# ICI-Free Channel Estimation and Wireless Gesture Recognition Based on
Cellular Signals

Rui Peng, , Yafei Tian, , and Shengqian Han,
This work was supported by the National Natural Science Foundation of China
under Grants 61971023.The authors are with the School of Electronics and
Information Engineering, Beihang University, Beijing 100191, P. R. China
(e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>sqhan@buaa.edu.cn).

###### Abstract

Device-free wireless sensing attracts enormous attentions since it senses the
environment without additional devices. While cellular signals are good
opportunistic radio sources, the influence of inter-cell interference (ICI) on
wireless sensing has not been adequately addressed. In this letter, we first
investigate the cause of ICI and its impact on wireless sensing. Then we
propose an ICI-free channel estimation method by reconstructing the broadcast
signals of adjacent cells and solving simultaneous equations. Wireless gesture
recognition can be greatly benefited by ICI mitigation. Finally, we build a
prototype system to receive the commercial 4G-LTE signals, and demonstrate the
accuracies of wireless gesture recognition under various conditions.

###### Index Terms:

Device-free wireless sensing, cellular signals, ICI, broadcast signals,
gesture recognition.

##  I Introduction

Integrated sensing and communication (ISAC) is a promising technology in the
sixth generation (6G) mobile communication system. Device-free wireless
sensing (DFWS) is one of the popular research areas in ISAC. Except for
regular data transmission, the communication system can also help to identify
human behaviors by capturing the variation of channel environment. The major
benefit of DFWS is that it optimizes the utilization of time-frequency
resources by integrating sensing tasks with communications, without any
additional devices [1].

A majority of DFWS researches exploit WiFi as radio source, since it is
convenient to extract the channel state information (CSI) by off-the-shelf
network interface card. On the contrary, currently, there is no CSI extraction
tool available for 4G/5G modem, so the acquisition of CSI relies on self-
developed software defined radio (SDR). From the opportunistic signal
perspective, cellular signal has broad coverage and continuous samplings of
the CSI [2], making it a better choice for DFWS.

Due to the limitation of CSI accessibility, only a few works studied DFWS with
4G long term evolution (LTE) signal. In the literature, [3] proposed a crowd
density estimation method through CSI, and proved that CSI can achieve better
performance than reference signal received power (RSRP). The respiration
detection was studied in [4]. By utilizing the coverage advantage of LTE and
combining the results measured in different frequency bands, the detection
error is decreased and the blind spot issue is solved. The gesture recognition
was explored in [5], where the direction of base station (BS) is first probed
by a preamble gesture, and then the user stands in the opposite direction of
the BS to improve the recognition performance.

Despite these works achieved wireless sensing with LTE signals, none of them
has investigated the influence of inter-cell interference (ICI). However, ICI
is a practical problem in cellular networks and has critical impact on
wireless sensing. Since the soft frequency reuse scheme is widely applied in
current networks, most areas are covered by multiple cells operated in the
same frequency band. Usually, CSI is estimated by cell-specific reference
signal (CRS). While the CRSs from adjacent cells are inserted on different
subcarriers to avoid collision, the CRS from one cell may still be
contaminated by the data transmitted from another cell. For fine sensing tasks
such as gesture recognition, the motion information is extracted from the
subtle variation of propagation channels, and is more vulnerable to ICI. The
recognition performance will be dramatically degraded whenever the strength of
ICI exceeds the reflection power of human hand.

In this letter, we propose an ICI-free channel estimation method for wireless
gesture recognition based on the time division duplexing (TDD) LTE cellular
signals, and validate its effectiveness by prototype experiments. The main
contributions of this work are as follows:

1) We analyze the source of ICI and its characteristics, and clarify its
impact on CRS-based channel estimation and subsequent gesture recognitions.

2) We propose to use the physical broadcast channel (PBCH) as reference
signal, so that the local sequences of ICI are known, and the channel
estimation is transformed from an interference-limited problem to a noise-
limited problem.

3) We develop the multi-cell PBCH local sequences reconstruction and joint
channel estimation methods, and the ICI-free CSI can thus be acquired.

4) We build a prototype system to receive commercial TDD-LTE signals and
implement gesture recognition in real-time. Experiment results exhibit
superior performance than existing methods.

##  II System Model

###  II-A LTE background

In LTE downlink (DL) signal, a system frame with duration 10ms is divided into
10 subframes. Each subframe comprises of 14 orthogonal frequency division
multiplexing (OFDM) symbols in normal cyclic prefix (CP) case. A resource
block (RB) is the basic scheduling unit, which consists of 12 subcarriers and
each subcarrier in one symbol is called a resource element (RE). The CRS is
inserted in each RB for channel estimation. The symbol index of CRS is fixed,
but the subcarrier offset inside each RB is determined by physical layer cell
identity (PCI), as

$\displaystyle k_{0}=(v+v_{\textrm{shift}})~{}\textrm{mod}~{}6,$ (1)

where $v$ is either 0 or 3 according to the antenna port and symbol number,
$v_{\textrm{shift}}=N_{\textrm{ID}}~{}\textrm{mod}~{}6$, and $N_{\textrm{ID}}$
denotes PCI. As a principle, the PCIs with different mod-6 value are assigned
to adjacent cells, so that their CRSs will not collide.

In every DL frame, PBCH are broadcasted from symbol 7 to 10 in subframe 0. It
occupies the rest resources other than CRS in the middle 72 subcarriers. PBCH
carries master information block (MIB), which includes necessary system
information such as system frame number (SFN) and bandwidth. The SFN is
circularly updated from 0 to 1023, while the rest information bits remain
constant. In addition, we can also implicitly obtain the antenna port number
during PBCH decoding.

TDD systems separates uplink (UL) and downlink in the time domain. Although
there are seven UL-DL subframe assignment configurations, subframe 0 and 5 are
always for DL transmission[6]. To prevent UL-DL interference, the timing of
adjacent TDD cells are well-synchronized, and they share the same UL-DL
configuration and SFN.

###  II-B Channel Model

We consider a time-variant wideband channel. The CSI of a single stream, i.e.,
from one transmit port to one receive port, consists of static paths and
dynamic paths. For symbol $n$ and subcarrier $k$, the CSI is denoted as

$\displaystyle{h}(k,n)$ $\displaystyle=$ $\displaystyle
e^{-j\theta_{\textrm{p}}(n)}\left[{h}_{\textrm{s}}(k)+{h}_{\textrm{d}}(k,n)\right],$
(2)

where ${h}_{\textrm{s}}(k)$ is the static channel response, caused by LoS path
and other paths reflected from static objects, ${h}_{\textrm{d}}(k,n)$ denotes
the time-variant dynamic channel response induced by moving objects. However,
the CSI is always polluted by the phase noise $\theta_{\textrm{p}}(n)$, which
is introduced by oscillator vibration. Even with accurate frequency-offset
tracking and compensation, the phase noise cannot be totally eliminated.

