spacelike joint terms. All of these joint terms take place near the singularities and are independent of the boundary time $$\begin{aligned}
    I_{J}^{X}&= \frac{\Omega_{d-1}~ r^{\ast}(\epsilon)^{d-1}}{16 \pi G_{N}} ~\log \big| f(r^{\ast}(\epsilon))\big|,
    \\
    I_{J}^{W}&=I_{J}^{Y}=I_{J}^{Z}=I_{J}^{X}.\end{aligned}$$

Putting all together we find $$\frac{d I}{d \mu}=0 \Rightarrow \dot{\mathcal{C}}=0.$$ Hence the rate of complexity growth for the initial time is zero.

Late time
---------

For the late times that are after critical time , the Penrose diagram is depicted in figure 2. Similar to the initial time we want to calculate on-shell action in a region of spacetime that is determined by the gray area in figure 2. This region is still the intersection of four WDW patches with UV and IR cutoffs. The distinct difference with the initial time is that the cutoff surface at the past singularity does not exist, and we have to consider the null joint term in the action at a new point $M$:

\[fig:a1\]

![ Late time for $d>3$. Gravitational action is evaluated on the gray region.](a2.png)

-   The only GHY term that contributes to our problem takes place near the future singularity $$\label{GHYlate}
    I_{GHY}^{sing}=\frac{\Omega_{d-1} ~d ~\omega^{d-2}}{16 \pi G_{N}}~ (\mu - \chi)$$

-   The contribution of points $P$ and $Q$ is similar 