We now can use apply the induction hypothesis and obtain that the formula is
in $\Sigma_{i}$.

Reachability formula 2. Let $\beta\in\Sigma_{i}$ and let $\tau\in\Pi_{i}$.
Then by induction hypothesis
$S\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\bcancel{T(\tau)}$}}^{\hbox{\set@color$\scriptstyle\,$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\bcancel{T(\tau)}$}}^{\hbox{\set@color$\scriptstyle\,$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\bcancel{T(\tau)}$}}^{\hbox{\set@color$\scriptstyle\,$}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\bcancel{T(\tau)}$}}^{\hbox{\set@color$\scriptstyle\,$}}}\mathrel{\mathchar
0\relax\symAMSa 04B}\\!\\!\scriptscriptstyle{}}}B\left(\beta\right)$ is in
$\Sigma_{i}$ and thus
$S\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\bcancel{B(\beta)}$}}^{\hbox{\set@color$\scriptstyle\,\text{\tiny
weak}$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\bcancel{B(\beta)}$}}^{\hbox{\set@color$\scriptstyle\,\text{\tiny
weak}$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\bcancel{B(\beta)}$}}^{\hbox{\set@color$\scriptstyle\,\text{\tiny
weak}$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\bcancel{B(\beta)}$}}^{\hbox{\set@color$\scriptstyle\,\text{\tiny
weak}$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\scriptscriptstyle{}}}T\left(\tau\right)$ is in $\Pi_{i}$.

Reachability formula 3. Let $\beta\in\Pi_{i}$ and let $\tau\in\Sigma_{i}$.
Note that under this assumption ${\mathbf{false}}$, $\eta$, and
$\rho\land{\mathbf{X}}\beta$ belong to $\Pi_{i}$ and that
$\sigma\land{\mathbf{X}}\tau$ belongs to $\Sigma_{i}$. Thus by applying the
induction hypothesis we obtain that $\langle
S,s\rangle\xrightarrow[\bcancel{\langle
B,b\rangle\left(\beta\right)}]{>0}\langle T,t\rangle\left(\tau\right)$ is in
$\Sigma_{i}$. Since $\neg\beta$ also belongs to $\Sigma_{i}$, we obtain that
$\langle S,s\rangle\xrightarrow[\bcancel{\langle
B,b\rangle\left(\beta\right)}]{}\langle T,t\rangle\left(\tau\right)$ is from
$\Sigma_{i}$.

Reachability formula 4. Let $\beta\in\Sigma_{i}$ and let $\tau\in\Pi_{i}$.
Note that under this assumption $\eta$ and $\rho\land{\mathbf{X}}\beta$ belong
to $\Sigma_{i}$ and that ${\mathbf{false}}$, $\neg\beta$, and
$\sigma\land{\mathbf{X}}\tau$ belong to $\Pi_{i}$. We then apply again the
induction hypothesis and obtain that $\langle
S,s\rangle\xrightarrow[\bcancel{\langle
B,b\rangle\left(\beta\right)}]{\text{\tiny weak},>0}\langle
T,t\rangle\left(\tau\right)$ and thus also $\langle
S,s\rangle\xrightarrow[\bcancel{\langle
B,b\rangle\left(\beta\right)}]{\text{\tiny weak}}\langle
T,t\rangle\left(\tau\right)$ belongs to $\Pi_{i}$.

Reachability formula 5. Let $\beta\in\Pi_{i}$ and let $\tau\in\Sigma_{i}$. By
an analogous argument to the two preceding cases we obtain that the formulas
in line (1) and (3) belong to $\Sigma_{i}$. Further, note that by the
induction hypothesis $\eta\land{\mathbf{X}}\Big{(}\delta(\langle
R,\cdot\rangle,\eta)\xrightarrow[\bcancel{\langle
T,t\rangle\left(\tau\right)}]{\text{\tiny weak}}\langle
B,b\rangle\left(\beta\right)\Big{)}$ belongs to $\Pi_{i}$ and thus the overall
reachability formula 5 belongs to $\Sigma_{i}$. ∎

###  0.A.4 Proofs from Section 4.3

*

###### Proof.

We first prove the upper bound on the depth of the formula and then move on to
the claim about the length of the formula.

Depth Analysis. We prove the claim by induction on the level $i$. For the base
case of $i=0$, the main reachability formula is just
$(\neg\beta){\mathbf{U}}\tau$, having $\mathsf{D}(0,d)=d+1$, which is equal to
$d+3^{0}$, as required. For the induction step of a general level $i>0$, we
will first establish a bound on $\mathsf{D}(i,d)$ that is relative to
$\mathsf{D}(i{-}1,d)$, and then get from it an absolute bound, using the
induction hypothesis.

Observe that $\mathsf{D}(i,d)$ is bounded in Table 2 to the maximum between
$\mathsf{D}_{3}(i,d)$ and $\mathsf{D}_{5}(i,d)$, while the bound on
$\mathsf{D}_{5}(i,d)$ is at least as on $\mathsf{D}_{3}(i,d)$. Hence:

$\displaystyle\mathsf{D}(i,d)\leavevmode\nobreak\ \leq\leavevmode\nobreak\
\mathsf{D}_{5}(i,d)\leavevmode\nobreak\ $ $\displaystyle\leq$
$\displaystyle\mathsf{D}(i{-}1,\max(1+\mathsf{D}_{3}(i,d),1+\mathsf{D}_{4}(i,d)))$
$\displaystyle\leq$ $\displaystyle\mathsf{D}(i{-}1,1+\mathsf{D}(i{-}1,d+1))$

Applying the induction hypothesis on $\mathsf{D}(i{-}1,d)$, we get an absolute
bound:

$\displaystyle\mathsf{D}(i,d)\leavevmode\nobreak\ $ $\displaystyle\leq$
$\displaystyle\mathsf{D}(i{-}1,1+(d+1+3^{i{-}1}))\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle\leq$ $\displaystyle
2+d+3^{i{-}1}+3^{i{-}1}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leq\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ d+3^{i}$

Length Analysis. We prove the claim by induction on the level $i$. For the
base case of $i=0$, the main reachability formula is just
$(\neg\beta){\mathbf{U}}\tau$, having $\mathsf{L}(0,l)=2+2l$, which is indeed
not larger than $l\cdot(10|\Sigma|^{2}n)^{4^{i}}$. For the induction step of a
general level $i>0$, we will first establish a bound on $\mathsf{L}(i,l)$
relative to $\mathsf{L}(i{-}1,\cdot)$, and then apply the induction hypothesis
to get an absolute bound.

Observe that $\mathsf{L}(i,l)$ is bounded in Table 2 with respect to
$\mathsf{L}_{3}(i,\cdot)$ and $\mathsf{L}_{5}(i,\cdot)$, and
$\mathsf{L}_{3}(i,l)$ is already bounded in Table 2 with respect to
$\mathsf{L}(i{-}1,\cdot)$. We simplify the $\mathsf{L}_{5}$ bound, and
substitute the bound on $\mathsf{L}_{3}$, getting:

$\displaystyle\mathsf{L}_{5}(i,l)$ $\displaystyle\leq$
$\displaystyle|\Sigma|n^{i-1}\cdot\big{(}\mathsf{L}_{1}(i-1,3+\mathsf{L}_{3}(i,l))+2+$
$\displaystyle|\Sigma|n^{i-1}\cdot\big{(}\mathsf{L}_{1}(i-1,\max(3+\mathsf{L}_{3}(i,l),3+\mathsf{L}_{4}(i,l)))+1\big{)}\big{)}+$
$\displaystyle 1+|\Sigma|n^{i-1}\cdot(1+\mathsf{L}_{3}(i,3+l))$
$\displaystyle\leq$
$\displaystyle|\Sigma|n^{i-1}\cdot\big{(}\mathsf{L}_{1}(i-1,3+\mathsf{L}_{3}(i,l))+2+$
$\displaystyle|\Sigma|n^{i-1}\cdot\big{(}\mathsf{L}_{1}(i-1,3+\mathsf{L}_{3}(i,l))+1\big{)}\big{)}+$
$\displaystyle 1+|\Sigma|n^{i-1}\cdot(1+\mathsf{L}_{3}(i,3+l))$
$\displaystyle\leq$
$\displaystyle|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i-1,3+\mathsf{L}_{3}(i,l))+$
$\displaystyle|\Sigma|n^{(i-1)}\mathsf{L}_{1}(i-1,3+\mathsf{L}_{3}(i,l))+$
$\displaystyle|\Sigma|n^{(i-1)}\mathsf{L}_{3}(i,3+l))+$
$\displaystyle|\Sigma|^{2}n^{2(i-1)}+$ $\displaystyle 3|\Sigma|n^{(i-1)}+1$
$\displaystyle\leq$ $\displaystyle
4|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i-1,3+\mathsf{L}_{3}(i,l+3))$
$\displaystyle\leq$ $\displaystyle
4|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i-1,12+2l+4|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i{-}1,l{+}3))$

We can now bound $\mathsf{L}(i,l)$ relative to $\mathsf{L}(i{-}1,\cdot)$,
getting:

$\displaystyle\mathsf{L}(i,l)$ $\displaystyle\leq$ $\displaystyle
1+\mathsf{L}_{3}(i,l)+\mathsf{L}_{5}(i,l)$ $\displaystyle\leq$ $\displaystyle
4+2l+4|\Sigma|n^{2(i-1)}\mathsf{L}_{1}(i{-}1,l{+}3)+$ $\displaystyle
4|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i-1,12+2l+4|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i{-}1,l{+}3))$
$\displaystyle\leq$ $\displaystyle
5|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i-1,12+2l+4|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i{-}1,l{+}3))$
$\displaystyle\leq$ $\displaystyle
5|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i-1,5|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i{-}1,l{+}3))$

Applying the induction hypothesis on $\mathsf{L}(i{-}1,\cdot)$, we get an
absolute bound:

$\displaystyle\mathsf{L}(i,l)$ $\displaystyle\leq$ $\displaystyle
5|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i-1,5|\Sigma|^{2}n^{2(i-1)}\mathsf{L}_{1}(i{-}1,l{+}3))$
$\displaystyle\leq$ $\displaystyle
5|\Sigma|^{2}n^{2(i-1)}(5|\Sigma|^{2}n^{2(i-1)}(l+3)(2|\Sigma|n)^{4^{i-1}})(10|\Sigma|^{2}n)^{4^{i-1}}$
$\displaystyle\leq$ $\displaystyle
25|\Sigma|^{4}n^{4(i-1)}(l+3)(10|\Sigma|^{2}n)^{2\cdot 4^{i-1}}$
$\displaystyle\leq$ $\displaystyle l(10|\Sigma|^{2}n)^{4^{i}}$

