![The rate of complexity growth for different $\mathfrak{a}$.](gr2)

 BMSFT Complexity Growth in $d=2$
=================================

Asymptotically flat black hole solutions, which were studied in the previous section, are given by taking the flat-space limit from the asymptotically AdS black holes. In the three-dimensional Einstein gravity, there is no black hole solution [@Ida:2000jh]. The flat space limit of BTZ black holes are three-dimensional cosmological spacetimes [@Cornalba:2002fi; @Cornalba:2003kd]. These solutions, which are known as flat space cosmology (FSC), have recently been studied in the context of flat-space holography [@Bagchi:2012xr; @Barnich:2012xq]. In this section, we find the exact formula for the growth rate of the BMSFT complexity in $d=2$ [^3]. To do so, we apply the CA conjecture for the nonrotating FSC given by $$\begin{aligned}
\label{FSCmetric}
ds^2 &= \hat{r}_{+}^2 d t^2 - \frac{d r^2}{\hat{r}_{+}^2} +r^2 d \phi^2,
\\
M&=\frac{\hat{r}_{+}^2}{8 G_{N}},\end{aligned}$$ where $M$ is the mass parameter. For this metric the tortoise coordinate is given by $$\label{tortoisefsc}
r^{\ast} = - \frac{r}{\hat{r}_{+}^2}.$$ The Penrose-Cartan diagram of FSC has been shown in figure 4. We note that all of the $r=\text{const}$ surfaces are spacelike in FSC. Similar to the higher dimensional cases, we impose a symmetric dynamic for the right- and left-hand sides of the Penrose-Cartan diagram. 