# Isotope purification induced reduction of spin relaxation and spin coherence
times in semiconductors

Oscar Bulancea-Lindvall  Department of Physics, Chemistry and Biology,
Linköping University, SE-581 83 Linköping, Sweden  Matthew Travis Eiles  Max-
Planck-Institut für Physik komplexer Systeme, Nöthnitzer Street 38, D-01187
Dresden, Germany  Nguyen Tien Son  Department of Physics, Chemistry and
Biology, Linköping University, SE-581 83 Linköping, Sweden  Igor A. Abrikosov
Department of Physics, Chemistry and Biology, Linköping University, SE-581 83
Linköping, Sweden  Viktor Ivády<EMAIL_ADDRESS>Department of Physics,
Chemistry and Biology, Linköping University, SE-581 83 Linköping, Sweden  Max-
Planck-Institut für Physik komplexer Systeme, Nöthnitzer Street 38, D-01187
Dresden, Germany

###### Abstract

Paramagnetic defects and nuclear spins are often the major sources of
decoherence and spin relaxation in solid-state qubits realized by optically
addressable point defect spins in semiconductors. It is commonly accepted that
a high degree of depletion of nuclear spins can enhance the coherence time by
reducing magnetic noise. Here we show that the isotope purification beyond a
certain optimal level becomes contra-productive, when both electron and
nuclear spins are present in the vicinity of the qubits. Using state-of-the-
art numerical tools and considering the silicon vacancy qubit in various spin
environments, we demonstrate that the coupling to spin-1/2 point defects in
the lattice can be significantly enhanced by isotope purification. The
enhanced coupling shortens the spin relaxation time that in turn may limit the
the coherence time of spin qubits. Our results can be straightforwardly
generalized to triplet point defect qubits, such as the NV center in diamond
and the divacancy in SiC.

##  I Introduction

Point defect qubits in semiconductors exhibit long coherence time at cryogenic
and room temperature.Balasubramanian  _et al._ (2009); Christle  _et al._
(2015); Simin  _et al._ (2017) Combining this feature with advanced magneto-
optical control of the qubit state has enabled these systems to become the
leading contender in several areas of quantum technology.Weber  _et al._
(2010); Degen  _et al._ (2017); Awschalom  _et al._ (2018) The properties of
point defect qubits depend to a very large degree on the host material. In
particular, magnetic fluctuations in the local spin environment of the defects
can profoundly influence the defect’s coherence time, in most cases reducing
it by several orders of magnitude from the theoretical upper limit set by the
spin relaxation time.Balasubramanian  _et al._ (2009); Childress  _et al._
(2006) As a result, a strategy of chemical and isotope purification in the
host material is commonly pursued in order to enhance the coherence time of
the point defect qubits. It is believed that this strategy can be applied in
most of the cases.Childress  _et al._ (2006); Balasubramanian  _et al._
(2009); Mizuochi  _et al._ (2009); Bar-Gill  _et al._ (2013); Yamamoto  _et
al._ (2013); Herbschleb  _et al._ (2019); Anderson  _et al._

In this Letter, we report on a counterintuitive effect that emerges when both
electron and nuclear spins are present in the vicinity of the qubits. In
particular, we show that isotope purification leads to a significant reduction
of the spin relaxation time, due to enhanced cross relaxation effects with
other paramagnetic defects of similar fine structure. This effect in turn sets
a reduced upper limit for the coherence time. We study the phenomenon
numerically for the quartet silicon vacancy center in SiC, where the
consequences may be of high importance. Our results can be easily extended to
other spin defects, such as the NV center in diamond and the divacancy in SiC.

The negatively charged silicon vacancy in SiC exhibits a quartet ground state
spin with long coherence timeWidmann  _et al._ (2015); Simin  _et al._ (2017).
This high spin state has been utilized in quantum sensing applicationsKraus
_et al._ (2014a); Lee  _et al._ (2015); Simin  _et al._ (2016); Niethammer
_et al._ (2016); Anisimov  _et al._ (2016); Soltamov  _et al._ (2019); Hoang
_et al._ (2021); Abraham  _et al._ (2021) and to implement a room temperature
maser Kraus  _et al._ (2014b). Furthermore, the defect’s favourable optical
propertiesUdvarhelyi  _et al._ (2020, 2019) and advanced fabrication
capabilitiesWidmann  _et al._ (2019); Lukin  _et al._ (2020); Babin  _et al._
(2022) make it potentially interesting for novel near-infra-red quantum
information processing applications.Baranov  _et al._ (2005); Riedel  _et al._
(2012); Kraus  _et al._ (2014b); Nagy  _et al._ (2019); Morioka  _et al._
(2020); Lukin  _et al._ (2020); Wang  _et al._ (2021); Babin  _et al._ (2022)
In 4H-SiC, the V1 and V2 photoluminescence lines and the Tv1-Tv2 electron spin
resonance (ESR) signalsWimbauer  _et al._ (1997); Mizuochi  _et al._ (2002);
Orlinski  _et al._ (2003); Mizuochi  _et al._ (2005); Son  _et al._ (2019) are
related to the negatively charged silicon vacancy. The V1 and V2 center were
assigned to the $h$ and $k$ silicon vacancy configurations, respectively, by
comparing with first principles resultsIvády  _et al._ (2017); Davidsson  _et
al._ (2019). In our numerical studies we consider the V2 center, which is the
most often studied configuration.

There are two main ingredients of the environmental spin bath in SiC. Natural
samples include 4.7% 29Si and 1.1% 13C spin-1/2 nuclear spins. In addition,
the structure of the host material incorporates various paramagnetic defects
and impurities, whose concentrations may vary over several orders of
magnitudes depending on the growth conditions, after growth sample
preparation, and nano-scale fabrication. Here, we consider the most common
intrinsic spin-1/2 defects, such as carbon vacancy and carbon antisite-vacancy
pair whose concentration can reach 1015 cm-3 in HPSI 4H-SiC.Nagy  _et al._
(2019); Son  _et al._ (2007) We note that this value may increase by 2-3
orders of magnitude due to irradiation and implantation that are frequently
used techniques to create silicon vacancy qubits.