The last inequality clearly holds for $i\geq 2$ as the
$(10|\Sigma|^{2}n)^{4^{i}}$ term becomes very large. For $i=1$, it also holds
as the $n^{4(i-1)}$ term disappears and we get $25\cdot 10^{2}(l+3)\leq
10^{4}ln^{2}$, which holds even for $l=n=1$. ∎

###  0.A.5 Proofs from Section 4.4

*

###### Proof.

Observe that $C$ is visited finitely often iff either $C$ is not visited at
all, which by Section 4.2 is formulated by
$\neg(\text{\Large$\iota$}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa 04B}\\!\\!\scriptscriptstyle{}}}C)$ or there exists a last
visit to $C$, namely a visit to $C$ after which there is no other visit to
$C$, which is formulated by
$\text{\Large$\iota$}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptscriptstyle{}}}C(\neg(C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\scriptscriptstyle{}}}C))$. Note that by Section 4.2 we have
$\text{\Large$\iota$}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptscriptstyle{}}}C,C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\scriptscriptstyle{}}}C\in\Sigma_{1}$. Thus
$\neg\text{\Large$\iota$}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa 04B}\\!\\!\scriptscriptstyle{}}}C,\neg
C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\scriptscriptstyle{}}}C\in\Pi_{1}$ and then again by Section 4.2 we
get
$\text{\Large$\iota$}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptscriptstyle{}}}C(\neg(C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
$}}^{\hbox{\set@color$\scriptstyle\,>0$}}}\mathrel{\mathchar 0\relax\symAMSa
04B}\\!\\!\scriptscriptstyle{}}}C))\in\Sigma_{2}$. Thus
${\mathit{Fin}}(C)\in\Sigma_{2}$. ∎

*

###### Proof.

We complete the proof of the Theorem and give a complete analysis of all six
acceptance conditions:

  * •

${\cal D}$ is a Muller automaton: the overall formula $\varphi$ is in
$\Delta_{2}$, since it is a Boolean combination of ${\mathit{Fin}}(C)$
formulas, which by Section 4.4 belong to $\Sigma_{2}$.

  * •

${\cal D}$ is a coBüchi automaton: we construct the formula $\varphi$ directly
from the coBüchi condition $\alpha$, having a conjunction of
${\mathit{Fin}}(C)$ formulas, over all configurations $C$ that are mapped to
states in $\alpha$. As ${\mathit{Fin}}(C)$ belongs to $\Sigma_{2}$, so does
$\varphi$.

  * •

${\cal D}$ is a Büchi automaton: we can complement it, apply the above
argument over the resulting coBüchi automaton, and negate the resulting
formula to obtain a formula from $\Pi_{2}$.

  * •

${\cal D}$ is a looping-coBüchi automaton: Let $s\in Q$ be the unique sink
state that all accepting runs end up in, and let $S$ be the set of
configurations mapped to $s$. We then define $\varphi$ as $\bigvee_{C\in
S}\text{\Large$\iota$}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.71559pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa 04B}\\!\\!\scriptscriptstyle{}}}C$. Note that $\varphi$
belongs to $\Sigma_{1}$ since every disjunct belongs to $\Sigma_{1}$ due to
Section 4.2.

  * •

${\cal D}$ is a looping-Büchi automaton: the dual of the previous argument.

  * •

${\cal D}$ is a weak automaton: Let $G\subseteq Q$ be an accepting SCC of
${\cal D}$ and $G^{\prime}\subseteq Q$ be all states that are reachable from
$G$, but are not in $G$. Let $H$ and $H^{\prime}$ be the set of configurations
that are mapped to $G$ and $G^{\prime}$, respectively. Then by Section 4.2, we
have that $(\bigvee_{C\in
H}\text{\Large$\iota$}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptscriptstyle{}}}C)\land(\bigwedge_{C^{\prime}\in
H^{\prime}}\neg\text{\Large$\iota$}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\displaystyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\textstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa
04B}\\!\\!\scriptstyle{}}{\mathrel{}\mathrel{\mathop{\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt\resizebox{}{}{{\leavevmode\hbox{\hskip-0.89449pt\hbox{\set@color{$\thicksim$}}}}}\hskip
0.2pt}\limits_{\hbox{\set@color$\scriptstyle\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}}}\mathrel{\mathchar
0\relax\symAMSa 04B}\\!\\!\scriptscriptstyle{}}}C^{\prime})$ exactly captures
all words that are accepted by eventually ending up in $G$. The overall
formula $\varphi$ is then the disjunction of these formulas constructed for
each accepting SCC of ${\cal D}$. The membership in $\Delta_{1}$ then follows
immediately from Section 4.2. ∎


# Near horizon linearized gravitaty and soft theorem

Pujian Mao, Kai-Yu Zhang, Bochen Zhou

Near horizon linearized gravity and soft theorem

Pujian Mao, Kai-Yu Zhang, Bochen Zhou

Center for Joint Quantum Studies and Department of Physics,

School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China

Abstract. In this paper, we study the linearized gravity theory in the near
horizon region of the Schwarzschild black hole in four dimensional spacetime.
Under the Newman-Unti gauge, we derive the most general near horizon symmetry
and solution space without any near horizon fall-off condition. There are four
towers of surface charges that are generic functions on the horizon associated
to the near horizon symmetry. With suitable near horizon fall-off conditions,
we reveal a soft graviton theorem from the Ward identity of the near horizon
supertranslation in both coordinates space and momentum space.

Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Near horizon linearized theory
    1. 2.1 Near horizon symmetries
    2. 2.2 Solution space
    3. 2.3 Near horizon surface charges
  3. 3 Near horizon soft graviton theorem
    1. 3.1 Fall-off conditions and charge
    2. 3.2 Commutation relations
    3. 3.3 Mode expansion
    4. 3.4 Soft graviton theorem in coordinates space
    5. 3.5 Soft graviton theorem in momentum space
  4. 4 Conclusion
  5. A Modification of stress tensor

##  1 Introduction

The existence of horizon is a very fascinating feature of geometries with
Lorentzian signature. A stationary black hole with a future event horizon
admits four mechanical laws which are very similar to the ordinary laws of
thermodynamics [1]. Then, the discovery of the Hawking radiation [2] makes the
laws of black hole physics real thermodynamical laws. Hence the process of
black hole formation equivalently is a process of thermalization. From a
quantum perspective, the process of black hole formation starts from pure
states and ends with a thermal state, which means that the evolution is not
unitary. This arises the black hole information paradox [3]. It is a long-
standing problem in black hole physics and quantum gravity. Recently, to
resolve the black hole information paradox, Hawking, Perry, and Strominger
(HPS) proposed that black holes can carry soft hairs [4]. The presence of soft
hairs are required from the charge conservation of asymptotic symmetries which
triggers quantum correlations between the outgoing Hawking quanta and the soft
hair configuration.

In the original proposal of HPS, it is assumed that the soft hairs are soft
gravitons or photons on the black hole horizon. Such observation is based on a
fascinating triangular equivalence [5], where soft theorems is nothing but the
quantum Ward identity of asymptotic symmetries [6, 7, 8, 9, 10]. Hence, an
asymptotic symmetry transformation on black hole spacetime is equivalent to
implanting soft hairs on the black hole [4, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22]. However, the soft theorems in the triangular relation is in
flat spacetime. The corrections from the presence of a black hole was not
considered in the original HPS proposal. Moreover, a black hole admits an
event horizon which can be considered as an inner boundary of the spacetime.
So the connections between symmetries at null infinity and soft theorems in
flat spacetime may not be complete for the black hole soft hair proposal. One
ongoing direction to fill in those gaps is the study of soft theorems in
curved spacetime derived from the near horizon symmetry [23, 24, 25]. In
particular, it is proven that the soft electric hairs of the Schwarzschild
spacetime are indeed implanted by soft photons on the horizon which are
derived from the Ward identity of the near horizon large gauge transformation
[23]. In this work, we will extend the interplay between near horizon symmetry
and soft theorems in [23, 24] to linearized gravity theory. The reason of
linearisation is twofold. On the one hand, the interplay that we are
interested is proposed for resolving the black hole information paradox. So we
need a fixed black hole background. On the other hand, one needs to involve
mode expansions to derive a soft theorem which is completely unclear how to
perform for a gravitational theory with the principle of equivalence. Hence
the obvious choice is to study linearized gravity theory in a black hole
spacetime.

In this paper, we study the linearized gravity theory in the near horizon
region of the Schwarzschild spacetime in four dimensions. The Newman-Unit (NU)
gauge [26] is adapted to the linearized theory. We derive the residual gauge
transformation that preserves the NU gauge conditions without imposing any
fall-off condition. The most general residual (near horizon) symmetry consists
of four arbitrary functions on the horizon which form a closed algebra for the
linearized theory. Then we compute the solution space of the linearized theory
and the surface charges that are associated to the near horizon symmetries.
The connection of asymptotic symmetry and soft theorem is subject to proper
fall-off conditions that the gauge or gravitational fields should satisfy [6,
7, 8, 9, 10, 5]. This is a very subtle point in the near horizon region.
Because there is no clear criterion to govern the near horizon behavior of the
physical fields. A crucial fact that such connection can be successfully
extended to the near horizon region for gauge theory in [23, 24] is that the
fall-off conditions at null infinity can be easily adapted to the near horizon
region in the sense that the near horizon solution space and the formula of
the near horizon surface charges are very similar to the null infinity case.
The situation is significantly changed for gravity. It is not obvious which
type of near horizon conditions should be imposed such that the null infinity
relations can be recovered from the near horizon analysis for the case of
Einstein gravity [27]. Such difficulty is inherited by the linearized theory.
Remarkably, it is shown that there is a proper set of near horizon fall-off
conditions such that it preserves the near horizon supertranslation [28], and
the supertranslation charge can naturally split into soft part and hard part
which will lead to a soft graviton theorem following the null infinity
scenario [8] with adaptions to the near horizon computations [23, 24, 25].

The organization of this paper is very simple. In the next section, we derive
the near horizon symmetry, solution space, and surface charges with respect to
the NU gauge. In Section 3, we introduce proper near horizon fall-off
conditions. A near horizon soft graviton theorem is derived from the Ward
identity of the near horizon supertranslation in the reduced solution space.
We conclude in the last section. There is one Appendix where we present the
details of the modified stress tensor.

