# Nonparametric Estimation of Matching Efficiency and Mismatch in Labor
Markets via Public Employment Security Offices in Japan, 1972-2024

Suguru Otani<EMAIL_ADDRESS>Market Design Center, University of
Tokyo

###### Abstract

I examine changes in matching efficiency and elasticities in Japan’s labor
market via Hello Work for unemployed workers from January 1972 to April 2024
using a nonparametric identification approach by Lange and Papageorgiou
(2020). I find a declining trend in matching efficiency, consistent with
decreasing job and worker finding rates. The implied match elasticity with
respect to unemployment is 0.5-0.9, whereas the implied match elasticity with
respect to vacancies is 0.1-0.4. Decomposing aggregate data into full-time and
part-time ones, I find that the sharp decline of matching efficiency after
2015 shown in the aggregate trend is driven by the decline of both full-time
and part-time ones. Second, I extend the mismatch index proposed by Şahin et
al. (2014) to the nonparametric version and develop the computational
methodology. I find that the mismatch across job categories is more severe
than across prefectures and the original Cobb-Douglas mismatch index is
underestimated.

Keywords: matching efficiency, mismatch
JEL code: E24, J61, J62, J64

##  1 Introduction

Understanding labor market matching efficiency and mismatch, which is defined
as the discrepancy between the distribution of vacant jobs and unemployed
workers, is crucial for comprehending labor market dynamics. However, few
studies have investigated the contributions of mismatch to unemployment
dynamics within the Japanese labor market. This paper seeks to address this
gap in the literature by quantifying matching efficiency via the novel
nonparametric approach of Lange and Papageorgiou (2020) and labor mismatch by
extending the mismatch index proposed by Şahin et al. (2014) to a
nonparametric version.

The matching function, which maps the number of hirings to the number of job
seekers and recruiters, incorporating matching efficiency that includes their
search effort, plays a central role in the literature on frictional labor
markets. A significant issue identified in the extensive literature on
estimating the matching function, as pointed out by Lange and Papageorgiou
(2020), is the reliance on strong functional form assumptions, such as the
Cobb-Douglas, which lack both empirical and theoretical grounding. If the
Cobb-Douglas specification is overly restrictive, it can lead to biased
estimates of matching efficiency and elasticity. To address this problem,
Lange and Papageorgiou (2020) relax the functional form restrictions on the
matching function, opting instead to non-parametrically identify the matching
function, thereby recovering matching efficiency. To the best of my knowledge,
this paper is the first to apply the nonparametric approach to the Japanese
labor market.

Relatedly, several papers quantified various dimensions of mismatch and their
contributions to the changes in the unemployment rates using the Cobb-Douglas
specification. Şahin et al. (2014) develop a methodology to measure mismatch
in various dimensions as the deviations of observed labor market outcomes from
the socially efficient outcomes. In the paper, the Cobb-Douglas specification
significantly simplifies the mismatch index by canceling out time-varying
matching efficiency and elasticities. As a methodological contribution, I
combine the nonparametric approach with the mismatch index allowing for
nonparametrically recovered matching efficiency and elasticities and provide
how to compute the index using Mathematical Programming with Equilibrium
Constraint (MPEC) proposed by Su and Judd (2012).

The main empirical findings of this paper are twofold. First, matching
efficiency (normalized to 1972) in the labor market via Hello Work shows a
declining trend with notable fluctuations, which is consistent with downward
trends of job and worker finding rates. This might be because matching
opportunity out of the government-operated platform has increased. Implied
match elasticity with respect to unemployment is 0.5-0.9, which is comparable
to previous worldwide findings such as Petrongolo and Pissarides (2001) (the
range in 0.5-0.7) and Japanese studies. On the other hand, implied match
elasticity with respect to vacancies is 0.1-0.4, which is comparable to Lange
and Papageorgiou (2020) (0.15-0.3) and Japanese studies.

Second, substantial mismatch are observed in both prefecture-level and job-
category-level and the trend is upward with fluctuations. The mismatch across
prefectures is 0.3 on average, whereas the mismatch across job categories is
0.4-0.65, larger than previous studies such as Shibata (2020) and Higashi and
Sasaki (2023). The nonparametric mismatch index highlights the downward bias
of the original Cobb-Douglas mismatch index through Monte Carlo simulation and
empirical application.

###  1.1 Related Literature

This paper contributes to the three strands of literature. First, I examine
the trend of matching efficiency in Japanese labor markets via Hello Work
nonparametrically using a novel approach (Lange and Papageorgiou 2020), which
shows how to nonparametrically identify the matching function and estimate the
matching function allowing for unobserved matching efficacy, without imposing
the usual independence assumption between matching efficiency and search on
either side of the labor market, allowing for multiple types of jobseekers.
Lange and Papageorgiou (2020) highlight positive correlations between
efficiency and market structure such as tightness and so on, which induces a
positive bias in the estimates of the vacancy elasticity whenever unobserved
matching efficacy is not controlled for, as is the case in the traditional
Cobb-Douglas matching function with constant elasticity
parameters.111Brancaccio et al. (2020a, 2023) apply the method to estimate
matching function in exporter-ship transactions in the global bulk shipping
market. Brancaccio et al. (2020b) summarize practical issues. Implementing
their approach, I add updated results from the more flexible approach to the
existing findings reported in Kano and Ohta (2005), Kambayashi and Ueno
(2006), Sasaki (2008), and Higashi (2018) using the traditional Cobb-Douglas
matching function with geographical and occupational category fixed effects to
capture geographical and occupational heterogeneity. Table 1 summarizes the
previous findings. My findings are also useful for comparison with other
countries’ results reported in Bernstein et al. (2022) and Petrongolo and
Pissarides (2001).

Table 1: Estimation results of the aggregate matching function in Japan

Paper | CRS | FE | Estimation | Sample | $\frac{d\log M}{dlogU}$ | $\frac{d\log M}{dlogV}$
---|---|---|---|---|---|---
Kano and Ohta (2005) | yes | yes | OLS | 1973–1999 | 0.56-0.59 | 0.29-0.3
Kambayashi and Ueno (2006) | no | yes | OLS | 1996–2001 | 0.81 | 0.30
Sasaki (2008) | yes | no | OLS | 1982–2016 |  | 0.38-0.45
Shibata (2013) | yes | no | GMM | 2000–2013 |  | 0.4
Shibata (2020) | yes | no | GMM | 2000–2019 |  | 0.34-0.4
Kawata et al. (2016) | yes | yes | IV | 2000–2009 |  | 0.51-0.6
Kawata (2019) | yes | yes | OLS | 2012–2016 |  | 0.52
Higashi (2018) | yes | no | OLS | 2008–2016 | 0.37-0.46 | 0.17-0.24
Higashi (2020) | yes | no | OLS | 2006–2016 | 0.48 | 0.17
Higashi (2021) | yes | no | OLS, 2SLS | 2008–2018 | 0.22-0.46 | 0.15-0.31
Higashi and Sasaki (2023) | yes | yes | OLS | 2016–2021 | Calibrate | Calibrate

Note: All models use a Cobb–Douglas matching function. Constant-returns-to-
scale (CRS) imposes restriction on matching elasticity parameters in
estimation. Fixed effect (FE) captures geographical-level and time-level
heterogeneity. Estimation Instrument Variable (IV), Two-Stage-Least-Square
(2SLS), and Generalized Method of Moments (GMM) based on Borowczyk-Martins et
al. (2013) use instruments in the specific context. Blank cells mean no
reported results.

Second, this paper contributes to the literature on mismatch Şahin et al.
(2014). The approach is widely used in a variety of contexts, for example,
United Kingdom in Patterson et al. (2016). In the context of Japanese labor
markets, several papers estimate mismatch at geographical- and job-category-
level using Cobb-Douglas specification (Shibata 2013, 2020, Kawata et al.
2016, Higashi 2018, Kawata 2019, Higashi 2020, 2021, Higashi and Sasaki 2023).
This paper points out that Cobb-Douglas specification cancels out time
variations of matching efficiency which induce bias of the mismatch index.
Then, this paper proposes the nonparametric version of the mismatch index, and
applies a novel computational method known as Mathematical Programming with
Equilibrium Constraint (MPEC) proposed by Su and Judd (2012), Dubé et al.
(2012) to calculate the geographical- and job-category-level mismatch index.

Third, this paper contributes to the literature on the impact of COVID-19 on
the matching efficiency of the worker-vacancy platform operated by the
government. Due to the extensive volume of literature relevant to this topic,
I will not provide a comprehensive list of all sources to maintain clarity and
conciseness. The most closely related papers in my context are Higashi and
Sasaki (2023) and Fukai et al. (2021). Higashi and Sasaki (2023) estimate
mismatch indices for local labor markets clustered in by occupations
vulnerable and not vulnerable to COVID-19 by assuming Cobb-Douglas matching
function with calibrated matching elasticity parameters. Fukai et al. (2021)
utilize the Labor Force Survey, a large-scale government statistic, to
estimate the group average treatment effect of COVID-19 on employment status
for each month from January to June 2020. To the best of my knowledge, this is
the first paper to describe changes in matching efficiency before and after
COVID-19 using a novel nonparametric approach.