###  II-C Impact of ICI

In urban area, the cells are small and densely covered, so that users receive
the superimposed signals from multiple cells. Usually, the cell with the
highest RSRP is chosen as the serving cell. The adjacent cells will reuse the
same time and frequency resources to improve the spectrum efficiency, thus
introduces ICI. Although CRSs from different cells will not collide, they may
collide with the physical downlink control channel (PDCCH) and physical
downlink shared channel (PDSCH) of neighbor cells, as shown in Fig.1(a) and
(b).

Figure 1: An example of CRS/PDCCH/PDSCH/PBCH allocation for different cells in
case of 4 antenna ports, normal CP. The cases shown in figures (a)(b) and
(c)(d) are in different RBs.

As a result, the received CRS will be interfered as

$\displaystyle y_{\textrm{CRS}}(k,n)$ $\displaystyle=$ $\displaystyle
x_{\textrm{CRS}}(k,n)h(k,n)+$ (3)
$\displaystyle\sum_{p=0}^{N_{\textrm{I}}-1}x^{p}_{\textrm{I}}(k,n)h^{p}_{\textrm{I}}(k,n)+w(k,n),$

where $x_{\textrm{CRS}}(k,n)$ and $x^{p}_{\textrm{I}}(k,n)$ denote the
sequence of CRS in serving cell and PDCCH/PDSCH payload in neighbor cell,
respectively, $p$ is the index of interference data streams. We assume that
there are $N_{\textrm{I}}$ interference streams in total. If all neighbor
cells configure with one antenna port, $N_{\textrm{I}}$ is equal to the number
of neighbor cells, $h(k,n)$ and $h^{p}_{\textrm{I}}(k,n)$ represent the
frequency domain channel of the serving cell and the $p^{\textrm{th}}$
interference stream, respectively, and ${w}(k,n)$ is the additive white
Gaussian noise.

The CRS is an always-on signal, while the PDCCH/PDSCH is transmitted when the
user equipment (UE) requests downlink data. In fact, the allocation of
PDCCH/PDSCH is determined by BSs, and it is hard to decode the PDCCH/PDSCH of
neighbor cells in the receiver, since the SINR is usually below zero.

Consequently, the ICIs on CRS subcarriers are random and time-variant, and
thus they are unknown and unpredictable. In the channel estimation results,
they behave as parts of the dynamic channel, i.e.,

$\displaystyle\hat{h}(k,n)=h(k,n)+\frac{\sum_{p=0}^{N_{\textrm{I}}-1}x^{p}_{\textrm{I}}(k,n)h^{p}_{\textrm{I}}(k,n)+w(k,n)}{x_{\textrm{CRS}}(k,n)}.$
(4)

Although the UE typically accesses the strongest cell, resulting in a
potentially weaker interference compared to the channel response of the
serving cell, the interference term can still be much stronger than the
reflection path of human hand, causing severe impact on dynamic CSI extraction
and subsequent gesture recognition.

##  III Methodology

###  III-A Multi-Cell Joint Channel Estimation

To eliminate the impact of ICI, we propose a multi-cell joint channel
estimation method. Instead of using CRS, we use PBCH as reference signal. The
PBCH of serving cell and neighbor cells are always overlapped in the same
$4\times 72$ RE grid in subframe 0, as shown in Fig.1(c) and (d). Although the
PBCHs from neighbor cells compose ICIs as well, the transmitted sequences of
all PBCHs can be predicted, since the generation of PBCH follows specific
rules. By solving simultaneous equations about the received signals and known
local sequences, we can acquire ICI-free channel estimation.

####  III-A1 Neighbor cells search and PBCH reconstruction

As introduced in Section II-A, PBCH carries MIB, wherein the only changed
information is the SFN bits. Due to the frame transmission mode in LTE, the
SFN bits in all frames can be derived by the SFN known in one frame. Once we
decode the MIB in any frame, the PBCH sequences can be reconstructed in all
frames. It means that we only need to decode the PBCHs for one time. Moreover,
the PBCH can be decoded in low SINR, since the coding rate is 1/3 and the
coded bits are repeated multiple times in rate matching.

To correctly decode PBCHs of all adjacent cells, a successive interference
cancellation (SIC) strategy is required. In TDD-LTE, not only the PBCH, but
also the primary synchronization signal (PSS) and secondary synchronization
signal (SSS) from different cells are all aligned. After cell search and
frequency offset compensation using the PSS and SSS, the UE can decode MIB of
the serving cell. To detect the neighbor cells, we need first reconstruct the
PSS/SSS/PBCH signals of the serving cell, and cancel them from the
corresponding REs. Then we can search and decode the next stronger neighbor
cell. In case that we successfully decode the MIB of a new cell, we list it as
a neighbor cell. The SIC procedures are repeated until no further cell can be
found.

Despite the complexity of SIC is high, it is only conducted in the preparation
stage, and it will not affect the real-time processing in channel estimation
and gesture recognition. The reconstruction of PBCH consists of convolutional
encoding, rate matching, scrambling, modulation and transmit diversity, as
elaborated in 3GPP specifications 36.211 [6] and 36.212 [7]. Due to space
limitations, we will not go into details here.

####  III-A2 Joint channel estimation

Considering a general scene, at one receiver, totally $N_{\textrm{s}}$ streams
of PBCH signals are received from adjacent cells. In frame $n$, symbol $l$ and
subcarrier $k$, the received signal at one RE can be represented as

$\displaystyle
y(k,l,n)=\sum_{p=0}^{N_{\textrm{s}}-1}x^{p}(k,l,n)h^{p}(k,l,n)+w(k,l,n),$ (5)

where $x^{p}(k,l,n)$, $h^{p}(k,l,n)$ and $w(k,l,n)$ denote the PBCH modulated
symbol, frequency domain channel response and noise, respectively.

In (5), although $x^{p}(k,l,n)$ is known after PBCH reconstruction, we have
$N_{\textrm{s}}$ unknown variables $h^{p}(k,l,n)$ to be estimated in each
equation. Fortunately, the channel of adjacent REs typically have only minor
changes. We consider that $h^{p}(k,l,n)$ keeps constant in a RE group, which
consists of $K$ contiguous subcarriers and $L$ symbols. To ensure that
equation (5) has a unique solution, the selection of $K$ and $L$ should
satisfy $KL\geq N_{\textrm{s}}$.

Since PBCH occupies 72 subcarriers and 4 symbols in a frame, it has plenty of
freedom to separate RE groups. We can slightly increase the number of $KL$, so
that the equations are overdetermined and the estimation error can be
decreased by least square (LS) method.