In our study, we divide the complex ground state spin Hamiltonian of the
quartet silicon vacancy-environmental spin bath system into two parts,

$H=H_{1}+H_{2}\text{,}$ (1)

where $H_{1}$ and $H_{2}$ describe one and two-spin interaction terms,
respectively. $H_{1}$ includes the zero-field splitting (ZFS) interaction of
the quartet silicon vacancy and Zeeman terms of all the spins in the system,

$\begin{gathered}H_{\text{1}}=D\left(S_{0,z}^{2}-\frac{5}{4}\right)+g_{3\parallel}\mu_{B}\frac{S_{0,+}^{3}-S_{0,-}^{3}}{4i}B_{z}+\\\
g_{e}\mu_{B}\sum_{j=0}^{M}S_{j,z}B_{z}+\mu_{\text{N}}\sum_{k=1}^{N}g_{\text{N},k}I_{k,z}B_{z}\text{,}\end{gathered}$
(2)

where $S_{j,z}$ is the $z$ component of the electron spin operator vector
$\mathbf{S}_{j}$ of defect $j$, $I_{k,z}$ is the $z$ component of the nuclear
spin operator vector $\mathbf{I}_{k}$ of nucleus $k$, $g_{e}$ is the g-factor
of the electron, $\mu_{B}$ is the Bohr magneton, $g_{\text{N},k}$ is the
nuclear g-factor of either 13C or 29Si, and $\mu_{\text{N}}$ is the nuclear
magneton. Terms with $j=0$ index label the silicon vacancy spin, while $j>0$
indices label the doublet paramagnetic defects in the environment. The ZFS
parameter $D$ is equal to $35.0$ MHz for the V2 silicon vacancy
configuration.(Ivády  _et al._ , 2017) The second term on the r.h.s. of Eq.
(2) accounts for a non-vanishing higher order term of the Zeeman interaction
of the quartet spin states in C3v symmetry, where $g_{3\parallel}=0.6$.Simin
_et al._ (2016); Ivády  _et al._ (2018) The second term in the Hamiltonian,
$H_{2}$ in Eq. (2), accounts for the hyperfine and the dipolar coupling of the
spins,

$\begin{gathered}H_{\text{2}}=\sum_{j=0}^{M}\sum_{k=1}^{N}\mathbf{S}_{j}A_{jk}\mathbf{I}_{k}+\\\
\sum_{i=0}^{M}\sum_{j>i}^{M}\frac{\mu_{0}}{4\pi}\frac{g_{e}^{2}\mu_{B}^{2}}{r_{ij}^{3}}\left(\mathbf{S}_{i}\mathbf{S}_{j}-3\left(\mathbf{S}_{i}\mathbf{r}_{ij}\right)\left(\mathbf{S}_{j}\mathbf{r}_{ij}\right)\right)+\\\
\sum_{k=1}^{N}\sum_{l>k}^{N}\frac{\mu_{0}}{4\pi}\frac{g_{\text{N},k}g_{\text{N},l}\mu_{\text{N}}^{2}}{r_{kl}^{3}}\left(\mathbf{I}_{k}\mathbf{I}_{l}-3\left(\mathbf{I}_{k}\mathbf{r}_{kl}\right)\left(\mathbf{I}_{l}\mathbf{r}_{kl}\right)\right)\text{,}\end{gathered}$
(3)

where $A_{jk}$ is the hyperfine tensor, $\mathbf{r}_{ij}$ is the position
vector pointing from spin $i$ to spin $j$, and
$r_{ij}=\left|\mathbf{r}_{ij}\right|$. $A_{jk}$ for $j>0$ are mostly unknown
as they depend on the paramagnetic defects found in the vicinity of the
silicon vacancy. For simplicity, we consider only the hyperfine interaction of
the silicon vacancy, i.e. $A_{jk}=0$ for $j>0$. This approximation does not
affect our qualitative results, however, our quantitative results are the most
accurate for environmental paramagnetic defects of small hyperfine
interactions, such the common carbon vacancy in SiC. The hyperfine coupling
tensors $A_{0k}$ are obtained from first principles density functional theory
calculations in Ref. Ivády (2020) and used here as well. Beyond 15 Å distance
from the silicon vacancy, the Fermi contact term is neglected and only the
dipolar hyperfine term is considered. Local inhomogeneities at the defect
sites due to the hyperfine interaction are included in calculations as an
effective magnetic field. Accordingly, the first term on the r.h.s. of Eq. (3)
is approximated as $H_{\text{inhomo}}=\Delta S_{0,z}$, where
$\Delta=\sum_{k}A_{0k,z}\left\langle I_{k,z}\right\rangle$ is the
inhomogeneous splitting, or the nuclear Overhauser field in other contexts.
Here, the angular bracket represents expectation value, while
$A_{z}=\sqrt{A_{xz}^{2}+A_{yz}^{2}+A_{zz}^{2}}$.

In order to numerically study the Hahn-echo coherence time (T2) of the quartet
silicon vacancy spin in natural and isotope purified SiC, we employ the second
order generalized cluster correlation expansion (gCCE2) methodOnizhuk  _et
al._ (2021). The converged models include $M\approx 1000$ nuclear spins within
a sphere of radius $r_{\text{Bath}}$ around the qubit. For natural nuclear
spin abundance models $r_{\text{Bath}}=50$ Å is used, while for lower
abundances $r_{\text{Bath}}$ is increased to keep the average number of
environmental spins $M$ fixed. Nuclear spin pairs are considered within the
cut-off radius $r_{\text{dip}}=6.0$ Å. The ensemble coherence function is
obtained by averaging over 500 randomly generated spin bath configurations and
fitted with an $A\text{exp}\\!\left(-\left(t/T_{2}\right)^{n}\right)$ function
to obtain the Hahn-echo coherence time $T_{2}$. To study the coherence
properties of the silicon vacancy, the $\left|+3/2\right\rangle$ and the
$\left|+1/2\right\rangle$ states are used to implement a qubit.

In order to quantify the dipolar spin relaxation time (T1) of the quartet
silicon vacancy spin states in a bath of spin-1/2 electron spins, we utilize
the method recently developed in Ref. Ivády (2020) and briefly reviewed in
Ref. Bulancea-Lindvall  _et al._ (2021). The spin bath models consist of
$N=32$ electron spins with varying concentration. We use the first order
cluster approximation, which is suitable for an electron spin bath of short
coherence timeIvády (2020); Bulancea-Lindvall  _et al._ (2021). The time step
of the propagation is set to 1 ps, while the simulation time is optimized for
the considered concentrations and vary between 0.05 and 1 ms. The coupling
strength between the qubit and the spin bath spins are obtained in the point-
spin density approximation, i.e. by using the second term on the r.h.s. of Eq.
(3).