##  2 Model

###  2.1 Nonparametric aggregate matching function

Our main interest is in matching efficiency and matching elasticity with
respect to the number of unemployed workers and vacancies in the labor market
via Public Employment Security Offices in Japan. A matching function based on
search models plays a central role in labor economics.222See Pissarides
(2000), Petrongolo and Pissarides (2001), and Rogerson et al. (2005) for
reference. The matching function relies on random search from both sides of
the market, that is, individuals seeking jobs represent the supply of labor
and recruiters represent the demand for labor. To estimate the matching
function and recover matching efficiency, I follow the novel approach proposed
by Lange and Papageorgiou (2020).333Lange and Papageorgiou (2020) additionally
incorporate search effort (Mukoyama et al. 2018) and recruitment index (Davis
et al. 2013). Unfortunately, our Hello Work data does not report the
information. The paper points out the endogeneity problem of matching
efficiency (Borowczyk-Martins et al. 2013) and the problem of too restrictive
specification of a Cobb-Douglas matching function with the fixed matching
elasticity then proposes nonparametric identification and estimation of
matching efficiency under some conditions introduced later.

Let unscripted capital letters $(A,U,V)$ denote random variables while
realizations are subscripted by time $t$. I consider the matching function
$m_{t}(\cdot,\cdot)$ that maps period-$t$ unemployed workers $U_{t}$, per-
capita search efficacy/matching efficiency of the unemployed workers $A_{t}$,
and vacancies $V_{t}$ into hires $H_{t}$. I assume that the underlying data
generating process is stationary and that I observe a long enough time-series
so that I can treat the joint distribution
$G:\mathbb{R}_{+}^{3}\rightarrow[0,1]$ of $\left(H_{t},U_{t},V_{t}\right)$ as
observed. Also, denote by $F(A,U)$ the joint distribution of $A$ and $U$.

I identify the matching function as well as unobserved, time-varying matching
efficiency, $A.$ First, I assume that $V$ and $A$ are independent conditional
on $U$, that is, $A\perp V\mid U$. Second, I assume that the matching function
$m(AU,V):\mathbb{R}_{+}^{2}\rightarrow\mathbb{R}$ has constant returns to
scale (CRS).444To follow the original model of Matzkin (2003), we can rewrite
$H=m(AU,V)$ into $H/U=m(A,V/U)$ under CRS. Here, $H/U$ and $V/U$ are known as
a job-finding rate and market tightness. Then, by applying nonparametric
identification results of Matzkin (2003), Proposition 1 of Lange and
Papageorgiou (2020) proves that $G(H,U,V)$ identifies $F(A,U)$ and
$m(AU,V):\mathbb{R}_{+}^{2}\rightarrow\mathbb{R}_{+}$ up to a normalization of
$A$ at one point denoted as $A_{0}$ of the support of $(A,U,V)$.555In
Appendix, I report finite sample performance and methodological extension with
Monte Carlo simulation. Based on simulation results with sample size $T=50$,
the sample size in this paper is enough to recover matching efficiency well.
The code is available on the author’s Github. Also, see Brancaccio et al.
(2020b) for practical issues. The approach is used to estimate a matching
function in a trade model (Brancaccio et al. 2020a, 2023).

###  2.2 Nonparametric mismatch

I construct a nonparametric version of a mismatch index which is a measure of
the fraction of hires foregone due to mismatch. Suppose there are $G$ local
markets in time $t$ with a given number of vacancies, unemployed workers, and
aggregate matching efficiencies in the economy. The number of hires $H_{gt}$
in market $g$ in time $t$ is determined by a matching function
$m(A_{gt}U_{gt},V_{gt})$. Note that, unlike Şahin et al. (2014), I allow any
specification of $m$ satisfying CRS and independence of $A_{gt}$ and $U_{gt}$
conditional on $V_{gt}$. Given matching efficiency $A_{gt}$ and the number of
vacancies $V_{gt}$, a social planner maximizes the total hires in the economy
by distributing a given number of unemployed workers to each labor market.

Following Şahin et al. (2014), I define mismatch as the deviation of the hires
in the data from the efficient allocation chosen by the social planner. Under
the assumptions of homogenous productivity and job separation rate across
labor markets, the optimal allocation of unemployed workers satisfies the
following equilibrium conditions:

$\displaystyle\frac{\partial m}{\partial
U}\left(A_{1t}U_{1t}^{*},V_{1t}\right)=\cdots=\frac{\partial m}{\partial
U}\left(A_{Gt}U_{Gt}^{*},V_{Gt}\right)$ (1)

where $*$ denotes the planner’s allocation. The equilibrium condition means
that keeping total unemployed workers $U_{t}=\sum_{g=1}^{G}U_{gt}$, the
planner allocates more unemployed workers to more effective labor markets,
that is, with more vacancies and higher matching efficiency until their
marginal contribution to the hires is equalized across markets. The main
difference from Şahin et al. (2014) is that equilibrium condition (1) does not
have an analytical formula like Cobb-Douglas specification. In Section 3.2, I
explain how to compute the optimal allocation of unemployed workers
numerically.

Using optimal allocation of unemployed workers $U_{gt}^{*}$, the aggregate
actual and optimal number of new hires can be expressed as

$\displaystyle H_{t}$ $\displaystyle=\sum_{g=1}^{G}m(A_{gt}U_{gt},V_{gt}),$
$\displaystyle H_{t}^{*}$
$\displaystyle=\sum_{g=1}^{G}m(A_{gt}U_{gt}^{*},V_{gt}).$

Using these expressions, I define mismatch index as

$\displaystyle\mathcal{M}_{t}=1-\frac{H_{t}}{H_{t}^{*}}$ (2)

where $\mathcal{M}_{t}$ measures the fraction of hires lost in period $t$
because of misallocation. This index accounts for the heterogeneity of
matching efficiencies across labor markets.

##  3 Estimation

###  3.1 Matching efficiency and elasticities

Following Lange and Papageorgiou (2020), I begin by estimating $F(A_{0}|U)$
across the support of $U$. To this end, we utilize the distribution of hires
conditional on unemployed, $U$, and observed vacancies, $V$. Specifically, we
have

$\displaystyle F(A_{0}|\psi U_{0})$ $\displaystyle=G_{H|U,V}(\psi H_{0}|\psi
U_{0},\psi V_{0})\quad\text{for any arbitrary scalar }\psi.$ $\displaystyle
F(\psi A_{0}|\lambda U_{0})$ $\displaystyle=G_{H|U,V}(\psi H_{0}|\lambda
U_{0},\psi V_{0})\quad\text{where }\lambda>0\text{ is a scaling factor}$

where $F(A_{0}|\psi U_{0})$ and $G_{H|U,V}$ are conditional distributions. By
varying $(\psi,\lambda)$, we can therefore trace out $F(A|U)$ across the
entire support of $(A,U)$.

Given our finite data, we rely on an estimate of $G_{H|U,V}$ for our
constructive estimator. Consider an arbitrary point
$(H_{\tau},U_{\tau},V_{\tau})$. To obtain $G(H_{\tau}|U_{\tau},V_{\tau})$, we
compute the proportion of observations with less than $H_{\tau}$ observed
hires among observations proximate to $(U_{\tau},V_{\tau})$ in $(U,V)$-space.
Practically, this is achieved by averaging across all observations in the
data, penalizing those with values $(U_{t},V_{t})$ using a kernel that weighs
down observations distant from $(U_{\tau},V_{\tau})$. Consequently, our
estimate is given by

$F(\psi A_{0}|\lambda U_{0})=G_{H|U,V}(\psi H_{0}|\lambda U_{0},\psi V_{0})$
$\hat{F}(\psi A_{0}|\lambda U_{0})=\sum 1(H_{t}<\psi
H_{0})\kappa(U_{t},V_{t},\lambda U_{0},\psi V_{0})$

where $\kappa(.)$ denotes a bivariate normal kernel with bandwidth 0.01.

Having recovered the distribution function $F(A|U)$, we invert
$F(A_{t}|U_{t})$ to derive $A_{t}$. This is achieved by

$A_{t}=F^{-1}(G(H_{t}|U_{t},V_{t})|U_{t})$

for all observations $(H_{t},U_{t},V_{t})$ in the dataset. Finally, we recover
the matching function as

$m(A_{t},U_{t})=G^{-1}(F(A_{t}|U_{t})|U_{t}).$

Finally, for calculating matching elasticities, I run a LASSO regression
projecting hires on the original and squared numbers of vacancies and
unemployed interacted with implied matching efficiency. The estimates
approximate the derivatives of the matching function with respect to vacancies
and unemployed interacted with implied matching efficiency, that is, an
estimate of the elasticity of the matching function.666The matching elasticity
with respect to unemployed $\frac{d\log m(AU,V)}{d\log
U}=\frac{dm(AU,V)}{dU}\frac{U}{H}=\frac{dm(AU,V)}{dAU}\frac{dAU}{dU}\frac{U}{H}=\frac{dm(AU,V)}{dAU}\frac{AU}{H}=\frac{d\log
m(AU,V)}{d\log AU}$ is obtained from the regression coefficient of $H$ on $AU$
and multiplying it by $\frac{AU}{H}$. Concretely, we approximate $m$ by the
second order polynomial
$m=\beta_{1}(AU)+\beta_{2}(AU)V+\beta_{3}V+\beta_{4}(AU)^{2}+\beta_{5}V^{2}$
and get $\frac{dm(AU,V)}{dAU}=\beta_{1}+\beta_{2}V+2\beta_{4}(AU)$ and
$\frac{d\log m}{d\log U}=\frac{d\log m}{d\log
AU}=(\beta_{1}+\beta_{2}V+2\beta_{4}(AU))\frac{U}{H}$ and $\frac{d\log
m}{d\log V}=(\beta_{2}(AU)+\beta_{3}+2\beta_{5}V)\frac{V}{H}$. For calculating
mismatch, we need $\frac{dm}{dU}=\frac{d\log m(AU,V)}{d\log
AU}\frac{H}{U}=\beta_{1}+\beta_{2}V+2\beta_{4}(AU)$.

