# On the specific relative entropy between martingale diffusions on the line

Julio Backhoff-Veraguas  Clara Unterberger

The specific relative entropy, introduced by N. Gantert, allows to quantify
the discrepancy between the laws of potentially mutually singular measures. It
appears naturally as the large deviations rate function in a randomized
version of Donsker’s invariance principle, as well as in a novel transport-
information inequality recently derived by H. Föllmer. A conjecture, put
forward by the aforementioned authors, concerns a closed form expression for
the specific relative entropy between continuous martingale laws in terms of
their quadratic variations. We provide a first partial result in this
direction, by establishing this conjecture in the case of well-behaved
martingale diffusions on the line.

##  1 Introduction

Let $(\mathcal{C},\mathcal{F})$ be the Wiener space
$\mathcal{C}=C([0,1],\mathbb{R})$ and $\mathcal{F}=\sigma(X_{t}:t\in[0,1])$
where $X_{t}\colon\mathcal{C}\to\mathbb{R}$ with $X_{t}(\omega)=\omega(t)$ is
the canonical process. Define also

$\mathcal{F}^{n}\coloneqq\sigma(X_{\frac{k}{n}}:k=0,1,\dots,n).$

The central object of study in this article is the concept of specific
relative entropy, introduced by N. Gantert in her PhD thesis [5] and recently
studied by the works of H. Föllmer [3, 2].

######  Definition 1.

Given $\mathbb{Q},\mathbb{P}$ probability measures on
$(\mathcal{C},\mathcal{F})$, the upper specific relative entropy of
$\mathbb{Q}$ with respect to $\mathbb{P}$ is defined as

$h^{u}\left(\mathbb{Q}\,|\,\mathbb{P}\right)\coloneqq\limsup_{n\to\infty}\frac{1}{n}\text{H}(\mathbb{Q}\,|\,\mathbb{P})|_{\mathcal{F}^{n}},$

where $\text{H}(\mathbb{Q}\,|\,\mathbb{P})|_{\mathcal{F}^{n}}$ is the relative
entropy of $\mathbb{Q}$ with respect to $\mathbb{P}$ restricted to
$\mathcal{F}^{n}$, namely

$\text{H}(\mathbb{Q}\,|\,\mathbb{P})|_{\mathcal{F}^{n}}\coloneqq\begin{cases}\int_{\mathcal{C}}\log\frac{d\mathbb{Q}}{d\mathbb{P}}|_{\mathcal{F}^{n}}\,d\mathbb{Q}&\text{if
$\,\mathbb{Q}|_{\mathcal{F}^{n}}\ll\mathbb{P}|_{\mathcal{F}^{n}}$}\\\
+\infty&\text{otherwise}.\end{cases}$

Similarly the lower specific relative entropy of $\mathbb{Q}$ with respect to
$\mathbb{P}$ is defined as

$h^{\ell}\left(\mathbb{Q}\,|\,\mathbb{P}\right)\coloneqq\liminf_{n\to\infty}\frac{1}{n}\text{H}(\mathbb{Q}\,|\,\mathbb{P})|_{\mathcal{F}^{n}}.$

Finally, if
$h^{\ell}(\mathbb{Q}\,|\,\mathbb{P})=h^{u}(\mathbb{Q}\,|\,\mathbb{P})$, we
denote this common value by $h(\mathbb{Q}\,|\,\mathbb{P})$ and call it the
specific relative entropy of $\mathbb{Q}$ with respect to $\mathbb{P}$.

######  Remark 1.

  1. 1.

The specific relative entropy is meaningful even in situations where measures
singular to each other are being compared. This is the case of continuous
martingale laws, which typically have infinite relative entropy but may still
have a finite specific relative entropy. In [5, Kapitel II.4] it was shown
that $h(\cdot|\mathbb{W})$, with $\mathbb{W}$ being Wiener measure, is the
rate function in a large deviations principle associated to a randomized
Donsker-type approximation of Brownian motion.

  2. 2.

We took the liberty to introduce the terminology of “upper/lower specific
relative entropies”. In fact in [3] already $h^{\ell}$ is called specific
relative entropy, whereas [5] restricts itself to the case when $h^{\ell}$ and
$h^{u}$ coincide. The works [3, 2] considers rather the subsequence of dyadic
sigma-algebras $\mathcal{F}^{2^{n}}$ in the definitions of specific relative
entropy.

  3. 3.

Both [3, 2] and [5] mostly study the specific relative entropy for
$\mathbb{Q},\mathbb{P}$ being general martingale laws, or even more general
stochastic processes than martingales. In the present work we will specialize
the discussion to $\mathbb{Q},\mathbb{P}$ being the laws of very well-behaved
_martingale diffusions_.

Let $B=(B_{t})_{t\in[0,1]}$ be a one-dimensional Brownian motion on some
filtered probability space
$(\Omega,\mathcal{S},(\mathcal{S}_{t})_{t\in[0,1]},\mathbb{S})$. Let
$M^{x}=(M^{x}_{t})_{t\in[0,1]}$ be the solution of

$\begin{cases}dM^{x}_{t}=\sigma(M^{x}_{t})\,dB_{t}\\\ M^{x}_{0}=x,\end{cases}$

and likewise let $N^{x}=(N^{x}_{t})_{t\in[0,1]}$ be the solution of

$\begin{cases}dN^{x}_{t}=\eta(N^{x}_{t})\,dB_{t}\\\ N^{x}_{0}=x.\end{cases}$

Throughout this article we assume rather strong regularity assumptions on the
diffusion coefficients $\sigma,\eta$. They will guarantee a unique strong
solution to the above stochastic differential equations in particular, but
more importantly, they will allow us to gauge the small-time behaviour of
$M^{x}$ and $N^{x}$. The precise assumptions are:

######  Assumption 1.

The coefficients $\sigma,\eta\colon\mathbb{R}\to\mathbb{R}_{+}$ are twice
continuously differentiable and such that

$\sigma(\cdot),\eta(\cdot)\in(\delta,{1}/{\delta}),$

for some $0<\delta\leq 1$. Moreover, for some $L\in\mathbb{R}$ we have

$\max\\{\lVert\sigma^{\prime}\rVert_{\infty},\lVert\sigma^{\prime\prime}\rVert_{\infty},\lVert\eta^{\prime}\rVert_{\infty},\lVert\eta^{\prime\prime}\rVert_{\infty}\\}<L.$

The main result of this note establishes the existence of the specific
relative entropy between the laws of $M^{x}$ and $N^{x}$, and provides a
closed form expression for this quantity:

######  Theorem 1.

Under Assumption 1, let $M^{x}$ and $N^{x}$ be as above and call
$\mathbb{Q}^{x}$ and ${\mathbb{P}^{x}}$ their respective laws in
$(\mathcal{C},\mathcal{F})$. Then

  * (i)

the specific relative entropy of $\mathbb{Q}^{x}$ with respect to
$\mathbb{P}^{x}$ exists, and

  * (ii)

it has the closed form

$h(\mathbb{Q}^{x}\,|\,{\mathbb{P}^{x}})=\frac{1}{2}\mathbb{E}\left[\int_{0}^{1}\left\\{\frac{\sigma(M^{x}_{s})^{2}}{\eta(M^{x}_{s})^{2}}-1-\log\frac{\sigma(M^{x}_{s})^{2}}{\eta(M^{x}_{s})^{2}}\right\\}\,ds\right].$
(1)

Recalling that $X$ stood for the canonical process, Formula (1) becomes

$h(\mathbb{Q}^{x}\,|\,{\mathbb{P}^{x}})=\frac{1}{2}\mathbb{E}_{\mathbb{Q}^{x}}\left[\int_{0}^{1}\left\\{\frac{\sigma(X_{s})^{2}}{\eta(X_{s})^{2}}-1-\log\frac{\sigma(X_{s})^{2}}{\eta(X_{s})^{2}}\right\\}ds\right].$

A formula of this very kind had been conjectured in [5, Kapitel I.1] in the
case that $\eta\equiv 1$. Specifically, calling $\mathbb{W}$ the Wiener
measure and $\mathbb{Q}$ a martingale law (w.l.o.g. both started at zero), the
conjecture asserted the equality

$h(\mathbb{Q}\,|\,{\mathbb{W}})=\frac{1}{2}\mathbb{E}_{\mathbb{Q}}\left[\int_{0}^{1}\left\\{\sigma(X_{s})^{2}-1-\log\sigma(X_{s})^{2}\right\\}ds\right].$

In fact, it was already established in [5, 3, 2] that the r.h.s. here is
always the smaller one, and this fact has already found an application in [3,
2] were a novel transport-information inequality was derived. Yet to the best
of our knowledge the conjecture (i.e. the equality of left- and right-hand
sides) had only been confirmed for Gaussian martingales, i.e. when $\sigma$ is
deterministic (that is to say, not a function of space). Our contribution is
hence to provide a first corroboration of this conjecture beyond the Gaussian
case. Furthermore we also allowed for $\eta$ being non-constant. This seems
relevant to us, as one could be possibly interested in other reference
martingales than Brownian motion. The proof of Formula (1) is achieved by
careful expansion of the transition densities of the involved martingales.
This goes some way into explaining why we restrict ourselves to martingale
diffusions with fairly regular coefficients.