##  2 Near horizon linearized theory

We consider linearized gravitational theory coupled to a conserved stress
tensor in the near horizon region of Schwarzschild black hole. The line
element of Schwarzschild black hole in the retarded coordinates $(u,r,x^{A})$
is

$\displaystyle\text{d}s^{2}=-f(r)\text{d}u^{2}-2\text{d}u\text{d}r+\Omega^{2}\gamma_{AB}\text{d}x^{A}\text{d}x^{B}\,,$
(1) $\displaystyle f(r)=\frac{r}{\Omega}\,,\quad\Omega=r+r_{s}\,.$ (2)

Here $x^{A}=(z,\bar{z})$ are the complex stereographic coordinates which are
related to the usual angular variables $(\theta,\phi)$ by
$z=\cot\frac{\theta}{2}e^{i\phi}$, and $r_{s}$ is the Schwarzschild radius
which is related to the mass of black hole by $r_{s}=2M$. The metric of unit
sphere in this coordinate system is

$\displaystyle\gamma_{AB}=\begin{pmatrix}0&\gamma_{z\overline{z}}\\\
\gamma_{z\overline{z}}&0\end{pmatrix}\,,\gamma_{z\bar{z}}=\frac{2}{(1+z\bar{z})^{2}}\,.$
(3)

The $r=0$ hypersurface is the horizon ${\cal H}$ with topology
$S^{2}\times\mathbb{R}$. Actually, the retarded coordinates just cover half of
the horizon ${\cal H}^{-}$ as illustrated in Figure 1. Hence we will only pay
attention in this work to the ${\cal H}^{-}$ part while everything can be
similarly repeated on ${\cal H}^{+}$ by introducing the advanced coordinates
$(v,r,z,\bar{z})$. Those two parts are connected at the bifurcation point $B$.
The near bifurcation symmetries derived in [29] provide a natural way of
connecting the near horizon symmetries on ${\cal H}^{-}$ and ${\cal H}^{+}$.

Figure 1: The Penrose diagram of the outside region of the Schwarzschild black
hole.

Under a background metric $g_{\mu\nu}$, the action of a spin-2 theory of field
$h_{\mu\nu}$ with zero cosmological constant is given by the quadratic part of
the expansion of the Einstein-Hilbert action

$\displaystyle S_{h}[h;g]$
$\displaystyle=\int\frac{\text{d}^{4}x\sqrt{-g}}{4}\left[\frac{1}{2}(\nabla_{\rho}h)^{2}-\frac{1}{2}(\nabla_{\rho}h_{\mu\nu})^{2}+\nabla_{\rho}h_{\mu\nu}\nabla^{\mu}h^{\nu\rho}-\nabla_{\nu}h\nabla_{\mu}h^{\mu\nu}\right]\,,$
(4) $\displaystyle S_{M}[h;g]$
$\displaystyle=\int\frac{\text{d}^{4}x\sqrt{-g}}{4}h^{\mu\nu}T_{\mu\nu}\,,$
(5)

where all indices are lowered and raised by $g_{\mu\nu}$ and $g^{\mu\nu}$
which is a solution of a vacuum Einstein field equation,
$h=h_{\mu\nu}g^{\mu\nu}$, $\nabla_{\mu}$ is the covariant derivative
corresponding to the background metric $g_{\mu\nu}$, $T_{\mu\nu}$ is a generic
conserved stress tensor satisfying $\nabla_{\mu}T^{\mu\nu}=0$, and coupled to
the spin-2 gauge field. Inherited from the diffeomorphism invariance of
Einstein-Hilbert action, the theory described by the action (4) is invariant
under the gauge transformations [30]

$\displaystyle\delta_{\zeta}h_{\mu\nu}=\nabla_{\mu}\zeta_{\nu}+\nabla_{\nu}\zeta_{\mu}\,,$
(6)

with an arbitrary vector $\zeta^{\mu}(x)$.

The equations of motion of the linearized theory are the Pauli-Fierz equations
in a covariant way,

$E_{\mu\nu}=T_{\mu\nu}.$ (7)

where

$E_{\mu\nu}\equiv\nabla_{\mu}\nabla_{\nu}h+\nabla^{\tau}\nabla_{\tau}h_{\mu\nu}-\nabla^{\tau}\nabla_{\mu}h_{\nu\tau}-\nabla^{\tau}\nabla_{\nu}h_{\mu\tau}-g_{\mu\nu}(\nabla^{\tau}\nabla_{\tau}h-\nabla^{\tau}\nabla^{\rho}h_{\tau\rho}),$
(8)

and we are using natural units with $8\pi G_{\textrm{N}}=1\,$.

We will work in the adapted Newman-Unti gauge [26], for which we set

$\displaystyle h_{rr}=h_{rz}=h_{r\bar{z}}=h_{ru}=0\,,$ (9) $\displaystyle
T_{rr}=T_{rz}=T_{r\bar{z}}=T_{ru}=0.$

Here, we set all the radial components of the stress tensor $T_{r\mu}$ to zero
to adapt to the gauge conditions of the perturbative metric [31]. This can be
done using an auxiliary conserved symmetric 2-tensor as was detailed in
Appendix A. Since the linearized fields can be resided on the horizon in any
form, there is no clear physical criteria to impose near horizon fall-off
conditions. Here, we will not impose any fall-off condition for the linearized
metric fields and derive the most general residual gauge transformations and
the solution space.

We decompose the perturbative metric and stress tensor in the NU gauge as
follows

$\displaystyle h_{\mu\nu}=\begin{pmatrix}\mathcal{C}&0&\mathcal{U}_{A}\\\
0&0&0\\\
\mathcal{U}_{A}&0&\mathcal{N}_{AB}+\gamma_{AB}\mathcal{T}/2\end{pmatrix}\,,\quad
T_{\mu\nu}=\begin{pmatrix}T_{uu}&0&T_{uA}\\\ 0&0&0\\\
T_{uA}&0&\mathbf{T}_{AB}+\gamma_{AB}\mathbf{T}/2\end{pmatrix}\,,$ (10)

where $\mathcal{N}_{AB}$ satisfying $\mathcal{N}_{AB}\gamma^{AB}=0$ is the
traceless part of $h_{AB}$, and $\mathcal{T}$ denotes the trace part. The same
type of decomposition is also applied for $T_{AB}$.

The gauge invariance of the linearized theory indicates that not all the
equations of motion are independent. The constraints among them are inherited
from the Bianchi identity of the full Einstein theory, which is now given by
$\nabla_{\mu}(E^{\mu\nu}-T^{\mu\nu})=0$. Taking into account those
constraints, we can arrange the ten equations of motion as follows:

  * •

Four hypersurface equations:

$\displaystyle E_{rr}=T_{rr},\quad E_{rz}=T_{rz},\quad
E_{r\bar{z}}=T_{r\bar{z}},\quad E_{ru}=T_{ru},.$ (11)

  * •

Two standard equations:

$\displaystyle E_{zz}=T_{zz},\quad E_{\bar{z}\bar{z}}=T_{\bar{z}\bar{z}}.$
(12)

  * •

One trivial equation:

$\displaystyle E_{z\bar{z}}=T_{z\bar{z}}.$ (13)

  * •

Three supplementary equations:

$\displaystyle E_{uz}=T_{uz},\quad E_{u\bar{z}}=T_{u\bar{z}},\quad
E_{uu}=T_{uu}.$ (14)

The advantage of such arrangement is well explained in the literature [32, 33,
34, 31]. Once the hypersurface equations and the standard equations are
satisfied, the trivial equation is fulfilled automatically and all
supplementary equations are left with only one order in the near horizon
(small $r$) expansions.

###  2.1 Near horizon symmetries

The linearized gravity theory is invariant under the gauge transformation
$\delta_{\zeta}h_{\mu\nu}=\nabla_{\mu}\zeta_{\nu}+\nabla_{\nu}\zeta_{\mu}$. We
are interested in the residual gauge transformations that preserve the gauge
conditions in (9).

We start from the first constraint $h_{rr}=0$ which leads to
$\partial_{r}\zeta_{r}=0$. Hence $\zeta_{r}=-\mathcal{F}$ where $\mathcal{F}$
is an arbitrary function of $(u,z,\bar{z})$. The minus sign is for later
convenience. We continue with $h_{rA}=0$, which yields

$\displaystyle\partial_{r}\zeta_{A}-\frac{2\zeta_{A}}{\Omega}=D_{A}\mathcal{F}\,.$
(15)

One can easily read the general solution of $\zeta_{A}$ from the above
equation,

$\displaystyle\zeta_{A}=\Omega^{2}Y_{A}-\Omega D_{A}\mathcal{F}\,,$ (16)

where $Y_{z},Y_{\bar{z}}$ are arbitrary functions of $(u,z,\bar{z})$ and
$D_{A}$ is the covariant derivative associated with the spherical metric
$\gamma_{AB}$. Finally, from the last condition $h_{ru}=0$, we obtain

$\displaystyle\partial_{r}\zeta_{u}=\partial_{u}\mathcal{F}-\frac{r_{s}\mathcal{F}}{\Omega^{2}}\,.$
(17)

Hence,

$\displaystyle\zeta_{u}=r\partial_{u}\mathcal{F}-f\mathcal{F}-Z\,,$ (18)

where $Z$ is a arbitrary function of $(u,z,\bar{z})$. The general solution of
$\zeta$ in vector form by raising the index with $g^{\mu\nu}$ is:

$\zeta^{\mu}=\left[\mathcal{F},\,Z-r\partial_{u}\mathcal{F},\,Y^{A}-\frac{D^{A}\mathcal{F}}{\Omega}\right].$
(19)

We have thus established the form of the residual gauge transformations of the
linearized theory. Obviously, the near horizon symmetry includes both near
horizon supertranslations and superrotations uncovered in [28].

The symmetry algebra is defined through
$[\delta_{\zeta_{1}},\delta_{\zeta_{2}}]h_{\mu\nu}=\delta_{[\zeta_{1},\zeta_{2}]_{{}_{l}}}h_{\mu\nu}$
. One can easily check that in the linearized case, the near horizon symmetry
algebra is not the standard Lie algebra between vectors, but an Abelian
algebra such that $[\zeta_{1},\zeta_{2}]_{{}_{l}}=0$ where the subscript “$l$”
denotes the algebra in linearized theory. This is another effect of
linearisation.