###  3.2 Mismatch

Unlike the original Cobb-Douglas specification of Şahin et al. (2014),
equilibrium condition (1) does not have an analytical formula. Instead, I
employ the Mathematical Programming with Equilibrium Constraint (MPEC)
approach proposed by Su and Judd (2012) and Dubé et al. (2012). The
constrained optimization problem in time $t$ is defined as follows.

$\displaystyle\max_{U_{1t},\cdots,U_{Gt}}$
$\displaystyle\sum_{g=1}^{G}m_{gt}\left(A_{gt}U_{gt},V_{gt}\right),$ (3)
$\displaystyle\text{subject to }\frac{\partial m}{\partial
U}\left(A_{1t}U_{1t}^{*},V_{1t}\right)$ $\displaystyle=\cdots=\frac{\partial
m}{\partial U}\left(A_{Gt}U_{Gt}^{*},V_{Gt}\right),$ $\displaystyle U_{t}$
$\displaystyle=\sum_{g=1}^{G}U_{gt}.$

Maximizing the objective function is equivalent to solving equilibrium
conditions (1) so that the central planner’s problem represents a nonlinear
system of equations with linear equilibrium constraints. I solve the model by
using Ipopt.jl and JuMP.jl which is often used for the MPEC approach. Note
that the formulation allows us to incorporate a rich set of additional
equilibrium constraints and elements such as the government’s costs of moving
unemployed workers from some markets to other markets for calibration studies,
which is out of the scope of this paper.

##  4 Data

First, I use the Report on Employment Service (Shokugyo Antei Gyomu Tokei) for
the month-level aggregate data from January 1972 to April 2024. These datasets
include the number of job openings, job seekers, and successful job
placements, primarily sourced from the Ministry of Health, Labour and Welfare
(MHLW) of Japan, which publishes monthly reports and statistical data on the
Public Employment Security Office, commonly known as Hello Work. Hello Work is
a government-operated institution in Japan that provides job seekers with
employment counseling, job placement services, and vocational training,
playing a critical role in Japan’s labor market. The data is often used as in
Kano and Ohta (2005), Kambayashi and Ueno (2006), Sasaki (2008), and Higashi
(2018) estimating the traditional Cobb-Douglas matching function. Using up-to-
date Hello Work data, Kawata and Sato (2021) construct a simple framework to
quantitatively measure the impacts of an economic shock of COVID-19 on
unemployed workers’ welfare in their companion
project.777https://www.crepe.e.u-tokyo.ac.jp/material/crepecl12.html: Accessed
2024 June 6. The period for my dataset is selected to ensure the longest
consistent timeframe available at the time of writing this paper. In Appendix
A, we provide additional analysis using the year-level aggregate data in more
extended periods, available from January 1963 to April 2024, instead of month-
level one.

The three types of number of job openings, job seekers, and successful job
placements are reported in the data. The first is the number for full-time and
part-time workers and jobs. The second is the number for full-time workers and
jobs. The third is the number for part-time workers and jobs. The first is the
sum of the second and third. Using the three types of data, I can decompose
the labor market features into full-time and part-time ones.

Second, for estimating nonparametrically worker-job mismatch across regions
and occupation categories, I use these submarket data in 2012-2024. The period
limitation shared with Kawata (2019) and Higashi (2018) is due to the revision
of job classifications before and after 2012, which makes it difficult to
accurately connect the data. Finally, I use all 47 prefectures and 67 job
categories in each month as submarkets.

##  5 Empirical Results

We apply the above estimation approach for each dataset. Before presenting the
results, I test the assumption that vacancies are independent of matching
efficiency, conditional on overall labor supply, i.e., $V\perp A\mid U$.
Specifically, we use the residuals from a regression of vacancies $V$ on the
unemployed $U$, and similarly, the residuals of implied matching efficiency
$A$ on $S$. For the aggregate data, the correlation between these two
residuals is close to zero (-0.04), indicating no systematic relationship
between them. Conversely, for the full-time and part-time data, the
correlations between the residuals are 0.12 and -0.14. Because the patterns
appear to be influenced by some outliers as in Figure 1, the effect of
violation of independence seems limited.

(a)

(b)

(c)

Figure 1: Residual plot

###  5.1 Aggregate trends in 1972-2023

Figures 2 (a)-(d) provide monthly data patterns of unemployed individuals,
vacancies, labor market tightness ($V/U$), hires, the ($U$,$V$) relationship,
and job and worker finding rates ($M/U$ and $M/V$) in aggregate labor markets.
In summary, the numbers of unemployed individuals and vacancies increase with
fluctuations, corresponding with market tightness, while the number of hires
shows a noticeable decline around the late 1970s, peaks and troughs from the
mid-1980s to the late 1990s, peaks around mid-2000s and 2010, and a sharp
decline towards recent years. Although the ($U$,$V$) relationship does not
corresponds with Beveridge curve because these numbers are not divided by the
corresponding number of labor force, the prominent rightward shifts are
consistent with the previous studies (Elsby et al. 2015).

Figures 2 (e) and (f) present the estimation results of the matching function
along with matching efficiency and elasticities ($\frac{d\ln M}{d\ln AU}$ and
$\frac{d\ln M}{d\ln V}$). Notably, matching efficiency (normalized to 1972)
shows a declining trend with notable fluctuations, which is consistent with
the downward trends of job and worker finding rates. In particular, the
significant decline after 2015 is remarkable. This seems due to an increase in
matching opportunities outside of the government-operated platform.

The implied match elasticity with respect to unemployment is 0.5-0.9, which is
comparable to previous worldwide findings such as Petrongolo and Pissarides
(2001) (range: 0.5-0.7) and Japanese studies such as Higashi (2018) (0.38 for
2000-2014 monthly), Kawata (2019) (0.48 for 2012-2017 prefecture-month-level),
Kano and Ohta (2005) (0.56 for 1972-1999 prefecture-year-level), Sasaki (2007)
(about 0.6 for 1998-2007 prefecture-quarter-level), and Kambayashi and Ueno
(2006) (about 0.8 for 1996-2001 prefecture-month-level).888For reference,
Petrongolo and Pissarides (2001) summarize early aggregate studies in many
countries based on a Cobb-Douglas matching function with the flow of hires on
the left-hand side and the stock of unemployment and job vacancies on the
right-hand side. In short, match elasticity with respect to unemployment is in
the range of 0.5–0.7. Bernstein et al. (2022) review the recent empirical
literature that estimates the matching elasticity in the U.S. by four types of
specification; Cobb-Douglas, Cobb-Douglas with endogeneity correction, a
constant elasticity of substitution (CES), and nonparametric (Lange and
Papageorgiou 2020). The broad range of estimates is partly due to variations
in data choices and periods. Higher estimates are typically derived from JOLTS
data (Borowczyk-Martins et al. 2013, Şahin et al. 2014), while lower estimates
are often obtained from CPS flows data (Barnichon and Figura 2015) or job-
category-level hires from CPS data (Şahin et al. 2014). On the other hand, the
implied match elasticity with respect to vacancies is 0.1-0.4, which is
comparable to Lange and Papageorgiou (2020) (range: 0.15-0.3) and Japanese
studies such as Higashi (2018) (0.24 for 2000-2014 monthly), Kawata (2019)
(0.52 for 2012-2017 prefecture-month-level), Kano and Ohta (2005) (0.3 for
1972-1999 prefecture-year-level), Sasaki (2007) (about 0.2 for 1998-2007
prefecture-quarter-level), and Kambayashi and Ueno (2006) (about 0.3 for
1996-2001 prefecture-month-level).

Figures 2 (g) and (h) illustrate some correlation patterns between matching
efficiency and market structure variables such as labor market tightness,
worker finding rate, and job finding rate. Consistent with Lange and
Papageorgiou (2020), these highlight positive correlations between efficiency
and market structure, such as tightness, which induce a positive bias in the
estimates of the vacancy elasticity whenever unobserved matching efficiency is
not controlled for, as is the case in traditional estimators.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 2: Month-level aggregate results 1972-2024

###  5.2 Decomposition of part-time and full time trends in 1972-2023

Next, we decompose the aggregate trends into full-time and part-time labor
markets’ trends. Figure 3 provide full-time and part-time labor markets’
trends corresponding Figures 2 (a)-(f). Panel (a) shows that full-time
unemployment and vacancies show higher counts and more significant
fluctuations compared to part-time data and that part-time unemployment and
vacancies exhibit a more gradual and steady trend over the years. Both full-
time and part-time labor markets experience periods of high tightness during
the same general periods, notably in the early 1990s. Part-time labor market
tightness shows more pronounced peaks compared to the full-time labor market.
Panel (b) shows that the sharp decline in full-time hires post-2008 suggests
greater sensitivity to economic downturns and structural changes in the labor
market. The steady rise in part-time hires indicates increasing reliance on
part-time employment, possibly due to greater flexibility and adaptability in
the labor market. Panel (c) illustrates that full-time employment data covers
a wider range of both unemployed and vacancies, indicating higher variability
in the full-time labor market in contrast to a more stable part-time labor
market. Panel (d) shows that full-time finding rates show higher initial
values and more significant declines compared to part-time finding rates.
Part-time finding rates demonstrate a more gradual decline and stable
fluctuations over time. Full-time finding rates exhibit larger fluctuations
and more pronounced declines, indicating greater sensitivity to economic
changes. Part-time finding rates, while also showing initial declines,
stabilize earlier and display less volatility, suggesting a more resilient
part-time labor market.

Given estimation results in each dataset, Figure 4 (e) depicts the trend of
implied matching efficiency. The downward trends are common. Full-time
matching efficiency fluctuates from 80% to 300%, whereas part-time matching
efficiency remains stable at 60% compared to January 1972. This concludes that
the sharp decline of matching efficiency after COVID-19 shown in the aggregate
trend is driven by the decline of full-time matching efficiency.