As a corollary to Theorem 1 we are able to obtain an interpretation of the
specific relative entropy as an almost sure limit. This is the content of our
second main result:

######  Corollary 1.

Under Assumption 1, let $M^{x}$ and $N^{x}$ be as before and call
$\mathbb{Q}^{x}$ and ${\mathbb{P}^{x}}$ their respective laws in
$(\mathcal{C},\mathcal{F})$. Let $\\{M^{x,n}\\}_{n\in\mathbb{N}}$ be iid
samples from $\mathbb{Q}^{x}$. Then we have the almost sure convergence

$\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{k\leq
n}\frac{1}{k}\log\left(\frac{d\mathbb{Q}^{x}}{d\mathbb{P}^{x}}\Big{|}_{\mathcal{F}^{k}}\Big{.}\big{(}M^{x,k}_{\frac{0}{k}},M^{x,k}_{\frac{1}{k}},\dots,M^{x,k}_{\frac{k}{k}}\big{)}\right)=h(\mathbb{Q}^{x}\,|\,{\mathbb{P}^{x}}).$
(2)

##  2 Proof of Theorem 1

The proof of Theorem 1 relies on the following rather precise estimates on the
transition density function of martingale diffusions as above. We believe that
the content of Lemma 1 is known to experts, as the study of asymptotic
expansions for the heat kernel is a classical subject, see e.g. [6, 1]. On the
other hand Lemma 1 is non-asymptotic and of global nature, which we find
useful. For the sake of completeness, we provide our own proof. In the
following we denote

$d_{\sigma}(a,b):=\left|\int_{a}^{b}\frac{du}{\sigma(u)}\right|.$

It can be seen that $d_{\sigma}(a,b)$ is the geodesic distance induced by
$\sigma$ (more precisely, by $\sigma^{2}$) on $\mathbb{R}$.

######  Lemma 1.

Let $\sigma$ and $M^{x}$ be as above and let $p(t,x,y)$ be the transition
density function of $M^{x}$ from $x$ at time $0$ to $y$ at time $t$. Under
Assumption 1 there are constants $C_{1}$ and $C_{2}$, depending only on
$\delta$ and $L$, such that, for all $t\in(0,1]$ and $x,y\in\mathbb{R}$:

$e^{-C_{2}t}\frac{1}{\sqrt{2\pi
t}}\sqrt{\frac{\sigma(x)}{\sigma(y)}}\frac{1}{\sigma(y)}e^{\left(-d_{\sigma}(x,y)^{2}/2t\right)}\leq
p(t,x,y)\leq e^{C_{1}t}\frac{1}{\sqrt{2\pi
t}}\sqrt{\frac{\sigma(x)}{\sigma(y)}}\frac{1}{\sigma(y)}e^{\left(-d_{\sigma}(x,y)^{2}/2t\right)}.$

###### Proof.

Let $g\colon\mathbb{R}\to\mathbb{R}$ be the strictly increasing and surjective
function defined by

$g(y)\coloneqq\int_{x}^{y}\frac{1}{\sigma(u)}\,du,$

and $g^{-1}$ its inverse. By Itô’s lemma we have

$g(M_{t}^{x})=B_{t}-\frac{1}{2}\int_{0}^{t}\sigma^{\prime}(M_{s}^{x})\,ds.$

Hence $Y_{t}\coloneqq g(M_{t}^{x})$ satisfies
$Y_{t}=B_{t}+\int_{0}^{t}b(Y_{s})ds$, where
$b\coloneqq-\frac{1}{2}\sigma^{\prime}\circ g^{-1}$.

For any $\Phi\geq 0$ measurable and bounded, we have

$\displaystyle\mathbb{E}\left[\Phi(M_{t}^{x})\right]$
$\displaystyle=\mathbb{E}\left[\Phi(g^{-1}(Y_{t}))\right]$
$\displaystyle=\mathbb{E}\left[\Phi(g^{-1}(B_{t}))\exp{\left(\int_{0}^{t}b(B_{s})\,dB_{s}-\frac{1}{2}\int_{0}^{t}b(B_{s})^{2}\,ds\right)}\right]$
$\displaystyle=\mathbb{E}\left[\Phi(g^{-1}(B_{t}))\exp{\left(\int_{0}^{B_{t}}b(u)\,du-\frac{1}{2}\int_{0}^{t}b^{\prime}(B_{s})\,ds-\frac{1}{2}\int_{0}^{t}b(B_{s})^{2}\,ds\right)}\right]$
(3)

where the second equality is due to Girsanov’s theorem (applicable as
$\sigma^{\prime}$ is bounded), whereas the last equality follows from

$\int_{0}^{B_{t}}b(u)\,du=\int_{0}^{t}b(B_{s})\,dB_{s}+\frac{1}{2}\int_{0}^{t}b^{\prime}(B_{s})\,ds,$

which is just Itô’s lemma applied to the function
$f(z)\coloneqq\int_{0}^{z}b(u)\,du$.

To get an upper bound for $\eqref{ErwartungswertPhi}$ note that,

$\displaystyle-b^{\prime}-b^{2}\leq-b^{\prime}=\frac{1}{2}(\sigma\cdot\sigma^{\prime\prime})\circ
g^{-1}\leq\frac{L}{2\delta}$ (4)

which implies,

$\exp{\left(-\frac{1}{2}\int_{0}^{t}b^{\prime}(B_{s})\,ds-\frac{1}{2}\int_{0}^{t}b(B_{s})^{2}\,ds\right)}\leq\exp{(t\frac{L}{4\delta})}.$

Inserting in $\eqref{ErwartungswertPhi}$ with
$C_{1}\coloneqq\frac{L}{4\delta}$ yields,

$\mathbb{E}\left[\Phi(M_{t}^{x})\right]\leq\exp{(tC_{1})}\mathbb{E}\left[\Phi(g^{-1}(B_{t}))\exp{\left(\int_{0}^{B_{t}}b(u)\,du\right)}\right].$
(5)

Note that

$b=-\frac{1}{2}\sigma^{\prime}\circ g^{-1}=-\frac{1}{2}\frac{(\sigma\circ
g^{-1})^{\prime}}{\sigma\circ g^{-1}}=-\frac{1}{2}(\log\circ\sigma\circ
g^{-1})^{\prime}$

and $g^{-1}(B_{0})=g^{-1}(0)=x$. Therefore

$\int_{0}^{B_{t}}b(u)\,du=-\frac{1}{2}\log\left(\frac{\sigma(g^{-1}(B_{t}))}{\sigma(g^{-1}(B_{0}))}\right)=\log\sqrt{\frac{\sigma(x)}{\sigma(g^{-1}(B_{t}))}}$

We can now rewrite the expectation on the right-hand side of (5) by using this
observation and then expressing it in terms of the density of $B_{t}$. The
desired representation then follows from a change of variables with
$y=g^{-1}(z)$, namely:

$\displaystyle\mathbb{E}\left[\Phi(g^{-1}(B_{t}))\exp{\left(\int_{0}^{B_{t}}b(u)\,du\right)}\right]$
$\displaystyle=\mathbb{E}\left[\Phi(g^{-1}(B_{t}))\sqrt{\frac{\sigma(x)}{\sigma(g^{-1}(B_{t}))}}\right]$
$\displaystyle=\int_{\mathbb{R}}\Phi(g^{-1}(z))\sqrt{\frac{\sigma(x)}{\sigma(g^{-1}(z))}}\frac{1}{\sqrt{2\pi
t}}\exp{\left(-\frac{z^{2}}{2t}\right)}\,dz$
$\displaystyle=\int_{\mathbb{R}}\Phi(y)\sqrt{\frac{\sigma(x)}{\sigma(y)}}\frac{1}{\sqrt{2\pi
t}}\exp{\left(-\frac{g(y)^{2}}{2t}\right)}g^{\prime}(y)\,dy$
$\displaystyle=\int_{\mathbb{R}}\Phi(y)\sqrt{\frac{\sigma(x)}{\sigma(y)}}\frac{1}{\sqrt{2\pi
t}}\exp{\left(-\frac{1}{2t}\left(\int_{x}^{y}\frac{1}{\sigma(u)}\,du\right)^{2}\right)}\frac{1}{\sigma(y)}\,dy.$

We conclude that

$\mathbb{E}\left[\Phi(M_{t}^{x})\right]\leq\exp{(tC_{1})}\int_{\mathbb{R}}\Phi(y)\sqrt{\frac{\sigma(x)}{\sigma(y)}}\frac{1}{\sqrt{2\pi
t}}\exp{\left(-\frac{1}{2t}d_{\sigma}(x,y)^{2}\right)}\frac{1}{\sigma(y)}\,dy.$

As this holds for all $\Phi\geq 0$ measurable and bounded, this proves the
desired upper bound.

The lower bound follows analogously by replacing (4) with

$-b^{\prime}-b^{2}=\frac{1}{2}(\sigma\sigma^{\prime\prime})\circ
g^{-1}-(\frac{1}{2}\sigma^{\prime}\circ
g^{-1})^{2}\geq-\left(\frac{L\delta}{2}+\frac{L^{2}}{4}\right).$

This implies

$\exp{\left(-\frac{1}{2}\int_{0}^{t}b^{\prime}(B_{s})\,ds-\frac{1}{2}\int_{0}^{t}b(B_{s})^{2}\,ds\right)}\geq\exp{(-tC_{2})},$

where $C_{2}\coloneqq\frac{L\delta}{4}+\frac{L^{2}}{8}$. ∎

We are now in a position to prove Theorem 1.