The transformation law of the metric fields in the decomposition (10) under
the residual gauge transformation can be obtained easily as

$\displaystyle\delta_{\zeta}\mathcal{C}$
$\displaystyle=2r\partial_{u}^{2}\mathcal{F}+\left(r\partial_{r}f-2f\right)\partial_{u}\mathcal{F}-\partial_{r}fZ-2\partial_{u}Z\,,$
(20) $\displaystyle\delta_{\zeta}\mathcal{U}_{A}$
$\displaystyle=-r_{s}\partial_{u}D_{A}\mathcal{F}-fD_{A}\mathcal{F}+\Omega^{2}\partial_{u}Y_{A}-D_{A}Z\,,$
(21) $\displaystyle\delta_{\zeta}\mathcal{T}$ $\displaystyle=-2\Omega
D^{A}D_{A}\mathcal{F}-4r\Omega\partial_{u}\mathcal{F}+2\Omega^{2}D^{A}Y_{A}+4\Omega
Z\,,$ (22) $\displaystyle\delta_{\zeta}\mathcal{N}_{AB}$
$\displaystyle=2\Omega D_{A}Y_{B}-2\Omega D_{A}D_{B}\mathcal{F}\,.$ (23)

###  2.2 Solution space

In this section, we want to solve the equations of motion in the near horizon
region by assuming that the metric fields are given in a small $r$ expansion.
We define a dimensionless quantity $\lambda=\frac{r}{r_{s}}$, and assume the
metric and the stress tensor can be expanded in the form of series expansion
as

$h_{\mu\nu}=h_{\mu\nu}^{0}+\sum_{n=1}^{\infty}\lambda^{n}h_{\mu\nu}^{(n)}\,,\quad
T_{\mu\nu}=T_{\mu\nu}^{0}+\sum_{n=1}^{\infty}\lambda^{n}T_{\mu\nu}^{(n)}\,.$
(24)

We begin with the simplest hypersurface equation $E_{rr}=0$, which yields

$\displaystyle
0=\partial_{\lambda}^{2}\left(\frac{\mathcal{T}}{\xi}\right)\,,$ (25)

where we introduce $\xi=(1+\lambda)$ for notational convenience. The solution
for the trace part is

$\displaystyle\mathcal{T}=\mathcal{T}^{0}+\lambda\mathcal{T}^{(1)}+\lambda^{2}(\mathcal{T}^{(1)}-\mathcal{T}^{0})\,,$
(26)

which implies that, under the NU gauge, the trace part of spin-2 field
truncates in the small $r$ expansion.

Then, the second group $E_{rA}=0$ leads to

$\displaystyle\frac{1}{r_{s}^{2}\xi^{2}}\partial_{\lambda}\left[\xi^{4}\partial_{\lambda}\left(\frac{\mathcal{U}_{A}}{\xi^{2}}\right)\right]-\frac{1}{r_{s}^{3}}\partial_{\lambda}\left(\frac{D^{B}\mathcal{N}_{BA}}{\xi^{2}}\right)-\frac{1}{r_{s}^{3}}\partial_{\lambda}\left(\frac{D_{A}\mathcal{T}}{2\xi^{2}}\right)=0\,.$
(27)

Considering $\mathcal{N}_{AB}$ and $\mathcal{T}$ as known functions,
$\mathcal{U}_{A}$ is completely fixed up to four integration constants of $r$
which are the two leading orders $\mathcal{U}_{A}^{(0)}$ and
$\mathcal{U}_{A}^{(1)}$. Suppose that $\mathcal{N}_{AB}$ is given as initial
data as

$\mathcal{N}_{AB}=\sum_{m=0}^{\infty}\mathcal{N}_{AB}^{(m)}r^{m}.$ (28)

The higher orders in the expansion of $\mathcal{U}_{A}$ for $n\geq 0$ are
determined as follows

$\displaystyle\mathcal{U}_{A}^{(n+2)}=\frac{1}{(n+2)(n+1)}\left\\{2\sum_{m=0}^{n}(-1)^{m}(1+m)\mathcal{U}_{A}^{(n-m)}\right.$
(29)
$\displaystyle\left.+\sum_{m=0}^{n}\frac{(-1)^{m}(1+m)}{r_{s}}\left[(n+1-m)D^{B}\mathcal{N}_{BA}^{(n+1-m)}-(2+m)D^{B}\mathcal{N}_{BA}^{(n-m)}\right]\right.$
$\displaystyle\left.+\sum_{m=0}^{n}\frac{(-1)^{m}(1+m)}{2r_{s}}\left[(n+1-m)D_{A}\mathcal{T}^{(n+1-m)}-(2+m)D_{A}\mathcal{T}^{(n-m)}\right]\right\\},$

where the series expansion on the right hand side of the above relation is
from that the expansion of $\frac{1}{\xi}$ in powers of $\lambda$ does not
truncate, which is

$\displaystyle\frac{1}{\xi^{n}}=\sum_{m=0}^{\infty}(-1)^{m}\frac{\Gamma(m+n)}{\Gamma(m)\Gamma(1+n)}\lambda^{m}\,.$
(30)

This is the main difference of the solution space compared to the null
infinity case [31]. An equivalent fact is that linearisation from the full
theory is more complicated on a curved background than a flat spacetime
background. Consequently, there is no advantage to obtaining the solution
space of the linearized theory in Schwarzschild spacetime from the full theory
in [27]. This is the main reason we study the linearized theory from scratch
in this work.111See also [35] for the study of the linearized theory in
Schwarzschild spacetime in radiation gauge.

We continue with the last hypersurface equation $E_{ru}=0$, which gives

$\displaystyle-\frac{2\partial_{\lambda}\left(\xi\mathcal{C}\right)}{r_{s}^{2}\xi^{2}}$
$\displaystyle+\frac{\partial_{\lambda}\left(\xi^{2}D^{A}\mathcal{U}_{A}\right)}{r_{s}^{3}\xi^{4}}-\frac{D^{A}D^{B}\mathcal{N}_{AB}}{r_{s}^{4}\xi^{4}}-\frac{D^{A}D_{A}\mathcal{T}}{2r_{s}^{4}\xi^{4}}$
(31)
$\displaystyle\quad\quad-\frac{\partial_{u}\partial_{\lambda}\mathcal{T}}{r_{s}^{3}\xi^{2}}+\frac{f\mathcal{T}}{r_{s}^{4}\xi^{4}}+\frac{f}{r_{s}^{4}\xi}\partial_{\lambda}\left(\frac{\partial_{\lambda}\mathcal{T}}{\xi}\right)+\frac{\partial_{\lambda}\mathcal{T}}{2r_{s}^{4}\xi^{4}}=0\,,$

which controls $\mathcal{C}$ up to an integration constant at the leading
order $\mathcal{C}^{(0)}$. The higher orders are determined by

$\displaystyle\mathcal{C}^{(1)}$
$\displaystyle=\frac{2D^{A}\mathcal{U}_{A}^{0}+D^{A}\mathcal{U}_{A}^{(1)}}{2r_{s}}-\frac{D^{A}D^{B}\mathcal{N}_{AB}^{0}}{2r_{s}^{2}}-\frac{D^{A}D_{A}\mathcal{T}^{0}}{4r_{s}^{2}}-\frac{\partial_{u}\mathcal{T}^{(1)}}{2r_{s}}+\frac{\mathcal{T}^{(1)}}{4r_{s}^{2}}-\mathcal{C}^{0}\,,$
(32)

and

$\displaystyle\mathcal{C}^{(n+1)}=\frac{1}{n+1}\left\\{\sum_{m=0}^{n}(-1)^{m+1}\mathcal{C}^{(n-m)}\right.$
(33)
$\displaystyle+\sum_{m=0}^{n}\frac{(-1)^{m+1}(1+m)(2+m)}{4r_{s}^{2}}\left[D^{A}D^{B}\mathcal{N}_{AB}^{(n-m)}+D^{A}D_{A}\mathcal{T}^{(n-m)}/2\right]$
$\displaystyle+\sum_{m=0}^{n}\frac{(-1)^{m}}{2r_{s}}\left[(n+1-m)D^{A}\mathcal{U}_{A}^{(n+1-m)}+2(1+m)D^{A}\mathcal{U}_{A}^{(n-m)}\right]$
$\displaystyle-\sum_{m=0}^{n}\frac{(-1)^{m}(n+1-m)}{2r_{s}}\partial_{u}\mathcal{T}^{(n+1-m)}$
$\displaystyle+\sum_{m=0}^{n}\frac{(-1)^{m}(1+m)(2+m)(n+1-m)}{8r_{s}^{2}}\mathcal{T}^{(n+1-m)}$
$\displaystyle+\sum_{m=0}^{n-1}\frac{(-1)^{m}(1+m)(2+m)}{12r_{s}^{2}}$
$\displaystyle\quad\times\left[\frac{6(n+1-m)(n-m)}{(2+m)}\mathcal{T}^{(n+1-m)}-3(n-m)\mathcal{T}^{(n-m)}+(3+m)\mathcal{T}^{(n-1-m)}\right]\bigg{\\}}\,,$

for $n\geq 1$.

We now turn to the standard equations. The standard equation $E_{zz}=T_{zz}$
is reduced to

$\displaystyle\partial_{u}\partial_{\lambda}\mathcal{N}_{AB}=\partial_{\lambda}D_{A}\mathcal{U}_{B}+\frac{\partial_{\lambda}\mathcal{N}_{AB}}{2r_{s}\xi^{2}}+\frac{\lambda\xi}{2r_{s}}\partial^{2}_{\lambda}\left(\frac{\mathcal{N}_{AB}}{\xi}\right)-\frac{\mathcal{N}_{AB}}{r_{s}\xi^{3}}+\frac{\partial_{u}\mathcal{N}_{AB}}{\xi}-\frac{r_{s}}{2}\mathbf{T}_{AB}\,.$
(34)

This equation controls the time evolution of the initial data
$\mathcal{N}_{AB}$ except its leading order $\mathcal{N}_{AB}^{(0)}$.
Following the terminology of [32, 27], we refer to the time derivative of the
components at this order as news functions. The precise information that is
extracted from (34) is

$\displaystyle\partial_{u}\mathcal{N}_{AB}^{(1)}=D_{A}\mathcal{U}_{B}^{(1)}+\partial_{u}\mathcal{N}_{AB}^{0}+\frac{\mathcal{N}_{AB}^{(1)}-2\mathcal{N}_{AB}^{0}}{2r_{s}}-\frac{\mathbf{T}^{0}_{AB}}{2r_{s}}\,,$
(35)

and, for $n\geq 1$,

$\displaystyle\partial_{u}$
$\displaystyle\mathcal{N}_{AB}^{(n+1)}=D_{A}\mathcal{U}_{B}^{(n+1)}+\frac{1}{(n+1)}\left\\{\sum_{m=0}^{n}(-1)^{m}\partial_{u}\mathcal{N}_{AB}^{(n-m)}-\frac{\mathbf{T}_{AB}^{(n)}}{2r_{s}}\right.$
(36)
$\displaystyle+\sum_{m=0}^{n}\frac{(-1)^{m}(1+m)}{2r_{s}}\left[(n+1-m)\mathcal{N}_{AB}^{(n+1-m)}-(2+m)\mathcal{N}_{AB}^{(n-m)}\right]$
$\displaystyle+\sum_{m=0}^{n-1}\frac{(-1)^{m}}{r_{s}}\left[(1+m)\mathcal{N}_{AB}^{(n-1-m)}-(n-m)\mathcal{N}_{AB}^{(n-m)}\right]$
$\displaystyle\left.+\sum_{m=0}^{n-1}\frac{(n-m)(n+1-m)}{2r_{s}}\mathcal{N}_{AB}^{(n+1-m)}\right\\}\,.$