Figure 4 (f) depicts the trends of matching elasticities. For matching
elasticity with respect to unemployed, the full-time trend exhibits
significant fluctuations, especially before the 1980s, ranging from 0.2 to
around 1.0, indicating varying sensitivity of matching efficiency to changes
in unemployment over time. In contrast, the part-time trend peaks, ranging
from 0.9 to 1.0 after 2020, which may indicate the post-COVID effect on the
elasticity. For matching elasticity with respect to vacancies, the full-time
trend shows a relatively stable trend, starting around 0.2, with slight
increases in recent years, whereas the part-time trend maintains lower and
more stable values around 0.1 to 0.3, suggesting that vacancies have a less
pronounced effect on matching efficiency in the part-time labor market.

Figure 4 illustrates some correlation patterns between matching efficiency and
market structure variables corresponding with Figures 2 (g)-(h). I find the
stronger and more consistent relationships in the full-time labor market. This
confirms that positive correlations between efficiency and market structure
driven mainly by full-time markets.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Month-level full-time and part-time results 1972-2024

(a)

(b)

Figure 4: Month-level full-time and part-time results 1972-2024 (Continued)

###  5.3 Prefecture-level aggregate results in 2012-2024

Figure 5 displays the month-prefecture level matching efficiency, normalized
to 2013 January Tokyo, across different regions of Japan from 2012 to 2024.
Each panel shows the efficiency trends within a specific region, highlighting
notable regional variations. For instance, the Tohoku region (panel b) and
Chubu region (panel d), which are rural areas, exhibit relatively high
volatility and generally higher efficiency values compared to other regions,
suggesting greater variability in matching efficiency. In contrast, regions
such as Kanto (panel c) and Kansai (panel e) show more stable and lower
efficiency levels. The Tohoku area is about twice as efficient as the Tokyo
area. These patterns suggest that regional economic factors and labor market
conditions significantly influence matching efficiency across Japan.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 5: Month-prefecture level matching efficiency 2012-2024

Figures 6 and 7 illustrate month-prefecture-level matching elasticities with
respect to unemployed and vacancies in 2012-2024. There is significant
regional heterogeneity within the specific region. The matching elasticities
with respect to unemployed and vacancies are 0.6-0.8 and 0.05-0.3 on average.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 6: Month-prefecture level matching elasticity with respect to
unemployed 2012-2024

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 7: Month-prefecture level matching elasticity with respect to vacancies
2012-2024

###  5.4 Job category level results in 2012-2024

Figure 8 illustrates significant disparities in matching efficiency across job
categories, normalized to 2013 January for clerical jobs. Primary and
secondary industries show relatively higher efficiency than tertiary
industries. For example, categories like ”Agriculture, Forestry, and Fishing”
(panel f) and ”Construction and Mining” (panel i) in primary and secondary
industries exhibit high volatility and generally higher efficiency levels. In
contrast, job categories such as ”Managerial” (panel a) and ”Service” (panel
e) display more stable and lower efficiency values. These variations highlight
the diverse nature of labor market dynamics across different sectors,
influenced by factors such as skill specialization, economic cycles, and
industry-specific demand fluctuations.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Figure 8: Month-job category level matching efficiency results 2012-2024

Figures 9 and 10 illustrate matching elasticities with respect to unemployed
and vacancies in each job category in 2012-2024. As in the prefecture-level
results, there is significant regional heterogeneity within the specific job
area. The matching elasticities with respect to unemployed and vacancies
fluctuates with a range generally between 0.1 and 6.0, which is larger than
prefecture-level results. This indicates that matching heterogeneity depends
on the definition of markets and matters in job-category level rather than
geographical level.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Figure 9: Month-job category level matching elasticity with respect to
unemployed 2012-2024

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Figure 10: Month-job category level matching elasticity with respect to
vacancies 2012-2024

###  5.5 Mismatch in 2012-2024

Finally, I compute the estimated nonparametric mismatch index in 2012-2024.
Panel (a) of Figure 11 illustrates the estimated nonparametric mismatch index
across 47 prefectures from 2014 to 2024. The index, representing the deviation
of the actual number of hires from a social planner’s optimal one,
demonstrates significant seasonal fluctuations throughout the period. Mismatch
across prefectures increases to around 0.3. Notably, the mismatch index shows
a cyclical pattern with peaks and troughs, suggesting seasonal factors or
regional economic cycles influencing labor market dynamics. The mismatch level
across prefectures is larger than previous Cobb-Douglas specification results
such as Shibata (2020) and Higashi and Sasaki (2023).

Panel (b) of Figure 11 shows the mismatch index across different job
categories over a similar period. Unlike the prefecture-based mismatch, this
index exhibits a more consistent upward trajectory, with less pronounced
seasonal variations. The mismatch index across job categories increases to
0.6, which is significantly larger than Shibata (2020) and Higashi and Sasaki
(2023), which indicates that the standard Cobb-Douglas specification
underestimates the mismatch index due to the absence of time-varying matching
efficiency and elasticity. The steady increase in the mismatch index across
job categories points to a growing dissonance between the skills offered by
job seekers and the skills demanded by employers. This trend may reflect
evolving industry demands, technological advancements, and the lag in the
workforce’s adaptation to new skills. The sharper rise in the mismatch index
post-2020 further suggests that the pandemic may have exacerbated skill
mismatches, possibly due to shifts in industry dynamics, such as the
acceleration of digital transformation and the differing impacts on various
sectors.

(a)

(b)

Figure 11: Estimated nonparametric mismatch index

##  6 Conclusion

I investigate how matching efficiency in the labor market via Public
Employment Security Offices in Japan for unemployed workers changed during the
period 1972-2024. By applying a novel nonparametric identification approach
proposed by Lange and Papageorgiou (2020) to monthly data, I find that
matching efficiency (normalized to 1972) exhibits a declining trend with
notable fluctuations, consistent with the downward trends in job and worker
finding rates. Finally, I extend the mismatch index proposed by Şahin et al.
(2014) to the nonparametric version and find that the mismatch across job
categories is more severe than across prefectures and the original Cobb-
Douglas mismatch index is biased.

#### Acknowledgments

I thank Fuhito Kojima, Kosuke Uetake, Akira Matsushita, Kazuhiro Teramoto, Ryo
Kambayashi, Higashi Yudai for their valuable advice. This work was supported
by JST ERATO Grant Number JPMJER2301, Japan.

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##  Appendix A Appendix

###  A.1 Year-level trends in 1966-2023

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 12: Year-level results 1966-2024

Figures 12 provide a year-level counterpart to Figure 2. The findings in the
main text remain valid.


# Horizons and Null Infinity: A Fugue in 4 voices

Abhay Ashtekar<EMAIL_ADDRESS>Institute for Gravitation and the
Cosmos, Pennsylvania State University, University Park, PA 16802, USA
Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, ON
N2L 2Y5, Canada  Simone Speziale<EMAIL_ADDRESS>Aix
Marseille Univ., Univ. de Toulon, CNRS, CPT, UMR 7332, 13288 Marseille, France

###### Abstract

Black hole horizons in equilibrium and null infinity of asymptotically flat
space-times are null 3-manifolds but have very different physical
connotations. We first show that they share a large number of geometric
properties, making them both weakly isolated horizons. We then use this new
unified perspective to unravel the origin of the drastic differences in the
physics they contain. Interestingly, the themes are woven together in a manner
reminiscent of voices in a fugue.

Introduction: Two types of boundaries commonly arise in general relativity:
future horizons $\Delta^{\\!{}^{+}}$ of black holes in equilibrium and
Penrose’s future null infinity $\mathscr{I}^{+}$. While it is obvious that
both these boundaries are null 3-surfaces, it is not well appreciated that
they share other, more subtle geometric structures as well. In particular each
of them inherits an intrinsically defined derivative operator with certain
properties that ensure that they are both _Weakly Isolated Horizons_ (WIHs)
[1, 2, 3]. At first this appears surprising –even impossible– because while
WIHs are generally thought of as representing black hole boundaries in strong
field regions, $\mathscr{I}^{+}$ lies in asymptotic, near-flat regions. More
importantly, WIHs are often associated with _‘stationary’_ black holes since
there is no flux of gravitational waves (or matter fields) across them, while
$\mathscr{I}^{+}$ is the quintessential arena to calculate the fluxes of
energy and momenta carried by gravitational waves (as well as other massless
fields). The purpose of this Letter is to summarize recent work [1] that
compares and contrasts their geometry and physics, and resolves this apparent
tension.

As we recall below, a WIH horizon $\mathfrak{h}$ can be defined purely in
geometrical terms, _without reference_ to any field equations. Nonetheless,
WIHs carry a universal structure which is very similar to that of
$\mathscr{I}^{+}$. As a result, the symmetry group $\mathfrak{G}$ of WIHs
$\mathfrak{h}$ is closely related to the BMS group [4]. Second, the intrinsic
(degenerate) metric $\bar{q}_{ab}$ is non-dynamical on $\mathfrak{h}$ just as
it at $\mathscr{I}^{+}$. The third common element seems counter intuitive at
first: _the intrinsic connection $\bar{D}$ on a general $\mathfrak{h}$ is
dynamical_ just as it is at $\mathscr{I}^{+}$. This important point is often
overlooked in the literature since WIHs have been often regarded as
‘stationary boundaries’.

