## Abstract

We evaluate the full time dependence of holographic complexity in various eternal black hole backgrounds using both the complexity=action (CA) and the complexity=volume (CV) conjectures. We conclude using the CV conjecture that the rate of change of complexity is a monotonically increasing function of time, which saturates from below to a positive constant in the late time limit. Using the CA conjecture for uncharged black holes, the holographic complexity remains constant for an initial period, then briefly decreases but quickly begins to increase. As observed previously, at late times, the rate of growth of the complexity approaches a constant, which may be associated with Lloyd’s bound on the rate of computation. However, we find that this late time limit is approached from above, thus violating the bound. For either conjecture, we find that the late time limit for the rate of change of complexity is saturated at times of the order of the inverse temperature. Adding a charge to the eternal black holes washes out the early time behaviour, i.e. complexity immediately begins increasing with sufficient charge, but the late time behaviour is essentially the same as in the neutral case. We also evaluate the complexity of formation for charged black holes and find that it is divergent for extremal black holes, implying that the states at finite chemical potential and zero temperature are infinitely more complex than their finite temperature counterparts.

Original language | English |
---|---|

Article number | 188 |

Journal | Journal of High Energy Physics |

Volume | 2017 |

Issue number | 11 |

DOIs | |

State | Published - 1 Nov 2017 |

Externally published | Yes |

### Bibliographical note

Funding Information:We would like to thank Alice Bernamonti, Adam Brown, William Cottrell, Josiah Couch, Bartek Czech, Lorenzo Di-Pietro, Federico Galli, Robie Hennigar, Javier Martinez, Henry Maxfield, Mark Mezei, Miguel Montero, Djordje Radicevic, Dan Roberts, Jamie Sully, Lenny Susskind, Todd Sierens, Brian Swingle, Tadashi Takayanagi and Ying Zhao for useful comments and discussions. We also would like to thank Ipsita Mandal for collaboration on the initial stages of this project. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. SC acknowledges support from an Israeli Women in Science Fellowship from the Israeli Council of Higher Education. RCM is supported by funding from the Natural Sciences and Engineering Research Council of Canada, from the Canadian Institute for Advanced Research and from the Simons Foundation through the “It from Qubit” collaboration. SS thanks Perimeter Institute for their hospitality during this project. The work of SS is supported in part by the Grant-in-Aid for JSPS Research Fellow, Grant Number JP16J01004.

Publisher Copyright:

© 2017, The Author(s).

## Keywords

- AdS-CFT Correspondence
- Black Holes
- Gauge-gravity correspondence

## ASJC Scopus subject areas

- Nuclear and High Energy Physics