###### Proof of Theorem 1.

Let $q(t,x,y)$ and $p(t,x,y)$ be the transition density functions of $M^{x}$
and $N^{x}$ respectively. To ease notation we now drop the superscript $x$
from $M$ and $N$. Since $M$ and $N$ are Markov and time-homogeneous, we have
by the tensorization property of the relative entropy that

$\displaystyle\text{H}(\mathbb{Q}^{x}\,|\,{\mathbb{P}^{x}})|_{\mathcal{F}^{n}}$
$\displaystyle=\text{H}\left(\mathcal{L}(M_{0},M_{\frac{1}{n}},\dots,M_{1})\mid\mathcal{L}(N_{0},N_{\frac{1}{n}},\dots,N_{1})\right)$
$\displaystyle=\sum_{k=1}^{n}\int_{\mathbb{R}^{2}}\log\frac{d\mathcal{L}(M_{\frac{k}{n}}|M_{\frac{k-1}{n}}=x_{k-1})(x_{k})}{d\mathcal{L}(N_{\frac{k}{n}}|N_{\frac{k-1}{n}}=x_{k-1})(x_{k})}\,d\mathcal{L}(M_{\frac{k-1}{n}},M_{\frac{k}{n}})(x_{k-1},x_{k})$
$\displaystyle=\sum_{k=1}^{n}\int_{\mathbb{R}^{2}}\log\frac{q(\frac{1}{n},x_{k-1},x_{k})}{p(\frac{1}{n},x_{k-1},x_{k})}\,d\mathcal{L}(M_{\frac{k-1}{n}},M_{\frac{k}{n}})(x_{k-1},x_{k})$
$\displaystyle=\sum_{k=1}^{n}\mathbb{E}\left[\log\frac{q(\frac{1}{n},M_{\frac{k-1}{n}},M_{\frac{k}{n}})}{p(\frac{1}{n},M_{\frac{k-1}{n}},M_{\frac{k}{n}})}\right]$

By Lemma 1 for the upper bound of $q$ and the lower bound of $p$, we derive
the existence of a constant $C$ such that for all
$x,y\in\mathbb{R},t\in(0,1]$:

$\log\frac{q(t,x,y)}{p(t,x,y)}\leq
Ct+\frac{1}{2}\log\frac{\sigma(x)\eta(y)}{\sigma(y)\eta(x)}-\log\frac{\sigma(y)}{\eta(y)}-\frac{1}{2t}d_{\sigma}(x,y)^{2}+\frac{1}{2t}d_{\eta}(x,y)^{2}.$

Therefore, for all $n\in\mathbb{N}$ and any $k=1,\dots,n$:

$\displaystyle\mathbb{E}\left[\log\frac{q(\frac{1}{n},M_{\frac{k-1}{n}},M_{\frac{k}{n}})}{p(\frac{1}{n},M_{\frac{k-1}{n}},M_{\frac{k}{n}})}\right]$
$\displaystyle\leq\mathbb{E}\Bigg{[}\frac{C}{n}+\frac{1}{2}\log\frac{\sigma(M_{\frac{k-1}{n}})\eta(M_{\frac{k}{n}})}{\sigma(M_{\frac{k}{n}})\eta(M_{\frac{k-1}{n}})}-\log\frac{\sigma(M_{\frac{k}{n}})}{\eta(M_{\frac{k}{n}})}-\frac{n}{2}d_{\sigma}(M_{\frac{k-1}{n}},M_{\frac{k}{n}})^{2}+\frac{n}{2}d_{\eta}(M_{\frac{k-1}{n}},M_{\frac{k}{n}})^{2}\Bigg{]}.$
(6)

Summing over $k=1,\dots,n$ the
$\log\frac{\sigma(M_{\frac{k-1}{n}})\eta(M_{\frac{k}{n}})}{\sigma(M_{\frac{k}{n}})\eta(M_{\frac{k-1}{n}})}$
terms form a telescopic sum. So for all $n\in\mathbb{N}$:

$\displaystyle\frac{1}{n}\text{H}(\mathbb{Q}^{x}\,|\,{\mathbb{P}^{x}})|_{\mathcal{F}^{n}}$
$\displaystyle\leq\frac{1}{n}\sum_{k=1}^{n}\mathbb{E}\Bigg{[}\frac{C}{n}+\frac{1}{2}\log\frac{\sigma(M_{\frac{k-1}{n}})\eta(M_{\frac{k}{n}})}{\sigma(M_{\frac{k}{n}})\eta(M_{\frac{k-1}{n}})}-\log\frac{\sigma(M_{\frac{k}{n}})}{\eta(M_{\frac{k}{n}})}-\frac{n}{2}d_{\sigma}(M_{\frac{k-1}{n}},M_{\frac{k}{n}})^{2}+\frac{n}{2}d_{\eta}(M_{\frac{k-1}{n}},M_{\frac{k}{n}})^{2}\Bigg{]}$
$\displaystyle=\frac{C}{n}+\mathbb{E}\Bigg{[}\frac{1}{2n}\log\frac{\sigma(M_{0})\eta(M_{1})}{\sigma(M_{1})\eta(M_{0})}-\frac{1}{n}\sum_{k=1}^{n}\log\frac{\sigma(M_{\frac{k}{n}})}{\eta(M_{\frac{k}{n}})}-\frac{1}{2}\sum_{k=1}^{n}d_{\sigma}(M_{\frac{k-1}{n}},M_{\frac{k}{n}})^{2}+\frac{1}{2}\sum_{k=1}^{n}d_{\eta}(M_{\frac{k-1}{n}},M_{\frac{k}{n}})^{2}\Bigg{]}$
(7)

We now show that $\eqref{obereSchrankeRelEntr}$ converges to the right-hand
side of $\eqref{equformula}$. Firstly, since $\sigma$ and $\eta$ are bounded
away from $0$ and bounded from above, it is clear that,

$\lim_{n\to\infty}\mathbb{E}\left[\frac{1}{2n}\log\frac{\sigma(M_{0})\eta(M_{1})}{\sigma(M_{1})\eta(M_{0})}\right]=0$

Furthermore for almost every $\omega\in\Omega$ the map
$t\mapsto\log(\frac{\sigma(M_{t}(\omega))}{\eta(M_{t}(\omega))})$ is
continuous, so that

$\sum_{k=1}^{n}\log\frac{\sigma(M_{\frac{k}{n}}(\omega))}{\eta(M_{\frac{k}{n}}(\omega))}\mathbbm{1}_{\left(\frac{k-1}{n},\frac{k}{n}\right]}(t)\xrightarrow{n\to\infty}{}\log\frac{\sigma(M_{t}(\omega))}{\eta(M_{t}(\omega))}\qquad\text{Leb}\otimes\mathbb{P}-a.e.$

Since the left-hand side above is bounded uniformly in $n$, we have by
dominated convergence

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[\frac{1}{n}\sum_{k=1}^{n}\log\frac{\sigma(M_{\frac{k}{n}})}{\eta(M_{\frac{k}{n}})}\right]$
$\displaystyle=\lim_{n\to\infty}\mathbb{E}\left[\int_{0}^{1}\sum_{k=1}^{n}\log\frac{\sigma(M_{\frac{k}{n}})}{\eta(M_{\frac{k}{n}})}\mathbbm{1}_{\left(\frac{k-1}{n},\frac{k}{n}\right]}(s)\,ds\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{1}\log\frac{\sigma(M_{s})}{\eta(M_{s})}\,ds\right].$

Next, Itô’s lemma applied to the function
$F(z):=\int_{0}^{z}\frac{1}{\eta(u)}\,du$ yields,

$F(M_{t})=\int_{0}^{M_{t}}\frac{1}{\eta(u)}\,du=\int_{0}^{t}\frac{\sigma(M_{u})}{\eta(M_{u})}\,dB_{u}-\frac{1}{2}\int_{0}^{t}\frac{\eta^{\prime}(M_{u})}{\eta(M_{u})^{2}}\sigma(M_{u})^{2}\,du.$

Thus we have

$\displaystyle\mathbb{E}\left[\frac{1}{2}\sum_{k=1}^{n}d_{\eta}(M_{\frac{k-1}{n}},M_{\frac{k}{n}})^{2}\right]=\mathbb{E}\left[\frac{1}{2}\sum_{k=1}^{n}\left(\int_{M_{\frac{k-1}{n}}}^{M_{\frac{k}{n}}}\frac{1}{\eta(u)}\,du\right)^{2}\right]=\mathbb{E}\left[\frac{1}{2}\sum_{k=1}^{n}\left(F(M_{\frac{k}{n}})-F(M_{\frac{k-1}{n}})\right)^{2}\right].$

At this point one can directly deduce

$\lim_{n\to\infty}\mathbb{E}\left[\frac{1}{2}\sum_{k=1}^{n}d_{\eta}(M_{\frac{k-1}{n}},M_{\frac{k}{n}})^{2}\right]=\mathbb{E}\left[\int_{0}^{1}\frac{1}{2}\left(\frac{\sigma(M_{u})}{\eta(M_{u})}\right)^{2}\,du\right],$
(8)

from the fact that
$\mathbb{E}\left[\frac{1}{2}\sum_{k=1}^{n}\left(F(M_{\frac{k}{n}})-F(M_{\frac{k-1}{n}})\right)^{2}\right]\to\mathbb{E}[\langle
F(M)\rangle_{1}]$, as follows directly per properties of the quadratic
variation of Itô processes with bounded coefficients. To be self-contained we
provide now a detailed argument:

$\displaystyle\mathbb{E}\left[\frac{1}{2}\sum_{k=1}^{n}\left(F(M_{\frac{k}{n}})-F(M_{\frac{k-1}{n}})\right)^{2}\right]$
$\displaystyle=\frac{1}{2}\sum_{k=1}^{n}\mathbb{E}\Bigg{[}\left(\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\sigma(M_{u})}{\eta(M_{u})}\,dB_{u}\right)^{2}-\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\sigma(M_{u})}{\eta(M_{u})}\,dB_{u}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\eta^{\prime}(M_{u})}{\eta(M_{u})^{2}}\sigma(M_{u})^{2}\,du+$
$\displaystyle\hskip
256.0748pt\frac{1}{4}\left(\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\eta^{\prime}(M_{u})}{\eta(M_{u})^{2}}\sigma(M_{u})^{2}\,du\right)^{2}\Bigg{]}$
$\displaystyle=\mathbb{E}\Bigg{[}\frac{1}{2}\int_{0}^{1}\left(\frac{\sigma(M_{u})}{\eta(M_{u})}\right)^{2}\,du-\frac{1}{2}\sum_{k=1}^{n}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\sigma(M_{u})}{\eta(M_{u})}\,dB_{u}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\eta^{\prime}(M_{u})}{\eta(M_{u})^{2}}\sigma(M_{u})^{2}\,du+$
$\displaystyle\hskip
256.0748pt\frac{1}{2}\sum_{k=1}^{n}\frac{1}{4}\left(\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\eta^{\prime}(M_{u})}{\eta(M_{u})^{2}}\sigma(M_{u})^{2}\,du\right)^{2}\Bigg{]}.$

