# Spatial Context based Angular Information Preserving Projection for
Hyperspectral Image Classification

Minshan Cui, and Saurabh Prasad The authors are with the Department of
Electrical & Computer Engineering, University of Houston, Houston, TX, 77004
USA. Contact author: Saurabh Prasad (email: saurabh.prasad@ieee.org).

###### Abstract

Dimensionality reduction is a crucial preprocessing for hyperspectral data
analysis - finding an appropriate subspace is often required for subsequent
image classification. In recent work, we proposed supervised angular
information based dimensionality reduction methods to find effective
subspaces. Since unlabeled data are often more readily available compared to
labeled data, we propose an unsupervised projection that finds a lower
dimensional subspace where local angular information is preserved. To exploit
spatial information from the hyperspectral images, we further extend our
unsupervised projection to incorporate spatial contextual information around
each pixel in the image. Additionally, we also propose a sparse representation
based classifier which is optimized to exploit spatial information during
classification - we hence assert that our proposed projection is particularly
suitable for classifiers where local similarity and spatial context are both
important. Experimental results with two real-world hyperspectral datasets
demonstrate that our proposed methods provide a robust classification
performance.

##  I Introduction

Hyperspectral imaging techniques have been widely used for a variety of
applications such as mineral mapping, surveillance, and environmental
monitoring. With its rich spectral information, hyperspectral data provide
unique spectral patterns of the materials present in the scene. This
capability makes hyperspectral data especially suitable for ground cover
classification problems. Although hyperspectral data forms a high dimensional
feature space, the information is often effectively represented in a much
lower dimensional subspace — hence, finding an appropriate subspace is crucial
for effective hyperspectral data classification. .

Several linear and non-linear dimensionality reduction methods have been
proposed in literature to address this issue [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12]. The objective function of such dimensionality reduction methods are
typically utilize Euclidean distance information. Different from such
Euclidean-based dimensionality reduction methods, angle or correlation-based
dimensionality reduction methods have been proposed in literature to exploit
specific benefits of angular information — something that is particularly
relevant for hyperspectral data. In [13], correlation discriminant analysis
(CDA) is proposed to transform the data so that the ratio of between-class and
within-class sample correlation is maximized. In [14], the authors propose
several supervised spectral angle based dimensionality reduction methods for
robust hyperspectral data classification.

In typical remote sensing data classification applications, the process of
collecting labeled training data is often very time consuming and expensive.
Unlabeled data on the other hand is easily available and hence unsupervised
dimensionality reduction methods that effectively learn the most appropriate
subspace can hence be readily utilized. With that in mind, we propose an
unsupervised counterpart of our recently proposed supervised subspace
algorithm, the angular discriminant Analysis (ADA). This unsupervised
counterpart is referred to as local similarity preserving projection (LSPP) in
this paper. Unlike ADA which requires labeled training samples, LSPP does not
require labeled training samples to learn the projection matrix. Additionally,
it is well-known that utilizing spatial information in hyperspectral data can
dramatically improve the classification accuracies because any such method
accounts for the spatial variability of spectral content in local spatial
neighborhoods. This follows from the observation that spatially neighboring
pixels are highly likely to belong to the same class and have similar spectral
characteristics. To incorporate such spatial information of hyperspectral data
into our unsupervised projection, we develop a spatial information driven
variant of LSPP (SLSPP) which effectively uses the spatial contextual
information around each pixel in hyperspectral images to learn the _optimal
projection_.

Recently, sparse representation-based classification (SRC) has been proposed
for face recognition [15] and achieved a great amount of attention in various
other applications such as digit recognition [16], speech recognition [17],
gesture recognition [18], vehicle classification [19] as well as remote
sensing data classification [20, 21, 22]. The central idea of SRC is to
represent a test sample as a sparse linear combination of atoms in a
dictionary which is formed by all of the available training samples. The class
label of the test sample is assigned to a class whose reconstruction error is
the lowest. In [21], the authors propose a join sparsity model to incorporate
the contextual information of test samples to improve the classification
performance of SRC. However, the contextual information of training samples
have not been used. Besides the proposed dimensionality reduction methods, we
also propose a sparse representation based classifier which takes into account
the spatial information for both training and test samples in this work.
Experimental results through two real-world hyperspectral datasets demonstrate
that our proposed methods outperform other existing approaches.

The rest of the paper is organized as follows. In Sec. II-A, we briefly
introduce the recently developed supervised dimensionality reduction methods
(ADA and LADA), and Sec. II-B and Sec. II-C describe the proposed LSPP and
SLSPP dimensionality reduction methods respectively. Sec. III describes the
proposed classification method. In Sec. IV, the two practical hyperspectral
benchmarking datasets employed to validate the proposed method are described,
and the experimental results illustrating the benefits of the proposed work is
described in Sec. V.

##  II Spectral Angle-Based Dimensionality Reduction

###  II-A ADA and LADA

Recently, supervised angular discriminant analysis (ADA) was proposed in [14].
ADA finds a subspace where the angular separation of between-class samples is
maximized while the angular separation of within-class samples is
simultaneously minimized. It is mainly used to improve the classification
performance of NN with cosine angle distance and SRC with OMP as the recovery
method.

Let us define
$\\{\boldsymbol{x}_{i}\in\mathbb{R}^{d},y_{i}\in\\{1,2,\ldots,c\\},i=1,2,\ldots,n\\}$
to be the $d$-dimensional $i$-th training sample with an associated class
label $y_{i}$, where $c$ is the number of classes and $n$ is the total number
of training samples. $n=\sum_{l=1}^{c}{n_{l}}$ where $n_{l}$ denotes the
number of training samples from class $l$. Let
$\boldsymbol{\mathit{T}}\in\mathbb{R}^{d\times r}$ be the projection matrix,
where $r$ denotes the reduced dimensionality. The projection
$\mathit{T}_{\textit{ADA}}$ in ADA can obtained by solving

$\displaystyle\boldsymbol{\mathit{T}}_{\textit{ADA}}$
$\displaystyle\approx\operatornamewithlimits{argmin}_{\boldsymbol{\mathit{T}}\in\mathbb{R}^{d\times
r}}\left[\operatorname{tr}\big{(}(\boldsymbol{\mathit{T}}^{t}\boldsymbol{\mathit{O}}^{(\text{w})}\boldsymbol{\mathit{T}})^{-1}\boldsymbol{\mathit{T}}^{t}\boldsymbol{\mathit{O}}^{(\text{b})}\boldsymbol{\mathit{T}}\big{)}\right].$
(1)

where $\boldsymbol{\mathit{O}}^{(\text{w})}$ and
$\boldsymbol{\mathit{O}}^{(\text{b})}$ are defined as

$\displaystyle\boldsymbol{\mathit{O}}^{(\text{w})}$
$\displaystyle=\sum^{c}_{l=1}\sum_{i:y_{i}=l}\boldsymbol{\mu}_{l}\boldsymbol{x}_{i}^{t},$
(2) $\displaystyle\boldsymbol{\mathit{O}}^{(\text{b})}$
$\displaystyle=\sum^{c}_{l=1}n_{l}\boldsymbol{\mu}\boldsymbol{\mu}_{l}^{t}.$
(3)

The problem in (1) can be converted to a generalized eigenvalue problem.

The authors in [14] also extend ADA into its localized variant named local
angular discriminant analysis (LADA) by incorporating a locality preserving
property through an affinity matrix. LADA was shown to handle more complex,
potentially multimodal data distributions. The projection matrix
$\mathit{T}_{\textit{LADA}}$ of LADA can be obtained by solving

$\displaystyle\boldsymbol{\mathit{T}}_{\textit{LADA}}$
$\displaystyle=\operatornamewithlimits{argmin}_{\boldsymbol{\mathit{T}}\in\mathbb{R}^{d\times
r}}\left[\operatorname{tr}\big{(}(\boldsymbol{\mathit{T}}^{t}\boldsymbol{\mathit{O}}^{(\text{lw})}\boldsymbol{\mathit{T}})^{-1}\boldsymbol{\mathit{T}}^{t}\boldsymbol{\mathit{O}}^{(\text{lb})}\boldsymbol{\mathit{T}}\big{)}\right].$
(4)

The within-class and between-class outer product matrices of LADA are defined
as

$\displaystyle\boldsymbol{\mathit{O}}^{(\text{lw})}$
$\displaystyle=\sum_{i,j=1}^{n}\mathit{W}_{ij}^{(\text{lw})}\boldsymbol{x}_{i}\boldsymbol{x}_{j}^{t},$
(5) $\displaystyle\boldsymbol{\mathit{O}}^{(\text{lb})}$
$\displaystyle=\sum_{i,j=1}^{n}\mathit{W}_{ij}^{(\text{lb})}\boldsymbol{x}_{i}\boldsymbol{x}_{j}^{t},$
(6)

where the weight matrices are defined based on heat kernel $\mathit{A}_{ij}$
as

$\displaystyle\mathit{W}^{(\text{lw})}_{ij}$
$\displaystyle=\begin{cases}\mathit{A}_{ij}/n_{l},&\text{if
$y_{i},y_{j}=l$},\\\ 0,&\text{if $y_{i}\neq y_{j}$},\end{cases}$ (7)
$\displaystyle\mathit{W}^{(\text{lb})}_{ij}$
$\displaystyle=\begin{cases}\mathit{A}_{ij}(1/n-1/n_{l}),&\text{if
$y_{i},y_{j}=l$},\\\ 1/n,&\text{if $y_{i}\neq y_{j}$}.\end{cases}$ (8)

###  II-B LSPP

In this paper, we seek to make two related contributions within the context of
angular discriminant analysis — (1) developing an unsupervised approach to
spectral angle based subspace learning, where local spectral angles are
preserved following this unsupervised project, and (2) developing a projection
that incorporates spatial information when learning such an _optimal_
projection. We first form a unsupervised version of ADA which we refer to as
local similarity preserving projection (LSPP). It seeks a lower dimensional
space where the correlation or angular relationship between samples that are
neighbors in the feature space are preserved. We can also think of it as an
angular equivalent of the commonly employed locality preserving projection
(LPP) [1].

Let $\textit{x}_{i}$ be the $i$-th training sample and P be the $d\times r$
projection matrix, where $r$ is the reduced dimensionality. The objective
function of LSPP can be simplified as

I
$\displaystyle=\sum_{ij}\textit{W}_{ij}(\textit{P}^{t}\textit{x}_{i})^{t}(\textit{P}^{t}\textit{x}_{j})$
$\displaystyle=\sum_{ij}\operatorname{\emph{tr}}\big{[}\textit{W}_{ij}(\textit{P}^{t}\textit{x}_{i})^{t}(\textit{P}^{t}\textit{x}_{j})\big{]}$
$\displaystyle=\sum_{ij}\operatorname{\emph{tr}}\big{[}\textit{W}_{ij}\textit{P}^{t}\textit{x}_{j}(\textit{P}^{t}\textit{x}_{i})^{t}\big{]}$
$\displaystyle=\sum_{ij}\operatorname{\emph{tr}}\big{[}\textit{P}^{t}\textit{W}_{ij}\textit{x}_{i}\textit{x}_{j}^{t}\textit{P}\big{]}$
$\displaystyle=\operatorname{\emph{tr}}\big{[}\textit{P}^{t}\textit{X}\textit{W}\textit{X}^{t}\textit{P}\big{]}$
(9)

The heat kernel $\textit{W}_{ij}\in[0,1]$ between $\textit{x}_{i}$ and
$\textit{x}_{j}$ is defined as

$\textit{W}_{ij}=\exp\left(-\frac{\|\textit{x}_{i}-\textit{x}_{j}\|^{2}}{\sigma}\right),$
(10)

where $\sigma$ is the parameter in the heat kernel.

We impose a constraint
($\textit{P}^{t}\textit{X}\textit{D}\textit{X}^{t}\textit{P}=1$ where
$D_{ii}=\sum_{j}\textit{W}_{ij}$) to avoid biases caused by different samples.
The larger the value $D_{ii}$ which is corresponding to $i$-th training
sample, the more important $i$-th training sample is.

The final objective function is defined as

$\operatornamewithlimits{argmax}_{\textit{P}}{\
\operatorname{\emph{tr}}\big{[}\textit{P}^{t}\textit{X}\textit{W}\textit{X}^{t}\textit{P}}\big{]}\qquad\operatorname{s.t.}\quad\textit{P}^{t}\textit{X}\textit{D}\textit{X}^{t}\textit{P}=I$
(11)

The problem in 11 can be solved as a generalized eigenvalue problem as

$\textit{X}\textit{W}\textit{X}^{t}\textit{P}=\lambda\textit{X}\textit{D}\textit{X}^{t}\textit{P}$
(12)

The projection matrix P are the eigenvectors corresponding to the $r$ largest
eigenvalues.

###  II-C SLSPP

It is well understood from many recent works [23, 21, 24, 25], that by aking
into account the spatial neighborhood information, hyperspectral image
classification accuracy can be significantly increased. This is based on the
observation that spatially neighboring samples in hyperspectral data often
consist of similar materials,and hence they are spectrally correlated. In
order to utilize spatially neighboring samples in a lower-dimensional
subspace, the spatial neighborhood relationship of hyperspectral data should
be preserved. To address this problem, we propose a spatial variant of LSPP
(SLSPP) in this work to further improve the classification accuracies. Let
$\\{\textit{z}_{k},k\in\Omega_{i}\\}$ be the spatial neighborhood samples
around a training sample $\textit{x}_{i}$, then the objective function of
SLSPP can be reduced to

I
$\displaystyle=\sum_{i}\sum_{k\in\Omega_{i}}\textit{W}_{ik}(\textit{P}^{t}\textit{x}_{i})^{t}(\textit{P}^{t}\textit{z}_{k})$
$\displaystyle=\sum_{i}\sum_{k\in\Omega_{i}}\operatorname{\emph{tr}}\big{[}\textit{W}_{ik}(\textit{P}^{t}\textit{x}_{i})^{t}(\textit{P}^{t}\textit{z}_{k})\big{]}$
$\displaystyle=\sum_{i}\sum_{k\in\Omega_{i}}\operatorname{\emph{tr}}\big{[}\textit{W}_{ik}\textit{P}^{t}\textit{z}_{k}(\textit{P}^{t}\textit{x}_{i})^{t}\big{]}$
$\displaystyle=\sum_{i}\sum_{k\in\Omega_{i}}\operatorname{\emph{tr}}\big{[}\textit{P}^{t}\textit{W}_{ik}\textit{z}_{k}\textit{x}_{i}^{t}\textit{P}\big{]}$
$\displaystyle=\operatorname{\emph{tr}}\big{[}\textit{P}^{t}M\textit{P}\big{]}$
(13)

where
$M=\sum_{i}\sum_{k\in\Omega_{i}}\textit{W}_{ik}\textit{z}_{k}\textit{x}_{i}^{t}$.