Both black hole horizons $\Delta^{\\!{}^{+}}$ and null infinity
$\mathscr{I}^{+}$ are examples of geometric WIHs $\mathfrak{h}$. However,
while $\Delta^{\\!{}^{+}}$ is a WIH with respect to the _physical_ metric
$g_{ab}$ that satisfies Einstein’s equations, $\mathscr{I}^{+}$ is a WIH with
respect to the rescaled metric $\hat{g}_{ab}$ used in the Penrose completion
that satisfies _conformal_ Einstein’s equations. Einstein’s equations imply
that the time dependence of the connection $D$ on $\Delta^{\\!{}^{+}}$ is
extremely constrained: $D$ is completely determined on the entire 3-manifold
$\Delta^{\\!{}^{+}}$ by ‘data’ that can be specified on a 2-dimensional cross-
section [3]. Therefore, in spite of its time dependence, $D$ does not carry
_any_ 3-d degrees of freedom. Physically this means that ‘radiative modes’ are
absent on $\Delta^{\\!{}^{+}}$; the connection $D$ on $\Delta^{\\!{}^{+}}$
encodes only ‘Coulombic information’. The situation is just the opposite at
$\mathscr{I}^{+}$: conformal Einstein’s equations imply that the intrinsic
connection $\hat{D}$ captures precisely the two radiative modes of the
gravitational field [5]; it carries no ‘Coulombic information’! We will see
that this striking difference emerges from the _same_ evolution equation
satisfied by the connection on any WIH $\mathfrak{h}$. It is remarkable that
both the close geometrical similarities and the diametrically opposite physics
emerges from the common WIH framework.

This structure is also subtle. For example, one may envisage using isolated
horizons (IHs) [6] in place of WIHs. This strengthening of conditions may seem
innocuous since the notion of IHs is also considerably weaker than that of
Killing horizons. (For example, the Robinson-Trautman and Kastor-Traschen
solutions admit IHs but no Killing horizons [7, 8].) But passage from WIHs to
IHs is already unacceptably strong for our purposes because while
$\mathscr{I}^{+}$ of any asymptotically flat space-time is a WIH, it would be
an IH only if the Bondi news vanishes, i.e., there is no radiation across
$\mathscr{I}^{+}$!

As discussed in [1] all our considerations apply also to past horizons and
past null infinity and, in presence of a positive $\Lambda$, also to
cosmological horizons $\Delta$ which may not be associated with black holes.
In this brief report we focus on $\Delta^{+}$ and $\mathscr{I}^{+}$ because
physically they are the most interesting cases. Also, for simplicity we assume
that the physical metric satisfies Einstein’s equations in a neighborhood of
$\Delta^{\\!{}^{+}}$ and $\mathscr{I}^{+}$; one can allow appropriate sources.

Geometric WIHs $\mathfrak{h}$: Fix a 4-manifold $\bar{M}$ with a -,+,+,+
metric $\bar{g}_{ab}$. Consider a null, 3-dimensional sub-manifold
$\mathfrak{h}$ of $\bar{M}$ with topology $\mathbb{S}^{2}\times\mathbb{R}$.
Recall that it is said to be a Non-Expanding Horizon (NEH) if [4]:
(i) Every null normal ${\bar{k}}^{a}$ to $\mathfrak{h}$ is expansion free:
$\theta_{({\bar{k}})}\,\hat{=}\,0$ where $\,\hat{=}\,$ _stands for equality at
points of $\mathfrak{h}$_. For definiteness, we will assume that the null
normals ${\bar{k}}^{a}$ are all future pointing; and,
(ii) The Ricci tensor $\bar{R}_{a}{}^{b}$ of $\bar{g}_{ab}$ satisfies
$\bar{R}_{a}{}^{b}{\bar{k}}^{a}\,\,\hat{=}\,\,\alpha{\bar{k}}^{b}$ for some
function $\alpha$.There is extensive literature on NEHs (for reviews, see
e.g., [9, 10, 11, 12]) and their key properties are summarized in [1, 4]. In
particular, the 4-metric $\bar{g}_{ab}$ induces on $\mathfrak{h}$, an
intrinsic (degenerate) metric $\bar{q}_{ab}$ and provides a family of null
normals $\bar{k}^{a}$ that are tangent to _affinely parametrized_ null
geodesics. Any two of these normals are related by
$\bar{k}^{\prime\,a}\,\hat{=}\,f\,\bar{k}^{a}$ where the positive function $f$
satisfies $\mathcal{L}_{\bar{k}}f\,\hat{=}\,0$. The derivative operator
$\bar{\nabla}$ defined by $\bar{g}_{ab}$ also induces a unique derivative
operator $\bar{D}$ on $\mathfrak{h}$ with the following properties:

$\bar{D}_{a}\bar{q}_{bc}\,\hat{=}\,0,\quad{\rm
and}\quad\bar{D}_{a}\bar{k}^{b}\,\hat{=}\,\bar{\omega}_{a}\bar{k}^{b},\qquad{\hbox{\rm
where $\bar{\omega}_{a}$
satisfies}}\quad\bar{\omega}_{a}\bar{k}^{a}\,\hat{=}\,0\quad{\rm
and}\quad\mathcal{L}_{\bar{k}}\bar{\omega}_{a}\,\hat{=}\,0\,.$ (1)

($\bar{\omega}_{a}$ depends on the choice of the null normal $\bar{k}^{a}$ but
we will suppress this dependence for notational simplicity.) One can eliminate
the functional rescaling freedom in $\bar{k}^{a}$ by requiring that the
rotation 1-form $\bar{\omega}_{a}$ be divergence free: One is then left with a
small class $[\bar{k}]$ in which any two null normals are related by a
$\bar{k}^{\prime\,a}=c\bar{k}^{a}$, for some positive _constant_ $c$ [3].

An NEH $\mathfrak{h}$ equipped with such an equivalence class $[\bar{k}^{a}]$
of null normals is said to be a _Weakly Isolated Horizon_. (Our WIHs are
called _extremal_ because the acceleration of preferred null normals
$[\bar{k}^{a}]$ vanishes.) Eq. (1) implies that the intrinsic metric
$\bar{q}_{ab}$ is time independent;
$\mathcal{L}_{\bar{k}}\,\bar{q}_{ab}\,\hat{=}\,0$. However, on generic WIHs,
the connection $\bar{D}$ is time _dependent_.

It is easy to verify that if a 1-form $\bar{f}_{a}$ is ‘horizontal’, i.e.,
satisfies ${\bar{k}^{a}}\bar{f}_{a}=0$, then
$\dot{\bar{D}}_{a}\bar{f}_{b}:=(\mathcal{L}_{\bar{k}}\bar{D}_{a}-\bar{D}_{a}\mathcal{L}_{\bar{k}})\,\bar{f}_{b}=0$.
Therefore the ‘time derivative’ $\dot{\bar{D}}$ is completely determined by
$\dot{\bar{D}}_{a}\bar{\jmath}_{b}$ for any 1-form $\bar{\jmath}_{a}$
satisfying $\bar{\jmath}_{a}\bar{k}^{a}=-1$. Since
$\bar{R}_{a}{}^{b}{\bar{k}}^{a}\,\hat{=}\,\alpha{\bar{k}}^{b}$ on any WIH
$\mathfrak{h}$, it follows that this action is given by [1, 3]

$\dot{\bar{D}}_{a}\bar{\jmath}_{b}\,\,\hat{=}\,\bar{D}_{a}\bar{\omega}_{b}+\bar{\omega}_{a}\bar{\omega}_{b}+{\bar{k}}^{c}\,\bar{C}_{c\,{\hbox{\lower
1.50694pt\hbox{${}_{\leftarrow}$}}\kern-5.72635pt{ab}}}{}^{d}\,\bar{\jmath}_{d}+\textstyle{\frac{1}{2}}\,\big{(}\bar{S}_{\hbox{\lower
1.50694pt\hbox{${}_{\leftarrow}$}}\kern-5.72635pt{ab}}+\alpha\,q_{ab}\big{)}\,$
(2)

where ${\bar{C}}_{abc}{}^{d}$ and ${\bar{S}}_{ab}$ are the Weyl and Schouten
tensors, and indices $a,b$ are pulled back to $\mathfrak{h}$. We will see that
the drastic difference in the physics of black hole horizons
$\Delta^{\\!{}^{+}}$ and null infinity $\mathscr{I}^{+}$ can be directly
traced back to Eq. (2). On a generic WIH, none of the terms on the right side
are zero, whence _$\bar{D}$ has genuine time dependence_. By contrast,
$\dot{\bar{D}}\,\hat{=}\,0$ on _isolated_ horizons (IHs) by their very
definition [6]; this is why IHs are much more restrictive. In particular,
while one can always choose a null normal on an NEH to endow it with the
structure of a WIH, generically there is no null normal that can endow it with
an IH structure.