The expression above converges to
$\mathbb{E}\left[\frac{1}{2}\int_{0}^{1}\left(\frac{\sigma(M_{u})}{\eta(M_{u})}\right)^{2}\,du\right]$
as $n\to\infty$. Indeed, since $\lVert\eta^{\prime}\rVert_{\infty}<L$ and
$\sigma,\eta\in(\delta,\frac{1}{\delta})$, we have that,

$\displaystyle\mathbb{E}\left[\left\lvert\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\sigma(M_{u})}{\eta(M_{u})}\,dB_{u}\right\rvert\right]\leq\left(\mathbb{E}\left[\left(\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\sigma(M_{u})}{\eta(M_{u})}\,dB_{u}\right)^{2}\right]\right)^{\frac{1}{2}}=\left(\mathbb{E}\left[\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\sigma(M_{u})^{2}}{\eta(M_{u})^{2}}\,du\right]\right)^{\frac{1}{2}}\leq\sqrt{\frac{1}{n}}\frac{1}{\delta^{2}},$

and therefore

$\left|\sum_{k=1}^{n}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\sigma(M_{u})}{\eta(M_{u})}\,dB_{u}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\eta^{\prime}(M_{u})}{\eta(M_{u})^{2}}\sigma(M_{u})^{2}\,du\right|\leq\frac{L}{\delta^{4}}\frac{1}{n}\sum_{k=1}^{n}\left|\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\sigma(M_{u})}{\eta(M_{u})}\,dB_{u}\right|\xrightarrow{L^{1}}0\quad\text{as
$n\to\infty$}.$

Moreover,

$\sum_{k=1}^{n}\left(\int_{\frac{k-1}{n}}^{\frac{k}{n}}\frac{\eta^{\prime}(M_{u})}{\eta(M_{u})^{2}}\sigma(M_{u})^{2}\,du\right)^{2}\leq\sum_{k=1}^{n}\frac{L^{2}}{\delta^{8}}\frac{1}{n^{2}}=\frac{L^{2}}{\delta^{8}}\frac{1}{n}\xrightarrow{L^{1}}0\quad\text{as
$n\to\infty$}.$

All in all, we have established (8). Applying the same arguments with $\eta$
replaced by $\sigma$, shows

$\lim_{n\to\infty}\mathbb{E}\left[\frac{1}{2}\sum_{k=1}^{n}d_{\sigma}(M_{\frac{k-1}{n}},M_{\frac{k}{n}})^{2}\right]=\frac{1}{2}.$

Finally we conclude that

$h^{u}({\mathbb{Q}^{x}}|{\mathbb{P}^{x}}):=\limsup_{n\to\infty}\frac{1}{n}\text{H}(\mathbb{Q}^{x}\,|\,{\mathbb{P}^{x}})|_{\mathcal{F}^{n}}\leq\frac{1}{2}\mathbb{E}\left[\int_{0}^{1}\frac{\sigma(M_{s})^{2}}{\eta(M_{s})^{2}}-1-\log\frac{\sigma(M_{s})^{2}}{\eta(M_{s})^{2}}\,ds\right].$

Analogously, by the lower bound for $q$ and the upper bound for $p$ from Lemma
1, applied in $\eqref{inequ}$, we get:

$h^{\ell}({\mathbb{Q}^{x}}|{\mathbb{P}^{x}}):=\liminf_{n\to\infty}\frac{1}{n}\text{H}(\mathbb{Q}^{x}\,|\,{\mathbb{P}^{x}})|_{\mathcal{F}^{n}}\geq\frac{1}{2}\mathbb{E}\left[\int_{0}^{1}\frac{\sigma(M_{s})^{2}}{\eta(M_{s})^{2}}-1-\log\frac{\sigma(M_{s})^{2}}{\eta(M_{s})^{2}}\,ds\right],$

which finishes the proof. ∎

######  Remark 2.

We find the limit in (8), equivalently

$\lim_{n\to\infty}\mathbb{E}_{\mathbb{Q}^{x}}\left[\sum_{k=1}^{n}d_{\eta}(X_{\frac{k-1}{n}},X_{\frac{k}{n}})^{2}\right]=\mathbb{E}_{\mathbb{Q}^{x}}\left[\int_{0}^{1}\eta(X_{u})^{-2}d\langle
X\rangle_{u}\right],$

curious and worth highlighting. We suspect that there is a geometric
interpretation to this result, but we do not pursue this line of reasoning
here.

##  3 Proof of Corollary 1

In this part it will be convenient to apply a weaker version of Lemma 1,
namely:

######  Lemma 2.

Let $\sigma$, $M^{x}$ and $p(t,x,y)$ as in Lemma 1. Under Assumption 1 there
is a constant $C$, depending only on $\delta$ and $L$, such that, for all
$t\in(0,1]$ and $x,y\in\mathbb{R}$:

$e^{-Ct}\frac{\delta^{2}}{\sqrt{2\pi
t}}e^{\left(-(x-y)^{2}/2t\delta^{2}\right)}\leq p(t,x,y)\leq
e^{Ct}\frac{1}{\delta^{2}\sqrt{2\pi
t}}e^{\left(-\delta^{2}(x-y)^{2}/2t\right)}.$

This follows directly from Lemma 1 and Assumption 1 on $\sigma$, so we omit
the proof. We remark that at least the upper bound in Lemma 2 can be obtained
in far more generality with PDE methods (see Theorems 4.5 and 5.4 in [4,
Chapter 6]).

We put ourselves now in the setting of Corollary 1, and introduce some
notation here: If $\mathbb{Q}$ is a probability measure on $\mathcal{C}$ and
if $\mathbb{Q}|_{\mathcal{F}^{n}}$ possesses a density with respect to
$n$-dimensional Lebesgue measure111To be fully precise: the first marginal of
$\mathbb{Q}|_{\mathcal{F}^{n}}$ is a Dirac measure. But as all martingales we
consider are started at a fixed $x$ we may “drop” the first marginal and
consider the density of the remaining arguments w.r.t. Lebesgue., we denote
this density by the lower-case symbol $q_{n}$. If $X\in\mathcal{C}$ we denote
$X_{(n)}:=(X_{i/n})_{i=0}^{n}$. Written in this terms, our aim is to establish
the almost sure convergence

$\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{k\leq
n}\frac{1}{k}\log\left(\frac{q^{x}_{k}}{p^{x}_{k}}(M^{x,k}_{(k)})\right)=h(\mathbb{Q}^{x}\,|\,{\mathbb{P}^{x}}).$
(9)

We now omit the $x$-dependence everywhere, in order to keep notation at bay.

###### Proof of Corollary 1.

We will use the following Kolmogorov’s version of the strong law of large
numbers (KSLLN) for independent non-identically distributed sequences: If
$(Y_{k})_{k}$ are independent, with finite second moments, and such that
$\sum_{k}\frac{1}{k^{2}}\text{Var}(Y_{k})<\infty$, then
$\frac{1}{n}\sum_{k\leq n}(Y_{k}-\mathbb{E}[Y_{k}])\to 0$ almost surely as
$n\to\infty$. This can be found in [7, Theorem 2.3.10]. For us

$Y_{k}:=\frac{1}{k}\log\left(\frac{q_{k}}{p_{k}}(M^{k}_{(k)})\right),$

so by definition the $(Y_{k})_{k}$ are independent and
$\mathbb{E}[Y_{k}]=\frac{1}{k}\text{H}(\mathbb{Q}\,|\,\mathbb{P})|_{\mathcal{F}^{k}}$.
Per Theorem 1 we know
$\frac{1}{k}\text{H}(\mathbb{Q}\,|\,\mathbb{P})|_{\mathcal{F}^{k}}\to
h(\mathbb{Q}\,|\,{\mathbb{P}})$, hence $\frac{1}{n}\sum_{k\leq
n}\mathbb{E}[Y_{k}]\to h(\mathbb{Q}\,|\,{\mathbb{P}})$. Thus if we can apply
KSLLN then we will have derived (9) as desired. Thus it remains to verify that
$\sum_{k}\frac{1}{k^{2}}\text{Var}(Y_{k})<\infty$, but as
$\sum_{k}\frac{1}{k^{2}}\mathbb{E}[Y_{k}]^{2}<\infty$ in our case, it is
enough to verify

$\displaystyle\sum_{k}\frac{1}{k^{2}}\mathbb{E}[Y_{k}^{2}]<\infty.$ (10)

To this end remark that
$Y_{k}^{2}\leq\frac{2}{k^{2}}\left\\{\log(q_{k}(M^{k}_{(k)}))^{2}+\log(p_{k}(M^{k}_{(k)}))^{2}\right\\}$.
We now establish

$\displaystyle\sum_{k}\frac{1}{k^{4}}\mathbb{E}[\log(\ell_{k}(M_{(k)}))^{2}]<\infty,$
(11)

where we let the symbol $\ell$ stand for either $p$ or $q$, implying then
(10). Applying the Markov property we have

$\ell_{k}(M_{(k)})=\prod_{i=1}^{k}\,\ell(1/k,M_{(i-1)/k},M_{i/k}),$

so taking logarithms, bounding the squared sum by a sum of squares, and
applying Lemma 2, we obtain