Figure 1:  Coherence of the quartet V2 center in a nuclear spin bath in SiC.
(a) Decay of the coherence function at various magnetic field values for
natural abundance of nuclear spins. The coherence time is provided for each
curve. For $B>200$ G the coherence time saturates and takes the value of T2 =
1.8 ms. (b) 29Si abundance dependence of the saturated coherence time at
$B=200$ G.

First, we study the coherence time of the V2 silicon vacancy qubit when _only
nuclear spins_ are included in the spin bath. The decay of the Hahn echo
coherence function due to a surrounding nuclear spin bath of natural isotope
abundance is depicted in Fig. 1(a). As can be seen, the coherence function
decays on two different time scales. Due to the hyperfine interaction driven
precession of the nuclear spins, see the first term on the r.h.s. of Eq. (3),
the coherence function partially collapses at first with a time scale
comparable with the inhomogeneous coherence time ($T_{2}^{\star}$). In
contrast to the NV center, the coherence function does not recover later and
no coherent beatings can be observed. This irregular behaviour is due to the
quartet spin state and further discussed in Ref. Yang  _et al._ (2014). The
long time scale decay, observable in Fig. 1(a), is due to the nuclear spin-
nuclear spin interaction induced magnetic field fluctuations, see the last
term on the r.h.s. of Eq. (3). The former effect dominates at small magnetic
field values, i.e. at strong hyperfine coupling, while the latter effect
dominates at high magnetic field values where the hyperfine interaction is
suppressed by the Zeeman splitting of the nuclear spin states. The coherence
time saturates above $B=200$ Gauss and takes the values of T2 = 1.8 ms for
natural abundance of paramagnetic nuclei. The strength of the nuclear spin
coupling and thus the saturated high magnetic field coherence time sensitively
depend on the abundance of the nuclear spins. As expected, the T2 time
significantly enhances as the nuclear spin bath is depleted, see Fig. 1(b).
These results are in accordance with the anticipated behaviour of the system
and previous results on the coherence time of the silicon vacancy.Yang  _et
al._ (2014)

Figure 2:  Resonant coupling of a quartet electron spin and a doublet electron
spin. (a) Energy levels of a quartet-doublet two electron spin system. The
corresponding states are provided in the
$\left|m_{\text{S}},m_{\text{S}^{\prime}}\right\rangle$ basis, where the
quantization axes is set parallel to the $c$-axis. (b) Thermalization of the
quartet spin states in a bath of spin-1/2 electron spins. The vertical axis
measures the amount of population transferred from the initial highly
polarized $m_{\text{S}}=+1/2$ state to the rest of the quartet states of the
V2 center under a fixed evolution time of $\Delta t=1$ $\mu$s in a bath of
spin-1/2 electron spin of $C=10^{18}$ cm-3 concentration. The figure depicts
polarization transfer curves for various inhomogeneous splitting $\Delta$ of
the silicon vacancy states.

However, the host material includes not only nuclear spins but also other
_electron spin defects_ in the local environment of the qubits. In order to
understand the behaviour of a quartet electron spin in a bath of spin-1/2
defects, we depict the magnetic field dependence of the energy levels of a
quartet-doublet two electron spin system in Fig. 2(a). For large magnetic
field values the Zeeman interaction is the strongest term in the spin
Hamiltonian in Eq. (1). Due to the magnetic splitting of both the quartet and
the doublet electron spins ($g_{e}\approx 2$ for both spins), the energy
levels form five distinct branches. The states that are included in the
branches are labelled in Fig. 2(a). Note that each of the three innermost
branches consist of a pair of states. These pairs include $\Delta
m_{\text{S}}=\pm 1$ and $\Delta m_{\text{S}^{\prime}}=\mp 1$ states of the
quartet and the doublet states and can be effectively coupled by the
$S_{0,+}S^{\prime}_{j,-}+S_{0,-}S^{\prime}_{j,+}$ term of the second term in
the r.h.s of Eq. (3). This interaction induces spin flip-flops of the electron
spins, shortens the spin state lifetime, and thus limits the coherence time of
the quartet silicon vacancy qubit states.

Since the dipolar coupling of the electron spins is generally small, any
splitting of the spin states within the branches has a significant effect on
the lifetime of the states. In this respect, it is noteworthy that the
$\left|+1/2,-1/2\right\rangle$ and the $\left|-1/2,+1/2\right\rangle$ states
are degenerate, when only ZFS and Zeeman interactions are taken into
consideration, see Fig. 2(a). In contrast, the zero-field interaction splits
the states in other branches. Due to the higher order terms of the Zeeman
interaction of the quartet silicon vacancy, the states within a branch may
cross each other, see for instance the region labelled by LAC in Fig. 2(a).

In order to study the thermalization of the silicon vacancy spin states in an
electron spin bath, we simulate the population transfer from the initially
highly polarized $\left|+1/2\right\rangle$ state to the rest of the quartet
spin states in Fig. 2(b). When the $\left|+1/2,-1/2\right\rangle$ and
$\left|-1/2,+1/2\right\rangle$ states are degenerate, the initial population
thermalizes within 1 $\mu$s irrespective of the external magnetic field, see
the uppermost curve in Fig. 2(b). On the other hand, by introducing an
effective inhomogeneous magnetic field acting on the quartet spin state and
causing a $\Delta$ splitting in the $m_{\text{S}}=\left\\{-1/2,+1/2\right\\}$
subspace of the quartet spin, the magnetic field independent component of the
spin relaxation reduces drastically, indicating the elongation of the spin
state lifetime at most magnetic field values. The remaining high population
transfer peaks, below 50 G and at around 400-500 G in Fig. 2(b), are related
to level anti-crossings (LACs) of the spin states that are studied in more
details in Ref. Bulancea-Lindvall  _et al._ (2022). Furthermore, we note that
the reduced lifetime of the $m_{\text{S}}=\left\\{-1/2,+1/2\right\\}$ state,
beyond the theoretical expectationsSoltamov  _et al._ (2019), has been
recently reported in Ref. Ramsay and Rossi (2020) and has been attributed to
the coupling to spin-1/2 defects. These results indicate that the
inhomogeneous splitting of the spin states plays a crucial role in elongating
the spin state lifetime in an electron spin bath.

Figure 3:  Distribution of the hyperfine splitting $\Delta$ of the
$m_{\text{S}}=\left\\{-1/2,+1/2\right\\}$ subspace of the quartet electron
spin in natural abundance. The inset depicts the variation of the median of
the hyperfine splitting distribution as a function of the paramagnetic 29Si
abundance.