The final objective function is defined as

$\operatornamewithlimits{argmax}_{\textit{P}}{\
\operatorname{\emph{tr}}\big{[}\textit{P}^{t}M\textit{P}}\big{]}\qquad\operatorname{s.t.}\quad\textit{P}^{t}\textit{P}=I$
(14)

Similar to LSPP, the projection matrix P are the eigenvectors corresponding to
the $r$ largest eigenvalues.

Note that both LSPP and SLSPP are unsupervised projections in the sense that
they do not require labels when learning the projections. We also note that a
projection such as SLSPP is particularly beneficial when the backend
classifier utilizes the spatial structure in the spatially neighboring pixels
for each test pixel during classification. Towards that end, we next propose a
modified sparse representation based classifier that utilizes sparse
representations of the entire spatial neighborhood of a test pixel
simultaneously when making a decision.

##  III Sparse Representation-Based Classification via SBOMP

###  III-A SBOMP

The simultaneous orthogonal matching pursuit (SOMP) [26] and block orthogonal
matching pursuit (BOMP) [27] are all variants of orthogonal matching pursuit
(OMP) that explore the _block structure_ of test samples and training samples
respectively. In this work, we effectively combine these two recovery methods
and propose SBOMP to explore the block structure of both training and test
samples simultaneously. SBOMP is illustrated in Algorithm 1. Let $x_{t}$ be a
test sample. Assume $\textit{A}_{i}$ contains spatial neighborhood samples
around $x_{i}$ (inclusive of $x_{i}$), $S$ contains the spatial neighborhood
samples of $x_{t}$ (inclusive of $x_{t}$) and $K$ is the sparsity level. SBOMP
estimates the coefficient $\hat{\textit{C}}$ based on the $K$ mostly
correlated spatial training samples in A.

Algorithm 1 SBOMP

1: Input: A spectral-spatial training data
$\textit{A}=\\{\textit{A}_{i}\\}_{i=1}^{n}$, test data S and row sparsity
level $K$.

2: Initialize $\textit{R}^{0}=\textit{S}$, $\Lambda^{0}=\emptyset$, and the
iteration counter $m=1$.

3: while $m\leq K$ do

4: Update the support set $\Lambda^{m}=\Lambda^{m-1}\cup\lambda$ by solving
$\lambda=\operatornamewithlimits{argmax}_{i=1,2,\ldots,n}\|\textit{A}_{i}^{t}\textit{R}^{m-1}\|_{2,1}.$

5: Derive the coefficient matrix $C^{m}$ based on
$\textit{C}^{m}=\left(\textit{A}^{t}_{\Lambda^{m}}\textit{A}_{\Lambda^{m}}\right)^{-1}\textit{A}^{t}_{\Lambda^{m}}S$

6: Update the residual matrix $\textit{R}^{m}$
$\textit{R}^{m}=\textit{S}-\textit{A}_{\Lambda^{m}}\textit{C}^{m}$

7: $m\leftarrow m+1$

8: end while

9: Output: Coefficient matrix $\hat{\textit{C}}=\textit{C}^{m-1}$.

###  III-B Classification via SBOMP

The classification method employed after SLSPP is SBOMP-based Classification
(SBOMP-C), as described in Algorithm 2. Since SLSPP can preserve the spatial
neighboring samples for both training and test samples in a lower-dimensional
subspace, by using SBOMP-C, we can exploit the block structure relationship
effectively in the spatial domain between training and test samples. The block
diagram of proposed methods are illustrated in Fig. 1.

Algorithm 2 SBOMP-C

1: Input: A spectral-spatial training data
$\textit{A}=\\{\textit{A}_{i}\\}_{i=1}^{n}$, test data S and row sparsity
level $K$.

2: Calculate row-sparsity coefficient $\hat{\textit{C}}$ based on
$\hat{\textit{C}}=\text{SBOMP}\ (\textit{A},S,K)$

3: Calculate residuals for each class
$\textit{r}_{k}(\textit{S})=\|\textit{S}-\textit{A}\delta_{k}(\hat{\textit{C}})\|_{2},\quad
k=1,2,\ldots,c$

4: Determine the class label of S based on
$\omega=\operatornamewithlimits{argmin}_{k=1,2,\ldots,c}(\textit{r}_{k}(\textit{S})).$

5: Output: A class label $\omega$.

---
Figure 1: Block diagram of the proposed work.

##  IV Experimental Hyperspectral Data

We validate the proposed approaches with two hyperspectral imagery datasets —
a standard benchmarking dataset from the University of Pavia (an urban ground
cover classification problem), and a hyperspectral imagery data collected by
our lab to study wetland species composition.

###  IV-A University of Pavia Data

The first experimental hyperspectral dataset employed was collected using the
Reflective Optics System Imaging Spectrometer (ROSIS) sensor [28]. This image,
covering the University of Pavia, Italy, has 103 spectral bands with a spatial
coverage of $610\times 340$ pixels, and 9 classes of interests are considered
in this dataset. A three-band true color image and its ground-truth are shown
in Figure 2. This dataset represents a standardized benchmark as it is a well
understood and widely studied dataset that is used to quantify performance of
machine learning algorithms with hyperspectral data.

|
---|---
(a) | (b)

Figure 2:  (a) True color image and (b) ground-truth of the University of
Pavia hyperspectral data.

###  IV-B Wetland Data

The second hyperspectral data used in this work was acquired by us in
Galveston, Texas in October, 2014 which includes two different wetland scenes
captured at ground-level (side-looking views) over wetlands in Galveston. This
is an unprecedented dataset in that it represents “street view” type
hyperspectral images that have very different characteristics and are very
useful for on-ground sensing. The two image cubes are referred to as area 1,
and area 2, representing different regions of the wetlands that were imaged —
In addition to common wetland classes, area 2 has Black Mangrove (_Avicennia
germinans_) trees in the scene, a species which is of particular interest in
ecological studies of wetlands, in addition to Spartina. The images were
annotated with species information by an expert in Marine biology, who
identified the various species in these hyperspectral scenes. This data was
acquired using a Headwall Photonics hyperspectral imager which provides
measurements in 325 spectral bands with a spatial size of $1004\times 5130$.
The hyperspectral data uniformly spanned the visible and near-infrared
spectrum from $400nm-1000nm$. The objects of interests are primarily
vegetation species common in such wetlands. Six different classes were
identified in area-1 including soil, _symphyotrichum_ , _schoenoplectus_ ,
_spartina patens_ , _borrichia_ and _rayjacksonia_. The second area includes
_Avicennia germinans_ , _batis_ , _schoenoplectus_ , _spartina alterniflora_ ,
_soil_ , _water_ and _bridge_. Since soil and _schoenoplectus_ are included in
both areas, the total number of classes in the combined library are eleven.
This dataset represents a “real-world” application of our method for
hyperspectral imaging in the field on arbitrary problems of interest.

---
Wetland data, area - 1
Wetland data, area - 2

Figure 3: True color images of the wetland dataset inset with ground truth.

##  V Experimental Settings and Results

The efficacy of the proposed LSPP, SLSPP and SBOMP are evaluated as a function
of training samples per class using the two practical hyperspectral datasets
mentioned previously. SLSPP–SBOMP-C indicates the data is projected based on
SLSPP, and SBOMP-C is employed as the backend classifier. A nearest-neighbor
(NN) classifier with cosine angle distance is used after LSPP and LADA
projection, since they do not take spatial information into account when
deriving the projection matrix. Similar to SLSPP–SOMP-C, LSPP–NN and LADA–NN.
Each experiment is repeated 10 times using a repeated random subsampling
validation technique, and the average accuracy is reported. The number of test
samples per class is fixed to 100 for every random subsampling.

Fig. 4 shows the classification accuracies as a function of training samples
for the University of Pavia hyperspectral data. The parameter for each
algorithm is determined via searching through a wide range of the parameter
space and the accuracies reported in this plot is based on the optimal
parameter values. As can be seen from this figure, our proposed SLSPP followed
by SBOMP-C gives the highest classification accuracies consistently over a
wide range of the training sample size. LSPP–NN also gives better
classification result compared with LADA–NN.

We perform a similar analysis using the wetland hyperspectral data. Fig. 5 and
Fig. 6 plot the classification accuracies with respect to the training sample
size per class for wetland area - 1 and wetland area - 2 respectively. It can
be seen from the two plots that the proposed methods generally outperform
other baseline methods for the vegetation type of hyperspectral data which
demonstrate the diverse applicability of the proposed methods.

Figure 4: Overall classification accuracy (%) versus number of training
samples for the University of Pavia data.
Figure 5: Overall classification accuracy (%) versus number of training
samples for the wetland data, area - 1.
Figure 6: Overall classification accuracy (%) versus number of training
samples for the wetland data, area - 2.

The class specific accuracies for different datasets are shown in Table I,
Table II and Table III respectively. In this experiment, the training sample
size per class is fixed to 10 and the test sample size is 100 per class. Each
experiment is repeated 10 times and the average accuracy is reported. As can
be seen from the class-specific tables for the wetland data, the spatial
information plays a crucial role, especially for the species with complex
textures and shapes such as _Spartina-Patens_ , _Sedge_ and _Symphyotrichum_.

TABLE I: Class-specific accuracies (%) for the University of Pavia Data.

Class Name / Algorithm | LSPP–SBOMP-C | LSPP–SOMP-C | LSPP–NN | LADA–NN
---|---|---|---|---
Asphalt | 77.2 | 60.8 | 31.4 | 30.9
Meadows | 64.4 | 62.5 | 60.2 | 59.6
Gravel | 79.2 | 65.5 | 61.1 | 59
Trees | 88.2 | 79 | 94.4 | 94.1
Metal Sheets | 98.9 | 98.2 | 99.9 | 99.9
Soil | 67.6 | 59.1 | 57.1 | 53.4
Bitumen | 84.9 | 83 | 84.8 | 84.4
Bricks | 60.1 | 64.8 | 66.2 | 62.9
Shadows | 99.2 | 96.9 | 96.9 | 95.9
Overall Accuracy | 80.0 | 74.4 | 72.4 | 71.1

TABLE II: Class-specific accuracies (%) for the Wetland, Area-1 data.

Class Name / Algorithm | LSPP–SBOMP-C | LSPP–SOMP-C | LSPP–NN | LADA–NN
---|---|---|---|---
Soil | 99.8 | 100 | 99.8 | 99.9
Symphyotrichum | 86.8 | 85.5 | 78.1 | 76.1
Sedge | 93.4 | 93.3 | 91.6 | 92.1
Spartina-Paten | 68.4 | 59.1 | 64.2 | 56
Borrichia | 91.6 | 91.2 | 91.7 | 93.7
Rayjacksonia | 91.6 | 89 | 84.7 | 92.2
Overall Accuracy | 88.6 | 86.4 | 85.0 | 85.0

TABLE III: Class-specific accuracies (%) for the Wetland, Area-2 data.

Class Name / Algorithm | LSPP–SBOMP-C | LSPP–SOMP-C | LSPP–NN | LADA–NN
---|---|---|---|---
Soil | 96.1 | 96.8 | 87.7 | 86.7
Mangrove Tree | 97.1 | 95 | 97.1 | 98.5
Batis | 99.2 | 99.1 | 97.9 | 97.2
Sedge | 97 | 94.3 | 92.3 | 89.8
Spartina-Alterniflora | 89.7 | 90.2 | 75.3 | 48
Water | 99 | 99 | 98.5 | 98.7
Bridge | 99.9 | 99.4 | 94.3 | 85.7
Overall Accuracy | 96.9 | 96.3 | 91.9 | 86.4

Next, we analyze the effect of the window size (that defines the spatial
neighborhood) for the SLSPP method. Fig. 7 depicts the classification
accuracies as a function of different window size using the University of
Pavia dataset. From this figure, we can see that the optimal window size is 5.

Figure 7: Overall classification accuracy (%) versus different window size for
the University of Pavia and Wetland hyperspectral data.

In this work, we also visualize the data distributions after projection for
the proposed angle-based SLSPP and the Euclidean-based LPP methods. Figure 8
(a) shows the subset image for University of Pavia and Figure 8 (b) plots all
the training samples used in this experiment on an $\ell_{2}$ normalized
sphere. Figure 8 (c) and (d) show the same samples after an SLSPP and LPP
projection on an $\ell_{2}$ normalized sphere respectively. $U_{1}$, $U_{2}$
and $U_{3}$ are the three projections found by SLSPP and LPP corresponding to
the largest eigenvalues. As can be seen from this figure, SLSPP is much more
effective at preserving the inter-sample relationships in terms of spectral
angle in the lower-dimensional subspace compared to LPP.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 8: Illustrating the (a) subset image of the University of Pavia, (b)
original samples, (c) SLSPP projected samples and (d) LPP projected samples on
the sphere.

##  VI Conclusion

In this work, we have presented an unsupervised variant (LSPP) of the recently
developed supervised dimensionality reduction method — ADA, as well as its
spatial variant, SLSPP, that utilize spatial information around the samples
when learning the projections. By incorporating spatial information in the
dimensionality reduction projection, we are able to learn much more effective
subspaces that not only preserves angular information among the training
pixels in the feature space, but also their spatial neighbors. We also propose
a related sparse representation classifier (SBOMP-C) that is optimized for
feature spaces generated by SLSPP. Through experimental results based on two
practical real-world hyperspectral datasets, we note that the proposed methods
significantly outperform the baseline methods.