Neither the metric $\bar{q}_{ab}$ nor the connection $\bar{D}$ on
$\mathfrak{h}$ are universal – they change, for example, as one passes from
the Schwarzschild WIH to the Kerr WIH. However, because of the
$\mathbb{S}^{2}\times\mathbb{R}$ topology, each WIH admits a 3-parameter
family of _unit, round_ 2-sphere metrics $\mathring{q}_{ab}$ that are
conformally related to its $\bar{q}_{ab}$:
$\mathring{q}_{ab}\,\hat{=}\,\mathring{\psi}^{2}\bar{q}_{ab}$. While the
conformal factors $\mathring{\psi}$ relating the metrics $\mathring{q}_{ab}$
and $\bar{q}_{ab}$ vary from one WIH to another, the _relative_ conformal
factors $\mathring{\alpha}$ between any two round metrics _is universal_ :

$\mathring{q}^{\prime}_{ab}\,\hat{=}\,\mathring{\alpha}^{2}\mathring{q}_{ab}\quad{\hbox{\rm
where $\mathring{\alpha}$
satisfies}}\quad\mathring{D}^{2}\ln\mathring{\alpha}+1\,\,\hat{=}\,\,\alpha_{o}^{-2}.$
(3)

The general solution is a linear combination of the first 4 spherical
harmonics of $\mathring{q}_{ab}$:
$\alpha_{o}^{-1}=\alpha_{o}+{\sum_{m}}\alpha_{m}Y_{1,m}$ where the
coefficients are constrained to satisfy
$-\alpha_{o}^{2}+\sum_{m}|\alpha_{m}|^{2}=-1$. Motivated by geometric
considerations related to multipoles, one introduces a parallel rescaling of
the vector fields $[{\bar{k}}^{a}]$ and sets
$[\mathring{k}^{a}]\,\hat{=}\,[\mathring{\psi}^{-1}{\bar{k}}^{a}]$ [4]. Then
_every_ WIH is naturally equipped with a 3-parameter family of pairs
$(\mathring{q}_{ab},\,[\mathring{k}^{a}])$ related by:
$(\mathring{q}^{\prime}_{ab},\,[\mathring{k}^{\prime\,a}])\,\,\hat{=}\,\,(\mathring{\alpha}^{2}\,\mathring{q}_{ab},\,\mathring{\alpha}^{-1}\,[\mathring{k}^{a}])$.
The relation is the same for all WIHs. Thus, the pairs capture the
‘kinematical’ structure _that is universal_ , leaving out fields such as
$\bar{q}_{ab},[{\bar{k}}^{a}],\bar{\omega}_{a}$ that can carry physical
information that varies from one WIH to another.

The WIH symmetry group $\mathfrak{G}$ is the subgroup of the diffeomorphism
group of $\mathfrak{h}$ that preserves this universal structure. It is
generated by vector fields $\xi^{a}$ satisfying

$\mathcal{L}_{\xi}\,\mathring{q}_{ab}\,\hat{=}\,2\mathring{\beta}\,\mathring{q}_{ab}\qquad{\rm
and}\qquad\mathcal{L}_{\xi}{\mathring{k}}^{a}\,\hat{=}\,-(\mathring{\beta}+\mathring{\varpi}){\mathring{k}}^{a}$
(4)

where $\mathring{\varpi}$ is a constant and $\mathring{\beta}$ satisfies
$\big{(}\mathring{D}^{2}+2\big{)}\,\mathring{\beta}=0$; it is a linear
combination of the first three spherical harmonics of $\mathring{q}_{ab}$. The
BMS vector fields $\xi^{a}$ at $\mathscr{I}^{+}$ satisfy the same equation (4)
but with $\mathring{\varpi}=0$. This difference can be directly traced-back to
the fact that while $\mathfrak{h}$ is equipped only with an equivalence class
$[\bar{k}^{a}]$ of null normals, where any two are related by a constant
rescaling, in any one conformal completion, $\mathscr{I}^{+}$ is equipped with
a canonical null normal $\hat{n}^{a}$. Eq. (4) implies that $\mathfrak{G}$ is
a semi-direct product, $\mathfrak{G}=\mathfrak{B}\ltimes\mathfrak{D}$, of the
BMS group $\mathfrak{B}$ with the 1-dimensional group $\mathfrak{D}$ of
dilations, with $\mathfrak{B}$ serving as the normal subgroup and
$\mathfrak{D}$, the quotient of $\mathfrak{G}$ by $\mathfrak{B}$. An example
of a dilation vector field is
$d^{a}=\mathring{\varpi}\,\mathring{v}\,\mathring{k}^{a}=\mathring{\varpi}\,\bar{v}\,\bar{k}^{a}$.
Its action rescales $\mathring{k}^{a}$ (and $\bar{k}^{a}$) by a constant,
leaving $\mathring{q}_{ab}$ (and $\bar{q}_{ab}$) untouched. (Here
$\mathring{v}$ and $\bar{v}$ are affine parameters of $\mathring{k}^{a}$ and
$\bar{k}^{a}$ respectively.) Could we perhaps have strengthened the boundary
conditions defining WIHs $\mathfrak{h}$ to eliminate this dilation and reduce
$\mathfrak{G}$ to $\mathfrak{B}$? The answer is in the negative: On the
Schwarzschild horizon, for example, the time translation is realized as the
dilation in the Lie algebra of $\mathfrak{G}$ (while it is realized as a
supertranslation at $\mathscr{I}^{+}$). $\mathfrak{G}$ has a rich structure
that has been spelled out in [4], and is a subgroup of the symmetry group of a
general null hypersurface studied in [13] (see also [14, 15, 16, 17, 18]).

Black Hole WIHs: $\Delta^{\\!{}^{+}}$. Following the terminology used in the
literature for black hole WIHs, we will now drop the bars and denote the
space-time by $(M,g_{ab})$, the intrinsic metric on $\Delta^{\\!{}^{+}}$ by
$q_{ab}$, the intrinsic derivative operator by $D$. For the same reason, the
equivalence class of preferred null normals will now be denoted by
$[\ell^{a}]$ (in place of $[\bar{k}^{a}]$) and the 1-form conjugate to any
given $\ell^{a}\in[\ell^{a}]$ by $n_{a}$, so that $\ell^{a}n_{a}=-1$ (in place
of $\bar{k}^{a}\bar{\jmath}_{a}=-1$).

The universal structure and the symmetry group of $\Delta^{\\!{}^{+}}$ is the
same as that of a general WIH $\mathfrak{h}$. In particular, the symmetry
vector fields $\xi^{a}$ satisfy (4). Given a foliation $\mathring{v}={\rm
const}$ of $\Delta^{\\!{}^{+}}$ where $\mathring{v}$ is an affine parameter of
$\mathring{\ell}^{a}$, their explicit expressions are given by

$\xi^{a}\,\hat{=}\,\big{(}(\mathring{\varpi}+\mathring{\beta})\mathring{v}+\mathring{\mathfrak{s}})\big{)}\mathring{\ell}^{a}\,+\,\mathring{\epsilon}^{ab}\mathring{D}_{b}\mathring{\chi}\,+\,\mathring{q}^{ab}\mathring{D}_{b}\mathring{\beta}\,.$
(5)

Here $\mathring{\mathfrak{s}}(\theta,\phi)$ is a general function on the
2-sphere of null generators of $\Delta^{\\!{}^{+}}$; each of
$\mathring{\chi}(\theta,\varphi)$ and $\mathring{\beta}(\theta,\varphi)$ is a
linear combination of first three spherical harmonics defined by
$\mathring{q}_{ab}$; and $\mathring{\epsilon}^{ab},\,\mathring{q}^{ab}$ are
the inverses of the area 2-form and the metric on the $\mathring{v}={\rm
const}$ cross-sections, respectively. In this representation,
$d^{a}:=\mathring{\varpi}\mathring{v}\mathring{\ell}^{a}$ is a dilation,
$\mathring{\mathfrak{s}}\mathring{\ell}^{a}$ are supertranslations,
$\mathring{\epsilon}^{ab}\mathring{D}_{b}\mathring{\chi}\,$ are rotations, and
$\mathring{\beta}\,\mathring{v}\ell^{o}+\mathring{q}^{ab}\mathring{D}_{b}\mathring{\beta}$
are boosts.

Next, let us use Einstein’s equations to simplify the dynamical equation (1).
First, the Ricci tensor contribution to the right side vanishes. Second, only
the Newman-Penrose component $\Psi_{2}$ of the Weyl tensor contributes and, on
any WIH, it is completely determined by $q_{ab}$ and $D$ via
$\Psi_{2}=\frac{1}{4}\,\mathcal{R}+\frac{i}{2}\epsilon^{ab}D_{a}\omega_{b}$,
where $\mathcal{R}$ is the scalar curvature of $q_{ab}$. Therefore, (2)
becomes:

$\dot{D}_{a}n_{b}\,=\,D_{(a}\omega_{b)}+\omega_{a}\omega_{b}\,-\textstyle{\frac{1}{4}}\,\mathcal{R}\,q_{ab}\,.$
(6)

Note that every field on the right side is Lie dragged by $\ell^{a}$; they are
time-independent. Therefore the time dependence of $D$ is now tightly
constrained: it is entirely determined by the triplet
$(q_{ab},\omega_{a},D_{a}n_{b})$ on any 2-dimensional cross-section of
$\Delta^{\\!{}^{+}}$. In other words, there is no radiation passing across
$\Delta^{\\!{}^{+}}$; it represents a black hole boundary in equilibrium.

Asymptotic WIHs: $\mathscr{I}^{+}$. Recall (e.g., from [19]) that a physical
space-time $(M,g_{ab})$ is _asymptotically flat at null infinity_ if there
exists a Manifold $\hat{M}$ with boundary $\mathscr{I}^{+}$ that is
topologically $\mathbb{S}^{2}\times\mathbb{R}$, and equipped with a metric
$\hat{g}_{ab}$ such that $\hat{M}=M\cup\mathscr{I}^{+}$ and
$\hat{g}_{ab}=\Omega^{2}g_{ab}$ on $M$ where $\Omega$ is a nowhere vanishing
on $M$, and,
(i) $\Omega\,\hat{=}\,0$, while $\hat{\nabla}_{a}\Omega$ is nowhere vanishing
on $\mathscr{I}^{+}$; and,
(ii) The physical metric $g_{ab}$ satisfies vacuum Einstein’s equations
$R_{ab}=0$ in a neighborhood of $\mathscr{I}^{+}$.