With the assumption that channel is invariant inside a RE group
$[k_{\textrm{g}},l_{\textrm{g}}]$, (5) can be approximated as

$\displaystyle
y(k,l,n)\approx\sum_{p=0}^{N_{\textrm{s}}-1}x^{p}(k,l,n)h^{p}_{k_{\textrm{g}},l_{\textrm{g}}}(n)+w(k,l,n)$
$\displaystyle{\forall}k,l\in\mathbb{I}_{k_{\textrm{g}},l_{\textrm{g}}},$ (6)

where $h^{p}_{k_{\textrm{g}},l_{\textrm{g}}}(n)$ is the channel to be solved
for RE group $[k_{\textrm{g}},l_{\textrm{g}}]$, and
$\mathbb{I}_{k_{\textrm{g}},l_{\textrm{g}}}$ denotes the subcarrier and symbol
index set in RE group $[k_{\textrm{g}},l_{\textrm{g}}]$. Write (III-A2) in
matrix form,

$\displaystyle\mathbf{y}_{k_{\textrm{g}},l_{\textrm{g}}}(n)\approx\mathbf{X}_{k_{\textrm{g}},l_{\textrm{g}}}(n)\mathbf{h}_{k_{\textrm{g}},l_{\textrm{g}}}(n)+\mathbf{w}_{k_{\textrm{g}},l_{\textrm{g}}}(n),$
(7)

and we can obtain the LS estimation of
$\mathbf{h}_{k_{\textrm{g}},l_{\textrm{g}}}(n)$ as

$\displaystyle\hat{\mathbf{h}}_{k_{\textrm{g}},l_{\textrm{g}}}(n)=[\mathbf{X}^{H}_{k_{\textrm{g}},l_{\textrm{g}}}(n)\mathbf{X}_{k_{\textrm{g}},l_{\textrm{g}}}(n)]^{-1}\mathbf{X}^{H}_{k_{\textrm{g}},l_{\textrm{g}}}(n)\mathbf{y}_{k_{\textrm{g}},l_{\textrm{g}}}(n).$
(8)

It can be seen that this is a joint channel estimation method by utilizing the
PBCH signals of all adjacent cells. Except some estimation errors and
approximation errors, there is no longer influence of ICI. For simplicity, we
only give LS estimation here, actually more sophisticated estimation method
can be developed based on this idea, such as exploiting the channel
correlation in time and frequency domains. Furthermore, since the sensing unit
just listens the opportunistic signals on air and do not affect the transmit
procedure in BSs, there is no additional backhaul or signaling consumption.

The simultaneous equations can also be constructed by PSS/SSS, but PSS/SSS
occupy less REs and only support single stream transmission. Furthermore, this
method can also be used in 5G new radio (NR), where the synchronization signal
and PBCH block (SSB) have similar characteristics as the PBCH in LTE.

###  III-B CSI Processing and Gesture Recognition

After channel estimation, the gestures can be recognized by analyzing the CSI
variation of the serving cell. We combine the CSI in multiple subcarriers by
inverse Fourier transform (IFFT), and extract the strongest path, which can
significantly improve the signal to noise ratio (SNR) of the channel
estimation results[8]. However, as shown in (2), even without ICI the CSI is
also impacted by phase noise. CSI ratio is an effective way to eliminate the
phase noise [9]. If the serving cell is equipped with two or more antenna
ports, we can select one port as the reference channel. Otherwise, we use two
receive antennas and set one as the reference.

For PBCH-based channel estimation, we can update CSI every 10ms, which
corresponds to a sampling frequency of the consecutive CSI at 100Hz. According
to Nyquist sampling theory, the maximum Doppler frequency that can be
identified is 50Hz. Fortunately, most commercial LTE bands operate at
frequencies below 3GHz. Therefore, even if a hand moves at a speed of 1m/s,
the Doppler shift will not exceed 20Hz. To extract the dynamic path
$\hat{h}_{\textrm{d}}(n)$, a band pass filter is used to separate the channel
response with frequency shift range of 3-20Hz from the denoised channel.

In this work, we utilize two receivers to recognize the two-dimensional
gestures. As shown in Fig. 2, totally 6 candidate gestures are designed, and
the user locates in the central area between two receivers and faces to the
BS. The direction of the BS can be identified through the solution proposed in
[5].

Figure 2: Candidate gestures and user position.

Since the focus of this letter is on channel estimation and the gesture
patterns are simple, we use a training-free method to recognize different
gestures based on the change rule of Doppler shifts. We employ a sliding
window with duration $K_{\textrm{D}}$ to calculate the Doppler spectrogram by
short-time discrete Fourier transform (DFT). In the period of gesture
performing, we identify the frequency component with the highest amplitude in
each sliding window, and form the Doppler shift vector in two receivers,
represented by $\mathbf{f}_{1}$ and $\mathbf{f}_{2}$, respectively.

After removing the outliers in $\mathbf{f}_{1}$ and $\mathbf{f}_{2}$, we first
check whether there is a short pause in the motion according to the values of
$|\mathbf{f}_{1}|+|\mathbf{f}_{2}|$, where ‘$|\cdot|$’ is the element-wise
absolute operator. The gestures ‘v1/v2’ can be distinguished if it is true,
and otherwise the gesture should belongs to the other four possibilities. In
Fig. 2, it can be observed that a push or pull movement results in positive or
negative Doppler shifts in both R1 and R2, so the gesture will be identified
as a push or pull when all the non-zero elements in both $\mathbf{f}_{1}$ and
$\mathbf{f}_{2}$ have the same signs. For the movement of left or right slide,
the Doppler shifts of two receivers are in the opposite direction. Therefore,
we can differentiate the left or right slide by the sign of the accumulated
value of $(\mathbf{f}_{1}-\mathbf{f}_{2})$. The same criterion is applied to
differentiate ‘v1’ and ‘v2’.

##  IV Evaluations

###  IV-A Experiment Environment and Configuration

To evaluate the effectiveness of the proposed method, we conduct some
experiments in an apartment using commercial LTE signals. The layout of the
rooms are shown in Fig. 3, and the experiments are implemented in the living
room and bedroom. The commercial signals are from China Mobile TDD-LTE network
operated in Band 40, centering in 2.3498GHz with 20MHz bandwidth. Both rooms
share the same serving cell with PCI 252, and there are also three neighbor
cells with PCI 249, 253, 256. From the PBCH decoding result, we find that they
all utilize only one antenna port. We set the parameters of joint channel
estimation, $K$ and $L$, as 3 and 4, respectively. The signals are transmitted
by outdoor macro-BSs from hundreds meters away. There are no line-of-sight
paths from the receiver to all BSs, but we observe that the signal of serving
cell is mostly penetrated from the window.

We design a prototype system to achieve real-time channel estimation and
gesture recognition. The RF signal is received and sampled by a software radio
platform YunSDR Y550s, which equips 4 antennas and supports 4 streams of data
receiving with maximal bandwidth of 100MHz. The baseband processings and
gesture recognition are implemented in the host computer with an Intel Core
i7-8700 CPU working at 3.20GHz. We place two antennas at position R1 and the
other two at position R2, as if we have two receivers. Employing two antennas
at one position is to eliminate the influence of phase noise.

Figure 3: The layout of the experiment environment.

###  IV-B Experiment Results

In the experiments, we start with measuring the signal strength of all cells
using RSRP, which is defined as the average power of CRS. On one hand, RSRP
indicates the signal quality of the serving cell. On the other hand, the RSRP
of neighbor cells can also reflect the strength of ICI, since in LTE the
transmission powers of PDCCH and PDSCH have fixed ratio with that of CRS. The
RSRP results are shown in Table I. Due to the indoor multipath propagation,
the RSRP fluctuates across different test positions. But it can be seen that
the ICI is strong.