## References

  * [1] X. He and P. Niyogi, “Locality preserving projections,” in _Proc. Conf. Adv. Neural Inf. Process. Syst._ , Vancouver, Canada, 2003, pp. 234–241.
  * [2] S. T. Roweis and L. K. Saul, “Nonlinear dimensionality reduction by locally linear embedding,” _Science_ , vol. 290, no. 5500, pp. 2323–2326, 2000.
  * [3] J. B. Tenenbaum, V. De Silva, and J. C. Langford, “A global geometric framework for nonlinear dimensionality reduction,” _Science_ , vol. 290, no. 5500, pp. 2319–2323, 2000.
  * [4] M. Belkin and P. Niyogi, “Laplacian eigenmaps and spectral techniques for embedding and clustering.” in _NIPS_ , vol. 14, 2001, pp. 585–591.
  * [5] M. Sugiyama, “Dimensionaltity reduction of multimodal labeled data by local Fisher discriminant analysis,” _J. Mach. Learn. Research_ , vol. 8, no. 5, pp. 1027–1061, May 2007.
  * [6] Q. Gao, Y. Huang, H. Zhang, X. Hong, K. Li, and Y. Wang, “Discriminative sparsity preserving projections for image recognition,” _Pattern Reco._ , vol. 48, no. 8, pp. 2543–2553, 2015.
  * [7] L. Qiao, S. Chen, and X. Tan, “Sparsity preserving projections with applications to face recognition,” _Pattern Reco._ , vol. 43, no. 1, pp. 331–341, January 2010.
  * [8] B. Raducanu and F. Dornaika, “A supervised non-linear dimensionality reduction approach for manifold learning,” _Pattern Reco._ , vol. 45, no. 6, pp. 2432–2444, 2012.
  * [9] L. Zhang, S. Chen, and L. Qiao, “Graph optimization for dimensionality reduction with sparsity constraints,” _Pattern Reco._ , vol. 45, no. 3, pp. 1205–1210, 2012.
  * [10] Y. Cui and L. Fan, “A novel supervised dimensionality reduction algorithm: Graph-based fisher analysis,” _Pattern Reco._ , vol. 45, no. 4, pp. 1471–1481, 2012.
  * [11] C. Hou, C. Zhang, Y. Wu, and F. Nie, “Multiple view semi-supervised dimensionality reduction,” _Pattern Reco._ , vol. 43, no. 3, pp. 720–730, 2010.
  * [12] Y. Song, F. Nie, C. Zhang, and S. Xiang, “A unified framework for semi-supervised dimensionality reduction,” _Pattern Reco._ , vol. 41, no. 9, pp. 2789–2799, 2008.
  * [13] Y. Ma, S. Lao, E. Takikawa, and M. Kawade, “Discriminant analysis in correlation similarity measure space,” in _Proc. Int. Conf. Mach. Learn._ , 2007, pp. 577–584.
  * [14] M. Cui and S. Prasad, “Angular discriminant analysis for hyperspectral image classification,” _IEEE J. Sel. Topics Signal Process._ , vol. 9, no. 6, pp. 1003–1015, April 2015.
  * [15] J. A. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” _IEEE Trans. Pattern Anal. Mach. Intell._ , vol. 31, no. 2, pp. 210–227, February 2009.
  * [16] K. Labusch, E. Barth, and T. Martinetz, “Simple method for high-performance digit recognition based on sparse coding,” _IEEE Trans. Neural Networks_ , vol. 19, no. 11, pp. 1985–1989, 2008.
  * [17] J. F. Gemmeke, T. Virtanen, and A. Hurmalainen, “Exemplar-based sparse representations for noise robust automatic speech recognition,” _IEEE Trans. Audio Speech Lang. Process._ , vol. 19, no. 7, pp. 2067–2080, 2011.
  * [18] Y. Zhou, K. Liu, R. E. Carrillo, K. E. Barner, and F. Kiamilev, “Kernel-based sparse representation for gesture recognition,” _Pattern Reco._ , vol. 46, no. 12, pp. 3208–3222, 2013.
  * [19] X. Mei and H. Ling, “Robust visual tracking and vehicle classification via sparse representation,” _IEEE Trans. Pattern Anal. Mach. Intell._ , vol. 33, no. 11, pp. 2259–2272, 2011.
  * [20] M. Cui and S. Prasad, “Class–dependent sparse representation classifier for robust hyperspectral image classification,” _IEEE Trans. Geosci. Remote Sens._ , vol. 53, no. 5, pp. 2683–2695, September 2015.
  * [21] Y. Chen, N. M. Nasrabadi, and T. D. Tran, “Hyperspectral image classification using dictionary-based sparse representation,” _IEEE Trans. Geosci. Remote Sens._ , vol. 49, no. 10, pp. 3973–3985, October 2011.
  * [22] U. Srinivas, Y. Chen, V. Monga, N. M. Nasrabadi, and T. D. Tran, “Exploiting sparsity in hyperspectral image classification via graphical models,” _IEEE Geosci. Remote Sens. Lett._ , vol. 10, no. 3, pp. 505–509, May 2013\.
  * [23] Y. Tarabalka, J. A. Benediktsson, and J. Chanussot, “Spectral–spatial classification of hyperspectral imagery based on partitional clustering techniques,” _IEEE Trans. Geosci. Remote Sens._ , vol. 47, no. 8, pp. 2973–2987, 2009.
  * [24] J. Li, J. M. Bioucas-Dias, and A. Plaza, “Spectral–spatial hyperspectral image segmentation using subspace multinomial logistic regression and markov random fields,” _IEEE Trans. Geosci. Remote Sens._ , vol. 50, no. 3, pp. 809–823, 2012.
  * [25] M. Cui, S. Prasad, W. Li, and L. M. Bruce, “Locality preserving genetic algorithms for spatial-spectral hyperspectral image classification,” _IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens._ , vol. 6, no. 3, pp. 1688–1697, 2013.
  * [26] J. A. Tropp, A. C. Gilbert, and M. J. Strauss, “Algorithms for simultaneous sparse approximation. part i: Greedy pursuit,” _IEEE Trans. Signal Proces._ , vol. 86, no. 3, pp. 572–588, 2006.
  * [27] Y. C. Eldar, P. Kuppinger, and H. Bölcskei, “Block-sparse signals: Uncertainty relations and efficient recovery,” _IEEE Trans. Signal Proces._ , vol. 58, no. 6, pp. 3042–3054, 2010.
  * [28] P. Gamba, “A collection of data for urban area characterization,” in _Proc. Int. Geosci. Remote Sens. Symp._ , Anchorage, Alaska, September 2004, pp. 69–72.


# Asymptotic charges for spin-1 and spin-2 fields at the critical sets of null
infinity

Mariem Magdy Ali Mohamed 111E-mail<EMAIL_ADDRESS>School of
Mathematical Sciences, Queen Mary, University of London, Mile End Road, London
E1 4NS, United Kingdom.  Juan A. Valiente Kroon 222E-mail
<EMAIL_ADDRESS>School of Mathematical Sciences, Queen
Mary, University of London, Mile End Road, London E1 4NS, United Kingdom.

###### Abstract

The asymptotic charges of spin-1 and spin-2 fields are studied near spatial
infinity. We evaluate the charges at the critical sets where spatial infinity
meets null infinity with the aim of finding the relation between the charges
at future and past null infinity. To this end, we make use of Friedrich’s
framework of the cylinder at spatial infinity to obtain asymptotic expansions
of the Maxwell and spin-2 fields near spatial infinity, which are fully
determined in terms of initial data on a Cauchy hypersurface. Expanding the
initial data in terms of spin-weighted spherical harmonics, it is shown that
only a subset of the initial data, that satisfies certain regularity
conditions, gives rise to well-defined charges at the point where future
(past) infinity meets spatial infinity. Given such initial data, the charges
are shown to be fully expressed in terms of the freely specifiable part of the
data. Moreover, it is shown that there exists a natural correspondence between
the charges defined at future and past null infinity.

##  1 Introduction

The asymptotic structure of gravitational fields in General Relativity can be
studied at past/future null infinities or at spatial infinity. Within this
context, the subject of asymptotic symmetries in General Relativity has gained
increased interest in recent years, due to its relation to soft theorems [1,
2, 3], gravitational memory effect [4, 5, 6] and black hole physics [7, 8].
While the subject of asymptotic symmetries is abstract and theoretical in
nature, the association with the gravitational memory effect provides a
possible link to observations. One of the key concepts for studying the
asymptotic symmetries in General Relativity is that of conformal
compactification, in which points at infinity in physical spacetimes are
mapped to points at finite distances. In the standard conformal
compactification of Minkowski spacetime, points at infinite null separation
are mapped to a smooth null hypersurface, _null infinity_ , part of the
conformal boundary. Null infinity consists of two disjoint sets:
$\mathscr{I}^{+}$ and $\mathscr{I}^{-}$, the future and past null infinity,
respectively. Moreover, the future and past null infinities are joined by
spatial infinity point $i^{0}$. Penrose's notion of _asymptotic simplicity_ is
concerned with identifying spacetimes with an asymptotic structure similar to
that of Minkowski near null infinity. In particular, a spacetime is said to be
asymptotically simple if a null hypersurface $\mathscr{I}$ (null infinity)
with two disjoint sets: $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$, can be
attached to the spacetime, and all null geodesics acquire two distinct
endpoints on the sets $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$—see e.g. [9, 10]
for a precise definition.

In the null portion of the conformal boundary, it was expected that the
asymptotic symmetry group for asymptotically simple spacetimes was to be given
by the Poincaré group —the symmetry group of Minkowski spacetime. However, it
is now established that the asymptotic symmetry group for asymptotically
simple spacetimes is given by the infinite-dimensional symmetry group, the
Bondi-Metzner-Sachs (BMS) group [11, 12]. The infinite number of
supertranslations distinguishes the BMS group from the Poincaré group.
Supertranslations are angle-dependent translations along future or past null
infinity. The BMS symmetry group gives rise to the BMS charges, these charges
are generally not integrable within the context of full gravity theory [13,
14]. However, the prescription in [13] allows one to obtain a definition for
the conserved charges associated with asymptotic symmetries at null infinity.

The study of the asymptotic symmetries in the spatial regime is more
challenging. This is partly due to the singular nature of the conformal
structure near spatial infinity for spacetimes with non-vanishing Arnowitt-
Deser-Misner (ADM) mass. Different representations for spatial infinity have
been proposed in the literature to deal with this singular behaviour e.g. [15,
16, 17]. The asymptotic symmetry group for General Relativity at spatial
infinity had been studied in [20, 15, 38, 16, 18, 19]. Similar to the null
case, it turns out that one also obtains an infinite-dimensional asymptotic
symmetry group, the Spi-group [15, 16], and corresponding conserved charges
associated with these symmetries. The structure of the Spi-group turns out to
be similar to the BMS group. More specifically, the Spi-group is the semi-
direct product of the Lorentz group with the Spi-supertranslations [15].

An interesting problem that arises when looking at the asymptotic symmetries
is the so-called matching problem —i.e. the problem of the relation between
the asymptotic symmetries at past and future null infinity in the limit of
spatial infinity. In [2], Strominger proposed the conjecture stating that the
BMS symmetries at past and future null infinity can be related by antipodal
reflection near spatial infinity. The main technique utilised in proving the
matching problem is to make use of the field equations and charges defined at
spatial infinity to relate the charges at past and future null infinities. On
Minkowski spacetime, the matching problem had been investigated for the spin-1
field (Maxwell field) in [21, 22] and for the spin-2 field (linearised
gravity) in [23]. While on Minkowski spacetime, the transformation between the
adapted asymptotic coordinates at spatial infinity [24] and null infinity [25,
11] can be derived explicitly, the relation between the asymptotic coordinates
for general spacetimes is not explicitly known. This, in addition to the
singular nature of spatial infinity, makes the matching problem more
complicated in more general spacetimes. One way to address this is to make use
of a covariant formulation in which the singular nature of spatial infinity is
resolved. In [26, 27], Ashtekar and Hansen's formulation of asymptotic
flatness was used to show that the matching of supertranslation charges holds
for spin-1 and gravitational fields in general asymptotically flat spacetimes.
In this instance, the notion of asymptotic flatness is the one introduced in
[15]. More recently, the matching problem has been investigated for Lorentz
charges in General Relativity for spacetimes that satisfy Ashtekar and
Hansen's notion of asymptotic flatness [28]. That being said, in this paper,
we revisit the asymptotic symmetries and matching problem for spin-1 and
spin-2 fields on Minkowski spacetime. In contrast to [21, 29, 22, 30], our aim
is to express the asymptotic charges at the critical sets in terms of the
freely specifiable initial data on a Cauchy hypersurface. To do so, we use a
different formulation for spatial infinity —namely, Friedrich's cylinder at
spatial infinity. _The advantage of using Friedrich's formulation is that it
allows us to directly link the charges defined at past and future null
infinity to initial data prescribed on a Cauchy hypersurface._ The approach
followed in this paper will serve as a stepping stone to investigate the
asymptotic charge and the matching problem for the full theory of gravity in
Friedrich's formalism in later work.

Friedrich's conformal representation of spatial infinity was first introduced
in [17] with the aim of formulating a regular initial value problem for the
conformal Einstein field equations. In this representation, the spatial
infinity point is blown up to $(-1,1)\times\mathbb{S}^{2}$, which is known as
the cylinder at spatial infinity $\mathcal{I}$. The end 'points' of the
cylinder, known as the _critical sets_ $\mathcal{I}^{\pm}=\\{\pm
1\\}\times\mathbb{S}^{2}$, are the set of points at which the future and past
null infinity touches the cylinder $\mathcal{I}$. This representation of
spatial infinity proves to be useful in relating quantities at
$\mathcal{I}^{\pm}$ to initial data given on a Cauchy hypersurface. This
framework had been used to obtain expressions for the Newman-Penrose (NP)
constants in terms of initial data in [31]. More recently, it was shown that,
generically, there is no natural correspondence between the NP constants for
spin-1 and spin-2 fields on $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$ for
generic initial data [32], except for the particular case of time-symmetric
initial data.