As is well known, the conformal Einstein equations satisfied by $\hat{g}_{ab}$
imply that $\mathscr{I}^{+}$ is null and, without loss of generality, one can
choose $\Omega$ so that $\hat{\nabla}^{a}\hat{n}_{a}\,\hat{=}\,0$ where
$\hat{n}^{a}=\hat{\nabla}^{a}\Omega$ is the null normal to $\mathscr{I}^{+}$.
As is standard in the literature on null infinity, _we will restrict ourselves
to such divergence-free frames_. Then conformal Einstein’s equations imply
$\hat{\nabla}_{a}\hat{n}^{b}\,\hat{=}\,0$. Therefore integral curves of
$\hat{n}^{a}$ are affinely parametrized null geodesics with vanishing
expansion and shear; $\mathscr{I}^{+}$ is a WIH. Furthermore, the 1-form
$\omega_{a}$ vanishes identically, whence in particular its divergence
vanishes. Thus $\hat{n}^{a}\,\,\hat{=}\,\,\hat{\nabla}^{a}\Omega$ serves as
the _preferred_ normal to this WIH. Note that, while a generic WIH
$\mathfrak{h}$ is equipped with only an equivalence class $[\bar{k}^{a}]$ of
such null normals, the given conformal completion endows $\mathscr{I}^{+}$
with a specific preferred normal $\hat{n}^{a}$; the freedom to rescale by a
constant has disappeared.

As on $\Delta^{\\!{}^{+}}$, let us adapt notation to the standard usage in the
literature on $\mathscr{I}^{+}$. Thus, $(\bar{M},\bar{g}_{ab})$ of general
WIHS $\mathfrak{h}$ will now replaced by a divergence-free completion
$(\hat{M},\hat{g}_{ab})$; $(\bar{q}_{ab},\bar{k}^{a})$ by
$(\hat{q}_{ab},\hat{n}^{a})$; and the 1-form $\bar{\jmath}_{a}$ will now be
denoted by $\hat{\ell}_{a}$ so that $\hat{n}^{a}\,\hat{\ell}_{a}=-1$. Of
course, given a physical space-time, we can introduce another divergence-free
conformal frame $\Omega^{\prime}=\mu\Omega$ where $\mu$ is nowhere vanishing
on $\hat{M}$ and satisfies $\mathcal{L}_{\hat{n}}\mu\,\hat{=}\,0$. In the WIH
perspective, the resulting $(\hat{M},\hat{g}_{ab})$ and
$(\hat{M},\hat{g}^{\prime}_{ab})$ are to be regarded as distinct space-times,
in both of which $\mathscr{I}^{+}$ is a WIH but with distinct WIH geometries
$(\hat{q}_{ab},\hat{n}^{a},\hat{D})$ and
$(\hat{q}_{ab}^{\prime},\,\hat{n}^{\prime\,a},\hat{D}^{\prime})$. Put
differently, while the fact that $\mathscr{I}^{+}$ is an extremal WIH is a
property of the given physical space-time, the geometry of this WIH varies
from one (divergence-free) conformal completion to another. The physics of
this family of WIHs is the same; final results are all conformally invariant
as usual.

As in the case of $\Delta^{\\!{}^{+}}$, let us begin with a discussion of the
universal structure and symmetries of this WIH. They are essentially the same
as on a general WIH _but with a small variation_. As remarked above, while the
given metric $\bar{g}_{ab}$ endows geometrical WIHs $\mathfrak{h}$ only with
equivalence class $[\bar{k}^{a}]$ of null normals, $\hat{g}_{ab}$ endows
$\mathscr{I}^{+}$ with a _single_ null normal $\hat{n}^{a}$ through $\Omega$.
Therefore, the universal structure that descends on $\mathscr{I}^{+}$ from the
WIH perspective consists of pairs $(\mathring{q}_{ab},\mathring{n}^{a})$, a
unit round metric and a null normal, conformally related to
$(\hat{q}_{ab},\hat{n}^{a})$ as on $\mathfrak{h}$, so that any two pairs are
related by
$(\mathring{q}_{ab}^{\prime},\mathring{n}^{\prime\,a})\,=\,(\mathring{\alpha}^{2}\mathring{q}_{ab},\,\mathring{\alpha}^{-1}\mathring{n}^{a})$.
In the more familiar terminology, these pairs correspond to Bondi conformal
frames. Now, in coordinate-free discussions of $\mathscr{I}^{+}$, Bondi
conformal frames play only a marginal role; one works directly with the pairs
$(\hat{q}_{ab},\hat{n}^{a})$ (see, e.g., [19]). In the WIH perspective, on the
other hand, these pairs correspond to distinct WIH geometries; what is
universally available on all of them are the pairs
$(\mathring{q}_{ab}^{\prime},\mathring{n}^{\prime\,a})$ that correspond to
Bondi conformal frames. Thus, when looked at through the WIH lens, symmetries
are now those diffeomorphisms of $\mathscr{I}^{+}$ that map any one of these
Bondi frames to another one. This subgroup is the BMS group $\mathfrak{B}$
since $\mathfrak{B}$ can also be characterized in this way.

For symmetry vector fields, the WIH perspective informs us that $\xi^{a}$
represents an infinitesimal symmetry if it satisfies Eq. (4), but now with
$\mathring{\varpi}=0$ since there is no longer a freedom to rescale the
normals $\mathring{n}^{a}$ by constants. Thus in terms of (5), we no longer
have dilations that rescaled $\mathring{k}^{a}$ by constants, leaving the
metric $\mathring{q}_{ab}$ unchanged. Given a pair
$(\mathring{q}_{ab},\mathring{n}^{a})$, then, the symmetry vector fields at
$\mathscr{I}^{+}$ have the form:

$\xi^{a}\,\hat{=}\,\big{(}\mathring{\beta}\,\mathring{u}+\mathring{\mathfrak{s}})\big{)}\mathring{n}^{a}\,+\,\mathring{\epsilon}^{ab}\mathring{D}_{b}\mathring{\chi}\,+\,\mathring{q}^{ab}\mathring{D}_{b}\mathring{\beta}\,,$
(7)

where $\mathring{u}$ is now an affine parameter of $\mathring{n}^{a}$. These
are the standard expressions of BMS vector fields in a Bondi conformal frame
$(\mathring{q}_{ab},\mathring{n}^{a})$. Thus, the symmetry group
$\mathfrak{G}=\mathfrak{B}\ltimes\mathfrak{D}$ of a general WIH $\mathfrak{h}$
just ‘loses the dilation part’ and reduces to the BMS group $\mathfrak{B}$.

On the other hand, there is a strong contrast between the evolution equation
of $\hat{D}$ on $\mathscr{I}^{+}$ and that of $D$ on $\Delta^{\\!{}^{+}}$: the
Weyl tensor $\bar{C}_{abc}{}^{d}$ as well as the 1-form $\bar{\omega}_{a}$ on
the right side of (2) vanish on $\mathscr{I}^{+}$ and the evolution is
governed by the Ricci part, which was the only one that vanished on
$\Delta^{\\!{}^{+}}$! Thus, in contrast to the Eq. (2) on $\Delta^{\\!{}^{+}}$
at $\mathscr{I}^{+}$ have:

$\dot{\hat{D}}_{a}\hat{\ell}_{b}\,=\textstyle{\frac{1}{2}}\,\,\big{(}\hat{S}_{\hbox{\lower
1.50694pt\hbox{${}_{\leftarrow}$}}\kern-5.72635pt{ab}}+\alpha\,\hat{q}_{ab}\big{)}\,.$
(8)

Now, the Bondi news is given by $\hat{N}_{ab}=\hat{S}_{ab}-\rho_{ab}$, where
$\hat{\rho}_{ab}$ is the (kinematical) Geroch tensor field [20] that serves to
remove the unphysical information contained in the conformal factor in
$\hat{S}_{\hbox{\lower 1.50694pt\hbox{${}_{\leftarrow}$}}\kern-5.72635pt{ab}}$
(see, e.g., [20, 5, 19]). Thus, the time-dependence of $\hat{D}$ is dictated
by the Bondi News that carries two degrees of freedom per point of
3-dimensional $\mathscr{I}^{+}$. To summarize, $\mathscr{I}^{+}$ is a WIH
w.r.t. $\hat{g}_{ab}$ that satisfies conformal Einstein’s equations while
$\Delta^{\\!{}^{+}}$ is a WIH w.r.t. $g_{ab}$ that satisfies Einstein’s
equations. It is this difference that leads to diametrically opposite physics
on the two WIHs, even though one starts from the same fundamental dynamical
equation (1). Thus, the unified WIH framework endows $\Delta^{\\!{}^{+}}$ and
$\mathscr{I}^{+}$ with essentially the same universal structures and symmetry
groups and, at the same time, teases out diagonally opposite physics from
them.

Phase spaces of local DOF: We will now summarize a new Hamiltonian framework,
geared to local degrees of freedom, that serves as a common platform to
discuss fluxes of physical observables associated with symmetries at both
$\mathscr{I}^{+}$ as well as $\Delta^{\\!{}^{+}}$.