TABLE I: MEASURED RSRP RESULTS $N_{\textrm{ID}}$ Rx | Living Room | Bedroom
---|---|---
R1 | R2 | R1 | R2
252 | -77dBm | -71dBm | -82dBm | -76dBm
249 | -86dBm | -85dBm | -101dBm | -88dBm
253 | -85dBm | -81dBm | -92dBm | -86dBm
256 | -100dBm | -94dBm | -95dBm | -93dBm

To illustrate the impact of ICI, we present an example in Fig. 4, where a pull
gesture is performed. As shown in Fig. 4(a), if we obtain the CSI of serving
cell by CRS, we observe numerous spurs in frequency domain. After main path
extraction, denoising and filtering, the gesture pattern is still obscured by
the ICI, as depicted in Fig. 4(c). Consequently, the Doppler spectrogram is
messy as illustrated in Fig. 4(e). On the contrary, for the CSI acquired by
PBCH-based joint channel estimation, the frequency domain channel response is
smooth, the extracted dynamic path waveform and the corresponding Doppler
spectrogram are clear and evident, as can be seen in Fig. 4(b)(d)(f),
respectively. In Fig. 4, the length of sliding window is 100ms, and we
calculate the DFT at 1Hz interval.

Figure 4: An example of CRS-based and PBCH-based channel estimation and
gesture recognition. (a) The CRS-based channel estimation results in full
bandwidth. (b) The PBCH-based channel estimation results in the middle 72
subcarriers. (c-d) The CRS-based/PBCH-based dynamic path extraction. (e-f) The
Doppler spectrogram calculated from CRS-based/PBCH-based channel estimation.

Recognition accuracy is the most important performance indicator for wireless
gesture recognition. Next, let us compare the recognition accuracies according
to the CSIs acquired by the CRS-based and PBCH-based channel estimations,
which are collected in the same groups of tests. For CRS-based CSI
acquisition, we obtain the frequency domain CSI by LS estimation in each
subframe 0. Furthermore, we consider a subcarrier selection scheme (CRS-SS)
based on the raw CRS-based CSIs, which is proved to be effective in combating
the co-channel interference in WiFi sensing. The subcarriers are chosen with
the weakest correlations after dynamic time warping [10].

In the test, we perform each type of gesture for 50 times. The error
recognition, false alarm, and missed detection are all counted as errors. In
CRS-based approach, the full 20MHz bandwidth is utilized, while the PBCH-based
approach only utilizes 1.4MHz bandwidth it occupied. In the CRS-SS scheme, the
highest accuracies in the living room and bedroom are achieved by selecting
100 and 120 subcarriers, respectively, out of the 200 CRS subcarriers. The
error number statistics of each gesture is shown in Fig. 5.

Figure 5: A comparison of error number for each gesture between the PBCH-
based, CRS-based and CRS-SS based approaches. Recognition accuracies: (a)
Living room, PBCH $94.3\%$, CRS $77.3\%$, CRS-SS $79.3\%$; (b) Bedroom, PBCH
$96.3\%$, CRS $73.3\%$, CRS-SS $76.3\%$.

We can see that the proposed PBCH-based method comprehensively outperforms the
other two methods. Although the CSI estimated by CRS can be smoothed by 200
subcarriers in 20MHz bandwidth, the ICI severely deteriorates the performance.
With subcarrier selection, the accuracies show some improvement, but it is
still far from the PBCH-based method in both scenarios. This is because the
ICI typically spreads over the entire bandwidth, making it difficult to select
clean subcarriers.

##  V Conclusion

This paper proposed an ICI-free channel estimation and wireless gesture
recognition scheme by employing TDD-LTE signals. Firstly, we analyzed the
cause of ICI and its impact on wireless sensing. Then, we proposed a multi-
cell joint channel estimation method based on the reconstructed broadcast
signals from all adjacent cells. The proposed method can eliminate ICI with
only one receive antenna and the minimum LTE system bandwidth. Finally, we
developed a prototype system to receive commercial LTE signals and perform
gesture recognition. The experiment results demonstrated that the CRS-based
approach was susceptible to ICI, while the proposed method achieved high
recognition accuracies.

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# Brown-York energy in stationary spacetimes

Hongsheng Zhang${}^{1,2~{}}$111Electronic address<EMAIL_ADDRESS>1
School of Physics and Technology, University of Jinan,
336 West Road of Nan Xinzhuang, Jinan, Shandong 250022, China
2 State Key Laboratory of Theoretical Physics, Institute of Theoretical
Physics, Chinese Academy of Sciences, Beijing, 100190, China

###### Abstract

One usually defines the Brown-York energy for a 2-surface embedded in a
spacelike 3-slice as an integration of the mean curvature of the 2-surface
isometrically embedded into the 3-slice, with a proper reference 3-space. We
demonstrate that this naive definition is ill for stationary spacetimes. As an
example, we investigate the Kerr-Newman spacetime in detail. We show that the
naive definition of the Brown-York energy is not a component of the Brown-York
boundary stress tensor, and thus deviates from the original idea of Brown and
York. Furthermore, we present the exact form of the Brown-York energy for the
Kerr-Newman spacetime with the proper reference.

Brown-York energy; stationary spacetimes

††preprint: arXiv:

##  I Introduction

The Einstein field equation describes the gravitational fields yielded by some
stress-energy distributions of matters. Surprisingly, the stress-energy,
especially the energy (mass) of gravitational fields is an intricate problem.
Gravitational fields must be associated energy. In fact we have a well-defined
total energy on an asymptotically flat manifold, that is, the Arnowitt-Deser-
Misner (ADM) energy adm . But a total energy is not enough. In the problems of
gravitational collapse, we need study how much energy is transferred from
matter fields to gravitational fields. Note that the energy of matter fields
can be localized, so one has energy density of a matter field. In the
observations of gravitational waves, a wave may propagate a billion years to
arrive at the earth. We cannot say that a gravitational wave carries energy
from the binary star to us if the energy of the gravitational field can only
be defined globally. A wave is non-physical if it does not carry energy.
However, the energy of gravitational fields is non-localizable because of the
equivalence principle. As an inevitable compromise, people consider the quasi-
local energy of gravitational fields, i.e., the energy enclosed in a finite
2-surface.

The positive energy theorem says that the ADM energy is larger than or equal
to zero for a manifold satisfying some energy conditions (without black holes
on it) yao1 . Later, some other types of total energy, such as Bondi-Sachs
energy is also shown to be positive definite or zero bondi . The positive
energy theorem is profound since it ensures that the Minkowski space is
stable. Otherwise the Minkowski space may decay to some other curved
configurations. The Penrose conjecture is a natural generalization of the
positive energy theorem, in which a quasi-local energy is involved pen .
Numerical relativity plays a central role to generate the templates of
gravitational waves of the compact binaries ligo . In numerical calculations,
all the domains involved are finite, and thus the quantities in such domains
must be quasi-local.

