Action-complexity in GMMG and EGMG

Abstract

In this paper, we study the holographic complexity of rotating Banados–Teitelboim–Zanelli black hole in the context of the ”Complexity-Action” conjecture and in the framework of General Minimal Massive Gravity (GMMG) and Exotic General Massive Gravity (EGMG). In addition, we investigate the action growth rate based on two kinds of conserved charges of the theories and examine the Lloyd bound on the rate of quantum computation. Then, we study the complexity of the formation of black holes in both theories.

Appendix: Boundary term

The action principle can be written in a CS−like form

$$\begin{aligned} L=\frac{1}{2}{\hat{g}}_{rs}a^{r}.da^{s}+\frac{1}{6}{\hat{f}}_{rst}a^{r}.(a^{s}\times a^{t}). \end{aligned}$$

(90)

The variation of the Lagrangian can be given as

$$\begin{aligned} \delta L=\sum E_{\phi _{i}}\delta \phi _{i} +d\Theta (\phi _{i},\delta \phi _{i}) \end{aligned}$$

(91)

where $\phi _{i}$ indicates all the fields (metric and matter) which solve their corresponding equations of motion $E_{\phi _{i}}$ and the presymplectic potential $\Theta$ is defined up to add some ambiguities related to the corner terms in the action principle. For the CS—like theory of gravity, the presymplectic form is given by [18]

$$\begin{aligned} \Theta =-\dfrac{1}{2}g_{rs}\delta u^{r}.u^{s}. \end{aligned}$$

(92)

where the constant $g_{rs}$ can be interpreted as a symmetric tensor and for GMMG is given by [21]

$$\begin{aligned} {\hat{g}}_{\omega e}=-\sigma , {\hat{g}}_{eh}=1, {\hat{g}}_{f\omega }=\frac{-1}{m^2}, {\hat{g}}_{\omega \omega }=\frac{1}{\mu }. \end{aligned}$$

(93)

By considering $u=\lbrace e,\omega ,h,f\rbrace$, one can obtain the presymplectic form of GMMG

$$\begin{aligned} \Theta _{GMMG}=-\left( \sigma +\dfrac{c_{f}}{m^2}\right) \delta \omega .e+\dfrac{1}{2\mu }\delta \omega .\omega +c_{h}\delta e. e, \end{aligned}$$

(94)

and considering $u=\lbrace e,\omega ,h,f\rbrace$, one can obtain the presymplectic form of EGMG

$$\begin{aligned} \Theta _{EGMG}=\dfrac{l}{2}\left[ \left( c_{h}+\dfrac{c_{f}^2}{m^4}\right) \delta e.e+\left( 1-\dfrac{\nu }{m^2}\right) \delta \omega .\omega -\dfrac{c_{f}}{m^2}\delta e.\omega \right] . \end{aligned}$$

(95)

where we have used

$$\begin{aligned} {\hat{g}}_{e h}=l, {\hat{g}}_{ff}=\dfrac{l}{m^4}, {\hat{g}}_{\omega \omega }=l\left( 1-\dfrac{\nu }{m^2}\right) , {\hat{g}}_{f\omega }=-\frac{l}{m^2}. \end{aligned}$$

(96)

Then by removing $\delta$ from the $\Theta$ one can achieve the Eqs. (10) and (41) for boundary terms.