When the hypersurface equations and standard equations are satisfied, the
trivial equation is satisfied automatically, which can be verified from the
$r$ component of the Bianchi identity $\nabla^{\mu}(E_{\mu\nu}-T_{\mu\nu})=0$,
from which one can obtain

$\frac{2(E_{z\bar{z}}-T_{z\bar{z}})}{\Omega^{2}\gamma_{z\bar{z}}}=0\,.$ (37)

For the supplementary equations, only one order in the expansion is involved.
Three supplementary equations fix the time evolution of the integration
constants as

$\displaystyle\frac{2\partial_{u}\mathcal{C}^{0}}{r_{s}}+\frac{D^{A}D_{A}\mathcal{C}^{0}}{r_{s}^{2}}-\frac{2\partial_{u}\left(D^{A}\mathcal{U}_{A}^{0}\right)}{r_{s}^{2}}-\frac{D^{A}\mathcal{U}_{A}^{0}}{r_{s}^{3}}+\frac{\left(2\partial_{u}^{2}\mathcal{T}^{0}+\partial_{u}\mathcal{T}^{0}\right)}{2r_{s}^{2}}=T_{uu}^{0}\,,$
(38)
$\displaystyle\frac{2r_{s}\partial_{u}\mathcal{U}_{A}^{0}-r_{s}\partial_{u}\mathcal{U}_{A}^{(1)}+2\mathcal{U}_{A}^{0}}{r_{s}^{2}}+\frac{D_{A}\mathcal{C}^{(1)}}{r_{s}}-\frac{\partial_{u}D^{B}\mathcal{N}_{AB}^{0}}{r_{s}^{2}}$
$\displaystyle\hskip
170.71652pt+\frac{D^{B}D_{B}\mathcal{U}_{A}^{0}-D^{B}D_{A}\mathcal{U}_{B}^{0}}{r_{s}^{2}}=T_{uA}^{0}\,.$
(39)

Substituting $\mathcal{C}^{(1)}$ in the second equation yields

$\displaystyle\frac{2r_{s}\partial_{u}\mathcal{U}_{A}^{0}-r_{s}\partial_{u}\mathcal{U}_{A}^{(1)}+\mathcal{U}_{A}^{0}}{r_{s}^{2}}+\frac{D_{A}\mathcal{T}^{(1)}}{4r_{s}^{3}}-\frac{\partial_{u}D_{A}\mathcal{T}^{(1)}}{2r_{s}^{2}}-\frac{D_{A}D^{B}D_{B}\mathcal{T}^{0}}{4r_{s}^{3}}$
(40)
$\displaystyle+\frac{2D^{B}D_{B}\mathcal{U}_{A}^{0}+D_{A}D^{B}\mathcal{U}_{B}^{(1)}}{2r_{s}^{2}}-\frac{D_{A}\mathcal{C}^{0}}{r_{s}}-\frac{D_{A}D^{B}D^{C}\mathcal{N}_{BC}^{0}}{2r_{s}^{3}}-\frac{\partial_{u}D^{B}\mathcal{N}_{AB}^{0}}{r_{s}^{2}}=T_{uA}^{0}\,.$

###  2.3 Near horizon surface charges

For the (linearized) Einstein gravity, the conserved charge defined on
bifurcation 2-sphere $B$ is given by [36]

${\mathcal{Q}}_{\zeta}=\int_{B}\,\sqrt{-g}\Big{(}h^{\lambda[u}\nabla_{\lambda}\zeta^{r]}-\zeta^{\lambda}\nabla^{[u}{h^{r]}}_{\lambda}-\frac{1}{2}h\nabla^{[u}\zeta^{r]}+\zeta^{[u}\nabla_{\lambda}h^{r]\lambda}-\zeta^{[u}\nabla^{r]}h\Big{)}\text{d}z\text{d}\bar{z}.$
(41)

Inserting the solution space and the near horizon symmetries, one obtains

${\mathcal{Q}}_{\zeta(\mathcal{F},Z,Y^{A})}=\int_{B}\sqrt{-g}\left\\{\frac{\mathcal{F}\mathcal{C}^{0}}{r_{s}}+\frac{\mathcal{U}_{A}^{(1)}D^{A}\mathcal{F}}{2r_{s}^{2}}+\frac{Y^{A}\left(2\mathcal{U}_{A}^{0}-\mathcal{U}_{A}^{(1)}\right)}{2r_{s}}\right.\\\
\left.\frac{Z\left(\mathcal{T}^{0}-\mathcal{T}^{(1)}\right)}{2r_{s}^{3}}+\frac{\mathcal{F}\mathcal{T}^{0}}{2r_{s}^{3}}+\frac{\mathcal{F}\partial_{u}\mathcal{T}^{0}-\partial_{u}\mathcal{F}\mathcal{T}^{0}}{r_{s}^{2}}\right\\}\mathrm{d}z\mathrm{d}\bar{z}\,.$
(42)

The charge algebra of the linearized theory is defined by

$\displaystyle[\mathcal{Q}_{\zeta_{1}},\mathcal{Q}_{\zeta_{2}}]\equiv\delta_{\zeta_{1}}\mathcal{Q}_{\zeta_{2}}=\mathcal{Q}_{[\zeta_{1},\zeta_{2}]_{l}}+\mathcal{K}_{\zeta_{1},\zeta_{2}}\,,$
(43)

where $\mathcal{K}_{\zeta_{1},\zeta_{2}}$ denotes a possible 2-cocycle term.
To verify the charge algebra, one needs to apply the transformation law of the
metric fields at the leading order,

$\displaystyle\delta_{\zeta}\mathcal{C}=$
$\displaystyle-Z/r_{s}-2\partial_{u}Z+\mathcal{O}(\lambda)\,,$ (44)
$\displaystyle\delta_{\zeta}\mathcal{U}_{A}=$ $\displaystyle-
D_{A}Z+r_{s}^{2}\partial_{u}Y_{A}-r_{s}\partial_{u}D_{A}\mathcal{F}+\lambda(-D_{A}\mathcal{F}+2r_{s}^{2}\partial_{u}Y_{A})+\mathcal{O}(\lambda^{2})\,,$
(45) $\displaystyle\delta_{\zeta}\mathcal{T}=$ $\displaystyle
2r_{s}^{2}D^{A}Y_{A}-2r_{s}D^{A}D_{A}\mathcal{F}+4r_{s}Z$
$\displaystyle\quad\quad\quad+\lambda(4r_{s}^{2}D^{A}Y_{A}-2r_{s}D^{A}D_{A}\mathcal{F}-4r_{s}^{2}\partial_{u}\mathcal{F}+4r_{s}Z)+\mathcal{O}(\lambda^{2})\,,$
(46) $\displaystyle\delta_{\zeta}\mathcal{N}_{AB}=$ $\displaystyle
2r_{s}^{2}D_{A}Y_{B}-2r_{s}D_{A}D_{B}\mathcal{F}+\mathcal{O}(\lambda)\,.$ (47)

Inserting field variation generated by
$\zeta^{\mu}_{2}(\mathcal{F}_{2},Z_{2},Y^{A}_{2})$ for the linearized theory,
the algebra of the charge is given by

$\displaystyle\delta_{\zeta_{2}}\mathcal{Q}_{\zeta_{1}}=$
$\displaystyle\mathcal{K}_{\zeta_{1},\zeta_{2}}$ (48) $\displaystyle=$
$\displaystyle\int_{B}\sqrt{-g}\left\\{\frac{Z_{1}\partial_{u}\mathcal{F}_{2}+\mathcal{F}_{1}\partial_{u}Z_{2}}{r_{s}/2}+\frac{D^{A}\mathcal{F}_{1}\partial_{u}D_{A}\mathcal{F}_{2}}{r_{s}}\right.$
$\displaystyle\quad\quad\quad\left.+D_{A}Y_{1}^{A}\partial_{u}\mathcal{F}_{2}-\frac{Z_{1}D_{A}Y_{2}^{A}}{r_{s}}+\frac{Y_{1}^{A}D_{A}\mathcal{F}_{2}}{2r_{s}}-(1\leftrightarrow
2)\right\\}\mathrm{d}z\mathrm{d}\bar{z}\,.$

##  3 Near horizon soft graviton theorem

In this section we will derive a soft graviton theorem from the Ward identity
of the near horizon symmetry. The success of deriving soft theorem from
asymptotic/near horizon symmetry resides in a natural split of the symmetry
charges into soft and hard parts [6, 7, 8, 9, 10, 5, 23, 24, 25].
Unfortunately, it is completely unclear how one can organize the near horizon
charge (42) derived in previous section into soft and hard parts.
Nevertheless, we can impose proper near horizon fall-off conditions to reduce
the solution space and near horizon charges, such that a natural split will
emerge.

After solving the hypersurface and standard equations, the solution space is
spanned by a collection of boundary initial data. This set includes
$\\{\mathcal{N}_{AB}^{0},\mathcal{C}^{0},\mathcal{U}_{A}^{0},\mathcal{U}_{A}^{(1)},\mathcal{T}^{0},\mathcal{T}^{(1)}\\}$.
Among these, the two dimensional trace-less tensor $\mathcal{N}_{AB}^{0}$
represents the propagating degree of freedom associated with a graviton in the
bulk which are the flux falling into the horizon. The scalar field
$\mathcal{C}^{0}$ represents the energy of the linearized theory, see also
[37, 38] from a thermodynamical perspective. A two dimensional vector from the
combination of $\mathcal{U}_{A}^{0}$and $\mathcal{U}_{A}^{(1)}$ represents the
angular momentum perspective. The time evolution of those three quantities are
controlled by the supplementary equations. There are still four free functions
or degrees of freedom which consist of $\mathcal{T}^{0},\mathcal{T}^{(1)}$ and
one two dimensional vector from another type of combination of
$\mathcal{U}_{A}^{0}$and $\mathcal{U}_{A}^{(1)}$. They are fake propagating
degree of freedom [27] which can be eliminated by either imposing fall-off
conditions or redefining the symmetry parameters by involving field dependence
[39, 40, 41]. We will use the fake degree of freedom to reorganize the
solution space and recover a natural split of the charge.