$\displaystyle\log(\ell_{k}(M_{(k)}))^{2}$ $\displaystyle\leq 4k\sum_{i\leq
k}\left\\{\frac{C^{2}}{k^{2}}+\log(\delta^{2})^{2}+\frac{1}{4}\log(k/2\pi)^{2}+\frac{k^{2}}{4\delta^{4}}(M_{(i-1)/k}-M_{i/k})^{4}\right\\}$
$\displaystyle=4C^{2}+4k^{2}\log(\delta^{2})^{2}+k^{2}\log(k/2\pi)^{2}+\frac{k^{3}}{\delta^{4}}\sum_{i\leq
k}(M_{(i-1)/k}-M_{i/k})^{4}.$

To check (11), first observe
$\sum_{k}(4C^{2}+4k^{2}\log(\delta^{2})^{2}+k^{2}\log(k/2\pi)^{2})/k^{4}<\infty$,
as $\log(x)\leq x^{r}/r$ for all $x\geq 1$ and $r\in(0,1)$. Second, by the BDG
inequality, we have

$\mathbb{E}[(M_{(i-1)/k}-M_{i/k})^{4}]\leq
c\mathbb{E}\left[\left(\int_{(i-1)/k}^{i/k}\sigma(M_{t})^{2}\,dt\right)^{2}\right]\leq\frac{c}{\delta^{4}k^{2}},$

so $\sum_{i\leq
k}\mathbb{E}[(M_{(i-1)/k}-M_{i/k})^{4}]\leq\frac{c}{\delta^{4}k}$ and
$\sum_{k}\frac{1}{k^{4}}\sum_{i\leq
k}\frac{k^{3}}{\delta^{4}}\mathbb{E}[(M_{(i-1)/k}-M_{i/k})^{4}]\leq
c^{\prime}\sum_{k}\frac{1}{k^{2}}<\infty$. ∎

## References

  * [1] Robert Azencott, Paolo Baldi, André Bellaïche, Catherine Bellaïche, Philippe Bougerol, Mireille Chaleyat-Maurel, Laure Elie, and Jacques Granara.  Géodésiques et diffusions en temps petit.  Astérisque, 84, 1981.
  * [2] Hans Föllmer.  Doob decomposition, Dirichlet processes and entropies on Wiener space.  ArXiv preprint 2022.
  * [3] Hans Föllmer.  Optimal couplings on wiener space and an extension of talagrand’s transport inequality.  In Stochastic Analysis, Filtering, and Stochastic Optimization, pages 147–175. Springer, 2022.
  * [4] Avner Friedman.  Stochastic differential equations and applications.  In Stochastic differential equations, pages 75–148. Springer, 2010\.
  * [5] Nina Gantert.  Einige große Abweichungen der Brownschen Bewegung.  Number 224. Universität Bonn, 1991.
  * [6] Stanislav A Molchanov.  Diffusion processes and riemannian geometry.  Russian Mathematical Surveys, 30(1):1, 1975.
  * [7] Pranab K Sen and Julio M Singer.  Large sample methods in statistics: an introduction with applications, volume 25.  CRC press, 1994.


# Soft theorems in curved spacetime

Peng Cheng and Pujian Mao

Soft theorems in curved spacetime

Peng Chenga,b and Pujian Maoa

a Center for Joint Quantum Studies and Department of Physics,

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

b Lanzhou Center for Theoretical Physics,

Key Laboratory of Theoretical Physics of Gansu Province,

Lanzhou University, 222 South Tianshui Road, Lanzhou 730000, Gansu, China

Abstract. In this paper, we derive a soft photon theorem in the near horizon
region of the Schwarzschild black hole from the Ward identity of the near
horizon large gauge transformation. The flat spacetime soft photon theorem can
be recovered as a limiting case of the curved spacetime. The soft photons on
the horizon are indeed soft electric hairs. This accomplishes the triangle
equivalence on the black hole horizon.

##  1 Introduction

It has recently been shown that black holes can carry soft hairs [1] which
provides a possible resolution of the black hole information paradox, see,
e.g., [2] for a review and [3, 4] for more recent developments. The discovery
of the black hole soft hair is originally inspired by a fascinating
equivalence relation in the infrared sector of gravity, where the BMS
supertranslation Ward identity is equivalent to Weinberg’s soft graviton
theorem [5, 6]. In the language of S-matrix, an alternative interpretation of
such equivalence relation is that the soft graviton is generated by the action
of a supertranslation on the quantum state. If one implements the
supertranslation action on an asymptotically flat black hole, then soft
gravitons can be created on the horizon. This is the original proposal for
soft hairs on black holes [1].

Though supertranslations are well defined in any asymptotically flat
spacetime, the soft theorem is highly relying on a flat spacetime background,
where the scattering amplitudes are defined. In curved spacetime, for instance
the near horizon region of a stationary black hole, it is completely not clear
if a soft graviton theorem exists. Then it is questionable if a
supertranslation action can really create soft gravitons on black hole
horizons. The aim of the present work is to explore soft theorems in curved
spacetime from the near horizon perspective and to address their connections
to soft hairs.

In this paper, we derive a soft photon theorem from the Ward identity of the
near horizon large gauge transformation. In [7, 8], it was argued that the
asymptotic symmetries of gauge theories are symmetries of the quantum
S-matrix. We assume that this statement should be valid for any boundary of
the spacetime. Considering the horizon as the inner boundary of the
Schwarzschild black hole, the near horizon symmetries should be symmetries of
S-matrix in the curved spacetime, at least in the near horizon region. We
extend the connection of soft photon theorems and asymptotic symmetries in [8]
to the near horizon region of the Schwarzschild black hole with two crucial
adaptations. The first one is that we need to deal with the dispersion
relation of the wave vector in curved spacetime in the mode expansion of the
gauge fields. The second one is that the stationary phase approximation at
null infinity (see, e.g., [2] for details) can not be applied in the near
horizon region. We use the Feynman prescription to regularize the divergence
from the near horizon limit. We first write the mode expansion in the
isotropic coordinates. Then the integration of the momentum in the mode
expansion can be performed by the usual angular variables. Applying the
Feynman prescription for regularization, we obtain the action of the charge
associated to the near horizon large gauge transformation on the state without
using the stationary phase approximation. Hence the soft photon theorem in
position space can be derived from the Ward identity of the near horizon large
gauge transformation. We transform the soft factor into momentum space and
discuss its relation to the one in flat spacetime. We further show that the
near horizon soft photons have precisely the forms of soft electric hairs.
This is a direct proof of the statement in [1] that soft particles implant
soft hairs on black hole horizons. Moreover, the near horizon soft photon
theorem, if it is verified from amplitudes side in curved spacetime, would
accomplish the triangle relation at the horizon.

##  2 Near horizon Maxwell theory

We consider Maxwell theory coupled to a conserved matter current in the near
horizon region of the Schwarzschild black hole. We adopt the retarded
coordinates $(u,r,z,\bar{z})$, where $(z,\bar{z})$ are the complex
stereographic coordinates. The line element of the Schwarzschild black hole is

$\text{d}s^{2}=-f(r)\text{d}u^{2}-2\text{d}u\text{d}r+2\Omega^{2}\gamma_{z\bar{z}}\text{d}z\text{d}\bar{z}\
,$ (1)

where

$\gamma_{z\bar{z}}=\frac{2}{(1+z\bar{z})^{2}}\ ,\quad
f(r)=\frac{r}{r+2M},\quad\Omega=(r+2M).$

The horizon ${\cal H}$ is precisely the submanifold $r=0$ in the retarded
spherical coordinates, with topology $S^{2}\times\mathbb{R}$.

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

The retarded coordinates can only cover half of the horizon, i.e., ${\cal
H}^{-}$ as demonstrated in Figure 1. The other half ${\cal H}^{+}$ can be
described by advanced coordinates $(v,r,z,\bar{z})$. The crossing point of
${\cal H}^{-}$ and ${\cal H}^{+}$ is the bifurcation point $B$. In this work,
we will only pay attention to the ${\cal H}^{-}$ part, while everything can be
similarly repeated on ${\cal H}^{+}$. The near horizon symmetries on ${\cal
H}^{-}$ and ${\cal H}^{+}$ can be connected by the near bifurcation symmetries
[9].

We consider a gauge field $A_{\mu}$ and a conserved matter current $J_{\mu}$
with the radial gauge and the near horizon fall-off conditions as follows

$\begin{split}&A_{r}=0,\quad A_{u}={\cal O}(r),\quad A_{z}={\cal O}(1),\\\
&J_{r}=0,\quad J_{u}={\cal O}(1),\quad J_{z}={\cal O}(1).\end{split}$ (2)

The radial component of the current is set to zero by the ambiguities of a
conserved current.111When a conserved current is derived from a global
symmetry, it is naturally defined up to the equivalence $J^{\mu}\thicksim
J^{\mu}+\nabla_{\nu}k^{[\mu\nu]}$, so it makes more sense to consider
equivalence classes of currents $[J^{\mu}]$ (see [10, 11] for details). This
is a more consistent choice of working in the radial gauge. We assume the
following ansatz as initial data for the near horizon expansion of the gauge
fields

$A_{z(\bar{z})}=A^{0}_{z(\bar{z})}(u,z,\bar{z})+\sum\limits_{m=1}^{\infty}A^{m}_{z(\bar{z})}(u,z,\bar{z})r^{m}\
,$ (3)

and the components of the current

$J_{z(\bar{z})}=J^{0}_{z(\bar{z})}(u,z,\bar{z})+\sum\limits_{m=1}^{\infty}J^{m}_{z(\bar{z})}(u,z,\bar{z})r^{m}\
.$ (4)

From the Maxwell’s equations, $\nabla_{\mu}F^{\mu\nu}=J^{\nu}$, and the
current conservation condition, $\nabla_{\mu}J^{\mu}=0$, in an $r$-expansion,
we obtain