In the literature, the asymptotic charges are usually given in terms of the NP
gauge. For the purpose of this article, we make sure of frames that are more
suited for the framework of the cylinder at spatial infinity, the so-called
F-gauge. These frames are adapted to a congruence of conformal geodesics and
Cauchy hypersurfaces —see [31]. For Minkowski spacetime, the relation between
the NP gauge and F-gauge is given explicitly in [32]. One can then rewrite the
expressions for the asymptotic charges in terms of field components in the
F-gauge. Using the expansion given in Assumption 1 in the Maxwell case and
given a generic class of initial data, we show that the charges associated
with supertranslations at $\mathcal{I}^{+}$ and at $\mathcal{I}^{-}$ can be
written in terms of freely specifiable initial data on the initial
hypersurface $\mathcal{S}_{\star}\equiv\\{\tau=0\\}$. These charges are not in
general well-defined, and one requires extra conditions —see Lemma 2, for
well-defined charges at the critical sets $\mathcal{I}^{\pm}$. Finally, we
show that there is a natural correspondence between the charges at
$\mathcal{I}^{+}$ and $\mathcal{I}^{-}$ in the sense that they can be traced
back to the same piece of freely specifiable data on the initial Cauchy
hypersurface. Similarly, in the spin-2 case, the expansion given in Assumption
2 and the generic initial data are used to show that the charges associated
with supertranslations at the critical sets can be written in terms of initial
data on the initial hypersurface. The conditions given in Lemma 4 ensures that
the charges are well-defined at $\mathcal{I}^{\pm}$.

As will be seen in the main text of the article, in order to obtain non-
vanishing BMS charges for both the Maxwell and spin-2 fields, one needs to
consider initial data sets which are _boosted_ —that is, the leading terms of
the various components of the initial data contain spin-weighted harmonics
${}_{n}Y_{l,m}$ with $l\geq 1$. For non-boosted initial data sets, only the
$l=0$ harmonic gives a non-trivial BMS charge. The precise form of the notion
of boosted data for the Maxwell and spin-2 fields used in this article are
given in equations (15) and (30), respectively.

### Main results of this article

The main results of the analysis in this article can be summarised in the
following:

######  Theorem.

For generic boosted initial data for the Maxwell and spin-2 fields, the
asymptotic charges at $\mathcal{I}^{\pm}$ are well-defined if and only if the
freely specifiable data on the initial hypersurface $\mathcal{S}_{\star}$
satisfy certain regularity conditions —given by Lemma 2; Lemma 4,
respectively. Moreover, for a given harmonic $Y_{l,m}$ the charges at
$\mathcal{I}^{+}$ and $\mathcal{I}^{-}$ are expressible in terms of the same
piece of freely specifiable data —thus, in this sense, they can be identified.

The detailed formulation of the results in the above theorem is given in
Theorems 1 and 2.

### Outline of the article

This article is structured as follows. In Section 2, we introduce the
construction of the cylinder at spatial infinity and the F-gauge on Minkowski
spacetime. The Newman-Penrose gauge conditions and the relation between the
NP-gauge and the F-gauge for Minkowski spacetime is summarised in Section 3.
In Section 4, the Maxwell equations are expanded in the F-gauge, the initial
data for the Maxwell field are discussed, Finally, the asymptotic charges
associated with supertranslations are written in terms of initial data given
on a spacelike hypersurface. In Section 5, we follow a similar strategy to
expand the spin-2 field equation using the F-gauge and write the asymptotic
charges associated with supertranslations in terms of initial data.

### Notations and conventions

In what follows, the Latin letters $a,\,b,\,c,\ldots$ will denote spacetime
abstract tensorial indices, while $A,\,B,\,C,\ldots$ denote abstract spinorial
indices. The Greek letter $\mu,\,\nu,\dots$ are used as spacetime coordinate
indices while $\alpha,\,\beta,\ldots$ are used as spatial coordinate indices.
Boldface Latin letters ${\bm{a}},{\bm{b}},{\bm{c}},\ldots$ are used as
tensorial frame indices and boldface capital Latin letters
${\bm{A}},{\bm{B}},{\bm{C}},\ldots$ are used as spinorial frame indices. Using
these notations, given a basis ${\bm{e}}_{a}\equiv\\{{\bm{e}}_{{\bm{a}}}\\}$,
the components of a generic tensor $T_{ab}$ with respect to the basis
$\\{{\bm{e}}_{{\bm{a}}}\\}$ can be written as
$T_{{\bm{a}}{\bm{b}}}=T_{ab}{\bm{e}}_{{\bm{a}}}{}^{a}{}{\bm{e}}_{{\bm{b}}}{}^{b}{}$.
Similarly, given a spin basis $\\{o,\iota\\}$ that satisfy $\llbracket
o,\iota\rrbracket=1$, where $\llbracket.,.\rrbracket$ is the antisymmetric
product, the components of a generic spinor $\zeta_{A}$ can be written as
$\zeta_{{\bm{A}}}=\zeta_{A}{\bm{\epsilon}}_{{\bm{A}}}{}^{A}{}$ where

$\displaystyle{\bm{o}}^{A}={\bm{\epsilon}}_{{\bm{0}}}{}^{A}{}\hskip
5.69054pt,\hskip 5.69054pt\iota^{A}={\bm{\epsilon}}_{{\bm{1}}}{}^{A}{}$
$\displaystyle{\bm{o}}_{A}={\bm{\epsilon}}^{{\bm{1}}}{}_{A}{}\hskip
5.69054pt,\hskip 5.69054pt\iota_{A}=-{\bm{\epsilon}}^{{\bm{0}}}{}_{A}{}.$

The antisymmetric product $\llbracket.,.\rrbracket$ of two generic spinors can
be expressed as

$\llbracket\zeta,\lambda\rrbracket=\zeta_{B}\lambda^{B}={\bm{\epsilon}}_{AB}\zeta^{A}\lambda^{B},$

where ${\bm{\epsilon}}_{AB}$ is known as the ${\bm{\epsilon}}$-spinor, and it
is an antisymmetric spinor that can be regarded as an index raising/lowering
object for spinors. In the rest of the article, we express spacetime frames
$\\{{\bm{e}}_{{\bm{a}}}\\}$ in spinorial notation. The spinorial counterpart
of $\\{{\bm{e}}_{{\bm{a}}}\\}$ is given by

${\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}=\sigma^{{\bm{a}}}{}_{{\bm{A}}{\bm{A}}^{\prime}}{\bm{e}}_{{\bm{a}}}$

where $\sigma^{{\bm{a}}}{}_{{\bm{A}}{\bm{A}}^{\prime}}$ are the Infeld-van der
Waerden symbols. Finally, the signature convention for spacetime metrics used
in this article is $(+,-,-,-)$. Throughout, we mostly follow the notation and
conventions of Penrose & Rindler [9] —see also [10].

##  2 The cylinder at spatial infinity and the F-gauge

In this section, we introduce the conformal representation of Minkowski
spacetime, which gives rise to Friedrich's cylinder at spatial infinity. In
this conformal representation, spatial infinity $i^{0}$ is blown up to
$\mathbb{R}\times\mathbb{S}^{2}$. In the following, we will only introduce the
cylinder at spatial infinity for Minkowski spacetime. A more general
discussion of this representation is presented in the original paper [17].

###  2.1 The cylinder at spatial infinity

The Minkowski metric written in Cartesian coordinates
$(\tilde{t},\tilde{x}^{\alpha})$ is given by

$\tilde{{\bm{\eta}}}=\eta_{\mu\nu}{\bm{d}}{\tilde{x}^{\mu}}\otimes{\bm{d}}{\tilde{x}^{\nu}},$

where $\eta_{\mu\nu}=\text{diag}\left(1,-1,-1,-1\right)$. Defining
$\tilde{r}^{2}\equiv\delta_{\alpha\beta}\tilde{x}^{\alpha}\tilde{x}^{\beta}$,
the metric can be written in polar coordinates as

$\tilde{{\bm{\eta}}}={\bm{d}}{\tilde{t}}\otimes{\bm{d}}{\tilde{t}}-{\bm{d}}{\tilde{r}}\otimes{\bm{d}}{\tilde{r}}-\tilde{r}^{2}{\bm{\sigma}},$

where ${\bm{\sigma}}$ is the standard metric on $\mathbb{S}^{2}$. Now
introduce the inversion coordinates $(t,x^{\alpha})$ where

$x^{\mu}=-\frac{\tilde{x}^{\mu}}{\tilde{X}^{2}},\qquad\tilde{X}^{2}\equiv\tilde{\eta}_{\mu\nu}\tilde{x}^{\mu}\tilde{x}^{\nu}.$

Then one can introduce the unphysical metric $\hat{{\bm{\eta}}}\equiv
X^{4}\tilde{{\bm{\eta}}}$ where $X^{2}\equiv\eta_{\mu\nu}x^{\mu}x^{\nu}$.
Hence, we get

$\hat{{\bm{\eta}}}={\bm{d}}{t}\otimes{\bm{d}}{t}-{\bm{d}}{\rho}\otimes{\bm{d}}{\rho}-\rho^{2}{\bm{\sigma}},$

where $\rho^{2}\equiv\delta_{\alpha\beta}x^{\alpha}x^{\beta}$ . In this
conformal representation, the spatial infinity point $i^{0}$ is at the origin.
Introduce $\tau\equiv t/\rho$ and define

${\bm{\eta}}\equiv\frac{1}{\rho^{2}}\hat{{\bm{\eta}}}=\frac{X^{2}}{\rho^{2}}\tilde{{\bm{\eta}}}.$
(1)

Then, one has that

${\bm{\eta}}={\bm{d}}{\tau}\otimes{\bm{d}}{\tau}-\frac{(1-\tau^{2})}{\rho^{2}}{\bm{d}}{\rho}\otimes{\bm{d}}{\rho}+\frac{\tau}{\rho}\left({\bm{d}}{\tau}\otimes{\bm{d}}{\rho}+{\bm{d}}{\rho}\otimes{\bm{d}}{\tau}\right)-{\bm{\sigma}}.$
(2)

This metric gives the desired representation of spatial infinity. In
particular, spatial infinity corresponds to a set with the topology of
$\mathbb{R}\times\mathbb{S}^{2}$. Consider the conformal extension
$(\mathcal{M},{\bm{\eta}})$ where ${\bm{\eta}}$ is given by equation (1) and

$\mathcal{M}\equiv\\{p\in\mathbb{R}^{4}\ |\ -1\leq\tau(p)\leq 1,\ \rho(p)\geq
0\\}.$

Then, the future and past null infinities are located at

$\mathscr{I}^{\pm}\equiv\big{\\{}p\in\mathcal{M}\ |\ \tau(p)=\pm 1\big{\\}}.$

Moreover, the following sets can be introduced

$\displaystyle\mathcal{I}\equiv\big{\\{}p\in\mathcal{M}\ |\ |\tau(p)|<1,\
\rho(p)=0\big{\\}},$
$\displaystyle\mathcal{I}^{0}\equiv\big{\\{}p\in\mathcal{M}\ |\ \tau(p)=0,\
\rho(p)=0\big{\\}},$

and

$\displaystyle\mathcal{I}^{+}\equiv\big{\\{}p\in\mathcal{M}\ |\ \tau(p)=1,\
\rho(p)=0\big{\\}},$
$\displaystyle\mathcal{I}^{-}\equiv\big{\\{}p\in\mathcal{M}\ |\ \tau(p)=-1,\
\rho(p)=0\big{\\}},$

where $\mathcal{I}$ is the cylinder at spatial infinity, $\mathcal{I}^{\pm}$
are the critical sets at which spatial infinity touches $\mathscr{I}^{+}$ and
$\mathscr{I}^{-}$. Finally, $\mathcal{I}^{0}$ is the intersection of
$\mathcal{I}$ with the initial hypersurface
$\mathcal{S}_{\star}\equiv\\{\tau=0\\}$. A schematic representation of these
sets is shown in Figure 1.

$\mathcal{I}^{+}$$\mathcal{I}^{0}$$\mathcal{I}^{-}$$\mathscr{I}^{+}$$\mathscr{I}^{-}$$\mathcal{I}$$\mathscr{I}^{+}$$\mathscr{I}^{-}$

Figure 1: Schematic representation of the sets of the cylinder at spatial
infinity. Note that the two-dimensional representation on the right is not a
conformal diagram.

######  Remark 1.

In this conformal representation, null infinity coincides with the
hypersurfaces given by the condition $\tau=\pm 1$. However, this can be
generalised so that future and past null infinity do not correspond to
$\tau=\pm 1$ by considering the transformation $\tau\mapsto\tau/\kappa$ where
$\kappa=\varkappa\rho$ and $\varkappa=O(\rho^{0})$. For simplicity, we choose
$\varkappa=1$ for the rest of the paper. This representation with
$\varkappa=1$ is known as the F-gauge _horizontal representation_.