There are several different sources of local inhomogeneous fields that in
principle could induce a splitting of the degenerate subspace of the coupled
quartet-doublet spin states and suppress the mutual flip-flops of the spins.
Hyperfine coupling is typically the strongest interaction that can give rise
to local inhomogeneities on sub-nanomater scales. In order to quantify the
hyperfine interaction induced inhomogeneity, we study the distribution of the
hyperfine splitting of the $m_{\text{S}}=\left\\{-1/2,+1/2\right\\}$ subspace
of the quartet silicon vacancy. In Fig. 3(a), we depict the distribution of
the maximal hyperfine splitting $\sum_{k}A_{0k,z}$ obtained in an ensemble of
$10^{4}$ random nuclear spin bath configurations of natural isotope abundance.
There are several peaks in the probability distribution of the maximal
hyperfine splitting. The largest peak on the left corresponds to
configurations with no first and second nearest neighbour nuclear spins. Going
from left to right the second, third, and fourth peaks include configurations
with one, two, and three second nearest neighbour 29Si nuclear spins. This
series continues with vanishing peak heights. Furthermore, there are two
additional noticeable peaks beyond 20 MHz that correspond to one 13C and zero
29Si nuclear spin and one 13C and one 29Si nuclear spins in the first and
second neighbourhood shell of the silicon vacancy.

To study the nuclear spin abundance dependence of the hyperfine interaction
induced local inhomogeneity we take the median of the distribution, for which
we obtain 3.97 MHz in natural abundance. We note that a single value cannot
properly characterize a multi peak distribution observed in Fig. 3(a),
however, in paramagnetic isotope depleted samples the amplitude of the peaks
beyond the first peak are significantly reduced and the distribution converges
to a single peak-asymmetric distribution. In such cases, the median is a good
measure of the distribution of the maximal hyperfine splitting. The median as
a function of the 29Si abundance is depicted in Fig. 3(b). As can be seen, the
median of the hyperfine splitting polynomially approaches zero as the
paramagnetic silicon isotopes are depleted.

Figure 4:  Paramagnetic 29Si abundance dependence of the spin coherence time
at various spin-1/2 point defect concentrations. T${}_{2,\text{nuc}}$ (dashed
black line) is obtained at 200 G by including 29Si nuclear spins in the
environment only. The theoretical maximum of the coherence time
T${}_{2,\text{max}}$ is set by spin relaxation due to electron spins (plotted
by colored dashed lines). The coherence time T2 is obtained by combining these
effects (colored thick solid lines).

Combining our results presented so far, we conclude that the isotope
purification reduces the inhomogeneous splitting of the qubit states, see Fig.
3(b), which in turn can enhance the coupling and cross-relaxation effects
between the quartet silicon vacancy spin states and spin-1/2 defects in the
local environment, see Fig. 2(b). This counterintuitive phenomena may lead to
a drastically reduced spin state lifetime that sets the maximum for the
coherence time. To quantify this effect, we calculate the spin relaxation time
T1 of the $\left|+1/2\right\rangle$ state the silicon vacancy as a function of
the 29Si abundance for various spin-1/2 paramagnetic point defect
concentrations. The hyperfine interaction induced local inhomogeneity, i.e.
the first term on the r.h.s. of Eq. (3), is approximated by an inhomogeneous
magnetic field acting on the silicon vacancy and splits the
$m_{\text{S}}=\left\\{-1/2,+1/2\right\\}$ subspace by
$\Delta=\text{median}\\!\left(\sum_{k}A_{0k,z}\right)$, see Fig. 3(b). We
define the absolute maximum of the coherence time as
$\text{T}_{\text{2,max}}=2\text{T}_{1}$. Here, we note that in experiments
$\text{T}_{\text{2, max}}\approx 0.5\text{T}_{1}$ is found for the NV
centerBar-Gill  _et al._ (2013), therefore our results can be considered as an
upper bound. To obtain the coherence time when both nuclear spins and electron
spins are included in the local environment of the silicon vacancy, we use the
$\text{T}_{2}^{-1}=\text{T}_{2,\text{max}}^{-1}+\text{T}_{2,\text{nuc}}^{-1}$
relation, where the last term accounts for decoherence effects due to nuclear
spin flip-flops induced magnetic fluctuations. Finally, we note that direct
calculation of the T2 time with the gCCE method provides inaccurate results in
this case, as this method substantially overestimates relaxation effects of
strongly coupled spin systemsOnizhuk  _et al._ (2021).

The results on the ensemble averaged coherence time are depicted in Fig. 4. As
can be seen the coherence time can be significantly reduced both by the
increase of the electron spin concentration and the depletion of the
paramagnetic isotopes. In high spin-1/2 defect concentration ($\approx$1018
cm-3) the T2 is maximized by the paramagnetic defects and cannot reach higher
than $\approx$100 $\mu$s. As the defect concentration reduces, the theoretical
maximum of the coherence time rapidly increases and the fluctuation of the
nuclear spin bath starts to limit the coherence time in natural abundance, see
the calculated T2 time at, for instance, natural abundance of 29Si isotope in
Fig. 4. Isotope purification not only reduces magnetic field fluctuations but
also enhances cross relaxation effects that may become the major limiting
factor in the coherence time in nuclear spin depleted samples, see Fig. 4. We
note that even a very low concentration of spin-1/2 defect may have a dramatic
effect on the coherence time in highly isotope purifies samples.

Very recently a nanophotonic device integrating V2 qubit with excellent spin
properties was realized in Ref. Babin  _et al._ (2022). The spin coherence
time was found to be 1.39 ms in a high purity isotope purified sample. The
isotope abundance of the sample was estimated to be ${}^{28}\text{Si}>99.85$%
and ${}^{12}\text{C}>99.98$%. Considering only magnetic field fluctuations due
to the residual nuclear spin bath ($\mathtt{\sim}$0.15% 29Si), we would expect
a coherence time close to 25 ms. The order of magnitude difference indicates
that the coherence time is limited by an effect other that nuclear spin flip-
flops. Based on our results, interaction with electron spins in the lattice is
a possible source of decoherence in this experiment.

Our qualitative and quantitative results are obtained for the quartet silicon
vacancy in SiC. However, the spin Hamiltonian and the results in Fig. 2 can be
straightforwardly generalized to spin-1 point defect qubits interacting with
other spin-1 environmental defects. In this case one can also observe nearly
degenerate spin states that are coupled by the dipolar interaction. For spin-1
defects not only the inhomogeneous hyperfine interaction, but also the
differences of the ZFS can contribute to the splitting of the coupled states
and suppress cross relaxation effects. Relaxation of the spin states is the
most efficient when nearby spin-1 defects of the same kind are coupled to each
other, i.e. their ZFS parameters are the same.