###  3.1 Fall-off conditions and charge

According to (25), we can completely turn off all orders of the trace
component $\mathcal{T}$ by imposing the fall-off condition
$\mathcal{T}=\mathcal{O}(\lambda^{2})$. Due to the absence of
$\mathcal{T}^{0}$ and $\mathcal{T}^{(1)}$, both the conserved charge and the
equations of motion undergo a significant simplification. To incorporate with
this fall-off requirement, the symmetry parameters are constrained by
$\delta_{\zeta}\mathcal{T}=0$. From Eq. (22), we obtain

$\displaystyle\left(-D^{A}D_{A}\mathcal{F}+r_{s}D^{A}Y_{A}+2Z\right)+\lambda\left(-2r_{s}\partial_{u}\mathcal{F}+r_{s}D^{A}Y_{A}\right)=0\,.$
(49)

Hence,

$\displaystyle Z$
$\displaystyle=\frac{1}{2}D^{A}D_{A}\mathcal{F}-\frac{r_{s}}{2}D^{A}Y_{A}\,,$
(50) $\displaystyle\mathcal{F}$
$\displaystyle=T+\frac{1}{2}D^{A}\mathbf{Y}_{A}\,,$ (51)

where $T$ is an arbitrary function of $(z,\bar{z})$ and

$\displaystyle\mathbf{Y}_{A}=\int\text{d}uY_{A}\,.$ (52)

Here $T(z,\bar{z})$ generates the supertranslation of linearized gravitational
fields on the horizon, and $Y_{A}$ is the generalized near horizon
superrotation. We are mainly interested in the soft theorem associated to the
supertranslation charge, which in the reduced solution space becomes

$\displaystyle Q_{T}$
$\displaystyle=\int_{B}\gamma_{z\bar{z}}T\left(r_{s}\mathcal{C}^{0}-\frac{D^{A}\mathcal{U}_{A}^{(1)}}{2}\right)\text{d}^{2}z\,.$
(53)

The supertranslation charge can be evaluated on the horizon applying the
relations in (38) and (39),

$\displaystyle Q_{T}=$
$\displaystyle\frac{1}{2r_{s}}\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}T\partial_{u}\left(D^{A}D^{B}\mathcal{N}_{AB}^{0}\right)\text{d}^{2}z\text{d}u$
(54)
$\displaystyle+\frac{r_{s}}{2}\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}T\left(r_{s}T_{uu}^{0}+D^{A}T_{uA}^{0}\right)\text{d}^{2}z\text{d}u$
$\displaystyle-\frac{1}{2}\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}T\left(D^{A}\mathcal{U}_{A}^{0}/r_{s}+D^{A}D_{A}\mathcal{C}^{0}+D^{A}D_{A}\mathcal{C}^{(1)}\right)\text{d}^{2}z\text{d}u\,.$

The first line of this charge expression has exactly the desired forms as a
soft part of the charge, and the second line is in the form of a hard part.
The third line can be turned off by imposing proper fall-off conditions, for
which we set

$\displaystyle\mathcal{U}_{A}^{0}+r_{s}D_{A}\left(\mathcal{C}^{0}+\mathcal{C}^{(1)}\right)=0\,.$
(55)

In order to take the symmetry parameter $\zeta^{\mu}(\mathcal{F},Y^{A})$ to be
compatible with this fall-off condition, those parameters should satisfy

$\displaystyle\delta_{\zeta}\left[\mathcal{U}_{A}^{0}+r_{s}D_{A}\left(\mathcal{C}^{0}+\mathcal{C}^{(1)}\right)\right]=0.$
(56)

From (20) and (21), we have

$\displaystyle\delta_{\zeta}\mathcal{U}_{A}^{0}$
$\displaystyle=-D_{A}Z+r_{s}^{2}\partial_{u}Y_{A}-r_{s}\partial_{u}D_{A}\mathcal{F}\,,$
(57)
$\displaystyle\delta_{\zeta}\left[r_{s}D_{A}\left(\mathcal{C}^{0}+\mathcal{C}^{(1)}\right)\right]$
$\displaystyle=D_{A}Z-2r_{s}\partial_{u}D_{A}Z+2r_{s}^{2}\partial^{2}_{u}D_{A}\mathcal{F}-r_{s}\partial_{u}D_{A}\mathcal{F}\,.$
(58)

Then, Eq. (56) could be reorganized as

$\displaystyle\partial_{u}\left[r_{s}\left(\gamma_{AB}+2D_{A}D_{B}\right)Y^{B}-(2\gamma_{AB}+D_{A}D_{B})D^{B}\mathcal{F}\right]=0\,.$
(59)

Clearly, a pure supertranslation with $\mathcal{F}=T$ and $Y^{A}=0$ satisfies
this constraint. Finally, the near horizon supertranslation charge can be
split as

$\displaystyle Q_{T}=Q^{S}+Q^{H}\,,$ (60)

where $Q^{S},Q^{H}$ denote the soft and hard charge respectively,

$\displaystyle Q^{S}$
$\displaystyle=\frac{1}{2r_{s}}\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}T\partial_{u}\left(D^{A}D^{B}\mathcal{N}_{AB}^{0}\right)\text{d}^{2}z\text{d}u\,,$
(61) $\displaystyle Q^{H}$
$\displaystyle=\frac{r_{s}}{2}\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}T\left(r_{s}T_{uu}^{0}+D^{A}T_{uA}^{0}\right)\text{d}^{2}z\text{d}u\,.$
(62)

In the null infinity analysis [7, 8], it is normally assumed that there is no
long-range magnetic mass aspect. This assumption can be adapted to the near
horizon case by imposing

$\displaystyle\left[D_{z}D_{z}\mathcal{N}^{0}_{\bar{z}\bar{z}}-D_{\bar{z}}D_{\bar{z}}\mathcal{N}^{0}_{zz}\right]_{\mathcal{H}^{-}_{\pm}}=0\,,$
(63)

Then we can rewrite the soft charge as

$\displaystyle
Q^{S}=-\frac{1}{r_{s}}\int_{\mathcal{H}^{-}}\partial_{\bar{z}}T\partial_{u}\left(\frac{\partial_{\bar{z}}\mathcal{N}_{zz}^{0}}{\gamma_{z\bar{z}}}\right)\text{d}^{2}z\text{d}u\,,$
(64)

For the hard part of the charge, it consists only the contribution from the
stress tensor. We have used an auxiliary fields to modify the stress tensor to
make it more consistent with our gauge conditions. Basically, it will just
change the way of coupling the matter fields to gravity. In the null infinity
case, the modification of the stress tensor does not affect the leading soft
theorem derived from the Ward identity of the supertranslation [31] which
simply reflects the fact that the leading soft graviton theorem is universal
for any type of gravitational theory [42]. In contrast, the hard part of the
near horizon supertranslation charge is sensitive to the modification of the
stress tensor. Correspondingly, this indicates that the soft theorem derived
from the Ward identity of the near horizon supertranslation charge will be of
dependence on the way of the matter fields couplings. In this work, we are
interested in the case that the near horizon supertranslation charge has a
similar expression as the null infinity case. We will further use the freedom
in the auxiliary fields to turn off $T_{uA}^{0}$, and the precise modification
is presented in Appendix A. Thus, the hard part of the charge becomes

$Q^{H}=\frac{r_{s}}{2}\int_{\mathcal{H}^{-}}\gamma_{z\bar{z}}T\left(r_{s}T_{uu}^{0}\right)\text{d}^{2}z\text{d}u\,.$
(65)

In Section 3.4, we will demonstrate that the soft charge creates a low-energy
soft graviton in near horizon states, and the action of the hard charge in
those states leads to the soft factor, akin to the scenario in flat spacetime.

###  3.2 Commutation relations

Now we derive the commutation relation between the supertranslation charge and
gauge field using the covariant phase space approach. Following the standard
procedure in [43, 44], one can find the presymplectic potential of a free
spin-2 theory defined in (4) as follows

$\displaystyle\boldsymbol{\Theta}=\frac{1}{2}\bigg{(}\frac{1}{2}\delta
h\nabla^{\rho}h$ $\displaystyle-\frac{1}{2}\delta h\nabla^{\mu}h_{\mu}^{\
\rho}-\frac{1}{2}\delta h_{\mu}^{\ \rho}\nabla^{\nu}h$ (66)
$\displaystyle\left.-\frac{1}{2}\delta
h^{\mu\nu}\nabla^{\rho}h_{\mu\nu}+\delta h^{\mu\nu}\nabla_{\mu}h_{\nu}^{\
\rho}\right)\sqrt{-g}\text{d}x_{\rho}\,,$

where
$\text{d}x_{\rho}=\frac{1}{3!}\hat{\epsilon}_{\rho\mu\nu\sigma}\text{d}x^{\mu}\wedge\text{d}x^{\nu}\wedge\text{d}x^{\sigma}$.
The presymplectic form $\boldsymbol{\omega}$ is defined by

$\displaystyle\boldsymbol{\omega}\equiv\delta\boldsymbol{\Theta}\,.$ (67)

Note that, in our case, the Cauchy surface is $\mathcal{H}^{-}$, and thus
after imposing near horizon fall-off conditions and gauge fixing, the
presymplectic form is reduced to

$\displaystyle\boldsymbol{\omega}_{uz\bar{z}}|_{\mathcal{H}^{-}}$
$\displaystyle=\frac{\gamma_{z\bar{z}}\gamma^{AC}\gamma^{BD}}{4r_{s}^{2}}\delta\partial_{u}\mathcal{N}^{0}_{AB}\wedge\delta\mathcal{N}^{0}_{CD}\,.$
(68)

We can write down the canonical commutator from the presymplectic form as

$\displaystyle[\partial_{u}\mathcal{N}^{0}_{AB}(u,z,\bar{z}),\mathcal{N}_{CD}^{0}(g,w,\bar{w})]=-i\frac{4r_{s}^{2}\gamma_{AC}\gamma_{BD}}{\gamma_{z\bar{z}}}\delta(u-g)\delta^{2}(z-w)\,.$
(69)

After a direct computation, we arrive at

$\displaystyle[Q_{T},\mathcal{N}_{AB}^{0}]=[Q^{S},\mathcal{N}_{AB}^{0}]=i\left(-2r_{s}D_{A}D_{B}T\right)\,.$
(70)

As expected, the soft charge generates the supertraslation. A similar
computation can be done when matter fields coupled. For instance, for a
massless scalar field as described in Appendix A, one has

$\displaystyle[\partial_{u}\bar{\phi}(u,z,\bar{z}),\phi(g,z,\bar{z})]$
$\displaystyle=-\frac{i}{r_{s}^{2}\gamma_{z\bar{z}}}\delta(u-g)\delta^{2}(z-w)\,,$
(71) $\displaystyle[Q_{T},\phi]=[Q^{H},\phi]$
$\displaystyle=-\frac{i}{2}T\partial_{u}\phi\,.$ (72)

The second line above in the momentum space can be written as

$\displaystyle[Q_{T},\phi_{k}]=\frac{E_{k}}{2}T\phi_{k}\,.$ (73)

###  3.3 Mode expansion

We will now derive the mode expansions of the near-horizon gauge fields, which
are the key step for formulating a soft theorem from near horizon symmetry. It
is convenient to write down the mode expansion of the free field operators in
the isotropic coordinates $(t,x_{1},x_{2},x_{3})$, in which the line element
of the Schwarzschild spacetime is

$\displaystyle\text{d}^{2}s=-\left(\frac{4\rho-
r_{s}}{4\rho+r_{s}}\right)^{2}\text{d}^{2}t+\left(\frac{4\rho+r_{s}}{4\rho}\right)^{4}\text{d}\vec{x}\cdot\text{d}\vec{x}\,.$
(74)

The coordinate transformations from $(u,r,z,\bar{z})$ to
$(t,x_{1},x_{2},x_{3})$ are

$\displaystyle t=u+\Omega+r_{s}\ln\left(\frac{r}{r_{s}}\right)\,,$ (75)
$\displaystyle x_{1}=\rho\frac{z+\bar{z}}{z\bar{z}+1}\,,\
x_{2}=\rho\frac{z-\bar{z}}{i(z\bar{z}+1)}\,,\,x_{3}=\rho\frac{z\bar{z}-1}{z\bar{z}+1}\,,$
(76)

where $\rho$ is a function of $r$, defined by

$\displaystyle\rho=\frac{1}{4}\left[2r+r_{s}+2\sqrt{r(r+r_{s})}\right]\,,$
(77)

thus the near horizon limit is $\rho\rightarrow\frac{r_{s}}{4}$.