###  2.2 F-gauge

As mentioned in the introduction of this article, we will make use of the
F-gauge frames, which are more suited for the framework of the cylinder at
spatial infinity, to write the asymptotic charges for spin-1 and spin-2 fields
in terms of initial data. To construct these frames, let
$\mathcal{Q}_{\tau,\rho}$ denote the 2-dimensional surfaces in $\mathcal{M}$
of constant $\rho$ and $\tau$. Then the metric on $\mathcal{Q}_{\tau,\rho}$ is
the standard metric on $\mathbb{S}^{2}$. One can introduce a complex null
frame $\\{{\bm{\partial}}_{+},{\bm{\partial}}_{-}\\}$ on
$\mathcal{Q}_{\tau,\rho}$ and impose

$[{\bm{\partial}}_{\tau},{\bm{\partial}}_{\pm}]=0,\qquad[{\bm{\partial}}_{\rho},{\bm{\partial}}_{\pm}]=0.$

These conditions allows us to propagate
$\\{{\bm{\partial}}_{+},{\bm{\partial}}_{-}\\}$ off $\mathcal{Q}_{\tau\rho}$.
Now, introduce the frames [32]

$\displaystyle{\bm{e}}_{{\bm{0}}{\bm{0}}^{\prime}}=\frac{\sqrt{2}}{2}\left((1-\tau){\bm{\partial}}_{\tau}+\rho{\bm{\partial}}_{\rho}\right),$
$\displaystyle{\bm{e}}_{{\bm{1}}{\bm{1}}^{\prime}}=\frac{\sqrt{2}}{2}\left((1+\tau){\bm{\partial}}_{\tau}-\rho{\bm{\partial}}_{\rho}\right),$
$\displaystyle{\bm{e}}_{{\bm{0}}{\bm{1}}^{\prime}}=\frac{\sqrt{2}}{2}{\bm{\partial}}_{+},$
$\displaystyle{\bm{e}}_{{\bm{1}}{\bm{0}}^{\prime}}=\frac{\sqrt{2}}{2}{\bm{\partial}}_{-}.$

and the dual frames

$\displaystyle{\bm{\omega}}^{{\bm{0}}{\bm{0}}^{\prime}}=\frac{\sqrt{2}}{2}\left({\bm{d}}{\tau}-\frac{1}{\rho}(1-\tau){\bm{d}}{\rho}\right),$
$\displaystyle{\bm{\omega}}^{{\bm{1}}{\bm{1}}^{\prime}}=\frac{\sqrt{2}}{2}\left({\bm{d}}{\tau}+\frac{1}{\rho}(1+\tau){\bm{d}}{\rho}\right),$
$\displaystyle{\bm{\omega}}^{{\bm{0}}{\bm{1}}^{\prime}}=\sqrt{2}{\bm{\omega}}^{+},$
$\displaystyle{\bm{\omega}}^{{\bm{1}}{\bm{0}}^{\prime}}=\sqrt{2}{\bm{\omega}}^{-},$

where ${\bm{\omega}}^{\pm}$ are dual to ${\bm{\partial}}_{\pm}$ —that is, one
has

$\langle{\bm{\omega}}^{+},{\bm{\partial}}_{+}\rangle=1,\qquad\langle{\bm{\omega}}^{-},{\bm{\partial}}_{-}\rangle=1.$

In terms of the above fields, the metric ${\bm{\eta}}$ can be written as

${\bm{\eta}}={\bm{\epsilon}}_{{\bm{A}}{\bm{B}}}{\bm{\epsilon}}_{{\bm{A}}^{\prime}{\bm{B}}^{\prime}}{\bm{\omega}}^{{\bm{A}}{\bm{A}}^{\prime}}\otimes{\bm{\omega}}^{{\bm{B}}{\bm{B}}^{\prime}}.$

##  3 The Newman-Penrose gauge

In this section, we briefly summarise NP-gauge conditions satisfied by NP
frames as well as the relation between the F-gauge frames and the null frames
in Minkowski spacetime derived in [32]. In the NP gauge, the frame fields are
adapted to null infinity —the conditions listed here apply for NP frames at
$\mathscr{I}^{+}$. Analogous conditions can be formulated, _mutatis mutandi_ ,
for $\mathscr{I}^{-}$. Let
$\\{{\bm{e}}^{\prime}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ denote a frame
satisfying
${\bm{\eta}}({\bm{e}}^{\prime}_{{\bm{A}}{\bm{A}}^{\prime}},{\bm{e}}^{\prime}_{{\bm{B}}{\bm{B}}^{\prime}})={\bm{\epsilon}}_{{\bm{A}}{\bm{B}}}{\bm{\epsilon}}_{{\bm{A}}^{\prime}{\bm{B}}^{\prime}}$
in a neighbourhood $\mathcal{U}$ of $\mathscr{I}^{+}$. One requires a null
frame in the NP-gauge to satisfy the following conditions:

  * (i)

${\bm{e}}^{\prime}_{{\bm{1}}{\bm{1}}^{\prime}}$ is tangent to
$\mathscr{I}^{+}$ and satisfies

$\nabla_{{\bm{1}}{\bm{1}}^{\prime}}{\bm{e}}^{\prime}_{{\bm{1}}{\bm{1}}}=0.$

  * (ii)

There exits a smooth function $u$ on $\mathcal{U}$ that satisfies
${\bm{e}}^{\prime}_{{\bm{1}}{\bm{1}}^{\prime}}(u)=1$ at $\mathscr{I}$.

  * (iii)

The frame ${\bm{e}}^{\prime}_{{\bm{0}}{\bm{0}}^{\prime}}$ is defined as

${\bm{e}}^{\prime}_{{\bm{0}}{\bm{0}}^{\prime}}={\bm{\eta}}^{\sharp}({\bm{d}}{u},\cdot).$

  * (iv)

Define

$\mathcal{N}_{u_{\bullet}}\equiv\\{p\in\mathcal{U}\ |\ u(p)=u_{\bullet}\\},$

where $u_{\bullet}$ is constant. Then one requires the frame
${\bm{e}}^{\prime}_{{\bm{A}}{\bm{A}}^{\prime}}$, tangent to
$\mathcal{N}_{u_{\bullet}}\cup\mathscr{I}^{+}$, to satisfy

$\nabla_{{\bm{0}}{\bm{0}}^{\prime}}{\bm{e}}^{\prime}_{{\bm{A}}{\bm{A}}^{\prime}}=0\hskip
5.69054pt\text{on}\hskip 5.69054pt\mathcal{N}_{u_{\bullet}}.$

In [32], the relation between the NP-gauge frame
$\\{{\bm{e}}^{\prime}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ and the F-gauge
frame$\\{{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$ for Minkowski spacetime was
explicitly computed. This computation is summarised in the following:

######  Proposition 1.

The NP-gauge frame at $\mathscr{I}^{+}$ and F-gauge frame on Minkowski
spacetime are related via

${\bm{e}}^{\prime}_{{\bm{A}}{\bm{A}}^{\prime}}=\Lambda^{{\bm{B}}}{}_{{\bm{A}}}\bar{\Lambda}^{{\bm{B}}^{\prime}}{}_{{\bm{A}}^{\prime}}{\bm{e}}_{{\bm{B}}{\bm{B}}^{\prime}},$
(3)

and

$\displaystyle\Lambda^{{\bm{1}}}{}_{{\bm{0}}}=\frac{2e^{i\omega}}{\sqrt{\rho}(1+\tau)},\hskip
14.22636pt\Lambda^{{\bm{0}}}{}_{{\bm{1}}}=\frac{e^{-i\omega}\sqrt{\rho}(1+\tau)}{2},$
$\displaystyle\Lambda^{{\bm{1}}}{}_{{\bm{1}}}=\Lambda^{{\bm{0}}}{}_{{\bm{0}}}=0,$
(4)

where $\omega$ is an arbitrary real number that encodes the spin rotation of
the frames on $\mathbb{S}^{2}$. For the NP-gauge frame at $\mathscr{I}^{-}$,
the roles of the vectors ${\bm{e}}^{\prime}_{{\bm{0}}{\bm{0}}^{\prime}}$ and
${\bm{e}}^{\prime}_{{\bm{1}}{\bm{1}}^{\prime}}$ are interchanged, and the NP-
gauge frame is related to the F-gauge by equation (3) with
$\Lambda^{{\bm{A}}}{}_{{\bm{B}}}$ given by

$\displaystyle\Lambda^{{\bm{1}}}{}_{{\bm{0}}}=\frac{e^{-i\omega}\sqrt{\rho}(1-\tau)}{2},\hskip
14.22636pt\Lambda^{{\bm{0}}}{}_{{\bm{1}}}=\frac{2e^{i\omega}}{\sqrt{\rho}(1-\tau)},$
$\displaystyle\Lambda^{{\bm{1}}}{}_{{\bm{1}}}=\Lambda^{{\bm{0}}}{}_{{\bm{0}}}=0.$
(5)

##  4 The Maxwell field and asymptotic charges in the F-gauge

In this section, we discuss the Maxwell field on Minkowski spacetime. First,
we start by writing the Maxwell equations in the F-gauge and expanding the
Maxwell field in terms of spin-weighted spherical harmonics. We then use the
Maxwell equations to solve for the coefficients of the expansion. The analysis
here follows the discussion found in [32].

###  4.1 The spinorial Maxwell equations

Starting with the Maxwell equations in the 2-spinor formalism

$\nabla_{A^{\prime}}{}^{A}\phi_{AB}=0,$ (6)

applying $-2\nabla_{C}{}^{A^{\prime}}$, one can readily finds that the spinor
$\phi_{AB}$ satisfies the wave equation

$\square\phi_{AB}=0,$ (7)

where $\square$ is the D'Alembertian operator
$\square\equiv\nabla_{AA^{\prime}}\nabla^{AA^{\prime}}$. We define a spin-dyad
similar to the frame $\\{{\bm{e}}_{{\bm{A}}{\bm{A}}^{\prime}}\\}$.
Accordingly, define ${\bm{\epsilon}}_{{\bm{A}}}{}^{A}$ with
${\bm{\epsilon}}_{\bm{0}}{}^{A}={\bm{o}}^{A}$ and
${\bm{\epsilon}}_{\bm{1}}{}^{A}=\iota^{A}$ as the spin-dyad in the F-gauge.
Then equation (7) can be split into a system of equations written in terms of
the components of $\phi_{AB}$ with respect to
${\bm{\epsilon}}_{\bm{A}}{}^{A}$, where
$\phi_{0}\equiv\phi_{AB}\iota^{A}\iota^{B}$,
$\phi_{1}\equiv\phi_{AB}o^{A}\iota^{B}$ and
$\phi_{2}\equiv\phi_{AB}o^{A}o^{B}$ so as to obtain the scalar equations

$(1-\tau^{2})\ddot{\phi}_{0}+2\rho\tau\partial_{\rho}\dot{\phi}_{0}-\rho^{2}\partial^{2}_{\rho}\phi_{0}-\frac{1}{2}(\eth\bar{\eth}\phi_{0}+\bar{\eth}\eth\phi_{0})-2(1+\tau)\dot{\phi}_{0}+\phi_{0}=0,$
(8a)
$(1-\tau^{2})\ddot{\phi}_{1}+2\rho\tau\partial_{\rho}\dot{\phi}_{1}-\rho^{2}\partial^{2}_{\rho}\phi_{1}-\frac{1}{2}(\eth\bar{\eth}\phi_{1}+\bar{\eth}\eth\phi_{1})+2\tau(\rho-1)\dot{\phi}_{1}=0,$
(8b)
$(1-\tau^{2})\ddot{\phi}_{2}+2\rho\tau\partial_{\rho}\dot{\phi}_{2}-\rho^{2}\partial^{2}_{\rho}\phi_{2}-\frac{1}{2}(\eth\bar{\eth}\phi_{2}+\bar{\eth}\eth\phi_{2})+2(1-\tau)\dot{\phi}_{2}+\phi_{2}=0.$
(8c)

where $\eth$ and $\bar{\eth}$ are the Newman and Penrose operators [33, 9,
34]. The relation between the NP operators ($\eth$, $\bar{\eth}$) and
(${\bm{\partial}}_{+}$, ${\bm{\partial}}_{-}$) is given in Appendix B in [32].
To analyse the solutions of these equations, the following Ansatz is made:

######  Assumption 1.

The components of the Maxwell field admit an expansion around $\rho=0$ and for
$\tau\in[-1,1]$ of the form

$\phi_{n}=\sum_{l=|n-1|}^{\infty}\sum_{m=-l}^{l}a_{n;l,m}(\tau){}_{n-1}Y_{l,m}+o_{1}(\rho),$
(9)

where
$a_{n;l,m}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}\rightarrow\mathbb{C}$
and $n=0,\,1,\,2$ and ${}_{n}Y_{l,m}$ are spin-weighted spherical harmonics
and the symbol $o_{1}(\rho)$ indicates that the higher order terms and their
first derivatives remain bounded uniformly as $\rho\rightarrow 0$ and
$\tau\rightarrow\pm 1$ with the corresponding decay rate —i.e. $o(\rho)$ for
the function and $o(1)$ for the derivative.

######  Remark 2.

Making use of the estimates developed in [36, 35] it is possible to show that
there exist large classes of solutions to the Maxwell equations having an
expansion consistent with (9). More precisely, it is possible to construct
solutions of the form

$\phi_{n}=\sum_{k=|1-k|}^{p-1}\frac{1}{k!}\phi_{n}^{(k)}\rho^{k}+R_{p}[\phi_{n}],$

where the coefficients $\phi_{n}^{(k)}$ are explicitly known functions of
$\tau$ and the angular variables which are smooth for $\tau\in(-1,1)$ and
whose regularity at $\tau=\pm 1$ can be controlled in terms of the initial
data. Moreover, the reminder satisfies $R_{p}[\phi_{n}]\in C^{m}$ for $p\geq
m+5$ for $\rho$ near 0 and $\tau\in[-1,1]$. In particular, for the expansion
in Assumption 1, one needs $p\geq 6$. The coefficients $\phi_{n}^{(k)}$ are
determined by solving transport equations on the cylinder $\mathcal{I}$ in an
analogous manner to what is done below for the coefficients $a_{n;l,m}(\tau)$
in expansion (1).