$\displaystyle
J_{u}=\frac{J^{0}_{u}(u,z,\bar{z})}{\Omega^{2}}+\frac{1}{\Omega^{2}}\int_{0}^{r}\text{d}r^{\prime}\left[\gamma_{z\bar{z}}^{-1}(\partial_{z}J_{\bar{z}}+\partial_{\bar{z}}J_{z})\right]\
,$ (5) $\displaystyle
A_{u}=\frac{A^{0}_{u}(u,z,\bar{z})}{2M}-\frac{A^{0}_{u}(u,z,\bar{z})}{\Omega}$
$\displaystyle+\int_{0}^{r}\text{d}r^{\prime}\,\frac{1}{\Omega^{2}}\int_{0}^{r^{\prime}}\text{d}r^{\prime\prime}\left[\gamma_{z\bar{z}}^{-1}(\partial_{z}\partial_{r^{\prime\prime}}A_{\bar{z}}+\partial_{\bar{z}}\partial_{r^{\prime\prime}}A_{z})\right]\
,$ (6)
$\displaystyle\partial_{u}A^{0}_{u}=-\gamma_{z\bar{z}}^{-1}\partial_{u}(\partial_{z}A^{0}_{\bar{z}}+\partial_{\bar{z}}A^{0}_{z})-J^{0}_{u}\
,$ (7) $\displaystyle
2\partial_{u}\partial_{r}A_{z}=\partial_{z}\partial_{r}A_{u}+\partial_{r}(f\partial_{r}A_{z})+\partial_{z}\left[\frac{1}{\Omega\gamma_{z\bar{z}}}(\partial_{\bar{z}}A_{z}-\partial_{z}A_{\bar{z}})\right]-J_{z}\
,$ (8)

where $J^{0}_{u}(u,z,\bar{z})$ and $A^{0}_{u}(u,z,\bar{z})$ are integration
constants. We have omitted one equation that can be obtained from the last one
above by simply swapping $z\rightleftharpoons\bar{z}$. The time evolution of
the coefficients in the expansion of $A_{z(\bar{z})}$ are uniquely determined
except the leading $A^{0}_{z(\bar{z})}$. We refer to
$\partial_{u}A^{0}_{z(\bar{z})}$ as the electromagnetic “news” since they
reflect the propagation of electromagnetic waves.

The gauge and fall-off conditions in (2) leave residual gauge transformations
of the form $\delta A_{\mu}=\partial_{\mu}\varepsilon(z,\bar{z})$, where
$\varepsilon(z,\bar{z})$ is an arbitrary function on the horizon. We define
the associated charge at the bifurcation point on the horizon [10],

$\begin{split}Q_{\varepsilon}&=\int_{B}\text{d}z\text{d}\bar{z}\,\gamma_{z\bar{z}}\,\varepsilon(z,\bar{z})\,\Omega^{2}\,F_{ru}\\\
&=\int_{{\cal
H}^{-}}\text{d}z\text{d}\bar{z}\text{d}u\,\gamma_{z\bar{z}}\,\varepsilon(z,\bar{z})\,\Omega^{2}\,\partial_{u}F_{ru},\end{split}$
(9)

where in the second equality we assumed $A^{0}_{u}|_{i^{-}}=0$, meaning that
in the far past the system contains no bulk electric charge. Inserting the
solution (7) into the charge yields

$Q_{\varepsilon}=-\int_{{\cal
H}^{-}}\text{d}z\text{d}\bar{z}\text{d}u\,\varepsilon(z,\bar{z})\left(\gamma_{z\bar{z}}J_{u}^{0}+\partial_{u}\partial_{z}A_{\bar{z}}^{0}+\partial_{u}\partial_{\bar{z}}A_{z}^{0}\right)\
.$ (10)

##  3 Near horizon soft photon theorem

If a symmetry is generated by a charge $Q$, then the associated Ward identity
in the S-Matrix language reads as

$\langle\rm{out}|Q^{\textrm{out}}-Q^{\textrm{in}}|\rm{in}\rangle=0\ .$ (11)

For a spontaneously broken symmetry, the charge must act non-linearly on the
states. Otherwise it would annihilate the vacuum. Hence one can decompose the
charge into linear and non-linear pieces $Q=Q_{\rm{L}}+Q_{\rm{NL}}$. The Ward
identity for a broken charge can be written as

$\langle\rm{out}|Q_{\rm{NL}}^{\textrm{out}}-Q^{\textrm{in}}_{\rm{NL}}|\rm{in}\rangle=-\langle\rm{out}|Q^{\textrm{out}}_{\rm{L}}-Q^{\textrm{in}}_{\rm{L}}|\rm{in}\rangle\,.$
(12)

The analysis in the previous section is carried out only near ${\cal H}^{-}$.
We assume that all in states in the S-matrix are defined on ${\cal H}^{-}$.
Particles are created on ${\cal H}^{-}$ then propagated to the bulk and
finally ended on ${\cal H}^{+}$. In this sense, ${\cal H}^{-}$ and ${\cal
H}^{+}$ are very similar to the past and future null infinity $I^{-}$ and
$I^{+}$. We will only paid attention to the in part in the present work. The
analysis of the out part can be carried out analogously up to an
identification at the bifurcation point. For notational brevity, we suppress
the in label in the following text.

Let us now decompose the charge (10) into a piece containing the “news” and
the rest containing the sources222We drop the total minus sign in the charge
as it won’t affect the Ward identity.

$\displaystyle Q_{\textrm{NL}}=\int_{{\cal
H}^{-}}\text{d}z\text{d}\bar{z}\text{d}u\,\varepsilon\,\partial_{u}\partial_{\bar{z}}A^{0}_{z}\
,$ (13) $\displaystyle Q_{\textrm{L}}=\frac{1}{2}\int_{{\cal
H}^{-}}\text{d}z\text{d}\bar{z}\text{d}u\,\gamma_{z\bar{z}}\,\varepsilon\,J_{u}^{0}\
.$ (14)

These are, respectively, the non-linear and linear pieces above. We keep only
the anti-holomorphic terms for the “news” part. The sources are split into two
parts. That is where the half factor comes in (14). Such split is consistent
with the treatment in [8] where the authors consider only the sector of the
phase space with no long-range magnetic fields. In our case, such conditions
lead to $F_{z\bar{z}}=0$ at $B$ and $i^{0}$. Correspondingly, the helicity of
soft photon is decoupled. Hence one can only deal with either the holomorphic
part or the anti-holomorphic part.

The soft photon theorem is obtained from the Ward identity by a concrete
choice of $\varepsilon(z,\bar{z})$ at null infinity [8]. We follow the same
choice for the near horizon case,

$\varepsilon(z,\bar{z})=\frac{1}{w-z}\ ,$ (15)

for which $\partial_{\bar{z}}\varepsilon=-2\pi\delta^{2}(z-w)\,$.

###  3.1 Mode expansion in Schwarzschild spacetime

The mode expansion of fields in curved spacetime needs to be adapted to the
dispersion relation of the momentum of a moving particle in curved spacetime.
It is more convenient to work in the isotropic coordinates system $(t,x_{i})$
where the line element of the Schwarzschild black hole is

$\text{d}s^{2}=-\left(\frac{2\rho-M}{2\rho+M}\right)^{2}\text{d}t^{2}+\frac{(2\rho+M)^{4}}{16\rho^{4}}d\vec{x}^{2}\,.$
(16)

The isotropic coordinates are connected to the retarded coordinates in (1) by

$t=u+r+2M+2M\ln\frac{r}{2M},\quad(1+\frac{M}{2\rho})^{2}\rho=r+2M,$ (17)

and

$x^{1}=\rho\frac{(z+\bar{z})}{1+z\bar{z}},\quad
x^{2}=\rho\frac{-i(z-\bar{z})}{1+z\bar{z}},\quad
x^{3}=\rho\frac{1-z\bar{z}}{1+z\bar{z}}.$ (18)

The Schwarzschild solution (16) has one timelike Killing vector
$\frac{\partial}{\partial t}$. One can define the corresponding conserved
energy as $\omega=-p_{0}$. Thus, the dispersion relation for the massless
particle moving along a null trajectory is

$-\left(\frac{2\rho+M}{2\rho-M}\right)^{2}\omega^{2}+\frac{16\rho^{4}}{(2\rho+M)^{4}}{\vec{p}~{}}^{2}=0,$
(19)

where $\vec{p}$ is the three momenta of the conformally flat part.

For a free massless scalar field $\Phi(x)$, we can write the field operator in
isotropic coordinates as [12]

$\Phi(x^{\mu})=-\frac{1}{(2\pi)^{4}}\int\text{d}\omega\text{d}^{3}\vec{p}~{}\tilde{\phi}(p_{\mu})~{}e^{ip\cdot
x}.$ (20)

Inserting the dispersion relation as a delta function in the integral and
working out the integration of $\omega$

$-\int\text{d}\omega~{}2\pi\delta\left[-\left(\frac{2\rho+M}{2\rho-M}\right)^{2}\omega^{2}+\frac{16\rho^{4}}{(2\rho+M)^{4}}{\vec{p}~{}}^{2}\right]=\left(\frac{2\rho-M}{2\rho+M}\right)^{2}\frac{1}{2\omega},$
(21)

the expansion (20) can be further written as

$\displaystyle\Phi(x^{\mu})$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\left(\frac{2\rho-M}{2\rho+M}\right)^{2}\int\frac{\text{d}^{3}\vec{p}}{2\omega}~{}\tilde{\phi}(p_{\mu})e^{ip\cdot
x}\Big{|}_{-p_{0}=\omega}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\left(\frac{2\rho-M}{2\rho+M}\right)^{2}\int~{}\frac{\text{d}^{3}\vec{p}}{2\omega}~{}\left[\mathfrak{a}(p_{\mu})e^{ip\cdot
x}+\mathfrak{a}^{\dagger}(p_{\mu})e^{-ip\cdot x}\right]\Big{|}_{\omega>0}.$

In the flat spacetime case one only has $\frac{1}{2\omega}$ in the mode
expansion. Now we have an extra factor
$\frac{({2\rho-M})^{2}}{({2\rho+M})^{2}}$ because of the dispersion relation
(19).

