Journal of High Energy Physics

, 2019:112 | Cite as

Extended thermodynamics and complexity in gravitational Chern-Simons theory

  • Antonia M. Frassino
  • Robert B. MannEmail author
  • Jonas R. Mureika
Open Access
Regular Article - Theoretical Physics


We study several aspects of the extended thermodynamics of BTZ black holes with thermodynamic mass \( M=\alpha m+\gamma \frac{j}{\ell } \) and angular momentum J = αj + γℓm, for general values of the parameters (α, γ) ranging from regular (α = 1, γ = 0) to exotic (α = 0, γ = 1). We show that there exist two distinct behaviours for the black holes, one when α > γ (“mostly regular”), and the other when γ < α (“mostly exotic”). We find that the Smarr formula holds for all (α, γ). We derive the corresponding thermodynamic volumes, which we find to be positive provided α and γ satisfy a certain constraint. The dependence of pressure on volume is unremarkable and strictly decreasing when α > γ, but a maximum volume emerges for large JT when γ > α; consequently an exotic black hole of a given horizon circumference and temperature can exist in two distinct anti de Sitter backgrounds. We compute the reverse isoperimetric ratio, and study the Gibbs free energy and criticality conditions for each. Finally we investigate the complexity growth of these objects and find that they are all proportional to the complexity of the BTZ black hole. Somewhat surprisingly, purely exotic BTZ black holes have vanishing complexity growth.


Black Holes AdS-CFT Correspondence Models of Quantum Gravity 


Open Access

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Antonia M. Frassino
    • 1
    • 2
  • Robert B. Mann
    • 3
    • 4
    Email author
  • Jonas R. Mureika
    • 5
  1. 1.Departament de Física Quàntica i Astrofísica, Institut de Ciències del CosmosUniversitat de BarcelonaBarcelonaSpain
  2. 2.Département de Physique Théorique and Center for Astroparticle PhysicsUniversité de GenèveGeǹeve 4Switzerland
  3. 3.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  4. 4.Department of Physics and AstronomyUniversity of WaterlooWaterlooCanada
  5. 5.Department of PhysicsLoyola Marymount UniversityLos AngelesU.S.A.

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