• Open Access

Complexity-action duality of the shock wave geometry in a massive gravity theory

Yan-Gang Miao and Long Zhao
Phys. Rev. D 97, 024035 – Published 25 January 2018

Abstract

On the holographic complexity dual to the bulk action, we investigate the action growth for a shock wave geometry in a massive gravity theory within the Wheeler-DeWitt (WDW) patch at the late time limit. For a global shock wave, the graviton mass does not affect the action growth in the bulk, i.e., the complexity on the boundary, showing that the action growth (complexity) is the same for both the Einstein gravity and the massive gravity. Nevertheless, for a local shock wave that depends on transverse coordinates, the action growth (complexity) caused by the boundary disturbance (perturbation) is proportional to the butterfly velocity for the two gravity theories, but the butterfly velocity of the massive gravity theory is smaller than that of the Einstein gravity theory, indicating that the action growth (complexity) of the massive gravity is depressed by the graviton mass. In addition, we extend the black hole thermodynamics of the massive gravity and obtain the right Smarr formula.

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  • Received 14 August 2017

DOI:https://doi.org/10.1103/PhysRevD.97.024035

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Yan-Gang Miao1,2,3,* and Long Zhao1,†

  • 1School of Physics, Nankai University, Weijin Str. 94, Tianjin 300071, China
  • 2Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany
  • 3Institut für Theoretische Physik, Goethe-Universität Frankfurt am Main, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany

  • *miaoyg@nankai.edu.cn
  • longzhao@mail.nankai.edu.cn

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Issue

Vol. 97, Iss. 2 — 15 January 2018

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