The external Schwarzschild spacetime is static, with $\partial_{t}$ serving as
a timelike Killing vector everywhere which leads to a definition of positive
energy of a moving particle as $\omega=-p_{0}$. In the isotropic coordinates,
the dispersion relation for massless particles is

$\displaystyle-\left(\frac{4\rho+r_{s}}{4\rho-
r_{s}}\right)^{2}\omega^{2}+\left(\frac{4\rho}{4\rho+r_{s}}\right)^{4}\vec{p}^{2}=0\,.$
(78)

The covariant measure of one-particle phase space for the mode expansion is

$\displaystyle\int\frac{\mathrm{d}\omega\mathrm{d}^{3}\vec{p}}{(2\pi)^{3}}\delta(p^{\mu}p_{\mu})\Theta(\omega)=\left(\frac{4\rho-
r_{s}}{4\rho+r_{s}}\right)^{2}\int\frac{\mathrm{d}^{3}\vec{p}}{(2\pi)^{3}2\omega}\,.$
(79)

The mode expansion of a free graviton field is given by

$\displaystyle h_{\mu\nu}=\sum_{\alpha=\pm}\left(\frac{4\rho-
r_{s}}{4\rho+r_{s}}\right)^{2}\int\frac{\text{d}^{3}p}{(2\pi)^{3}}\frac{1}{2\omega}\left[\epsilon^{\alpha\star}_{\mu\nu}a_{\alpha}(\vec{p})e^{i\vec{p}\cdot\vec{x}}+\epsilon^{\alpha}_{\mu\nu}a^{\dagger}_{\alpha}(\vec{p})e^{-i\vec{p}\cdot\vec{x}}\right]\,.$
(80)

where $\epsilon_{\mu\nu}^{\alpha}$ is a product of the pair polarization
vectors
$\epsilon_{\mu\nu}^{\alpha}=\epsilon^{\alpha}_{\mu}\epsilon^{\alpha}_{\nu}$.
We will use a new set of null parameterizations for describing a massless
particle. The momentum of a graviton can be parameterized as follows:

$\displaystyle p_{\mu}=\frac{\omega}{1+z\bar{z}}$
$\displaystyle\frac{(4\rho+r_{s})^{3}}{16\rho^{2}(4\rho-r_{s})}$ (81)
$\displaystyle\times\left[\frac{16\rho^{2}(r_{s}-4\rho)}{(4\rho+r_{s})^{3}}(1+z\bar{z}),(z+\bar{z}),-i(z-\bar{z}),(1-z\bar{z})\right]\,.$

Correspondingly, the polarization vectors that are perpendicular to the
direction of propagation of graviton are reparametrized as

$\displaystyle\epsilon^{+\mu}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\frac{4\rho}{4\rho+r_{s}}\right)^{2}\left[\frac{(4\rho+r_{s})^{3}}{16\rho^{2}(4\rho-
r_{s})}\bar{z},1,-i,z\right]\,,$ (82) $\displaystyle\epsilon^{-\mu}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\frac{4\rho}{4\rho+r_{s}}\right)^{2}\left[\frac{(4\rho+r_{s})^{3}}{16\rho^{2}(4\rho-
r_{s})}z,1,i,\bar{z}\right]\,,$ (83)

with the following properties

$\displaystyle\epsilon^{\alpha}_{r}=0\,,\,p^{\mu}\epsilon^{\alpha}_{\mu}=0\,,\,\epsilon^{\alpha\mu}\epsilon^{\star\beta}_{\mu}=\delta^{\alpha\beta}\,.$
(84)

Projecting those two vectors on the sphere by
$\epsilon^{\pm}_{A}=\frac{\partial x^{\mu}}{\partial
x^{A}}\epsilon^{\pm}_{\mu}$, we obtain

$\displaystyle\epsilon_{A}^{+}=\delta_{zA}\frac{(4\rho+r_{s})^{2}}{4\sqrt{2}\rho(1+z\bar{z})}\,,\,\epsilon_{A}^{-}=\delta_{\bar{z}A}\frac{(4\rho+r_{s})^{2}}{4\sqrt{2}\rho(1+z\bar{z})}\,.$
(85)

Here, we introduce the parameter $R=\rho-\frac{r_{s}}{4}$ to position the
horizon at $R\rightarrow 0$. Following the treatment used in [23, 24], we
implement a near-horizon regularization by replacing $R$ with $R+i\mathcal{R}$
to regularize the divergence from the near-horizon limit. Eventually, the near
horizon field is related to the plane wave modes by

$\displaystyle\mathcal{N}^{0}_{zz}=$
$\displaystyle\gamma_{z\bar{z}}\frac{8}{\pi^{2}}\frac{(r_{s}+2R)^{6}}{(r_{s}+4R)^{5}}\int\text{d}\omega\left(\frac{4R^{2}}{r_{s}^{2}+2r_{s}R}\right)^{-ir_{s}\omega}$
(86)
$\displaystyle\times\sin\left[\frac{2\omega(R+r_{s}/2)^{3}}{R(2R+r_{s}/2)}\right]\left(a_{+}e^{-i\omega
u-i\omega\frac{2(R+r_{s}/2)^{2}}{2R+r_{s}/2}}+a_{-}^{\dagger}e^{i\omega
u+i\omega\frac{2(R+r_{s}/2)^{2}}{2R+r_{s}/2}}\right)\,.$

###  3.4 Soft graviton theorem in coordinates space

In this subsection, we derive the soft graviton theorem by applying the Ward
identity of supertranslation

$\displaystyle\left\langle{\rm out}\right|[Q_{T},\mathcal{S}]\left|{\rm
in}\right\rangle=0\Rightarrow Q^{S}\left|{\rm
in}\right\rangle=-Q^{H}\left|{\rm in}\right\rangle\,,$ (87)

where on the right side above we have omitted the out-part on
$\mathcal{H}^{+}$ by $\mathcal{CPT}$ invariance and one can easily restore
this part by the symmetry. We choose $T=\frac{1}{z-w}$, and thus

$\displaystyle\partial_{\bar{z}}T(z,\bar{z})=2\pi\delta^{2}(z-w)\,.$ (88)

From this choice, the soft charge (64) is reduced to

$\displaystyle Q^{S}$
$\displaystyle=-\frac{4\pi^{2}i}{r_{s}}\lim_{\omega\rightarrow
0^{+}}\left[\omega\frac{\partial_{\bar{z}}\tilde{\mathcal{N}}^{0}_{zz}}{\gamma_{z\bar{z}}}\right]\
,$ (89)

where $\tilde{\mathcal{N}}_{zz}$ is defined by a Fourier relation
$\mathcal{N}_{zz}=\int_{-\infty}^{+\infty}\text{d}ue^{i\omega
u}\tilde{\mathcal{N}}_{zz}^{0}$ . After some tedious but direct computations
involving the mode expansion (86), one can obtain

$\displaystyle
Q^{S}=-16i\frac{r_{s}^{2}}{R}\frac{\partial_{\bar{z}}\gamma_{z\bar{z}}}{\gamma_{z\bar{z}}}\lim_{\omega\rightarrow
0^{+}}\left[\omega^{2}a_{+}+\omega^{2}a_{-}^{\dagger}\right]\,.$ (90)

Thus $Q^{S}$ acts on the in-state as

$\displaystyle Q^{S}\left|{\rm
in}\right\rangle=-16i\frac{r_{s}^{2}}{R}\frac{\partial_{\bar{z}}\gamma_{z\bar{z}}}{\gamma_{z\bar{z}}}\lim_{\omega\rightarrow
0^{+}}\omega^{2}a_{-}^{\dagger}\left|{\rm in}\right\rangle.$ (91)

As expected, the soft charge creates a low-energy soft graviton in the near
horizon state.

Regarding to the hard charge, applying the commutation relation (73), we get

$\displaystyle Q^{H}\left|{\rm
in}\right\rangle=\frac{1}{2}\sum_{k=1}^{n}\frac{E_{k}}{(z-z_{k})}\left|{\rm
in}\right\rangle\,.$ (92)

Finally, we can obtain a soft graviton theorem in coordinate space from the
Ward identity (87) from the insertion of (91) and (92) as

$\displaystyle\partial_{\bar{z}}\left[\gamma_{z\bar{z}}\lim_{\omega\rightarrow
0^{+}}a^{\dagger}_{-}\left|{\rm
in}\right\rangle\right]=\lim_{\omega\rightarrow
0^{+}}\left[\frac{\pi}{32i}\frac{R}{r_{s}^{2}\omega}\gamma_{z\bar{z}}\sum_{k=1}^{n}\frac{E_{k}}{\omega(z-z_{k})}\left|{\rm
in}\right\rangle\right]\,.$ (93)

###  3.5 Soft graviton theorem in momentum space

Here we will use the null parametrization in (81) to rewrite the soft theorem
derived previously in momentum space. In particular, for each hard particle,
the momentum can be parametrized as

$\displaystyle q_{k\mu}=$
$\displaystyle\frac{E_{k}}{1+z_{k}\bar{z}_{k}}\frac{(4\rho+r_{s})^{3}}{16\rho^{2}(4\rho-
r_{s})}$ (94)
$\displaystyle\times\left[\frac{16\rho^{2}(r_{s}-4\rho)}{(4\rho+r_{s})^{3}}(1+z_{k}\bar{z}_{k}),(z_{k}+\bar{z}_{k}),-i(z_{k}-\bar{z}_{k}),(1-z_{k}\bar{z}_{k})\right]\,.$