Using equation (9) and substituting in equations (8a)-(8c), one readily finds
that the coefficients $a_{n;l,m}$ satisfy the equations

$(1-\tau^{2})\ddot{a}_{0;l,m}-2(1+\tau)\dot{a}_{0;l,m}+l(l+1)a_{0;l,m}=0,\\\ $
(10a) $(1-\tau^{2})\ddot{a}_{1;l,m}-2\tau\dot{a}_{1;l,m}+l(l+1)a_{1;l,m}=0,\\\
$ (10b)
$(1-\tau^{2})\ddot{a}_{2;l,m}+2(1-\tau)\dot{a}_{2;l,m}+l(l+1)a_{2;l,m}=0.$
(10c)

As will be seen later, the expression for the Maxwell charges is written in
terms of the coefficients $a_{1;l,m}$. Thus, we are interested in the general
solution of equation (10b). This is described by the following proposition:

######  Proposition 2.

For $l\geq 0$ and $-l\leq m\leq l$, the general solution to equation (10b) is
given by

$a_{1;l,m}=A_{l,m}P_{l}(\tau)+B_{l,m}Q_{l}(\tau),$

where $P_{l}(\tau)$ denotes the Legendre polynomial of order $l$ and
$Q_{l}(\tau)$ is the Legendre function of the second kind of order $l$.
Moreover, $A_{l,m}$ and $B_{l,m}$ are constants that can be expressed in terms
of the initial conditions $a_{1;l,m}(0)$ and $\dot{a}_{1;l,m}(0)$ as

$A_{l,m}=-(l+1)Q_{l+1}(0)a_{1;l,m}(0)-Q_{l}(0)\dot{a}_{1;l,m}(0),$ (11a)
$B_{l,m}=\frac{\sqrt{\pi}(l+1)}{\Gamma(-\frac{l}{2})\Gamma(\frac{l+3}{2})}a_{1;l,m}(0)+\frac{\sqrt{\pi}}{\Gamma(\frac{1}{2}-\frac{l}{2})\Gamma(1+\frac{l}{2})}\dot{a}_{1;l,m}(0).$
(11b)

######  Remark 3.

The analysis in [36, 35] has been restricted to expansions for which the
leading term in $\phi_{1}$ is of the form

$a_{1;0,0}(\tau)\,Y_{0,0}.$

As will be seen in the following, the higher order harmonics in (9) are
required to generate non-trivial generic BMS charges.

######  Remark 4.

For non-vanishing values of $B_{l,m}$, one can directly deduce from the
recurrence relation of the Legendre function of the second kind that the
solution given by Proposition 2 diverges logarithmically near $\tau=\pm 1$ in
the sense that

$Q_{l}(\tau)=\mathcal{C}_{l}\ln(1\pm\tau)+O(1),\mbox{for some constant
$\mathcal{C}_{l}$.}$

From equation (11b), one can deduce that requiring that $B_{l,m}=0$ gives the
following:

######  Lemma 1.

The solution given in Proposition 2 is regular if and only if

  * (i)

$a_{1;l,m}(0)=0$ for odd $l$, and

  * (ii)

$\dot{a}_{1;l,m}(0)=0$ for even $l$.

###  4.2 Initial data for the Maxwell equations

The coefficients of the expansion given in Assumption 1 are fully determined
by the evolution and constraint equations implied by equation (6). In this
section, we express the initial data conditions given in Lemma 1 in terms of
freely specifiable initial data. The space-spinor decomposition of equation
(6) gives the evolution and constraint equations which can be written,
respectively, as

$\mathcal{D}\phi_{AB}+2\mathcal{D}_{A}{}^{C}\phi_{CB}=0,\hskip
17.07164pt\text{(evolution equations)}$ (12a)
$\mathcal{D}^{AB}\phi_{AB}=0,\hskip 61.17342pt\text{(constraint equation)}$
(12b)

where

$\mathcal{D}\equiv\tau^{AA^{\prime}}\nabla_{AA^{\prime}},\qquad\mathcal{D}_{AB}\equiv\tau_{(A}{}^{C^{\prime}}\nabla_{B)C^{\prime}},$

and $\tau_{AA^{\prime}}$ is the spinorial counterpart of a future directed
timelike vector ${\bm{\tau}}$ satisfying

$\tau^{A}{}_{A^{\prime}}\tau^{BA^{\prime}}={\bm{\epsilon}}^{AB},\hskip
5.69054pt\tau_{AA^{\prime}}\tau^{AA^{\prime}}=2.$

In particular, ${\bm{\tau}}$ is orthogonal to the initial hypersurface
$\mathcal{S}_{\star}$ given by $\tau=0$. Expanding in terms of the spin-dyad
of the F-gauge, one can write the evolution equations as

$(1-\tau)\dot{\phi}_{0}+\rho\partial_{\rho}\phi_{0}+\bar{\eth}\phi_{1}=\phi_{0},\\\
$ (13a)
$\dot{\phi}_{1}+\frac{1}{2}\left(\eth\phi_{0}+\bar{\eth}\phi_{2}\right)=0,\\\
$ (13b)
$(1+\tau)\dot{\phi}_{2}-\rho\partial_{\rho}\phi_{2}+\eth\phi_{2}=-\phi_{2},$
(13c)

and the constraint equation as

$\tau\dot{\phi}_{1}-\rho\partial_{\rho}\phi_{1}+\frac{1}{2}\left(\eth\phi_{0}-\bar{\eth}\phi_{2}\right)=0.$
(14)

######  Remark 5.

Now, consistent with equation (9) in Assumption 1 one has that

$\phi_{n}|_{\mathcal{S}_{\star}}=\sum_{l=|n-1|}^{\infty}\sum_{m=-l}^{l}a_{n;l,m}(0){}_{n-1}Y_{l,m}+o(\rho).$
(15)

Initial data of this type will be called _boosted_. Boosted data can be
obtained from _non-boosted_ data of the form

$\displaystyle\phi_{1}|_{\mathcal{S}_{\star}}=0,$
$\displaystyle\phi_{1}|_{\mathcal{S}_{\star}}=a_{1;0,0,0}(0)Y_{0,0}+o(\rho),$
$\displaystyle\phi_{2}|_{\mathcal{S}_{\star}}=0,$

via a boost at infinity of the initial hypersurface $\mathcal{S}_{\star}$.

######  Remark 6.

In the following, we will omit the ;l,m labels for the coefficients of the
field expansion, e.g. $a_{1}\equiv a_{1;l,m}$ etc. For initial data, we write
$(a_{1})_{*}\equiv a_{1;l,m}(0)$ etc.

For $l\geq 1$, the expansion given by equation (15) can be used to write
equations (13b) and (14) at $\tau=0$ as

$(\dot{a}_{1})_{*}+\frac{1}{2}\sqrt{l(l+1)}\left((a_{0})_{*}-(a_{2})_{*}\right)=0,$
(16a) $\frac{1}{2}\sqrt{l(l+1)}\left((a_{0})_{*}+(a_{2})_{*}\right)=0.$ (16b)

Thus, for even $l\geq 1$, the condition $(\dot{a}_{1})_{*}=0$ translates into
$(a_{0})_{*}=(a_{2})_{*}=0$.

######  Remark 7.

For $l=0$, the condition $\dot{a}_{1,0;0,0}=0$ gives the well-known fact that
the Coulomb charges are conserved.

A systematic way to obtain solutions to the constraint equations with an
expansion of the form given by equation (15) is to make use of the theory
developed in [37]. One has that a symmetric spinor $\phi_{AB}$ satisfying the
constraint (16b) can be written as

$\phi_{AB}=(\mathcal{G}_{2}\Phi)_{AB}\equiv 2D_{(A}{}^{C}\Phi_{B)C},$ (17)

where $D_{AB}$ denotes the spinorial counterpart of the Levi-Civita connection
associated with the intrinsic metric on the initial hypersurface
$\mathcal{S}_{\star}$. Consistent with equation (15) we consider a spinorial
potential $\Phi_{AB}$ with components of the form

$\Phi_{n}=\sum_{l=|n-1|}^{\infty}\sum_{m=-l}^{l}b_{n;l,m}(0){}_{n-1}Y_{l,m}+o(\rho).$

A direct computation then gives the following:

######  Lemma 2.

The solution given in Proposition 2 is regular if and only if

  * (i)

$(b_{1})_{*}=0$ for even $l$, and

  * (ii)

$(b_{0})_{*}=(b_{2})_{*}$ for odd $l$,

where $(b_{0})_{*}\equiv b_{0;l,m}(0)$, $(b_{1})_{*}\equiv b_{1;l,m}(0)$ and
$(b_{2})_{*}\equiv b_{2;l,m}(0)$.

Thus, one can always choose freely specifiable initial data for the Maxwell
spinor in such a way that the coefficients $a_{1;l,m}$ in expansion (9) are
regular at the critical sets $\mathcal{I}^{\pm}$. As it will be seen in the
next subsection this regularity condition is a necessary requirement for the
BMS charges to be well-defined at $\mathcal{I}^{\pm}$.

###  4.3 The Maxwell charges

Let in the following $\tilde{F}_{ab}$ denote the physical Maxwell field
tensor. Then the source-free Maxwell equations can be written as

$\tilde{\nabla}_{b}\tilde{F}^{ba}=0\hskip
3.0pt,\qquad\tilde{\nabla}_{[a}\tilde{F}_{bc]}=0.$

These equations are invariant under the conformal transformation
${\bm{g}}=\Omega^{2}\tilde{{\bm{g}}}$ if one sets

$F_{ab}=\tilde{F}_{ab}.$

The spinorial counterpart of the Maxwell Field tensor $F_{ab}$ is related to
the Maxwell spinor $\phi_{AB}$ via

$F_{AA^{\prime}BB^{\prime}}=\phi_{AB}{\bm{\epsilon}}_{A^{\prime}B^{\prime}}+\bar{\phi}_{A^{\prime}B^{\prime}}{\bm{\epsilon}}_{AB}.$

Now, we define the self-dual Faraday tensor $\mathcal{F}_{ab}$ as

$\mathcal{F}_{ab}=F_{ab}+\mathrm{i}(*F)_{ab},$

where $(*F)_{ab}$ is the Hodge dual of $F_{ab}$ with respect to the volume
form of the unphysical metric ${\bm{g}}$.

The asymptotic symmetries at null infinity are associated with smooth
functions $\lambda$ on $\mathbb{S}^{2}$ sphere [26, 14]. For each of the
functions $\lambda$, the corresponding charge $\mathscr{Q}$ defined on some
cross-section $\mathcal{C}$ of $\mathscr{I}$ is given by

$\mathscr{Q}\equiv\int_{C}\lambda\mathcal{F}_{ab}l^{a}n^{b}\mathrm{d}S,$ (18)

where $l^{a}$ and $n^{a}$ are elements of the NP-gauge null tetrad and
$\mathrm{d}S$ denotes the area element of the cross-section $\mathcal{C}$. We
make the identification
$l^{a}\equiv{\bm{e}}^{\prime}_{{\bm{0}}{\bm{0}}^{\prime}}$ and
$n^{a}\equiv{\bm{e}}^{\prime}_{{\bm{1}}{\bm{1}}^{\prime}}$ at
$\mathscr{I}^{+}$.Thus, in terms of the Maxwell spinor $\phi_{AB}$, one can
verify that

$\mathscr{Q}=2\int_{C}\lambda\bar{\phi}_{1}\mathrm{d}S,$

with the understanding that $\bar{\phi}_{1}$ is evaluated at future null
infinity. The electric and magnetic parts of the charge are then given by

$\displaystyle\mathscr{Q}_{[e]}\equiv-\mathrm{Re}\,[\mathscr{Q}],$
$\displaystyle\mathscr{Q}_{[m]}\equiv\mathrm{Im}\,[\mathscr{Q}].$

Spherical harmonics $Y_{l,m}\equiv{}_{0}Y_{l,m}$ form a complete basis on
$\mathbb{S}^{2}$. Thus, for simplicity, one can choose $\lambda=Y_{l,m}$.
Then, using the expansion given by Assumption 1, one can show the BMS charges
at $\mathcal{I}^{+}$ ($\tau=1$, $\rho=0$) can be written as

$\mathscr{Q}|_{\mathcal{I}^{+}}=2\bar{a}_{1;l,m}(1).$

In the case of past null infinity $\mathscr{I}^{-}$, the roles of $l^{a}$ and
$n^{a}$ in (18) are reversed. In view of the symmetry of equation (18) in
$l^{a}$ and $n^{a}$ one readily finds that at $\mathcal{I}^{-}$ ($\tau=-1$,
$\rho=0$) one has

$\mathscr{Q}|_{\mathcal{I}^{-}}=2\bar{a}_{1;l,m}(-1).$

Then, for generic Cauchy initial data for the Maxwell equations, one can
deduce that the BMS charges at $\mathcal{I}^{+}$ are not well-defined unless
the free specifiable data satisfy the conditions given in Lemma 2 so that the
coefficients $a_{1;l,m}(\tau)$ contain no logarithmic singularity —see Remark
4. A similar argument applies for the charges at $\mathcal{I}^{-}$. Making use
of the results in Section 4.2, one can show:

######  Theorem 1.

Given initial conditions for the Maxwell equations satisfying the conditions
in Lemma 2 so that $B_{l,m}$ vanishes, the asymptotic charges for the Maxwell
field on Minkowski spacetime can be expressed in terms of the initial data on
some initial hypersurface. One has that:

  * (i)

at $\mathcal{I}^{+}$, the charges are given by

$\mathscr{Q}|_{\mathcal{I}^{+}}=\begin{cases}-2(l+1)Q_{l+1}(0)(a_{1})_{*}\qquad\text{for
even }l\geq 0,\\\
\sqrt{l(l+1)}Q_{l}(0)\left((a_{0})_{*}-(a_{2})_{*}\right)\qquad\text{for odd
}l.\end{cases}$ (19)

  * (ii)

At $\mathcal{I}^{-}$, one obtains

$\mathscr{Q}|_{\mathcal{I}^{-}}=\begin{cases}-2(l+1)Q_{l+1}(0)(a_{1})_{*}\qquad\text{for
even }l\geq 0,\\\
-\sqrt{l(l+1)}Q_{l}(0)\left((a_{0})_{*}-(a_{2})_{*}\right)\qquad\text{for odd
}l.\end{cases}$ (20)

If the conditions in Lemma 2 are not satisfied, then the BMS charges are not
well-defined at $\mathcal{I}^{\pm}$.

######  Remark 8.

The charges can be expressed in terms of freely specifiable data using
equation (17). For example, at $\mathcal{I}^{+}$ we get

$\mathscr{Q}|_{\mathcal{I}^{+}}=2\sqrt{2}l(l+1)Q_{l}(0)(b_{1})_{*}\hskip
17.07164pt\text{for odd }l.$

One can obtain expressions for $\mathscr{Q}|_{\mathcal{I}^{+}}$ for even $l$
in a similar fashion.

######  Remark 9.

Notice that the charges at $\mathcal{I}^{-}$ are equal to charges at
$\mathcal{I}^{-}$ multiplied by $(-1)^{l}$ factor. More specifically, the
charges at $\mathcal{I}^{\pm}$ are written in terms of the same piece of
initial data so that the vanishing of the initial data for a given $l$ implies
the vanishing of both charges.