We still do not reach a consensus about the quasi-local energy of the
gravitational field. Several different quasi-local energies have been
proposed, for a review stressing on the physical ideas of different quasi-
local energies, see review . Among all these forms, the Brown-York expression
is a leading one BR . The physical idea of Brown-York mass originates from an
observation of the classical Hamilton-Jacobi equation, that is, the energy of
a system can be expressed as a derivative of the action with respect to time
on the boundary. As an extension of this idea to the gravity theory, Brown and
York define the gravitational boundary stress and a variation of the action
with resect to the boundary metric. Thus the zero-zero component of boundary
stress should be the gravitational surface energy density. An integration
leads to the gravitational energy enclosed in the 2-surface. Such an energy
usually diverges for an infinite surface. One needs to introduce a reference
metric to counter this divergence, such that one can obtain a finite result.
In some cases, it is difficult to find proper reference space and reference
metric even if they satisfy the embedding theorems. For example, only
asymptotic form of the reference metric at large $r$ are obtained for the
Kerr-Newman metric kerr1 . The exact form is still in absence. The crucial
flaw in the previous calculations of Brown-York mass of the Kerr-Newman metric
is that the mean curvature of a 2-surface isometrically embedded into a
3-slice with $t$=constant is treated as the surface energy density. One will
see that this definition has little relation to the physical problems in
rotating spaces. This problematic definition of Brown-York mass appeared in
many previous references many .

The boundary Brown-York stress is extensively applied in AdS/CFT adscft ,
which is one of the most significant progress in theoretical physics in recent
years. To obtain a concrete proper form of the Brown-York stress is urgent and
essential to these problems. In the next section, we present the concrete form
of Brown-York energy in stationary spacetimes with proper reference metric. In
section III, we study the properties of this energy form. We demonstrate it
has the desired properties. For example it reduces to the ADM energy when
$r\to\infty$, larger than $m$ but less than $2m$ at the horizon. In section
IV, we present our conclusion.

##  II Brown-York energy in stationary spacetime

We start from a general axial symmetric spacetime,

$ds^{2}=g_{00}(t,r,\theta)dt^{2}+2g_{03}(t,r,\theta)dtd\phi+g_{11}(t,r,\theta)dr^{2}+g_{22}(t,r,\theta)d\theta^{2}+g_{33}(t,r,\theta)d\phi^{2},$
(1)

where $t,~{}r,~{}\theta,~{}\phi$ are four coordinates on the 4-manifold, and
$g_{\mu\nu}$ are components of the metric $g$ of this manifold. According to
the definition, $dr$ is the normal covector of the hypersurface $r=$constant.
Thus we obtain the normalized normal vector $n$ of this hypersurface,

$n^{a}=\alpha g^{ab}(dr)_{b},$ (2)

where $\alpha$ is a normalization factor. Requiring $n_{a}n^{a}=1$, we reach

$n^{a}=g_{11}^{-1/2}\left(\frac{\partial}{\partial r}\right)^{a}.$ (3)

According to definition, the Brown-York stress is the variation of the
boundary term with respect to the boundary metric,

$8\pi\tau_{\mu\nu}=-\left(K_{\mu\nu}-K\gamma_{\mu\nu}\right),$ (4)

where $K_{\mu\nu}$ is the second fundamental form of the hypersurface
$r=$constant, $K=K_{\mu\nu}g^{\mu\nu}$, and $\gamma$ is the induced boundary
metric,

$\gamma^{ab}=g^{ab}-n^{a}n^{b}(g_{cd}n^{c}n^{d}).$ (5)

Eq.(4) implies,

$8\pi\tau_{0}^{0}=-(K_{0}^{0}-(K_{0}^{0}+K_{2}^{2}+K_{3}^{3}))=K_{2}^{2}+K_{3}^{3},$
(6)

where we have used $K_{1}^{1}=0$. Eq.(6) shows that the boundary energy
density equals a sum of the last two components of the second fundamental form
of the hypersurface. However, this can $not$ be further simplified to the mean
curvature of the 2-surface isometrically embedded into the 3-slice with
$t=$constant, which is widely accepted as the definition of the Brown-York
energy density. We demonstrate this point in the spacetime (1). Directly
calculation presents,

$K_{2}^{2}=\frac{g_{22}^{\prime}}{2g_{22}\sqrt{g_{11}}},$ (7)

and

$K_{3}^{3}=\frac{g_{30}g^{\prime}_{30}-g_{00}g_{33}^{\prime}}{2(g_{30}^{2}-g_{00}g_{33})\sqrt{g_{11}}},$
(8)

where a prime denotes a derivative with resect to $r$. As a contrast, we
calculate the mean curvature of the 2-surface $r=$constant imbedded in a
3-slice with $t=$constant. First the induced metric of this 3-slice is,

$ds^{2}=g_{11}(t,r,\theta)dr^{2}+g_{22}(t,r,\theta)d\theta^{2}+g_{33}(t,r,\theta)d\phi^{2}.$
(9)

The following calculation is simple and straightforward. The components of the
second fundamental form of the 2-surface $r=$constant in this 3-slice
$k_{\mu\nu}$ read,

$k_{2}^{2}=\frac{g_{22}^{\prime}}{2g_{22}\sqrt{g_{11}}},$ (10)
$k_{3}^{3}=\frac{g_{33}^{\prime}}{2g_{33}\sqrt{g_{11}}},$ (11)

and $k_{1}^{1}=0$. It is clear $K_{2}^{2}=k_{2}^{2}$, but $K_{3}^{3}\neq
k_{3}^{3}$. So we have,

$8\pi\tau_{0}^{0}=K_{2}^{2}+K_{3}^{3}\neq k_{2}^{2}+k_{3}^{3}.$ (12)

This implies that the mean curvature of the 2-surface imbedded in the 3-slice
does not equal to the surface energy density, and is thus not a component of
the Brown-York stress. Our demonstrations disqualified the mean curvature of
such a 2-surface as the surface energy density. To obtain the proper form of
the surface energy density, we need to go back to the whole spacetime rather
than just starting from a 3-slice. Only for time-orthogonal spacetimes,
$K_{2}^{2}+K_{3}^{3}$ coincides with $k_{2}^{2}+k_{3}^{3}$. In such cases, for
example, the Schwarzschild spacetime, $k_{2}^{2}+k_{3}^{3}$ represents the
surface energy density. The above observation may lead to more extensive
results than what we pointed out in frame of Brown-York energy. In the
expressions of Liu-Yao yao2 , Wang-Yaoyao3 , and Kijowski energies kijo , the
mean curvature of the 2-surface imbedded in the 3-slice are involved. In
analogy to the above discussions, whether they correspond physical energies in
rotating spacetimes needs to be studied seriously.

