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Journal of High Energy Physics

, 2010:54 | Cite as

Thermodynamical metrics and black hole phase transitions

  • Haishan Liu
  • H. Lü
  • Mingxing Luo
  • Kai-Nan Shao
Article

Abstract

An important phase transition in black hole thermodynamics is associated with the divergence of the specific heat with fixed charge and angular momenta, yet one can demonstrate that neither Ruppeiner’s entropy metric nor Weinhold’s energy metric reveals this phase transition. In this paper, we introduce a new thermodynamical metric based on the Hessian matrix of several free energy. We demonstrate, by studying various charged and rotating black holes, that the divergence of the specific heat corresponds to the curvature singularity of this new metric. We further investigate metrics on all thermodynamical potentials generated by Legendre transformations and study correspondences between curvature singularities and phase transition signals. We show in general that for a system with n-pairs of intensive/extensive variables, all thermodynamical potential metrics can be embedded into a flat (n,n)-dimensional space. We also generalize the Ruppeiner metrics and they are all conformal to the metrics constructed from the relevant thermodynamical potentials.

Keywords

Black Holes Classical Theories of Gravity 

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Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  • Haishan Liu
    • 1
    • 2
  • H. Lü
    • 3
    • 4
  • Mingxing Luo
    • 1
  • Kai-Nan Shao
    • 1
    • 2
  1. 1.Zhejiang Institute of Modern Physics, Department of PhysicsZhejiang UniversityHangzhouChina
  2. 2.Kavli Institute for Theoretical Physics China, CASBeijingChina
  3. 3.China Economics and Management AcademyCentral University of Finance and EconomicsBeijingChina
  4. 4.Institute for Advanced StudyShenzhen UniversityShenzhenChina

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