##  5 The spin-2 field and the asymptotic charges in the F-gauge

In this section, we discuss spin-2 fields on Minkowski spacetime and carry out
an analysis analogous to that of the Maxwell field in the previous section. We
start by writing the spin-2 field in terms of spin-weighted spherical
harmonics and expanding the field equations in terms of the F-gauge. Finally,
we obtain a solution for the coefficient of the expansion that appears in the
definition of the asymptotic charges.

###  5.1 The spin-2 equation

Let $\psi_{ABCD}=\psi_{(ABCD)}$ denote a valence 4 symmetric spinor. The
massless spin-2 field equation in spinorial formalism can be written as

$\nabla^{A}{}_{A^{\prime}}\psi_{ABCD}=0.$ (21)

Applying $-2\nabla_{C}{}^{A^{\prime}}$, one readily obtains the wave equation
for the spin-2 field:

$\displaystyle\square\psi_{ABCD}=0.$ (22)

Using the F-gauge spin-dyad, we define the components of the spin-2 field as

$\displaystyle\psi_{0}\equiv\psi_{ABCD}o^{A}o^{B}o^{C}o^{D},$
$\displaystyle\psi_{1}\equiv\psi_{ABCD}o^{A}o^{B}o^{C}\iota^{D},$
$\displaystyle\psi_{2}\equiv\psi_{ABCD}o^{A}o^{B}\iota^{C}\iota^{D},$
$\displaystyle\psi_{3}\equiv\psi_{ABCD}o^{A}\iota^{B}\iota^{C}\iota^{D},$
$\displaystyle\psi_{4}\equiv\psi_{ABCD}\iota^{A}\iota^{B}\iota^{C}\iota^{D}.$

Expanding equation (22), it follows that in the F-gauge, one has

$\displaystyle(1-\tau^{2})\ddot{\psi}_{0}+2\rho\tau\partial_{\rho}\dot{\psi}_{0}-\rho^{2}\partial^{2}_{\rho}\psi_{0}-\frac{1}{2}\left(\eth\bar{\eth}\psi_{0}+\bar{\eth}\eth\psi_{0}\right)+2(2-\tau)\dot{\psi}_{0}+4\psi_{0}=0,$
(23a)
$\displaystyle(1-\tau^{2})\ddot{\psi}_{1}+2\rho\tau\partial_{\rho}\dot{\psi}_{1}-\rho^{2}\partial^{2}_{\rho}\psi_{1}-\frac{1}{2}\left(\eth\bar{\eth}\psi_{1}+\bar{\eth}\eth\psi_{1}\right)+2(1-\tau)\dot{\psi}_{1}+\psi_{1}=0,$
(23b)
$\displaystyle(1-\tau^{2})\ddot{\psi}_{2}+2\rho\tau\partial_{\rho}\dot{\psi}_{2}-\rho^{2}\partial^{2}_{\rho}\psi_{2}-\frac{1}{2}\left(\eth\bar{\eth}\psi_{2}+\bar{\eth}\eth\psi_{2}\right)-2\tau\dot{\psi}_{2}=0,$
(23c)
$\displaystyle(1-\tau^{2})\ddot{\psi}_{3}+2\rho\tau\partial_{\rho}\dot{\psi}_{3}-\rho^{2}\partial^{2}_{\rho}\psi_{3}-\frac{1}{2}\left(\eth\bar{\eth}\psi_{3}+\bar{\eth}\eth\psi_{3}\right)-2(1+\tau)\dot{\psi}_{3}+\psi_{3}=0,$
(23d)
$\displaystyle(1-\tau^{2})\ddot{\psi}_{4}+2\rho\tau\partial_{\rho}\dot{\psi}_{4}-\rho^{2}\partial^{2}_{\rho}\psi_{4}-\frac{1}{2}\left(\eth\bar{\eth}\psi_{4}+\bar{\eth}\eth\psi_{4}\right)-2(2+\tau)\dot{\psi}_{4}+4\psi_{4}=0.$
(23e)

To analyse the solutions for these equations, we make the following Ansatz:

######  Assumption 2.

The components of the spin-2 field can be expanded near $\rho=0$ and
$\tau\in[-1,1]$ as

$\displaystyle\psi_{n}=\sum_{l=|2-n|}^{\infty}\sum_{m=-l}^{l}a_{n;l,m}(\tau){}_{2-n}Y_{l,m}+o_{1}(\rho),$
(24)

where
$a_{n;l,m}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}\rightarrow\mathbb{C}$
and $n=0,...,4$.

######  Remark 10.

As in the case of the Maxwell field, making use of the estimates developed in
[36], it is possible to show that there exist large classes of solutions to
the spin-2 equations having an expansion consistent with (24). More precisely,
one has

$\psi_{n}=\sum_{k=|1-k|}^{p-1}\frac{1}{k!}\psi_{n}^{(k)}\rho^{k}+R_{p}[\psi_{n}],$

where the coefficients $\psi_{n}^{(k)}$ are explicitly known functions of
$\tau$ and the angular variables which are smooth for $\tau\in(-1,1)$ and
whose regularity at $\tau=\pm 1$ can be controlled in terms of the initial
data. Moreover, the reminder satisfies $R_{p}[\psi_{n}]\in C^{m}$ for $p\geq
m+6$ for $\rho$ near 0 and $\tau\in[-1,1]$. In particular, for expansion (24)
in Assumption 2 one requires $p\geq 7$.

Using the expansion Ansatz (24) and (23a)-(23e), one gets

$(1-\tau^{2})\ddot{a}_{0;l,m}+2(2-\tau)\dot{a}_{0;l,m}+l(l+1)a_{0;l,m}=0,\\\ $
(25a)
$(1-\tau^{2})\ddot{a}_{1;l,m}+2(1-\tau)\dot{a}_{1;l,m}+l(l+1)a_{1;l,m}=0,\\\ $
(25b) $(1-\tau^{2})\ddot{a}_{2;l,m}-2\tau\dot{a}_{2;l,m}+l(l+1)a_{2;l,m}=0,\\\
$ (25c)
$(1-\tau^{2})\ddot{a}_{3;l,m}-2(1+\tau)\dot{a}_{3;l,m}+l(l+1)a_{3;l,m}=0,\\\ $
(25d)
$(1-\tau^{2})\ddot{a}_{4;l,m}-2(2+\tau)\dot{a}_{4;l,m}+l(l+1)a_{4;l,m}=0.$
(25e)

We are interested in obtaining a solution to equation (25c) as the asymptotic
BMS charges for the spin-2 field are expressed in terms of $a_{2;l,m}$. The
solution to equation (25c) is similar to the solution to (10b) in the Maxwell
case. Thus, we have the following:

######  Proposition 3.

For $l\geq 0$ and $-l\leq m\leq l$, the solution to (25c) is given by

$a_{2;l,m}=A_{l,m}P_{l}(\tau)+B_{l,m}Q_{l}(\tau)$

where $A_{l,m}$ and $B_{l,m}$ are constants that can be expressed in terms of
initial conditions $a_{2;l,m}(0)$ and $\dot{a}_{2;l,m}(0)$ as

$A_{l,m}=-(l+1)Q_{l+1}(0)a_{2;l,m}(0)-Q_{l}(0)\dot{a}_{2;l,m}(0),$ (26a)
$B_{l,m}=\frac{\sqrt{\pi}(l+1)}{\Gamma(-\frac{l}{2})\Gamma(\frac{l+3}{2})}a_{2;l,m}(0)+\frac{\sqrt{\pi}}{\Gamma(\frac{1}{2}-\frac{l}{2})\Gamma(1+\frac{l}{2})}\dot{a}_{2;l,m}(0).$
(26b)

Similar to the case of Maxwell equations, the solution given by Proposition 26
diverges near $\tau=\pm 1$ if $B_{l,m}\neq 0$. Thus, we require $B_{l,m}$ to
be vanishing to get a regular solution at the critical sets. Let
$(a_{2})_{*}\equiv a_{2;l,m}(0)$ and
$(\dot{a}_{2})_{*}\equiv\dot{a}_{2;l,m}(0)$, one can show the following:

######  Lemma 3.

The solution in Proposition 26 is regular if and only if:

  * (i)

$(a_{2})_{*}=0$ for odd $l$, and

  * (ii)

$(\dot{a}_{2})_{*}=0$ for even $l$.

###  5.2 Initial data for the spin-2 equations

The spin-2 field equation implies a relation between the coefficients of the
expansions given in equation (24). In this section, our aim is to derive the
conditions on the freely specifiable data so that the solution for equation
(25c) is non-diverging. One can decompose equation (21) to obtain the
evolution and constraint equations as follows:

$\mathcal{D}\psi_{ABCD}+2\mathcal{D}_{A}{}^{E}\psi_{EBCD}=0,\hskip
8.53581pt\text{(evolution equations)}$ (27a)
$\mathcal{D}^{AB}\psi_{ABCD}=0.\hskip 61.17342pt\text{(constraint equation)}$
(27b)

Expanding in terms of the F-gauge, the evolution equations for the spin-2
field are given by

$(1+\tau)\dot{\psi}_{0}-\rho\partial_{\rho}\psi_{0}-\eth\psi_{1}=-2\psi_{0},\\\
$ (28a)
$(1+\frac{\tau}{2})\dot{\psi}_{1}-\frac{\rho}{2}\partial_{\rho}\psi_{1}-\frac{\bar{\eth}\psi_{0}}{4}-\frac{3\eth\psi_{2}}{4}=-\psi_{1},\\\
$ (28b)
$\dot{\psi}_{2}-\frac{\bar{\eth}\psi_{1}}{2}-\frac{\eth\psi_{3}}{2}=0,\\\ $
(28c)
$(1-\frac{\tau}{2})\dot{\psi}_{3}+\frac{\rho}{2}\partial_{\rho}\psi_{3}-\frac{3\bar{\eth}\psi_{2}}{4}-\frac{\eth\psi_{4}}{4}=\psi_{3},\\\
$ (28d)
$(1-\tau)\dot{\psi}_{4}+\rho\partial_{\rho}\psi_{4}-\bar{\eth}\psi_{3}=2\psi_{4},$
(28e)

and the constraint equations are given by

$\tau\dot{\psi}_{1}-\rho\partial_{\rho}\psi_{1}+\frac{\bar{\eth}\psi_{0}}{2}-\frac{\eth\psi_{2}}{2}=0,\\\
$ (29a)
$\tau\dot{\psi}_{2}-\rho\partial_{\rho}\psi_{2}+\frac{\bar{\eth}\psi_{1}}{2}-\frac{\eth\psi_{3}}{2}=0,\\\
$ (29b)
$\tau\dot{\psi}_{3}-\rho\partial_{\rho}\psi_{3}+\frac{\bar{\eth}\psi_{2}}{2}-\frac{\eth\psi_{4}}{2}=0.$
(29c)

Now, we consider an expansion which is consistent with Assumption 2 so that
$\psi_{n}|_{\mathcal{S}_{\star}}$ takes the form

$\psi_{n}|_{\mathcal{S}_{\star}}=\sum_{l=|2-n|}^{\infty}\sum_{m=-l}^{l}a_{n;l,m}(0){}_{2-n}Y_{l,m}+o(\rho).$
(30)

As in the case of the Maxwell field, we call this class of initial data
_boosted_ —see Remark 5. Making use of this expansion, equations (28c) and
(29b) evaluated at $\tau=0$, yield

$\displaystyle(\dot{a}_{2})_{*}+\frac{1}{2}\sqrt{l(l+1)}\left((a_{3})_{*}-(a_{1})_{*}\right)=0,$
$\displaystyle\frac{1}{2}\sqrt{l(l+1)}\left((a_{3})_{*}+(a_{1})_{*}\right)=0,$

where the index ;l,m have been omitted and we use the notation introduced in
Remark 6. In particular, from the above one can deduce that at $\tau=0$, the
condition $(\dot{a}_{2})_{*}=0$ translates to $(a_{1})_{*}=(a_{3})_{*}=0$ for
even $l\geq 2$.

In order to write the initial conditions in terms of freely specifiable data,
we make use of the theory developed in [37] for solutions to the Gauss
constraint (27b). Specifically, for a spin-2 field satisfying the constraint
equation, one can write

$\psi_{ABCD}=(\mathcal{G}_{4}\Psi)_{ABCD}=2\left(\text{}_{2}{\text{}_{4}{\text{}_{4}{\Psi}}}\right)_{ABCD}+8\left(\text{}_{4}{\text{}_{4}{\text{}_{4}{\Psi}}}\right)_{ABCD},$
(31)

where $\text{}_{4},\text{}_{4}$ and $\text{}_{2}$ are the divergence, curl and
twistor operators as defined in [37]. Consistent with the previous discussion,
we assume that the components of the Hertz potential $\Psi_{ABCD}$ can be
expanded in terms of spin-weighted spherical harmonics as

$\Psi_{n}=\sum_{l=|2-n|}^{\infty}\sum_{m=-l}^{l}b_{n;l,m}(0){}_{2-n}Y_{l,m}+o(\rho).$
(32)

Letting $(b_{0})_{*}\equiv b_{0;l,m}(0)$, $(b_{1})_{*}\equiv b_{1;l,m}(0)$,
$(b_{2})_{*}\equiv b_{2;l,m}(0)$, $(b_{3})_{*}\equiv b_{3;l,m}(0)$ and
$(b_{4})_{*}\equiv b_{4;l,m}(0)$, one can expand equation (31) in the F-gauge
and use (24) and (32) to obtain:

######  Lemma 4.

The solution given in Proposition 26 is regular if and only if:

  * (i)

$(b_{2})_{*}=C_{l}(b_{4})_{*}-D_{l}(b_{0})_{*}$ for even $l\geq 1$, and

  * (ii)

$(b_{1})_{*}=(b_{3})_{*}$ for odd $l$,

where $C_{l}$ and $D_{l}$ are constants that depend on $l$ and whose specific
form is not relevant for the subsequent discussion.

