_\textrm{\tiny dS})$ of the number of degrees of freedom in the putative boundary dual of dS$_5$, for the leading Gauss-Bonnet (GB) correction of eq. \ref{['eq:leading_a_SconfdS']}. Both the perturbative \ref{['eq:Ndof_dS']} and non-perturbative EGB curves as shown, along with the GR one. The general trend is that $\mathcal{N}_{\text{dof}}(\mathcal{P}\ell_\textrm{\tiny dS})$ is larger for greater values of $\tilde{\alpha}$. Again, the story is the same in higher dimensions.
• Figure 3: Plot of $\lambda^2(\tfrac{\mathfrak{r}}{\ell})$\ref{['eq:BH_lambda2']} for the charged AdS$_5$ black hole for $\mu=1, \ q=0.1$, and for various negative and positive values of the coupling $\tilde{\alpha}$. The gray vertical dashed line indicates the radius of the outer event horizon for $\tilde{\alpha}=0.24$. The qualitative features of the curves do not depend on the precise value of the charge $q$ or the dimension $d$.
• Figure 4: Plot of $\lambda^2(\mathfrak{r})$\ref{['eq:BH_lambda2']} for a charged black hole in $5$-dimensional flat space, where $\mu=1, \ q=0.1$ (the values of $d,q$ do not affect the qualitative aspects of the curves), for various negative and positive values of the coupling $\alpha$. The gray vertical dashed line on the left plot corresponds to the horizon position when $\alpha=0.6$.
• Figure 5: Plots of $\lambda^2(\tfrac{\mathfrak{r}}{\ell})$\ref{['eq:BH_lambda2']} for the charged AdS$_5$ black brane, when $\mu=1, \ q=0.01 \ \text{(left)} \ \text{and} \ q=0.5 \ \text{(right)}$, for the various indicated values of the coupling $\tilde{\alpha}$. The gray vertical dashed line indicates the radius of the outer event horizon when $\tilde{\alpha}=0.01$, while the curves in all cases blow-up at the position of the horizon. Again, the general picture is the same in higher dimensions.