flat spacetimes. To our knowledge, the holographic renormalization of asymptotically flat spacetimes is still an open problem.

Using we can calculate the terms of . The null boundary term vanishes because we can always choose a null parameter to be affine, and then the null boundary terms in do not contribute to the on-shell action. It remains the GHY term that has its contribution from the timelike or the spacelike surfaces $$\begin{aligned}
\label{GHYspace}
I_{GHY}^{spacelike} &= -\int dt~ \frac{r^{d-1}~ \Omega_{d-1}}{16 \pi G_{N}} \left( f'(r) +\frac{2(d-1)}{r} f(r) \right)\Bigg|_{\text{r=const}},
\\
\label{GHYtime}
I_{GHY}^{timelike} &= \int dt~ \frac{r^{d-1} ~\Omega_{d-1}}{16 \pi G_{N}} \left( f'(r) +\frac{2(d-1)}{r} f(r) \right)\Bigg|_{\text{r=const}}.\end{aligned}$$ In our calculation in the rest of this paper, all of the joint terms have at least one null part. Hence, it is adequate to compute the last term of , $$\begin{aligned}
\label{jointspace}
I_{J}^{spacelike} &= \frac{\Omega_{d-1} ~r^{d-1}}{16 \pi G_{N}} \log |f(r)|,
\\
\label{jointtime}
I_{J}^{timelike} &= \frac{-\Omega_{d-1}~ r^{d-1}}{16 \pi G_{N}} \log |f(r)|,
\\
\label{jointnull}
I_{J}^{null} &= \frac{-\Omega_{d-1} ~r^{d-1}}{8 \pi G_{N}} ~\text{Sign}(f(r)) ~\log \frac{\mathfrak{a}^2}{|f(r)|},\end{aligned}$$ where all of the joint points are labeled by their second non-null leg. The null joint term has an ambiguity due to the normalization constant of the null vectors $\mathfrak{a}$. This ambiguity is the same as the 