If $g_{\mu\nu}$ in (1) are not functions of $t$, it is a stationary metric. In
this case, we can reconstruct the above results in the conservation charge
picture. The gravitational energy is not an isolated problem. An ideal way is
to define it as a conserved charge of a conserved current, and thus we can
define the momentum, and further the angular momentum of the gravitational
field in a unified way. For example, the original form of Misner-Sharp energy
was presented only as a quasi-local energy of the gravitational field, while
the corresponding gravitational momentums are not considered. The subsequent
researches show that the Misner-Sharp mass can be treated as the conserved
charge of the conserved current in a static spacetime,

$J_{MS}^{a}=T^{ab}\xi_{b},$ (13)

where $T$ denotes the stress-energy of the spacetime, and $\xi$ is a time-like
Killing vector of the spacetime. The Misner-Sharp mass is defined in an
elegant way,

$M_{MS}=\int*J_{MS},$ (14)

where $*$ denotes the Hodge dual of a tensor, which performs in the total
spacetime. In this way, the momentum of a gravitational field also gets
natural definition. Similarly, the Brown-York energy can be defined as the
boundary conserved charge of a boundary conserved current,

$J_{BR}^{a}=\tau^{ab}\xi_{b},$ (15)

if a Killing vector $\xi=\frac{\partial}{\partial t}$ is permitted at the
boundary spacetime with $r=$constant,

$B_{ab}=g_{00}(r,\theta)dt^{2}+2g_{03}(r,\theta)dtd\phi+g_{22}(r,\theta)d\theta^{2}+g_{33}(r,\theta)d\phi^{2},$
(16)

the Brown-York energy reads,

$M_{BR}=\int*J_{BR}.$ (17)

One can show $J_{BR}$ is really a conserved current.

$\nabla_{a}J_{BR}^{a}=\nabla_{a}\tau^{ab}\xi_{b}+\tau^{ab}\nabla_{a}\xi_{b}.$
(18)

Both of the two terms in the right hand side of the above equation equal zero,
since $\tau_{ab}$ is conserved, and $\xi_{b}$ is Killing at the same time
$\tau_{ab}$ is symmetric. We thus derive $\nabla_{a}J_{BR}^{a}=0$. Note that
here the Hodge dual operator performs on the 3-boundary $B_{ab}$ shown in
(16). One calculates $*J_{BR}$ directly,

$*J_{BR}=-\sqrt{g_{22}g_{33}}\tau^{0}_{0}d\theta\wedge
d\phi=-\frac{1}{8\pi}\sqrt{g_{22}g_{33}}(K^{2}_{2}+K^{3}_{3})d\theta\wedge
d\phi.$ (19)

Thus, we exactly obtain the same result using the conserved charge method.

We apply the above result to the case of the Kerr-Newman metric. The Kerr-
Newman metric may be the unique asymptotically flat stationary metric with
electromagnetic charge in the Einstein gravity, which is a special case of the
metric (1). In the Boyer-Lindquist coordinates, the components of the Kerr-
Newman metric read,

$g_{00}=-(1-\frac{2mr-q^{2}}{\rho^{2}}),$ (20)
$g_{03}=-\frac{a(2mr-q^{2})}{\rho^{2}}\sin^{2}\theta,$ (21)
$g_{11}=\frac{\rho^{2}}{\Delta},$ (22) $g_{22}=\rho^{2},$ (23)
$g_{33}=\frac{(2mr-q^{2})a^{2}\sin^{4}\theta}{\rho^{2}}+(r^{2}+a^{2})\sin^{2}\theta.$
(24)

Here, as usual,

$\rho^{2}=r^{2}+a^{2}\cos^{2}\theta,$ (25)

and

$\Delta=r^{2}-2mr+a^{2}+q^{2}.$ (26)

$m,~{}q,~{}a$ are three parameters in the metric, denoting the total mass,
total charge, and angular momentum per mass respectively. The corresponding
electromagnetic potential reads,

$A=-\frac{qr}{\rho^{2}}(dt-a\sin^{2}\theta d\phi).$ (27)

Based on the above discussions one immediately sees that
$K_{2}^{2}+K_{3}^{3}\neq k_{2}^{2}+k_{3}^{3}$ for the Kerr-Newman case. The
proper surface energy density $\sigma$ reads,

$\sigma=-{\tau_{0}^{0}}=-\frac{K_{2}^{2}+K_{3}^{3}}{8\pi}.$ (28)

Substituting the metric components of Kerr-Newman into (7) and (8), we obtain
the surface density $\sigma$,

$\sigma=-\frac{1}{8\pi}\frac{L-a^{4}(m-r)\cos 4\theta+4a^{2}r\cos
2\theta\left(2a^{2}+3r(r-m)+q^{2}\right)}{\sqrt{2}\left(a^{2}\cos
2\theta+a^{2}+2r^{2}\right)^{5/2}}\sin\theta\sqrt{\frac{\left(a^{2}+r^{2}\right)^{2}-a^{2}\Delta\sin^{2}\theta}{\Delta}},$
(29)

where,

$L=a^{4}m+7a^{4}r-20a^{2}mr^{2}+12a^{2}q^{2}r+20a^{2}r^{3}-32mr^{4}+16q^{2}r^{3}+16r^{5}.$
(30)

It is easy to check that the surface density degenerates to

$\sigma=-\frac{1}{4\pi}\sin\theta\sqrt{r(r-2m)},$ (31)

when $m=0,~{}q=0$, which is exactly the unreferenced Brown-York energy density
of the Schwarzschild spacetime. All the previous studies on Kerr-Newman
sacetime treat $\sigma_{w}=-(k_{2}^{2}+k_{3}^{3})/(8\pi)$ as the surface
density. From (10) and (11), we obtain,

$\sigma_{w}=-\frac{1}{8\pi}\frac{r\sin\theta\left(2r\left(a^{2}+r^{2}\right)-a^{2}\sin^{2}\theta(r-m)\right)}{\sqrt{\left(a^{2}\cos^{2}\theta+r^{2}\right)\left(\left(a^{2}+r^{2}\right)^{2}-a^{2}\Delta\sin^{2}\theta\right)}}\sqrt{\frac{a^{2}+q^{2}}{r^{2}}-\frac{2m}{r}+1}.$
(32)

It is easy to check the above density is exactly the results in kerr1 .
Essentially, this surface density corresponds a hypothetical metric whose
components equal to the Kerr-Newman’s components besides $g_{03}$,

$ds^{2}_{i}=g_{00}dt^{2}+g_{11}dr^{2}+g_{22}d\theta^{2}+g_{33}d\phi^{2},$ (33)

where $g_{00}$, $g_{11}$, $g_{22}$, and $g_{33}$ are given by (20), (22),
(23), and (24), respectively. This hypothetical metric is not a solution of
the Einstein-Maxwell equation (we name it HMNEM later). One sees that the
Kerr-Newman spacetime has a different Brown-York surface energy density
compared with this HMNEW, although they share the same 3-slice. Thus, it is
far from enough if one uses a 3-manifold as the starting point when
investigating the Brown-York energy. The Brown-York energy depends on the
whole spacetime. This is our principle observation in this letter.

