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On non-uniform smeared black branes

Hideaki Kudoh, Umpei Miyamoto

TL;DR

We address the stability and thermodynamics of charged dilatonic smeared black branes on a transverse circle by applying the Harmark–Obers ansatz, which maps the problem to neutral black branes in an effective dimension $\mathcal{D}=D-p=d+1$. Using static perturbations up to third order, we construct non-uniform solutions and extract the GL critical wavelength $k_0$ and its higher-order correction $k_1$, with the analysis showing these quantities are largely independent of charge. Thermodynamic analysis reveals a non-universal critical dimension: in the microcanonical ensemble the non-uniform phase is entropically disfavored for $\mathcal{D}\le 13$, while in the canonical ensemble the phase becomes favorable only for $\mathcal{D}>12$ (vacuum) or $\mathcal{D}>14$ near extremality for smeared branes. Near extremality, reduced quantities indicate the critical dimension shifts further, underscoring ensemble- and extremality-dependent phase structure and supporting the GM conjecture in the explored regimes. Overall, the perturbative HO framework provides a tractable route to fully nonlinear solutions and a deeper understanding of the smeared brane phase diagram.

Abstract

We investigate charged dilatonic black $p$-branes smeared on a transverse circle. The system can be reduced to neutral vacuum black branes, and we perform static perturbations for the reduced system to construct non-uniform solutions. At each order a single master equation is derived, and the Gregory-Laflamme critical wavelength is determined. Based on the non-uniform solutions, we discuss thermodynamic properties of this system and argue that in a microcanonical ensemble the non-uniform smeared branes are entropically disfavored even near the extremality, if the spacetime dimension is $D \le 13 +p$, which is the critical dimension for the vacuum case. However, the critical dimension is not universal. In a canonical ensemble the vacuum non-uniform black branes are thermodynamically favorable at $D > 12+p$, whereas the non-uniform smeared branes are favorable at $D > 14+p$ near the extremality.

On non-uniform smeared black branes

TL;DR

We address the stability and thermodynamics of charged dilatonic smeared black branes on a transverse circle by applying the Harmark–Obers ansatz, which maps the problem to neutral black branes in an effective dimension . Using static perturbations up to third order, we construct non-uniform solutions and extract the GL critical wavelength and its higher-order correction , with the analysis showing these quantities are largely independent of charge. Thermodynamic analysis reveals a non-universal critical dimension: in the microcanonical ensemble the non-uniform phase is entropically disfavored for , while in the canonical ensemble the phase becomes favorable only for (vacuum) or near extremality for smeared branes. Near extremality, reduced quantities indicate the critical dimension shifts further, underscoring ensemble- and extremality-dependent phase structure and supporting the GM conjecture in the explored regimes. Overall, the perturbative HO framework provides a tractable route to fully nonlinear solutions and a deeper understanding of the smeared brane phase diagram.

Abstract

We investigate charged dilatonic black -branes smeared on a transverse circle. The system can be reduced to neutral vacuum black branes, and we perform static perturbations for the reduced system to construct non-uniform solutions. At each order a single master equation is derived, and the Gregory-Laflamme critical wavelength is determined. Based on the non-uniform solutions, we discuss thermodynamic properties of this system and argue that in a microcanonical ensemble the non-uniform smeared branes are entropically disfavored even near the extremality, if the spacetime dimension is , which is the critical dimension for the vacuum case. However, the critical dimension is not universal. In a canonical ensemble the vacuum non-uniform black branes are thermodynamically favorable at , whereas the non-uniform smeared branes are favorable at near the extremality.

Paper Structure

This paper contains 17 sections, 76 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Smeared black branes
  3. Harmark-Obers ansatz
  4. Thermodynamic quantities
  5. Static perturbation
  6. First-order perturbation $X_{1,0}$
  7. Second-order perturbation
  8. Zero modes $X_{0,1}$
  9. KK modes $X_{2,0}$
  10. Third-order perturbation $X_{1,1}$
  11. Thermodynamics
  12. Vacuum black branes
  13. Weakly charged smeared black branes
  14. Near-extremal smeared black branes
  15. Conclusion
  16. ...and 2 more sections

Figures (2)

  • Figure 1: The numerical plots of metric perturbations for $\mathcal{D}=6$. The event horizon is located at $y=1$. Each line is normalized by the absolute value of $b$ at the horizon. Only $b_{0,1}$ (and $a_{0,1}$) is the zero mode, so it shows slow decay at the asymptotics, compared to other KK massive modes.
  • Figure 2: Dependence of the coefficients on the effective spacetime dimension ${\mathcal{D}}$. $\rho_2$ and $\delta \rho_2$ are divided by $\mathcal{D}$ and ${\mathcal{D}}^2$, respectively. The sign change of $\sigma_2$ is at ${\mathcal{D}} > 13$, whereas the sign change of $\rho_2$ is at ${\mathcal{D}} > 12$.