## Acknowledgments

We acknowledge support from the Knut and Alice Wallenberg Foundation through
WBSQD2 project (Grant No. 2018.0071). Support from the Swedish Government
Strategic Research Area SeRC and the Swedish Government Strategic Research
Area in Materials Science on Functional Materials at Linköping University
(Faculty Grant SFO-Mat-LiU No. 2009 00971) is gratefully acknowledged. N. T.
S. acknowledges the support from the Swedish Research Council (Grant No. VR
2016-04068), the EU H2020 project QuanTELCO (Grant No. 862721). The
calculations were performed on resources provided by the Swedish National
Infrastructure for Computing (SNIC) at the National Supercomputer Centre
(NSC).

## References

  * Balasubramanian  _et al._ (2009) G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383 (2009).
  * Christle  _et al._ (2015) David J. Christle, Abram L. Falk, Paolo Andrich, Paul V. Klimov, Jawad Ul Hassan, Nguyen T. Son, Erik Janzén, Takeshi Ohshima, and David D. Awschalom, “Isolated electron spins in silicon carbide with millisecond coherence times,” Nat. Mater. 14, 160–163 (2015).
  * Simin  _et al._ (2017) D. Simin, H. Kraus, A. Sperlich, T. Ohshima, G. V. Astakhov, and V. Dyakonov, “Locking of electron spin coherence above 20 ms in natural silicon carbide,” Physical Review B 95, 161201 (2017), publisher: American Physical Society.
  * Weber  _et al._ (2010) J. R. Weber, W. F. Koehl, J. B. Varley, A. Janotti, B. B. Buckley, C. G. Van de Walle, and D. D. Awschalom, “Quantum computing with defects,” PNAS 107, 8513–8518 (2010).
  * Degen  _et al._ (2017) C. L. Degen, F. Reinhard, and P. Cappellaro, “Quantum sensing,” Rev. Mod. Phys. 89, 035002 (2017).
  * Awschalom  _et al._ (2018) David D. Awschalom, Ronald Hanson, Jörg Wrachtrup, and Brian B. Zhou, “Quantum technologies with optically interfaced solid-state spins,” Nature Photonics 12, 516–527 (2018), number: 9 Publisher: Nature Publishing Group.
  * Childress  _et al._ (2006) L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, “Coherent dynamics of coupled electron and nuclear spin qubits in diamond,” Science 314, 281–285 (2006).
  * Mizuochi  _et al._ (2009) N. Mizuochi, P. Neumann, F. Rempp, J. Beck, V. Jacques, P. Siyushev, K. Nakamura, D. J. Twitchen, H. Watanabe, S. Yamasaki, F. Jelezko, and J. Wrachtrup, “Coherence of single spins coupled to a nuclear spin bath of varying density,” Phys. Rev. B 80, 041201 (2009).
  * Bar-Gill  _et al._ (2013) N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nature Communications 4, 1743 (2013).
  * Yamamoto  _et al._ (2013) T. Yamamoto, T. Umeda, K. Watanabe, S. Onoda, M. L. Markham, D. J. Twitchen, B. Naydenov, L. P. McGuinness, T. Teraji, S. Koizumi, F. Dolde, H. Fedder, J. Honert, J. Wrachtrup, T. Ohshima, F. Jelezko, and J. Isoya, “Extending spin coherence times of diamond qubits by high-temperature annealing,” Phys. Rev. B 88, 075206 (2013).
  * Herbschleb  _et al._ (2019) E. D. Herbschleb, H. Kato, Y. Maruyama, T. Danjo, T. Makino, S. Yamasaki, I. Ohki, K. Hayashi, H. Morishita, M. Fujiwara, and N. Mizuochi, “Ultra-long coherence times amongst room-temperature solid-state spins,” Nature Communications 10, 3766 (2019), number: 1 Publisher: Nature Publishing Group.
  * (12) Christopher P. Anderson, Elena O. Glen, Cyrus Zeledon, Alexandre Bourassa, Yu Jin, Yizhi Zhu, Christian Vorwerk, Alexander L. Crook, Hiroshi Abe, Jawad Ul-Hassan, Takeshi Ohshima, Nguyen T. Son, Giulia Galli, and David D. Awschalom, “Five-second coherence of a single spin with single-shot readout in silicon carbide,” Science Advances 8, eabm5912, publisher: American Association for the Advancement of Science.
  * Widmann  _et al._ (2015) Matthias Widmann, Sang-Yun Lee, Torsten Rendler, Nguyen Tien Son, Helmut Fedder, Seoyoung Paik, Li-Ping Yang, Nan Zhao, Sen Yang, Ian Booker, Andrej Denisenko, Mohammad Jamali, S. Ali Momenzadeh, Ilja Gerhardt, Takeshi Ohshima, Adam Gali, Erik Janzén, and Jörg Wrachtrup, “Coherent control of single spins in silicon carbide at room temperature,” Nat. Mater. 14, 164–168 (2015).
  * Kraus  _et al._ (2014a) H. Kraus, V. A. Soltamov, F. Fuchs, D. Simin, A. Sperlich, P. G. Baranov, G. V. Astakhov, and V. Dyakonov, “Magnetic field and temperature sensing with atomic-scale spin defects in silicon carbide,” Scientific Reports 4, 5303 (2014a), number: 1 Publisher: Nature Publishing Group.
  * Lee  _et al._ (2015) Sang-Yun Lee, Matthias Niethammer, and Jörg Wrachtrup, “Vector magnetometry based on $s=\frac{3}{2}$ electronic spins,” Phys. Rev. B 92, 115201 (2015).
  * Simin  _et al._ (2016) D. Simin, V. A. Soltamov, A. V. Poshakinskiy, A. N. Anisimov, R. A. Babunts, D. O. Tolmachev, E. N. Mokhov, M. Trupke, S. A. Tarasenko, A. Sperlich, P. G. Baranov, V. Dyakonov, and G. V. Astakhov, “All-optical dc nanotesla magnetometry using silicon vacancy fine structure in isotopically purified silicon carbide,” Phys. Rev. X 6, 031014 (2016).