Then we continue to complete the Brown-York energy in a compact 2-surface with
$r=$constant and $t=$constant for the Kerr-Newman spacetime and present the
proper reference metric. The energy enclosed in this 2-surface is a simple
integral

$E_{BY}=\int_{S}\sigma,$ (34)

where $S$ denotes the 2-surface, and $\sigma$ is given by (29). The result is
a little lengthy,

$\displaystyle
E_{BY}=\frac{A_{0}}{6(1+A_{1})^{3/2}A_{1}^{3/2}(A_{1}-A_{3})(1+A_{3})^{1/2}}\left(iB_{1}E\left(i\ln\frac{1+\sqrt{A_{1}+1}}{\sqrt{A_{1}}}|\frac{A_{1}}{A_{3}}\right)\right.$
$\displaystyle\left.+\sqrt{\frac{1+A_{1}}{A_{1}}}\left(B_{2}+iB_{3}F\left(i\ln\frac{1+\sqrt{A_{1}+1}}{\sqrt{A_{1}}}|\frac{A_{1}}{A_{3}}\right)\right)\right),$
(35)

where $E$ denotes the elliptic integral of the second kind, which is defined
as,

$E(x|y)=\int_{0}^{x}(1-y\sin^{2}\theta)^{1/2}d\theta,$ (36)

and $F$ denotes the elliptic integral of the first kind, which is defined as,

$F(x|y)=\int_{0}^{x}(1-y\sin^{2}\theta)^{-1/2}d\theta.$ (37)

The variables are defined as follows,

$B_{1}=(1+A_{1}^{-1})^{1/2}(1+A_{3}^{-1})^{1/2}(1+A_{1})A_{3}\left(2A_{1}^{2}+A_{1}(A_{2}-A_{3})-2A_{2}A_{3}\right),$
(38)
$B_{2}=(1+A_{1})^{-1/2}(1+A_{3})(A_{1}^{3}+2A_{1}^{2}(1+A_{2})-2A_{2}A_{3}+A_{1}(A_{2}-A_{3}-3A_{2}A_{3})),$
(39) $B_{3}=A_{3}(1+A_{1})(A_{1}+2A_{2})(1+A_{3}^{-1})^{1/2}(A_{3}-A_{1}).$
(40)

Here $A_{0},~{}A_{1},~{}A_{2},~{}A_{3}$ are defined by,

$A_{0}=\frac{a^{2}(m-5r)-2r\left(3r(r-m)+q^{2}\right)}{2a^{2}},$ (41)
$A_{1}=\frac{r^{2}}{a^{2}},$ (42)
$A_{2}=\frac{a^{4}(r-m)-2a^{2}r\left(r(r-m)+q^{2}\right)-4r^{3}\left(r(r-2m)+q^{2}\right)}{a^{2}\left(a^{2}(m-5r)-2r\left(3r(r-m)+q^{2}\right)\right)},$
(43) $A_{3}=\frac{(a^{2}+r^{2})^{2}}{\Delta a^{2}}-1.$ (44)

When $a\to 0$, the Brown-York energy of Kerr-Newman degenerates to

$E_{BY}=-\sqrt{r^{2}-2mr+q^{2}}.$ (45)

This is exactly the unreferenced Brown-York energy of the Reissner-Nordstrom
black hole.

A proper reference metric is critical to obtain the exact form of the Brown-
York energy. For the Schwarzschild and Reissner-Nordstrom spacetimes, a
Minkowski space is the operable reference metric. However, even for the HMNEW,
a suitable reference metric is still does not found. Here we find the proper
reference metric for the Kerr-Newman spacetime as follows,

$ds^{2}=-dt^{2}+\frac{\rho^{2}}{r^{2}+a^{2}}dr^{2}+\rho^{2}d\theta^{2}+(r^{2}+a^{2})\sin^{2}\theta
d\phi^{2}.$ (46)

It is nothing but the Minkowski metric in the ellipsoidal coordinates. In
principle, the reference space should be Minkowskian one in ellipsoidal
coordinates, since here we write the Kerr-Newman metric in quasi-ellipsoidal
coordinates (Boyer-Lindquist). The Boyer-Lindquist coordinates reduce to the
ellipsoidal ones when $M=0$ and $q=0$ for Kerr-Newman. We thus deduce that the
proper reference is the Minkowski space in ellipsoidal coordinates. When one
needs the Brown-York energy in other coordinates, for example, in the Kerr-
Schild Cartesian coordinates, we should use a reference as the Minkowski in
Cartesian coordinates. In this sense, the reference metric is covariant.

Now we discuss the Brown-York energy in this reference metric. As we have
analysed in detail in the previous discussions, in this case (46)
$K_{2}^{2}+K_{3}^{3}$ equals $k_{2}^{2}+k_{3}^{3}$, since it is a time-
orthogonal space. Thus the surface energy density reads,

$8\pi\sigma=-(K_{2}^{2}+K_{3}^{3})=-(k_{2}^{2}+k_{3}^{3}).$ (47)

From (10) and (11), we obtain,

$\sigma=-\frac{1}{8\pi}\left(\frac{g_{22}^{\prime}}{2g_{22}\sqrt{g_{11}}}+\frac{g_{33}^{\prime}}{2g_{33}\sqrt{g_{11}}}\right).$
(48)

Substituting the components of (43), we thus arrive at the effective surface
energy density in the coordinates (43),

$\sqrt{g_{22}g_{33}}\sigma=-\frac{1}{8\pi}\frac{(3a^{2}+4r^{4}+a^{2}\cos
2\theta)r\sin\theta}{a^{2}+2r^{2}+a^{2}\cos 2\theta}.$ (49)

Integrating the density over the 2-space $(\theta,\phi)$, we obtain,

$E_{BY0}=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\left(-\frac{1}{8\pi}\frac{(3a^{2}+4r^{4}+a^{2}\cos
2\theta)r\sin\theta}{a^{2}+2r^{2}+a^{2}\cos
2\theta}\right)=-\frac{r}{2}-\frac{a^{2}+r^{2}}{2a}{\arctan}\frac{a}{r}.$ (50)

Then we obtain the exact form of the Brown-York energy enclosed in a 2-surface
with $r=$constant and $t$=constant for the Kerr-Newman spacetime,

$E_{BYKN}=E_{BY}-E_{BY0}.$ (51)

This equation enables us to calculate the Brown-York energy in analytical
form.

One can also define the Brown-York mass by using the asymptotic behaviour of
$E_{BY0}$. Expanding the unreferenced energy around $r\to\infty$,

$E_{BY0A}=-r-\frac{a^{2}}{3r}+\frac{a^{4}}{15r^{3}}+\mathcal{O}(1/r^{5}).$
(52)

At the leading order, it is exactly the same as of a Schwarzschild one. By
using this form of the unreferenced Brown-York energy, we obtain,

$E^{\prime}_{BYKN}=E_{BY}-E_{BY0}.$ (53)

$E^{\prime}_{BYKN}$ degenerates to $E_{BYKN}$ in (51) when $r\to\infty$.

