Linearized Einstein’s equation around pure BTZ from entanglement thermodynamics

  • Partha PaulEmail author
  • Pratik Roy
Research Article


It is known that the linearized Einstein’s equation around the pure AdS can be obtained from the constraint \( \Delta S = \Delta \langle H \rangle \), known as the first law of entanglement, on the boundary CFT. The corresponding dual state in the boundary CFT is the vacuum state around which the linear perturbation is taken. In this paper we revisit this question, in the context of \( AdS _3/ CFT _2 \), with the state of the boundary \( CFT _2\) as a thermal state. The corresponding dual geometry is a planar BTZ black hole. By considering the linearized perturbation around this black brane we show that Einstein’s equation follows from the first law of entanglement.


Ads/CFT correspondence Entanglement thermodynamics BTZ black hole Linearised Einstein equations 



We are grateful to Shamik Banerjee for suggesting this problem and guiding us throughout the project. We are also thankful to him for carefully reading the draft and for correcting some parts of it. We are thankful to Amitabh Virmani and Jyotirmay Bhattacharya for helpful discussions on related matters. We would also like to acknowledge the hospitality at Chennai Mathematical Institute during the workshop ‘Student Talks in Trending Topics in Theory \((\hbox {ST}^4)\)’ where part of this work was done. PR is grateful to Debangshu Mukherjee for his generous help and multiple discussions on this work. PR is partially supported by a grant to CMI from the Infosys Foundation.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of PhysicsBhubaneshwarIndia
  2. 2.Homi Bhabha National InstituteMumbaiIndia
  3. 3.Chennai Mathematical InstituteSiruseriIndia

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