Recall that the standard covariant phase space $\boldsymbol{\Gamma}_{\rm cov}$
consists of solutions $g_{ab}$ to field equations on the entire space-time $M$
satisfying the standard boundary conditions at $\mathscr{I}^{+}$ or/and
$\Delta^{\\!{}^{+}}$ (see, e.g., [21, 22, 13, 23]). One can extract from it
phase spaces $\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal{\hat{R}}}}$ and
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal R}}$ that are adapted to open
regions $\hat{\mathcal{R}}$ and $\mathcal{R}$ of $\mathscr{I}^{+}$ and
$\Delta^{\\!{}^{+}}$, respectively, each bounded by arbitrary 2-sphere cross-
sections [1]. $\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal{\hat{R}}}}$
consists of certain equivalence classes $\\{\hat{D}\\}$ of connections on
$\hat{\mathcal{R}}$ (where the equivalence relation serves just to get rid of
a redundant conformal freedom). They encode precisely the two radiative
degrees of freedom of the gravitational field in full non-linear general
relativity _that reside in_ ${\hat{\mathcal{R}}}$ [5]. In particular, they do
not capture any of the ‘Coulombic information’ contained in $g_{ab}$.
Similarly, $\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal R}}$ consists of
the (time independent) pair $(q_{ab},\,\omega_{a})$, evaluated at a cross-
section of $\Delta^{\\!{}^{+}}$. These fields encode just the Coulombic
information in $g_{ab}$; although there may be radiation arbitrarily close to
$\Delta^{\\!{}^{+}}$, as in Robinson-Trautman space-times [24], none of it
registers in $\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal R}}$! Thus,
there is an interesting complementarity. In addition,
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal{\hat{R}}}}$ and
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal R}}$ have an extremely simple
structure that greatly facilitates the subsequent analysis: while
$\boldsymbol{\Gamma}_{\rm cov}$ is a complicated non-linear space,
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal{\hat{R}}}}$ is an affine
space, and $\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal R}}$ is a convex
set in a vector space. (This contrast can be traced back to the fact that
elements of $\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal{\hat{R}}}}$ and
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal R}}$ are freely specifiable;
they are not subject to non-linear equations). Thanks to this ‘effective
linearity’, one can endow
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal{\hat{R}}}}$ and
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal R}}$ with a topology in which
two elements are close to one another if they and their first derivatives on
$\hat{\mathcal{R}}$ and $\mathcal{R}$ are close in the $L^{2}$-sense. This
topology is motivated by detailed considerations involving test fields on
asymptotically flat space-times [25, 1].

The symplectic current $\mathfrak{J}_{abc}$ on $\boldsymbol{\Gamma}_{\rm cov}$
is a 3-form whose integral over a Cauchy surface provides the symplectic
structure $\boldsymbol{\omega}$ of $\boldsymbol{\Gamma}_{\rm cov}$. Integrals
of $\mathfrak{J}_{abc}$ on $\hat{\mathcal{R}}$ and $\mathcal{R}$ provide
symplectic structures
$\boldsymbol{\omega}_{\scriptscriptstyle{\cal{\hat{R}}}}$ and
$\boldsymbol{\omega}_{\scriptscriptstyle{\cal R}}$ on
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal{\hat{R}}}}$ and
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal R}}$ respectively. With this
setup at hand, one can compute the Hamiltonians associated with symmetries.
Let us begin with $\mathscr{I}^{+}$. Then each BMS generator $\xi^{a}$ defines
a Hamiltonian vector field $\delta_{\xi}$. However, as is common on infinite
dimensional phase spaces (see, e.g., [26]), $\delta_{\xi}$ is only densely
defined on $\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal{\hat{R}}}}$.
Therefore one first calculates the corresponding Hamiltonian
$H_{\xi}(\\{\hat{D}\\})$ on this subspace and asks if it can be continuously
extended to the full phase space.111If one cannot, then one would have the
‘non-integrability’ problem discussed in in other approaches, see e.g. [22,
27, 28, 13, 29, 16, 30, 31]. The answer is in the affirmative for all BMS
vector fields $\xi^{a}$ and the result is:

$H_{\xi}(\\{\hat{D}\\})=\frac{1}{16\pi
G}\,\,\int_{\mathscr{I}^{+}}\big{[}(\mathcal{L}_{\xi}\hat{D}_{a}-\hat{D}_{a}\mathcal{L}_{\xi})\hat{\ell}_{b}+2\hat{\ell}_{(a}\hat{D}_{b)}\beta\big{]}\,\hat{N}_{cd}\,\hat{q}^{ac}\hat{q}^{bd}\,\,\hat{\epsilon}_{mnp}$
(9)

where $\beta$ is given by $\mathcal{L}_{\xi}\hat{q}_{ab}=2\beta\hat{q}_{ab}$
and $\hat{\epsilon}_{mnp}$ is the volume 3-form on $\mathscr{I}^{+}$. This
entire procedure can be applied to familiar, simple systems such as a massless
scalar field in asymptotically flat space-times, and the Hamiltonians thus
obtained agree with the standard expressions of BMS fluxes across 3-d regions
$\hat{\mathcal{R}}$ of $\mathscr{I}^{+}$, obtained using the stress-energy
tensor of the scalar field [25, 1]. These applications serve to motivate the
specific topology we use on
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal{\hat{R}}}}$ and lead one to
interpret $H_{\xi}(\\{\hat{D}\\})$ of (9) as the flux
$F_{\xi}[\hat{\mathcal{R}}]$ of the BMS momentum across the region
$\hat{\mathcal{R}}$ of $\mathscr{I}^{+}$. Indeed, the right side of (9) is the
standard expression of BMS fluxes, now obtained using the phase space
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal{\hat{R}}}}$ of degrees of
freedom local to the region $\hat{\mathcal{R}}$ of $\mathscr{I}^{+}$. Note
that the procedure does not require additional structures such as preferred
symplectic potentials or extensions of BMS vector fields away from
$\mathscr{I}^{+}$. Finally for open regions $\mathcal{R}$ of
$\Delta^{\\!{}^{+}}$, one can apply exactly the same procedure but now the
result is trivial: the pull-back of the symplectic current
$\mathfrak{J}_{abc}$ to $\Delta^{\\!{}^{+}}$ vanishes identically [13, 23]!
(This is a reflection of the fact that there are no 3-d degrees of freedom in
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal R}}$.) Consequently, fluxes
$F_{\xi}$ $[\mathcal{R}]$ also vanish identically for all generators $\xi^{a}$
of the $\Delta^{\\!{}^{+}}$-symmetry group $\mathfrak{G}$, just as one would
expect physically.

Thus, there is a single framework involving phase spaces of degrees of freedom
that reside in open regions of WIHs. But it yields very different results
depending on which of the two types of WIHs it is applied to. When applied to
$\Delta^{\\!{}^{+}}$, it predicts zero fluxes. When applied to
$\mathscr{I}^{+}$, it yields the expressions of BMS fluxes with all the
subtleties; e.g., including both the so-called ‘hard and soft terms’ in the
expression of the BMS supermomenta.

Finally, a natural extension of this framework leads to 2-surface charges
$Q_{\xi}[S]$ associated with cross-section $S$ of $\mathscr{I}^{+}$ or
$\Delta^{\\!{}^{+}}$. The starting point is the observation that (conformal)
Einstein’s equations (and Bianchi identities) imply that the flux 3-forms
–that constitute the integrand of the right side of (9)– are exact, whence the
integrals of their 2-forms potentials provide us with charges $Q_{\xi}[S]$.
However, this procedure requires us to go beyond the phase spaces of local
degrees of freedom use additional inputs. In particular, one has to return to
full $\boldsymbol{\Gamma}_{\rm cov}$ since one needs to use full field
equations, not just their pull-backs to the WIHs. In the case of
$\mathscr{I}^{+}$, $\boldsymbol{\Gamma}_{\rm cov}$ enables one to access the
‘Coulombic’ degrees of freedom that are absent in
$\boldsymbol{\Gamma}_{\\!\scriptscriptstyle{\cal{\hat{R}}}}$ but are needed to
obtain the explicit expressions of the charge 2-forms. These results, together
with a comparison with other procedures [22, 13] to define charges, appear in
the companion paper [1].

Discussion: To summarize, $\Delta^{\\!{}^{+}}$ and $\mathscr{I}^{+}$ share a
number of properties, inherited directly from geometric WIHs $\mathfrak{h}$.
In particular, their intrinsic metrics are non-dynamical and their symmetry
groups are almost the same. In both cases, the intrinsic connections $D$ and
$\hat{D}$ are dynamical, satisfying Eq. (2) that holds on any $\mathfrak{h}$.
But complementary terms on the right side of this equation trivialize in the
two cases: While the Ricci terms vanish in $\Delta^{\\!{}^{+}}$, the Weyl term
vanishes on $\mathscr{I}^{+}$. Thanks to this subtle and surprising
‘complementarity’ $\Delta^{\\!{}^{+}}$ and $\mathscr{I}^{+}$ can share a large
number of properties and, at the same time, display diametrically opposite
physics. $\Delta^{\\!{}^{+}}$ represents black hole horizons in equilibrium
that lie in the strong field regime, and no radiation crosses them.
$\mathscr{I}^{+}$ lies in the near-flat asymptotic regime and provides the
arena for discussions of fluxes carried away by gravitational waves! This
unified WIH framework can provide a foundation for ‘gravitational wave
tomography’ [32]– imaging horizon dynamics using black hole mergers using
waveforms at $\mathscr{I}^{+}$– and also have application the analysis of
black hole evaporation in quantum gravity.

This interwoven structure is reminiscent of a musical fugue in four voices.
The first voice is that of a geometric WIHs $\mathfrak{h}$. It introduces the
theme/subject. The second voice is that of black hole horizons
$\Delta^{\\!{}^{+}}$. It picks up on the theme and further develops it,
providing us with one interpretation of the main subject that conveys the
serenity of equilibrium. The third voice is that of null infinity
$\mathscr{I}^{+}$. It continues the main theme but is contrapuntal because it
exudes change and dynamism, with potential for bursts of energy and momentum.
The fourth voice returns to the central idea enunciated by the first: a single
theme –now, the phase space of local degrees of freedom– can display the deep-
down unity in spite of the diversity of manifestations.

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