Applying the null parameterization, one can easily check that

$\displaystyle\partial_{\bar{z}}\left(\gamma_{z\bar{z}}\sum_{k=1}^{n}\frac{[q_{k}\cdot\epsilon(p)]^{2}}{p\cdot
q_{k}}\right)$
$\displaystyle=-\gamma_{z\bar{z}}\sum_{k=1}^{n}\frac{E_{k}}{\omega(z-z_{k})}\left(1+\frac{\bar{z}_{k}(z-z_{k})}{1+z_{k}\bar{z}_{k}}\right)$
(95)
$\displaystyle=-\gamma_{z\bar{z}}\sum_{k=1}^{n}\frac{E_{k}}{\omega(z-z_{k})}\,,$

where we have used the fact that a combination of two components of momenta is
also conserved

$\displaystyle\sum_{k=1}^{n}\frac{E_{k}\bar{z}_{k}}{1+z_{k}\bar{z}_{k}}\propto\sum_{k=1}^{n}\left(q_{k1}-iq_{k2}\right)=0\,.$
(96)

Finally, inserting (95) into (93), we arrive at the soft graviton theorem in
the momentum space as

$\displaystyle\lim_{\omega\rightarrow 0^{+}}$ $\displaystyle\langle{\rm
out}|a_{+}\mathcal{S}-\mathcal{S}a^{\dagger}_{-}\left|{\rm in}\right\rangle$
(97) $\displaystyle=\lim_{\omega\rightarrow 0^{+}}i\frac{\pi R}{32\omega
r_{s}^{2}}\left[\sum_{l=1}^{m}\frac{(q_{l}^{\rm out}\cdot\epsilon)^{2}}{p\cdot
q_{l}^{\rm out}}-\sum_{k=1}^{n}\frac{(q_{k}^{\rm in}\cdot\epsilon)^{2}}{p\cdot
q_{k}^{\rm in}}\right]\langle{\rm out}|\mathcal{S}\left|{\rm
in}\right\rangle\,,$

where we have dropped the derivative from both sides.

##  4 Conclusion

To conclude, we study a linearized gravity theory in the near horizon region
of the Schwarzschild spacetime. We compute the near horizon symmetry and the
on shell surface charges associated to the near horizon symmetries. A soft
graviton theorem is obtained from the Ward identity of the near horizon
supertranslation. Together with the previous works for gauge theories [24,
23], they constitute a complete realization of the equivalence between the
soft theorem and near horizon symmetry in curved spacetime and confirm that
the black hole soft hairs are indeed soft particles on the horizon.

## Acknowledgments

The authors thank Peng Cheng for the early collaboration on this project and
discussions which helped us clarify some aspects of this work. This work is
supported in part by the National Natural Science Foundation of China (NSFC)
under Grants No. 11905156 and No. 11935009. K.Y.Z. is also supported in part
by NSFC Grant No. 11905158.

##  Appendix A Modification of stress tensor

In flat spacetime, a conserved stress tensor is ambiguous up to a rank four
tensor [31], which can be used to modify a stress tensor. In the curved
spacetime, such ambiguity disappears due to the existence of curvature.
Nevertheless, one can modify it by adding a divergence-free symmetric rank two
tensor. To our purpose, the modified stress tensor satisfies the gauge
condition (9). In this work, we are interested in the stress tensor of a
massless complex scalar field which is minimally coupled to gravity, thus

$\displaystyle\tilde{T}_{\mu\nu}\equiv\frac{1}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{M})}{\delta
g_{\mu\nu}}=\frac{1}{2}\left(\partial_{\mu}\Phi\partial_{\nu}\bar{\Phi}+\partial_{\nu}\Phi\partial_{\mu}\bar{\Phi}\right)-\frac{1}{2}g_{\mu\nu}\nabla^{\rho}\Phi\nabla_{\rho}\bar{\Phi}\,,$
(98)

where $\mathcal{L}_{M}=\nabla_{\mu}\Phi\nabla^{\mu}\bar{\Phi}\,$. We assume
those scalar fields are given in the form of near horizon expansions as

$\displaystyle\Phi=\phi+\sum_{n=1}^{\infty}\lambda^{n}\Phi^{(n)}\,,\quad\bar{\Phi}=\bar{\phi}+\sum_{n=1}^{\infty}\lambda^{n}\bar{\Phi}^{(n)}\,.$
(99)

We construct a stress tensor by
$T_{\mu\nu}=\tilde{T}_{\mu\nu}-\Upsilon_{\mu\nu}$, where $T_{r\mu}=0$ or
$\Upsilon_{r\mu}=\tilde{T}_{r\mu}$, and $\Upsilon_{\mu\nu}$ is an arbitrary
divergence-free symmetric 2 tensor. Now we use the divergence-free conditions
$\nabla^{\mu}\Upsilon_{\mu\nu}=0$ to fix it. We expand $\Upsilon_{\mu\nu}$ as

$\displaystyle\Upsilon_{\mu\nu}=\Upsilon^{0}_{\mu\nu}+\sum_{n=1}^{\infty}\lambda^{n}\Upsilon_{\mu\nu}^{(n)}\,.$
(100)

Begin by the simplest equation $\nabla^{\mu}\Upsilon_{\mu r}=0$, which yields

$\displaystyle\gamma^{AB}\Upsilon_{AB}=r_{s}\xi D^{A}\tilde{T}_{rA}$
$\displaystyle-
r_{s}^{2}\xi\partial_{\lambda}\left(\xi^{2}\tilde{T}_{ru}\right)$ (101)
$\displaystyle+r_{s}^{2}\xi\partial_{\lambda}\left(f\xi^{2}\tilde{T}_{rr}\right)+r_{s}^{2}\xi^{3}\partial_{\lambda}f\tilde{T}_{rr}/2-r_{s}^{3}\xi^{3}\partial_{u}\tilde{T}_{rr}\,.$

Thus the trace part of $\Upsilon_{\mu\nu}$ is totally fixed by
$\tilde{T}_{\mu\nu}$.

The transverse equations $\nabla^{\mu}\Upsilon_{\mu A}=0$ lead to

$\displaystyle\partial_{\lambda}\Upsilon_{uA}$
$\displaystyle=\frac{\partial_{\lambda}\left(f\xi^{2}\tilde{T}_{rA}\right)}{\xi^{2}}-r_{s}\partial_{u}\tilde{T}_{rA}-2\frac{\Upsilon_{uA}}{\xi}+\frac{D^{B}\Upsilon_{BA}}{r_{s}\xi^{2}}\,.$
(102)

All information that could be extracted from the above equations is

$\displaystyle\Upsilon^{(1)}_{uA}$
$\displaystyle=\tilde{T}_{rA}^{0}-r_{s}T_{rA}^{0}-2\Upsilon^{0}_{uA}+D^{B}\Upsilon_{BA}^{0}/r_{s}\,,$
(103)

and, for $n\geq 1$,

$\displaystyle\Upsilon^{(n+1)}_{uA}=$
$\displaystyle\frac{1}{n+1}\left\\{-r_{s}\partial_{u}\tilde{T}_{rA}^{(n)}+\sum_{m=0}^{n}(-1)^{m}\tilde{T}_{rA}^{(n-m)}\right.$
(104)
$\displaystyle+\sum_{m=0}^{n-1}(-1)^{m}\left[(1+m)\tilde{T}_{rA}^{(n-1-m)}+(n-m)\tilde{T}_{rA}^{(n-m)}\right]$
$\displaystyle+\left.\sum_{m=0}^{n}(-1)^{m}\left[(1+m)D^{B}\Upsilon_{BA}^{(n-m)}/r_{s}-2\Upsilon_{uA}^{(n-m)}\right]\right\\}\,.$

While the leading order $\Upsilon^{0}_{uA}$ is free. We continue with the
equation of the $u$ component, which gives

$\displaystyle\partial_{\lambda}\Upsilon_{uu}=\frac{\partial_{\lambda}(f\xi^{2}\tilde{T}_{ru})}{\xi^{2}}-r_{s}\partial_{u}\tilde{T}_{ru}-2\frac{\Upsilon_{uu}}{\xi}+\frac{D^{A}\Upsilon_{uA}}{r_{s}\xi^{2}}\,.$
(105)

This equation controls $\Upsilon_{uu}$ up to an initial data
$\Upsilon^{0}_{uu}$,

$\displaystyle\Upsilon^{(1)}_{uu}$
$\displaystyle=\tilde{T}_{ru}^{0}-r_{s}T_{ru}^{0}-2\Upsilon^{0}_{uu}+D^{A}\Upsilon_{uA}^{0}/r_{s}\,,$
(106)

and, for $n\geq 1$,

$\displaystyle\Upsilon^{(n+1)}_{uu}=$
$\displaystyle\frac{1}{n+1}\left\\{-r_{s}\partial_{u}\tilde{T}_{ru}^{(n)}+\sum_{m=0}^{n}(-1)^{m}\tilde{T}_{ru}^{(n-m)}\right.$
(107)
$\displaystyle+\sum_{m=0}^{n-1}(-1)^{m}\left[(1+m)\tilde{T}_{ru}^{(n-1-m)}+(n-m)\tilde{T}_{ru}^{(n-m)}\right]$
$\displaystyle+\left.\sum_{m=0}^{n}(-1)^{m}\left[(1+m)D^{A}\Upsilon_{uA}^{(n-m)}/r_{s}-2\Upsilon_{uu}^{(n-m)}\right]\right\\}\,.$

Finally, we find the traceless part of $\Upsilon_{AB}$ and the leading order
of $\Upsilon_{uu}$, $\Upsilon_{uA}$ are free. The $uu,uA,zz$ components of the
modified stress tensor at the leading orders are

$\displaystyle T_{uu}^{0}$
$\displaystyle=\partial_{u}\phi\partial_{u}\bar{\phi}-\Upsilon_{uu}^{0}\,,$
(108) $\displaystyle T_{uA}^{0}$
$\displaystyle=\frac{1}{2}\left(\partial_{u}\phi\partial_{A}\bar{\phi}+\partial_{A}\phi\partial_{u}\bar{\phi}\right)-\Upsilon_{uA}^{0}\,,$
(109) $\displaystyle T_{zz}^{0}$
$\displaystyle=\partial_{z}\phi\partial_{z}\bar{\phi}-\Upsilon_{zz}^{0}\,.$
(110)

The hard part of the near horizon supertranslation charge (62) in the main
text is sensitive to the modification of the stress tensor. In particular,
different choices of the leading orders of $\Upsilon_{uu}$, $\Upsilon_{uA}$
will yield different formulas of soft theorem. In this work, we set
$\Upsilon_{uA}^{0}=0$, and use the free initial data $\Upsilon_{uA}^{0}$ to
set $T_{uA}^{0}=0$.

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