  * Niethammer  _et al._ (2016) Matthias Niethammer, Matthias Widmann, Sang-Yun Lee, Pontus Stenberg, Olof Kordina, Takeshi Ohshima, Nguyen Tien Son, Erik Janzén, and Jörg Wrachtrup, “Vector magnetometry using silicon vacancies in $4h$-sic under ambient conditions,” Phys. Rev. Applied 6, 034001 (2016).
  * Anisimov  _et al._ (2016) A. N. Anisimov, D. Simin, V. A. Soltamov, S. P. Lebedev, P. G. Baranov, G. V. Astakhov, and V. Dyakonov, “Optical thermometry based on level anticrossing in silicon carbide,” Sci. Rep. 6, 33301 (2016).
  * Soltamov  _et al._ (2019) V. A. Soltamov, C. Kasper, A. V. Poshakinskiy, A. N. Anisimov, E. N. Mokhov, A. Sperlich, S. A. Tarasenko, P. G. Baranov, G. V. Astakhov, and V. Dyakonov, “Excitation and coherent control of spin qudit modes in silicon carbide at room temperature,” Nature Communications 10, 1678 (2019).
  * Hoang  _et al._ (2021) Tuan Minh Hoang, Hitoshi Ishiwata, Yuta Masuyama, Yuichi Yamazaki, Kazutoshi Kojima, Sang-Yun Lee, Takeshi Ohshima, Takayuki Iwasaki, Digh Hisamoto, and Mutsuko Hatano, “Thermometric quantum sensor using excited state of silicon vacancy centers in 4H-SiC devices,” Applied Physics Letters 118, 044001 (2021), publisher: American Institute of Physics.
  * Abraham  _et al._ (2021) John B. S. Abraham, Cameron Gutgsell, Dalibor Todorovski, Scott Sperling, Jacob E. Epstein, Brian S. Tien-Street, Timothy M. Sweeney, Jeremiah J. Wathen, Elizabeth A. Pogue, Peter G. Brereton, Tyrel M. McQueen, Wesley Frey, B. D. Clader, and Robert Osiander, “Nanotesla Magnetometry with the Silicon Vacancy in Silicon Carbide,” Physical Review Applied 15, 064022 (2021), publisher: American Physical Society.
  * Kraus  _et al._ (2014b) H. Kraus, V. A. Soltamov, D. Riedel, S. Väth, F. Fuchs, A. Sperlich, P. G. Baranov, V. Dyakonov, and G. V. Astakhov, “Room-temperature quantum microwave emitters based on spin defects in silicon carbide,” Nat. Phys. 10, 157–162 (2014b).
  * Udvarhelyi  _et al._ (2020) Péter Udvarhelyi, Gerg ő Thiering, Naoya Morioka, Charles Babin, Florian Kaiser, Daniil Lukin, Takeshi Ohshima, Jawad Ul-Hassan, Nguyen Tien Son, Jelena Vučković, Jörg Wrachtrup, and Adam Gali, “Vibronic states and their effect on the temperature and strain dependence of silicon-vacancy qubits in $4h$-$\mathrm{Si}\mathrm{C}$,” Phys. Rev. Applied 13, 054017 (2020).
  * Udvarhelyi  _et al._ (2019) Péter Udvarhelyi, Roland Nagy, Florian Kaiser, Sang-Yun Lee, Jörg Wrachtrup, and Adam Gali, “Spectrally stable defect qubits with no inversion symmetry for robust spin-to-photon interface,” Phys. Rev. Applied 11, 044022 (2019).
  * Widmann  _et al._ (2019) Matthias Widmann, Matthias Niethammer, Dmitry Yu. Fedyanin, Igor A. Khramtsov, Torsten Rendler, Ian D. Booker, Jawad Ul Hassan, Naoya Morioka, Yu-Chen Chen, Ivan G. Ivanov, Nguyen Tien Son, Takeshi Ohshima, Michel Bockstedte, Adam Gali, Cristian Bonato, Sang-Yun Lee, and Jörg Wrachtrup, “Electrical Charge State Manipulation of Single Silicon Vacancies in a Silicon Carbide Quantum Optoelectronic Device,” Nano Letters 19, 7173–7180 (2019).
  * Lukin  _et al._ (2020) Daniil M. Lukin, Constantin Dory, Melissa A. Guidry, Ki Youl Yang, Sattwik Deb Mishra, Rahul Trivedi, Marina Radulaski, Shuo Sun, Dries Vercruysse, Geun Ho Ahn, and Jelena Vučković, “4H-silicon-carbide-on-insulator for integrated quantum and nonlinear photonics,” Nature Photonics 14, 330–334 (2020), number: 5 Publisher: Nature Publishing Group.
  * Babin  _et al._ (2022) Charles Babin, Rainer Stöhr, Naoya Morioka, Tobias Linkewitz, Timo Steidl, Raphael Wörnle, Di Liu, Erik Hesselmeier, Vadim Vorobyov, Andrej Denisenko, Mario Hentschel, Christian Gobert, Patrick Berwian, Georgy V. Astakhov, Wolfgang Knolle, Sridhar Majety, Pranta Saha, Marina Radulaski, Nguyen Tien Son, Jawad Ul-Hassan, Florian Kaiser, and Jörg Wrachtrup, “Fabrication and nanophotonic waveguide integration of silicon carbide colour centres with preserved spin-optical coherence,” Nature Materials 21, 67–73 (2022), bandiera_abtest: a Cg_type: Nature Research Journals Number: 1 Primary_atype: Research Publisher: Nature Publishing Group Subject_term: Quantum information;Quantum physics;Single photons and quantum effects Subject_term_id: quantum-information;quantum-physics;single-photons-and-quantum-effects.
  * Baranov  _et al._ (2005) P.G. Baranov, I.V. Il’in, E.N. Mokhov, M.V. Muzafarova, S.B. Orlinskii, and J. Schmidt, “Epr identification of the triplet ground state and photoinduced population inversion for a si-c divacancy in silicon carbide,” JETP Letters 82, 441–443 (2005).
  * Riedel  _et al._ (2012) D. Riedel, F. Fuchs, H. Kraus, S. Väth, A. Sperlich, V. Dyakonov, A. A. Soltamova, P. G. Baranov, V. A. Ilyin, and G. V. Astakhov, “Resonant addressing and manipulation of silicon vacancy qubits in silicon carbide,” Phys. Rev. Lett. 109, 226402 (2012).
  * Nagy  _et al._ (2019) Roland Nagy, Matthias Niethammer, Matthias Widmann, Yu-Chen Chen, Péter Udvarhelyi, Cristian Bonato, Jawad Ul Hassan, Robin Karhu, Ivan G. Ivanov, Nguyen Tien Son, Jeronimo R. Maze, Takeshi Ohshima, Öney O. Soykal, Ádám Gali, Sang-Yun Lee, Florian Kaiser, and Jörg Wrachtrup, “High-fidelity spin and optical control of single silicon-vacancy centres in silicon carbide,” Nature Communications 10, 1954 (2019).