######  Remark 11.

It follows then that, as in the case of the Maxwell spin field, the conditions
ensuring solutions that are regular at the critical sets can be encoded in
terms of freely specifiable data.

###  5.3 The asymptotic BMS charges for the spin-2 field

In this section, we consider the asymptotic charges for the spin-2 field
associated with supertranslations. For the spin-2 field, the asymptotic
symmetries at null infinity are given by the smooth functions $\lambda$ on
$\mathbb{S}^{2}$. The associated BMS charges defined on some cross-section
$\mathcal{C}$ of null infinity can be written as

$\mathscr{P}=\int_{\mathcal{C}}\lambda\mathcal{W}_{abcd}l^{a}n^{b}m^{c}\bar{m}^{d}\mathrm{d}S,$

where $\mathcal{W}_{abcd}$ is given by

$\mathcal{W}_{abcd}=W_{abcd}+\mathrm{i}(*W)_{abcd}.$

Here, $W_{abcd}$ denotes a Weyl-like tensor. The spinorial counterpart
$W_{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}$ satisfies

$W_{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}=-\psi_{ABCD}{\bm{\epsilon}}_{A^{\prime}B^{\prime}}{\bm{\epsilon}}_{C^{\prime}D^{\prime}}-\bar{\psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}{\bm{\epsilon}}_{AB}{\bm{\epsilon}}_{CD}.$

The BMS charges can then be written in terms of the spin-2 spinor
$\psi_{ABCD}$. At $\mathscr{I}^{+}$, if one makes the identification
$l^{a}\equiv{\bm{e}}^{\prime}_{{\bm{0}}{\bm{0}}^{\prime}}$,
$n^{a}\equiv{\bm{e}}^{\prime}_{{\bm{1}}{\bm{1}}^{\prime}}$ and
$m^{a}\equiv{\bm{e}}_{{\bm{0}}{\bm{1}}^{\prime}}$, then the charges can be
written as

$\mathscr{P}=-2\int_{\mathcal{C}}\lambda\bar{\psi}_{2}\mathrm{d}S.$

Using the expansion given in Assumption 2 and assuming $\lambda=Y_{l,m}$, the
expression for the charges at $\mathcal{I}^{+}$ can be written as

$\mathscr{P}|_{\mathcal{I}^{+}}=-2\bar{a}_{2;l,m}(1).$

At $\mathcal{I}^{-}$, one can show that

$\mathscr{P}|_{\mathcal{I}^{-}}=-2\bar{a}_{2;l,m}(-1).$

Making use of the results of Section 5.2, one readily obtains the following:

######  Theorem 2.

Given initial conditions for the spin-2 field equations satisfying Lemma 4 so
that $B_{l,m}$ vanishes, the asymptotic charges for the spin-2 field on
Minkowski spacetime can be expressed in terms of the initial data on some
initial hypersurface. More precisely, one has that:

  * (i)

At $\mathcal{I}^{+}$, the charges are given by

$\mathscr{P}|_{\mathcal{I}^{+}}=\begin{cases}2(l+1)Q_{l+1}(0)(a_{2})_{*}\qquad\text{for
even }l\geq 0,\\\
\sqrt{l(l+1)}Q_{l}(0)\left((a_{1})_{*}-(a_{3})_{*}\right)\qquad\text{for odd
}l.\end{cases}$

  * (ii)

At $\mathcal{I}^{-}$, the charges are given by

$\mathscr{P}|_{\mathcal{I}^{-}}=\begin{cases}2(l+1)Q_{l+1}(0)(a_{2})_{*}\qquad\text{for
even }l\geq 0,\\\
-\sqrt{l(l+1)}Q_{l}(0)\left((a_{1})_{*}-(a_{3})_{*}\right)\qquad\text{for odd
}l.\end{cases}$

If the conditions of Lemma 4 are not satisfied then the BMS charges are not
well-defined.

######  Remark 12.

As in the case of the Maxwell field, the BMS asymptotic charges for the spin-2
field can be fully expressed in terms of freely specifiable data using
equation (31). The resulting expressions are much more involved and not
particularly illuminating.

######  Remark 13.

Similar to the case of the Maxwell field, for a given harmonic $Y_{l,m}$, the
charges at $\mathcal{I}^{+}$ and $\mathcal{I}^{-}$ can be traced back to the
same piece of freely specifiable data. In this sense, one can say they contain
the same information.

##  6 Conclusions

In this article, we have analysed the relation between the asymptotic charges
for the Maxwell and spin-2 fields at $\mathcal{I}^{\pm}$ and initial data
given on some Cauchy hypersurface $\mathcal{S}_{\star}$ of the Minkowski
spacetime. The study of the asymptotic charges in GR is of physical importance
as it is directly related to the continuous symmetries of the system under
consideration. In our analysis, we have focused on charges associated with
smooth functions defined on $\mathbb{S}^{2}$. In full General Relativity,
these charges would correspond to the supertranslation charges of the BMS
group.

For the Maxwell field, our analysis shows that the asymptotic charges are not
in general defined unless one imposes extra conditions on the initial data. To
show this, we expanded the initial data in terms of spin-weighted spherical
harmonics, and we showed that the asymptotic charges at $\mathcal{I}^{+}$ are
well-defined and are fully written in terms of the initial data if and only if
the freely specifiable initial data satisfy the conditions given in Lemma 2.
Similarly, the asymptotic charges at $\mathcal{I}^{-}$ can be written in terms
of the freely specifiable initial data on $\mathcal{S}$ and we show that they
correspond to the same initial data used for the asymptotic charges at
$\mathcal{I}^{+}$.

A similar analysis with analogous conclusions was carried out for the spin-2
field. In particular, we show that the asymptotic charges at
$\mathcal{I}^{\pm}$ are well-defined and can be written in terms of the freely
specifiable initial data if and only if the conditions given in Lemma 4 are
satisfied. Given these conditions, we show that there is a correspondence
between the charges at $\mathcal{I}^{+}$ and $\mathcal{I}^{-}$ in the sense
that both charges are written in terms of the same initial data.

In summary, our analysis shows that the asymptotic BMS charges at
$\mathcal{I}^{\pm}$ are not defined for generic initial data unless we choose
initial data that satisfy the conditions given in Lemma 2 in the Maxwell case
and Lemma 4 in the spin-2 case. Moreover, the resulting expressions for the
charges in terms of the freely specifiable data show that, for a given
harmonic, there is a natural correspondence between the future and past
charges without the need for any further assumptions. The analysis in this
article relies on the leading order structure of the cylinder at spatial
infinity —i.e. its Minkowskian part. As this is a universal aspect of the
structure of spatial infinity for more general spacetimes (e.g. the
Schwarzschild spacetime or, more generally, any stationary spacetime) it is
conjectured that the results in this article can be extended to massless test
fields propagating on stationary spacetimes —the particular details, however,
need to be verified. For more general (i.e. non-stationary) spacetimes, the
situation is less clear because of the generic non-smooth behaviour of the
spacetime near spatial infinity.

The approach followed in this article can be generalised to the case of the
asymptotic BMS charges on asymptotically flat spacetimes. The general
properties of the asymptotic expansions of the conformal Einstein field
equations required for this type of analysis have been studied in [17, 31].
Notice, however, that the analysis in these two references is restricted to
the case of spacetimes arising from time-symmetric initial data conditions
—i.e. vanishing extrinsic curvature. Thus, an important part of the extension
of the analysis in the present article to the vacuum Einstein field equations
(or the Einstein-Maxwell system) is to obtain a suitably large class of
initial data sets for the Einstein field equations giving rise to non-
vanishing BMS charges. The analysis for the Einstein vacuum field equations
will be discussed elsewhere.

##  7 Acknowledgements

M.M.A.Mohamed would like to thank Mahdi Godazgar for the useful discussions
while working on this project. We also thank Kartik Prabhu for interesting
conversations. Some calculations in this project used the computer algebra
system Mathematica with the package xAct for tensor manipulations —see [40].

## References

  * [1] S. Weinberg,  Infrared Photons and Gravitons,  Phys. Rev. 140, B516 (1965).
  * [2] A. Strominger,  On BMS invariance of gravitational scattering,  J. High Energ. Phys. 07, 152 (2014).
  * [3] T. He, V. Laysov, P. Mitra & A. Strominger,  BMS supertranslations and Weinberg's soft graviton theorem,  J. High Energ. Phys. 5, 151 (2015).
  * [4] M. Favata,  The gravitational-wave memory effect,  Class. Quant. Grav. 27, 084036 (2010).
  * [5] D. Christodoulou,  Nonlinear nature of gravitation and gravitational-wave experiments,  Phys. Rev. Lett. 67, 1486 (1991).
  * [6] L. Blanchet & T. Damour,  Hereditary effects in gravitational radiation,  Phys. Rev. D46, 4304 (1992).
  * [7] S. W. Hawking, M. J. Perry & A. Strominger  Soft Hair on Black Holes,  Phys. Rev. Lett. 116, 231301 (2016).
  * [8] S. W. Hawking, M. J. Perry & A. Strominger,  Superrotation Charge and Supertranslation Hair on Black Holes,  J. High Eenerg. Phys. 05, 161 (2017).
  * [9] R. Penrose & W. Rindler,  Spinors and space-time. Volume 2. Spinor and twistor methods in space-time geometry,  Cambridge University Press, 1986.
  * [10] J. A. Valiente Kroon,  Conformal Methods in General Relativity,  Cambridge University Press, 2016.
  * [11] H. Bondi H, M. G. J. Van der Burg & A. W. K. Metzner,  Gravitational waves in general relativity VII. Waves from axisymmetric isolated systems,  Proc. Roy. Soc. A 269, 21 (1962).
  * [12] R. K. Sachs,  Gravitational waves in general relativity VIII. Waves in asymptotically flat spacetimes,  Proc. Roy. Soc. A 270, 103 (1962).
  * [13] R. M. Wald & A. Zoupas,  A general definition of 'conserved quantities' in general relativity and other theories of gravity,  Phys. Rev. D 61, 084027 (2000).
  * [14] G. Barnich & C. Troessaert,  BMS charge algebra,  J. High Energ. Phys. 12, 105 (2011).
  * [15] A. Ashtekar & R. O. Hansen,  A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity,  J. Math. Phys. 19, 1542 (1978).
  * [16] A. Ashtekar & J. D. Romano,  Spatial infinity as a boundary of spacetime,  Class. Quantum Grav. 9, 1069 (1992).
  * [17] H. Friedrich,  Gravitational fields near space-like and null infinity,  J. Geom. Phys. 24, 83 (1998).
  * [18] M. Henneaux & C. Troessaert  BMS group at spatial infinity: the Hamiltonian (ADM) approach J. High Energ. Phys. 147 (2018).
  * [19] M. Henneaux & C. Troessaert  Hamiltonian structure and asymptotic symmetries of the Einstein-Maxwell system at spatial infinity J. High Energ. Phys. 171 (2018).
  * [20] R. Geroch  Structure of the Gravitational Field at Spatial infinity,  J. Math. Phys. 13, 956 (1972).
  * [21] M. Campiglia & R. Eyheralde,  Asymptotic U(1) charges at spatial infinity,  J. High Energ. Phys., 168 (2017).
  * [22] M. Henneaux & C. Troessaert  Asymptotic symmetries of electromagnetism at spatial infinity J. High Energ. Phys. 137 (2018).
  * [23] C. Troessaert  The BMS4 algebra at spatial infinity,  Class. Quantum Grav. 35, 074003 (2018).
  * [24] R. Beig & B. G. Schmidt,  Einstein's equation near spatial infinity,  Comm. Math. Phys. 87, 65 (1982).
  * [25] H. Bondi,  Gravitational Waves in General Relativity,  Nature 186, 535 (1960).
  * [26] K. Prabhu,  Conservation of asymptotic charges from past to future null infinity: Maxwell fields,  J. High Energ. Phys. 113, (2018).
  * [27] K. Prabhu,  Conservation of asymptotic charges from past to future null infinity: supermomentum in general relativity,  J. High Energ. Phys. 148, (2019).
  * [28] K. Prabhu & I. Shehzad  Conservation of asymptotic charges from past to future null infinity: Lorentz charges in general relativity,  arXiv:2110.04900, (2021).
  * [29] K. Prabhu & I. Shehzad  Asymptotic symmetries and charges at spatial infinity in general relativity Class. Quantum Grav. 37 (2020).
  * [30] M. Henneaux & C. Troessaert  The asymptotic structure of gravity at spatial infinity in four spacetime dimensions arXiv:1904.04495 (2019).
  * [31] H. Friedrich & J. Kánnár,  Bondi-type systems near space-like infinity and the calculation of the NP-constants,  J. Math. Phys. 41, 2195 (2000).
  * [32] E. Gasperín & J. A. Valiente Kroon,  Zero rest-mass fields and the Newman-Penrose constants on flat space,  J. Math. Phys. 61, 122503 (2020).
  * [33] E. Newman & R. Penrose  An Approach to Gravitational Radiation by a Method of Spin Coefficients J. Math. Phys. 3, 566 (1962).
  * [34] J. Stewart  Advanced General Relativity,  Cambridge University Press (1991).
  * [35] J. A. Valiente Kroon,  Estimates for the Maxwell field near the spatial and null infinity of the Schwarzschild spacetime,  J. Hyp. Diff. Eqns. 6, 229 (2009).
  * [36] H. Friedrich,  Spin-2 fields on Minkowski space near space-like and null infinity,  Class. Quantum Grav. 20, 101 (2003).
  * [37] L. Andresson, T. Bäckdahl & J. Joudioux,  Hertz Potentials and Asymptotic Properties of Massless Fields,  Commun.Math.Phys., 331, 775 (2014).
  * [38] A. Ashtekar,  Asymptotic structure of the gravitational field at spatial infinity,  in General Relativity and Gravitation: One hundred years after the birth of Albert Einstein, edited by A. Held, volume 2, page 37, Plenum Press, 1980.
  * [39] R. Beig,  Integration of Einstein's equations near spatial infinity,  Proc. Roy. Soc. Lond. A 391, 295 (1984).
  * [40] J. M. Martín-García, https://www.xact.es