For a $U(1)$ gauge field $A_{\mu}$, the mode expansion will involve two
polarization vectors orthogonal to the propagating direction. The mode
expansion of $A_{\mu}$, with respect to the dispersion relation (19), is

$A_{\mu}(x)=\sum_{\alpha=\pm}\frac{1}{(2\pi)^{3}}\left(\frac{2\rho-M}{2\rho+M}\right)^{2}\int~{}\frac{\text{d}^{3}\vec{p}}{2\omega}~{}\left[\epsilon_{\mu}^{*\alpha}\mathfrak{a}_{\alpha}(p)e^{ip\cdot
x}+\epsilon_{\mu}^{\alpha}\mathfrak{a}_{\alpha}^{\dagger}(p)e^{-ip\cdot
x}\right].$ (22)

The polarization vectors satisfy the normalization condition
$\epsilon_{\alpha}^{\mu}\epsilon^{*}_{\beta\mu}=\delta_{\alpha\beta}$. It is
of convenience to parametrize the photon four-momenta by

$p_{\mu}=\frac{\omega}{1+z\bar{z}}\frac{(2\rho+M)^{3}}{4\rho^{2}(2\rho-M)}\left(\frac{4\rho^{2}(M-2\rho)}{(2\rho+M)^{3}}(1+z\bar{z}),(z+\bar{z}),-i(z-\bar{z}),(1-z\bar{z})\right).$
(23)

The polarization vector can be written as

$\begin{split}\epsilon^{+\mu}=&\frac{1}{\sqrt{2}}\frac{4\rho^{2}}{(2\rho+M)^{2}}\left(\frac{(2\rho+M)^{3}}{4\rho^{2}(2\rho-M)}\bar{z},1,-i,-\bar{z}\right),\\\
\epsilon^{-\mu}=&\frac{1}{\sqrt{2}}\frac{4\rho^{2}}{(2\rho+M)^{2}}\left(\frac{(2\rho+M)^{3}}{4\rho^{2}(2\rho-M)}z,1,i,-z\right).\end{split}$
(24)

In the $(z,\bar{z})$ coordinates, we have

$\epsilon^{+}_{z}=\frac{(M+2\rho)^{2}}{2\sqrt{2}\rho(1+z\bar{z})},\quad\epsilon^{-}_{\bar{z}}=\frac{(M+2\rho)^{2}}{2\sqrt{2}\rho(1+z\bar{z})}.$
(25)

Inserting those expressions and using the fact that the spatial part of the
metric is conformally flat, the mode expansion can be rewritten as

$\begin{split}A_{\mu}=&\sum_{\alpha=\pm}\frac{(2\rho+M)^{7}}{(2\rho)^{6}(2\rho-M)}\int\frac{\omega~{}\text{d}\omega}{8\pi^{2}}\int_{0}^{\pi}\text{d}\Theta\\\
&\times\left(\sin\Theta\epsilon^{\alpha}_{\mu}\mathfrak{a}_{\alpha}e^{-i\omega
t+i\omega\frac{(2\rho+M)^{3}}{4\rho(2\rho-M)}\cos\Theta}+c.c.\right),\end{split}$
(26)

where $\Theta$ is the angle between $\vec{p}$ and $\vec{x}$. The mode
expansion in the flat spacetime (see, e.g., in [8, 13, 2]) can be recovered by
setting $M=0$. Near the horizon, we have $\rho\to\frac{M}{2}$. Let
$R=\rho-\frac{M}{2}$. Hence $R=r+O(r^{2})$ in the near horizon region. The
$\Theta$ integration in the mode expansion can be worked out directly with the
help of the relation

$\int_{0}^{\pi}\text{d}\Theta\sin\Theta~{}e^{i\omega\frac{2(M+R)^{3}}{R(M+2R)}\cos\Theta}=\frac{R(M+2R)}{\omega(M+R)^{3}}\sin\left[\frac{2\omega(M+R)^{3}}{R(M+2R)}\right].$
(27)

The exponential in the integrand blows up as $R\to 0$. At null infinity, one
can apply a stationary-phase approximation in this step to control the
divergence when $r\to\infty$, see, e.g., in [8, 13, 2]. Here we follow the
Feynman prescription to introduce a small imaginary part to the radial
coordinate $R\to R+i\tau$. This will keep the radial coordinate $R$ continuous
on the real axis. Then we will trace all the contributions without any
approximation. For notational brevity, we will momentarily suppress the
imaginary part $i\tau$. In terms of $R$, we obtain the mode expansion as

$A_{\mu}=\sum_{\alpha=\pm}\frac{8(M+R)^{4}}{\pi^{2}(M+2R)^{5}}\int\text{d}\omega~{}\left(\frac{R^{2}}{M^{2}+2MR}\right)^{-2iM\omega}\\\
\times\epsilon^{\alpha}_{\mu}\mathfrak{a}_{\alpha}e^{-i\omega
u-i\omega\frac{2(M+R)^{2}}{M+2R}}\sin\left[\frac{2\omega(M+R)^{3}}{R(M+2R)}\right]+c.c..$
(28)

Note that we have adopted the retarded time $u$ using the transformation in
(17). The near horizon field $A_{z}^{0}$ is related to the plane wave modes by

$A_{z}^{0}=\frac{16\sqrt{2}}{\pi^{2}(1+z\bar{z})}\frac{(M+R)^{6}}{(M+2R)^{6}}\int\text{d}\omega~{}\left(\frac{R^{2}}{M^{2}+2MR}\right)^{-2iM\omega}\\\
\times\sin\left[\frac{2\omega(M+R)^{3}}{R(M+2R)}\right]\left(\mathfrak{a}_{+}e^{-i\omega
u-i\omega\frac{2(M+R)^{2}}{M+2R}}+\mathfrak{a}^{\dagger}_{-}e^{i\omega
u+i\omega\frac{2(M+R)^{2}}{M+2R}}\right).$ (29)

###  3.2 Soft photon theorem in position space

The non-linear part of the conserved charge in (13) can be expressed as

$Q_{\textrm{NL}}=\lim_{R\to 0}\int_{{\cal
H}^{-}}\text{d}z\text{d}\bar{z}\text{d}u\,\varepsilon(z,\bar{z})\,\partial_{u}\partial_{\bar{z}}A_{z}\,.$
(30)

Inserting $\varepsilon(z,\bar{z})=\frac{1}{w-z}$ and using the Fourier
relation

$\int_{-\infty}^{\infty}\text{d}u\,\partial_{u}F(u)=2\pi i\lim_{\omega\to
0}\left[\omega\tilde{F}(\omega)\right],$ (31)

where we define $F(u)=\int_{-\infty}^{\infty}\text{d}\omega\,e^{i\omega
u}\tilde{F}(\omega)$, the non-linear part of charge with the mode expansion
(29) is reduced to

$\displaystyle Q_{\textrm{NL}}$ $\displaystyle=$
$\displaystyle\frac{-128\sqrt{2}i}{(1+w\bar{w})}\frac{M^{2}}{R}\lim_{\omega\to
0}~{}\left[\omega^{2}\mathfrak{a}_{+}+\omega^{2}\mathfrak{a}^{\dagger}_{-}\right].$
(32)

Hence, the action of this piece of the charge on the in state is

$\displaystyle\langle\text{out}|Q_{\rm{NL}}|\text{in}\rangle=\frac{-128\sqrt{2}i}{(1+w\bar{w})}\frac{M^{2}}{R}\lim_{\omega\to
0}~{}\langle\text{out}|\omega^{2}\mathfrak{a}^{\dagger}_{-}|\text{in}\rangle$
(33)

Regarding the linear piece (14), for simplicity, we restrict ourselves to
scalar charged (with charge $Q_{e}$) matter with current
$J_{\mu}=iQ_{e}(\bar{\Phi}\nabla_{\mu}\Phi-\Phi\nabla_{\mu}\bar{\Phi})$. Since
the derivatives act on complex scalar field $\Phi$, it makes no difference
whether using covariant derivative or normal derivative. With respect to (5),
the current at leading order is
$J^{0}_{u}=i\Omega^{2}Q_{e}(\bar{\Phi}\partial_{u}\Phi-\Phi\partial_{u}\bar{\Phi})$.
Then one can apply the canonical commutation relation similar to the flat
spacetime case [13]

$[\bar{\Phi}^{0}(u,z,\bar{z}),\Phi^{0}(u^{\prime},w,\bar{w})]=\frac{i}{4}\Omega^{-2}\gamma_{w\bar{w}}^{-1}\,\Theta(u-u^{\prime})\delta^{2}(z-w)\,,$
(34)

to get the action of the linear charge on the in state as

$\langle\textrm{out}|Q^{(0)}_{\textrm{L}}|\textrm{in}\rangle=\sum_{k=1}^{n}\frac{Q_{k}}{4(w-w_{k})}\langle\textrm{out}|\textrm{in}\rangle,$
(35)

where $Q_{k}$ is the electric charge of the $k$-th particle. We have assumed
there are $n$ incoming particles.

Finally the Ward identity (12) will yield a soft photon theorem in
Schwarzschild spacetime as

$\lim_{\omega\to
0}~{}\langle\text{out}|\mathfrak{a}^{\dagger}_{-}|\text{in}\rangle=\frac{1}{512}\frac{R}{i\omega
M^{2}}\left(\frac{1+w\bar{w}}{\sqrt{2}\omega}\sum_{k=1}^{n}\frac{Q_{k}}{w-w_{k}}\right)\langle\textrm{out}|\textrm{in}\rangle.$
(36)

We have several remarks on this soft theorem as follows:

  * •

A priori, one can not deduce that the Ward identity of the near horizon
symmetry must yield a soft theorem, though it does reveal some relations from
the S-Matrix. The derivation of the soft theorem is particularly based on the
form of the charge (10) from near horizon analysis.