  * Morioka  _et al._ (2020) Naoya Morioka, Charles Babin, Roland Nagy, Izel Gediz, Erik Hesselmeier, Di Liu, Matthew Joliffe, Matthias Niethammer, Durga Dasari, Vadim Vorobyov, Roman Kolesov, Rainer Stöhr, Jawad Ul-Hassan, Nguyen Tien Son, Takeshi Ohshima, Péter Udvarhelyi, Gergő Thiering, Adam Gali, Jörg Wrachtrup, and Florian Kaiser, “Spin-controlled generation of indistinguishable and distinguishable photons from silicon vacancy centres in silicon carbide,” Nature Communications 11, 2516 (2020), number: 1 Publisher: Nature Publishing Group.
  * Wang  _et al._ (2021) Jun-Feng Wang, Fei-Fei Yan, Qiang Li, Zheng-Hao Liu, Jin-Ming Cui, Zhao-Di Liu, Adam Gali, Jin-Shi Xu, Chuan-Feng Li, and Guang-Can Guo, “Robust coherent control of solid-state spin qubits using anti-Stokes excitation,” Nature Communications 12, 3223 (2021), number: 1 Publisher: Nature Publishing Group.
  * Wimbauer  _et al._ (1997) T. Wimbauer, B. K. Meyer, A. Hofstaetter, A. Scharmann, and H. Overhof, “Negatively charged si vacancy in $4h$ sic: A comparison between theory and experiment,” Phys. Rev. B 56, 7384–7388 (1997).
  * Mizuochi  _et al._ (2002) N. Mizuochi, S. Yamasaki, H. Takizawa, N. Morishita, T. Ohshima, H. Itoh, and J. Isoya, “Continuous-wave and pulsed epr study of the negatively charged silicon vacancy with $s=\frac{3}{2}$ and ${C}_{3v}$ symmetry in $n$-type $4h-\mathrm{SiC}$,” Phys. Rev. B 66, 235202 (2002).
  * Orlinski  _et al._ (2003) S. B. Orlinski, J. Schmidt, E. N. Mokhov, and P. G. Baranov, “Silicon and carbon vacancies in neutron-irradiated sic: A high-field electron paramagnetic resonance study,” Phys. Rev. B 67, 125207 (2003).
  * Mizuochi  _et al._ (2005) N. Mizuochi, S. Yamasaki, H. Takizawa, N. Morishita, T. Ohshima, H. Itoh, T. Umeda, and J. Isoya, “Spin multiplicity and charge state of a silicon vacancy $({T}_{V2a})$ in $4h$-sic determined by pulsed endor,” Phys. Rev. B 72, 235208 (2005).
  * Son  _et al._ (2019) Nguyen Tien Son, Pontus Stenberg, Valdas Jokubavicius, Takeshi Ohshima, Jawad Ul Hassan, and Ivan G. Ivanov, “Ligand hyperfine interactions at silicon vacancies in 4H-SiC,” Journal of Physics: Condensed Matter 31, 195501 (2019), publisher: IOP Publishing.
  * Ivády  _et al._ (2017) Viktor Ivády, Joel Davidsson, Nguyen Tien Son, Takeshi Ohshima, Igor A. Abrikosov, and Adam Gali, “Identification of si-vacancy related room-temperature qubits in $4h$ silicon carbide,” Phys. Rev. B 96, 161114 (2017).
  * Davidsson  _et al._ (2019) Joel Davidsson, Viktor Ivády, Rickard Armiento, Takeshi Ohshima, N. T. Son, Adam Gali, and Igor A. Abrikosov, “Identification of divacancy and silicon vacancy qubits in 6H-SiC,” Applied Physics Letters 114, 112107 (2019), publisher: American Institute of Physics.
  * Son  _et al._ (2007) N. T. Son, P. Carlsson, J. ul Hassan, B. Magnusson, and E. Janzén, “Defects and carrier compensation in semi-insulating $4h\text{$-$}\mathrm{Si}\mathrm{C}$ substrates,” Phys. Rev. B 75, 155204 (2007).
  * Ivády  _et al._ (2018) Viktor Ivády, Igor A. Abrikosov, and Adam Gali, “First principles calculation of spin-related quantities for point defect qubit research,” npj Computational Materials 4, 76 (2018).
  * Ivády (2020) Viktor Ivády, “Longitudinal spin relaxation model applied to point-defect qubit systems,” Phys. Rev. B 101, 155203 (2020).
  * Onizhuk  _et al._ (2021) Mykyta Onizhuk, Kevin C. Miao, Joseph P. Blanton, He Ma, Christopher P. Anderson, Alexandre Bourassa, David D. Awschalom, and Giulia Galli, “Probing the Coherence of Solid-State Qubits at Avoided Crossings,” PRX Quantum 2, 010311 (2021), publisher: American Physical Society.
  * Bulancea-Lindvall  _et al._ (2021) Oscar Bulancea-Lindvall, Nguyen T. Son, Igor A. Abrikosov, and Viktor Ivády, “Dipolar spin relaxation of divacancy qubits in silicon carbide,” npj Computational Materials 7, 1–11 (2021), bandiera_abtest: a Cc_license_type: cc_by Cg_type: Nature Research Journals Number: 1 Primary_atype: Research Publisher: Nature Publishing Group Subject_term: Semiconductors;Spintronics Subject_term_id: semiconductors;spintronics.
  * Yang  _et al._ (2014) Li-Ping Yang, Christian Burk, Matthias Widmann, Sang-Yun Lee, Jörg Wrachtrup, and Nan Zhao, “Electron spin decoherence in silicon carbide nuclear spin bath,” Phys. Rev. B 90, 241203 (2014).
  * Bulancea-Lindvall  _et al._ (2022) Oscar Bulancea-Lindvall, Matthew T. Eiles, Nguyen Tien Son, Igor A. Abrikosov, and Viktor Ivády, “Low-field microwave-free sensors using dipolar spin relaxation of quartet spin states in silicon carbide,” arXiv:2201.03953 [cond-mat] (2022), arXiv: 2201.03953.
  * Ramsay and Rossi (2020) A. J. Ramsay and A. Rossi, “Relaxation dynamics of spin-$\frac{3}{2}$ silicon vacancies in $4H$-SiC,” Physical Review B 101, 165307 (2020), publisher: American Physical Society.


