include the main part of our calculations, and we directly evaluate the rate of complexity growth in BMSFTs by using flat space holography in, respectively, $d\geq 3$ and $d=2$ dimensions. The last section is devoted to discussions.

Preliminaries
=============

In this section we use the flat-space holography to compute the rate of complexity growth of BMSFT. We use the CA proposal for BMSFT$_2$ and BMSFT$_3$, which requires computation of the on-shell action for, respectively, three- and four- dimensional asymptotically flat geometries. In this paper we consider static solutions with line element $$ds^2 = - f(r) dt^2+\frac{dr^2}{f(r)} +r^2 d \Omega^2_{d-1},$$ where $f(r)=-8 G_N M$ for $d=2$ and for $d\geq 3$ it is given by $$\label{schwmet}
f(r) = 1 - \frac{\omega^{d-2}}{r^{d-2}}, \qquad M= \frac{d-1}{16 \pi G_{N}} \Omega_{d-1} \omega^{d-2},$$ where M is the mass parameter and $\Omega_{d-1}$ is the volume of a $(d-1)$-dimensional unit sphere. It will prove convenient to use $(u,r^{\ast})$ or $(v,r^{\ast})$ coordinates instead of $(t,r)$ where

$$\label{tort}
r^{\ast}(r) = \int \frac{dr}{f(r)},\qquad v= t + r^{\ast}, \qquad u= t-r^{\ast}.$$

$v$ and $u$ are, respectively, the advanced and retarded times, and $r^{\ast}$ is the tortoise coordinate. It is important to note here that at $r=r_{h}$ where $r_h$ is the root of $f(r)$, 