  * •

We use the Feynman prescription to regularize the radial coordinate $R$ on the
horizon. A similar treatment is not needed at null infinity because the null
infinity is conformally introduced. The divergence from $r\to\infty$ can be
absorbed into the conformal factor. Hence the quantities defined at the null
infinity should be finite, but the horizon is part of the spacetime, and
particles can freely cross it. So the divergence from the horizon should be
regularized. On the horizon, we set $R=i\tau$ and $\tau$ being infinitesimal.

  * •

Let $k=\frac{\tau}{512\omega M^{2}}$ be a finite constant. The soft factor in
(36) recovers the flat spacetime soft photon factor when $k=1$.

###  3.3 Soft photon theorem in momentum space

We consider the in states (the hard particles) are created on the horizon.
Then the hard momenta in the near horizon region can be parametrized as

$q_{k\mu}=\frac{E_{k}}{1+z_{k}\bar{z}_{k}}\frac{4M}{R}\left(-\frac{R}{4M}(1+z_{k}\bar{z}_{k}),(z_{k}+\bar{z}_{k}),-i(z_{k}-\bar{z}_{k}),(1-z_{k}\bar{z}_{k})\right).$
(37)

Using the soft momentum in (23) and the polarization vectors in (24), one can
show that

$\sum_{k=1}^{n}\frac{Q_{k}q_{k}\cdot\epsilon}{q_{k}\cdot p}=\frac{R}{\omega
M}\left(\frac{1+z\bar{z}}{\sqrt{2}}\sum_{k=1}^{n}\frac{Q_{k}}{z-z_{k}}\right).$
(38)

Applying this relation, we obtain the soft photon theorem in momentum space as

$\lim_{\omega\to
0}~{}\langle\text{out}|\mathfrak{a}^{\dagger}_{-}|\text{in}\rangle=\frac{1}{512}\frac{1}{i\omega
M}\sum_{k=1}^{n}\frac{Q_{k}q_{k}\cdot\epsilon}{q_{k}\cdot
p}\langle\textrm{out}|\textrm{in}\rangle.$ (39)

To close this section, we will comment on two limits of the black hole mass
parameter $M$. One can read from (39) that the soft factor is singular when
$M=0$. Though setting $M=0$ will not bring any divergence in the computation,
it will lead to the fact that $R=0$ is just a point rather than a three
dimensional null hypersurface. There is no horizon at all at $R=0$ in such
case. Somehow the regularization in $R\to R+i\tau$ when $R=0$ prevents one
taking the limit $M\to 0$. Consequently, we need to introduce a corresponding
regularization $M\to M+im$, where $m$ is infinitesimal. Another limit is to
set $M\to\infty$. Of course, this case may not be physically sound, it can
provide a mathematical consistency check for the soft theorem. In this limit,
one can arrange that $512\omega M=1$. Hence the flat space soft theorem is
recovered. Naively, it is a curious fact that the flat space result is not
recovered from the flat space limit $(M=0)$. Nevertheless, the radius of the
horizon is proportional to the mass $M$. Technically, a large mass limit
requires a large radius to compensate. This makes the near horizon analysis
very close to the null infinity analysis. Since the flat space soft theorem
can be derived from asymptotic symmetry at null infinity, it is reasonable to
see the flat space result is from the large mass limit.

##  4 Soft photons as soft electric hair on the horizon

###  4.1 Near horizon electromagnetic memories and soft electric hairs

We follow the prescription in [14] to formulate the near horizon
electromagnetic memories. Consider that an electromagnetic shockwave passes
through the horizon of the black hole. The information about the
electromagnetic wave is encoded in the change of the surface charge. Suppose
that there is initially no electromagnetic field and the system finally
settles into a stationary solution with flat gauge connections on the horizon
$\partial_{z}A_{\bar{z}}^{0}=\partial_{\bar{z}}A_{z}^{0}$ [8]. According to
(10), the change of the surface charge is just the charge evaluated at a later
time,

$\Delta
Q_{\varepsilon}=-\int_{u=u_{f}}\text{d}z\text{d}\bar{z}\,\varepsilon(z,\bar{z})\left(\partial_{z}A_{\bar{z}}^{0}+\partial_{\bar{z}}A_{z}^{0}\right)\
,$ (40)

where we only consider the contribution from electromagnetic shockwave. The
permanent change of the surface charge (40) is called the near horizon
electromagnetic memory.

Alternatively, the near horizon electromagnetic memory can be derived from the
near horizon large transformation as a precise example of the equivalence
between memories and asymptotic symmetries [1, 15, 16]. The action of the
infinitesimal near horizon large transformation, denoted by $\chi(z,\bar{z})$,
on the electromagnetic fields is

$\delta_{\chi}A_{z}=\partial_{z}\chi(z,\bar{z}),\quad\delta_{\chi}A_{\bar{z}}=\partial_{\bar{z}}\chi(z,\bar{z}).$
(41)

For the case of initially no electromagnetic field, the final electromagnetic
fields are just

$A_{z}^{0}=\partial_{z}\chi(z,\bar{z}),\quad
A_{\bar{z}}^{0}=\partial_{\bar{z}}\chi(z,\bar{z}).$ (42)

The change of the surface charge (40) is

$\Delta
Q_{\varepsilon}=-2\int_{u=u_{f}}\text{d}z\text{d}\bar{z}\,\varepsilon(z,\bar{z})\partial_{z}\partial_{\bar{z}}\chi.$
(43)

Since large gauge transformations create soft hairs on the black hole horizon
[1, 15, 16], the electromagnetic shockwave can indeed implant soft electric
hairs on the black hole horizon. As we will show in the next subsection, the
soft electric hairs can be implanted by soft photons on the horizon.

###  4.2 Near horizon soft photons as soft hairs

Soft photon theorems connect an $n+1$ particles state to an $n$ particles
state in a low-energy expansion of the extra photon in the form

$M_{n+1}\big{(}p_{1},\ldots,p_{n},\left[q;\epsilon^{\pm}\right]\big{)}=\frac{1}{\omega}S^{(0)\pm}M_{n}(p_{1},\ldots,p_{n})+{\cal
O}\left(1\right).$ (44)

In the low-energy limit, the soft theorem should have certain relevance to
classical computation. Following the proposal in [17, 18] at null infinity,
one can read from the soft theorem that the soft factors can be interpreted as
the expectation value of near horizon fields fluctuation,

$S^{(0)\pm}=\lim_{\omega\rightarrow 0}\omega\frac{M_{n+1}^{\pm}}{M_{n}}.$ (45)

Then the expectation value of near horizon fields fluctuation with soft photon
emission is simply related to the change of classical fields as

$A_{z}=\epsilon^{*+}_{z}S^{(0)+}+\epsilon^{\ast-}_{z}S^{(0)-},$ (46)

where we assume the system initially has no electromagnetic field.

Inserting the soft photon factors (36) into (46) (recall the complex
conjugate), we obtain

$A_{z}=2k\sum_{k=1}^{n}\frac{Q_{k}}{z-z_{k}},\quad
A_{\bar{z}}=2k\sum_{k=1}^{n}\frac{Q_{k}}{\bar{z}-\bar{z}_{k}}.$ (47)

Consider $k=\frac{\tau}{512\omega M^{2}}$ as a constant factor, the above
electromagnetic fields are exactly the form of a memory discussed previously
in this section. Hence the soft photons in the near horizon region naturally
have the form of soft electric hairs. Inserting the gauge fields into the
charge (40), one obtains

$\Delta Q_{\varepsilon}=-8\pi
k\sum_{k=1}^{n}Q_{k}\varepsilon(z_{k},\bar{z}_{k}),$ (48)

where $(z_{k},\bar{z}_{k})$ are the locations of the hard particles projected
on the horizon, which generate the electromagnetic shockwave.

##  5 Conclusion and discussion

We derive a soft photon theorem in the near horizon region of the
Schwarzschild black hole. The soft factor can recover the flat spacetime soft
photon factor in the zero mass limit $M\to\infty$. The soft factor interpreted
as the expectation value of near horizon fields fluctuation is in the form of
soft electric hairs on the black hole horizon. Hence this verifies that the
soft particles on black hole horizons can implant soft hairs.

In the derivation of the soft photon theorem, we restrict ourselves to the
case that all particles are in the near horizon region, but we expect our
analysis to be generic to the formalism and relevant issues can be addressed
elsewhere. We point out some for future directions. The most direct one is to
extend our derivation to other curved spacetime with horizon, such as a Kerr
black hole or cosmological solution. Another glaring one is to study the
scattering with particles in both a near horizon region and asymptotic region.
In particular, one may need to enhance the near horizon or asymptotic
symmetries to symplectic symmetries [19]. Inspired by the sub-leading soft
photon theorem [20] (see also [13]) in the flat spacetime, one may wonder if
there are more soft photon theorems in the near horizon region. It is more
promising in the case of gravity, where one has two types of near horizon
symmetries, namely near horizon supertranslations and superrotations [21].
Naively, one should be able to derive two soft graviton theorems in the near
horizon region. Another interesting point in the gravity case is to include
higher derivative terms. They are order-suppressed in the asymptotic region,
see also discussions in [22] from the soft theorem side. However in the near
horizon region, the higher derivative terms have significant influence on the
near horizon charges [23]. If they can affect the near horizon soft graviton
theorem, then that would be a very meaningful investigation.

## Acknowledgments

The authors thank Laura Donnay, Zhengwen Liu, Shahin Sheikh-Jabbari, and Jun-
Bao Wu for useful discussions. The authors thank Kai-Yu Zhang for pointing out
the mistake in the commutation relation of the scalar fields. This work is
supported in part by the National Natural Science Foundation of China (NSFC)
under Grants No. 11905156 and No. 11935009. P.C. is also supported in part by
NSFC Grant No. 12047501.

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