Let’s just recapitulate the essential difference of the definitions of Brown-
York energy in previous literatures and in present letter. As we have
mentioned, all the previous discussions of the Brown-York energy treated the
mean curvature of the 2-space as the surface energy density kerr1 . It can be
written as,

$\sigma_{w}=-\frac{k_{2}^{2}+k_{3}^{3}}{8\pi}.$ (54)

By using (10) and (11), we obtain,

$\sigma_{w}=-\frac{1}{8\pi}\left(\frac{g_{22}^{\prime}}{2g_{22}\sqrt{g_{11}}}+\frac{g_{33}^{\prime}}{2g_{33}\sqrt{g_{11}}}\right).$
(55)

Thus, the BY energy enclosed in such a 2-surface reads,

$E_{BYw}=\int_{S}\sigma_{w}=-\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sqrt{g_{22}g_{33}}\frac{1}{8\pi}\left(\frac{g_{22}^{\prime}}{2g_{22}\sqrt{g_{11}}}+\frac{g_{33}^{\prime}}{2g_{33}\sqrt{g_{11}}}\right).$
(56)

In the above expression, neither $g_{00}$ nor $g_{03}$ appear. To obtain the
energy, only information of the metric of the 3-slice is necessary. As
comparison, in our expression, the BY energy reads,

$E_{BY}=-\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sqrt{g_{22}g_{33}}\frac{1}{8\pi}\left(\frac{g_{22}^{\prime}}{2g_{22}\sqrt{g_{11}}}+\frac{g_{30}g^{\prime}_{30}-g_{00}g_{33}^{\prime}}{2(g_{30}^{2}-g_{00}g_{33})\sqrt{g_{11}}}\right).$
(57)

Clearly, one sees that all the components of the full metric are involved, and
especially the essential character of the stationary space $g_{03}$ is
involved. But $E_{BYw}$ does not consider the rotation effects of spacetime.
This is to say, according to the previous definition in literatures of BY
energy, spacetime with same spacelike 3-slice,

$dl^{2}=g_{11}dr^{2}+g_{22}d\theta^{2}+g_{33}d\phi^{2},$ (58)

but different rotations even without rotation will share the same gravity
energy. We think this is not reasonable. In our refined definition, the BY
energy needs the information of the full metric. We think this is more
reasonable.

This is an abstract discussion of the two definitions, i.e., we don’t require
the metric is an on-shell configuration of some gravity theory. For a concrete
gravity theory, for example the Einstein gravity, the field equation imposes
relations among $g_{\mu\nu}$. Thus they are not independent. In this sense,
$g_{11},g_{22},g_{33}$ can carry the information of the full metric.

##  III Properties of the Brown-York energy in stationary spacetimes

Because the expression of Brown-York energy in (34) is rather lengthy. We try
to show some limits to investigate its asymptotic behaviour. And one will see
that it has all the desired properties. One can obtain the familiar results in
time orthogonal spacetimes. First, when $q=0$ and $m=0$ (34) degenerates to

$E_{BY}=-\frac{r}{2}-\frac{a^{2}+r^{2}}{2a}{\arctan}\frac{a}{r}.$ (59)

That is exactly $E_{BY0}$. The outer horizon of Kerr-Newman reads,

$r_{h}=m+\sqrt{m^{2}-a^{2}-q^{2}}.$ (60)

At the horizon we have

$\lim_{r\to r_{h}}E_{BY}=0.$ (61)

Still this property is the same as the Schwarzschild black hole. Thus the
horizon mass of Kerr-Newman black hole with reference spacetime,

$E_{HKN}=\frac{r_{h}}{2}+\frac{a^{2}+r_{h}^{2}}{2a}{\arctan}\frac{a}{r_{h}}.$
(62)

We find that,

$M<E_{HKN}\leq 2M.$ (63)

When $q=a=0$, it reduces to the case of the Schwarzschild , and thus the
equality holds. The extreme black hole is an interesting topic. A
Schwarzschild black hole cannot be extreme. Here we explore Brown-York energy
for extreme Kerr-Newman black holes. For an extreme black hole, the horizon
mass reads

$E_{HKNE}=\lim_{r_{h}\to
M}\left(\frac{r_{h}}{2}+\frac{a^{2}+r_{h}^{2}}{2a}{\arctan}\frac{a}{r_{h}}\right).$
(64)

$E_{HKNE}\leq M$, and the equality holds iff $a=0$.

Then we explore its behaviour at large distance by two plots.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt]{X1.eps}&\includegraphics[width=195.12767pt]{X2.eps}\end{array}$

Figure 1: In this figure, we show the Brown-York energy of a Kerr-Newman black
hole with $a/m=0.9,~{}q/m=0.3$. This figure displays the behaviour of
referenced Brown-York energy $E_{BYKN}$ from the horizon to some large
distance. Note that $E_{BYKN}$ is not divergent at the horizon, as shown in
(62). Here it is 1.506m.

From the figure (1), one finds that the referenced Brown-York energy has a
nice property. It converges to the ADM energy at large distance, which is
never obtained in the previous studies.

##  IV conclusion

Now we present a short summary of this letter. The problem of gravitational
energy is an important problem in both theoretical and practical aspects.
Unexpectedly, the gravitational is essentially nonlocal even at the classical
level. However, it is impossible to make out the energy loss from a binary
bodies in the gravitational radiation problems if one only has a global
definition of gravitational energy. Thus several quasilocal forms of
gravitational energy are developed. Based on the Hamilton-Jaccobi approach,
the Brown-York energy is presented. Traditionally, the Brown-York energy
enclosed in a 2-surface embedded in a spacelike 3-slice is calculated by an
integration of the mean curvature of the 2-surface isometrically embedded into
the 3-slice, with substraction by a proper reference 3-space (usually a flat
space). We find that the mean curvature of the 2-surface does not correspond
to the surface energy density for the stationary spacetimes. And thus it is
improper to calculate the Brown-York energy through an integration of the mean
curvature of a 2-surface isometrically embedded into a 3-slice. In the
problems of the Brown-York energy to start from a 3-slice is misleading. The
proper Brown-York energy depends on the total spacetime. We demonstrate this
point in a general axial symmetric spacetime, and confirm it by the conserved
charge method if there exists a time-like Killing vector. Applying our result
to the Kerr-Newman spacetime, we obtain the analytical form of the Brown-York
energy for the Kerr-Newman spacetime, which is different from the traditional
result kerr1 . Similar problems in Liu-Yao, Wang-Yao, and Kijowski energies
deserve to study further. We conclude that this refined form is more
appropriate to realize the original physical idea of Brown and York, i.e., the
”00”-component of the boundary stress-energy. And thus it should be as the
proper Brown-York energy in stationary spacetimes. We suggest to use this
refined form in the related progresses, such as the proof of positive energy
theorem, and AdS/CFT. When a cosmological constant is introduced, we should
still treat the ”00”-component of the boundary stress-energy as the proper
Brown-York energy, rather than the mean curvature of a two-sphere.

Acknowledgments. This work is supported in part by the National Natural
Science Foundation of China (NSFC) under grant Nos. 11575083, and Shandong
Province Natural Science Foundation under grant No. ZR201709220395.

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