Quasilocal energy-momentum for tensor V in small regions

Lau Loi So

Department of Physics, National Central University, Chung-Li 320, Taiwan

Department of Physics, Tamkang University, Tamsui 251, Taiwan

###### Abstract

The Bel-Robinson tensor $B$ and the tensor $V$ have the same quasilocal
energy-momentum in a small sphere. Using a pseudotensor approach to evaluate
the energy-momentum in a half-cylinder, we find that $B$ and $V$ have
different values, not proportional to the ‘Bel-Robinson energy-momentum’.
Furthermore, even if we arrange things so that we do get the same ‘Bel-
Robinson energy-momentum’ value, the angular momentum gives different values
using $B$ and $V$ in half-cylinder. In addition, we find that $B$ and $V$ have
a different number of independent components. The fully trace free property of
$B$ and $V$ implies conservation of pure ‘Bel-Robinson energy-momentum’ in
small regions, and vice versa.

##  1 Introduction

In attempts to identify a good physical expression for the local distribution
of gravitational energy-momentum there have been many different approaches
which are similar to Einstein’s [1]. For example, those of Landau-Lifshitz
[2], Bergmann-Thomson [3], Papapetrou [4] and Weinberg [5]. Most of them deal
with the Einstein equation: $G_{\mu\nu}=\kappa{}T_{\mu\nu}$, where $\kappa$ is
a constant, $G_{\mu\nu}$ and $T_{\mu\nu}$ are the Einstein and stress tensors.
One can define a superpotential with a suitable anti-symmetry
$U_{\alpha}{}^{\mu\nu}\equiv{}U_{\alpha}{}^{[\mu\nu]}$ and remove a divergence
of $U_{\alpha}{}^{\mu\nu}$ from $G_{\mu\nu}$ to define the gravitational
energy-momentum density

$2\kappa{}\mathbf{t}_{\alpha}{}^{\mu}:=\partial_{\nu}U_{\alpha}{}^{[\mu\nu]}-2\sqrt{-g}\,G_{\alpha}{}^{\mu}.$
(1)

Note that $\mathbf{t}_{\alpha}{}^{\mu}$ is a pseudotensor [6]. Using the
Einstein equation, we have a total energy-momentum density which satisfies

$\partial_{\nu}U_{\alpha}{}^{[\mu\nu]}=2\kappa{\cal{}T}_{\alpha}{}^{\mu}=2\kappa(\mathbf{T}_{\alpha}{}^{\mu}+\mathbf{t}_{\alpha}{}^{\mu}),$
(2)

where $\mathbf{T}_{\alpha}{}^{\mu}=\sqrt{-g}\,T_{\alpha}{}^{\mu}$ and hence,
due to the antisymmetry of $U_{\alpha}{}^{[\mu\nu]}$, is automatically
conserved, i.e., has a vanishing divergence.

The proposed criteria for testing quasilocal expressions include: (i) limit to
good weak field values (i.e., linearized gravity). (ii) good asymptotic values
both at spatial and null infinity. We emphasize that the criteria for these
two are not very restrictive; they only test the quasilocal expression to
linear order. (iii) positivity (i.e., globally) is a strong test but is not
easy to achieve, (iv) small region inside of matter: the quasilocal energy-
momentum expression should, by the equivalence principle, reduce to the
material source terms. Most classical pseudotensors pass this test. (v) small
region in vacuum: positivity for the first non-vanishing parts of the
quasilocal expression. This depends on the gravitational field non-linearly,
and hence it can give a discriminating test of the expression; it is quite
non-trivial but not impossibly difficult.

Positive quasilocal gravitational energy should hold not only on a large scale
but also on the small scale [7]. However it is generally not at all easy to
prove that a particular expression enjoys this property. A good test case is
the small region limit. This will be our concern in this work. Here we
consider specifically the pseudotensor expressions. For a small region, one
can expand the energy-momentum density in Riemann normal coordinates (RNC)
about the origin:

$\displaystyle{\cal{}T}_{\alpha}{}^{\beta}(x)$ $\displaystyle=$
$\displaystyle{\cal{T}}_{\alpha}{}^{\beta}|_{0}+\partial_{\mu}{\cal{}T}_{\alpha}{}^{\beta}|_{0}x^{\mu}+\frac{1}{2}\partial^{2}_{\mu\nu}{\cal{T}}_{\alpha}{}^{\beta}|_{0}x^{\mu}x^{\nu}+...$
(3) $\displaystyle=$
$\displaystyle\mathbf{T}_{\alpha}{}^{\beta}|_{0}+\partial_{\mu}\mathbf{T}_{\alpha}{}^{\beta}|_{0}x^{\mu}+...+\mathbf{t}_{\alpha}{}^{\beta}|_{0}+\partial_{\mu}\mathbf{t}_{\alpha}{}^{\beta}|_{0}x^{\mu}+\frac{1}{2}\partial^{2}_{\mu\nu}\mathbf{t}_{\alpha}{}^{\beta}|_{0}x^{\mu}x^{\nu}+....$

By construction $\mathbf{t}_{\alpha}{}^{\beta}|_{0}$ and
$\partial_{\mu}\mathbf{t}_{\alpha}{}^{\beta}|_{0}$ vanish in vacuum.
Consequently, for small $x^{\mu}$ inside of matter the
$\mathbf{T}_{\alpha}{}^{\beta}$ and
$\partial_{\mu}\mathbf{T}_{\alpha}{}^{\beta}$ terms dominate (this is a
reflection of the equivalence principle). In vacuum regions all the
$\mathbf{T}_{\alpha}{}^{\beta}$ terms vanish, then the lowest order non-
vanishing term is
$\frac{1}{2}\partial^{2}_{\mu\nu}\mathbf{t}_{\alpha}{}^{\beta}|_{0}x^{\mu}x^{\nu}$.
This is the object on which we focus our attention in this work. It turns out
that for all the proposed pseudotensors and quasilocal energy-momentum
expressions this fourth rank tensor is quadratic in the Riemann (equivalent in
empty space regions to the Weyl) tensor. That is why the quadratic curvature
expressions become interesting and important (i.e.,
$\partial^{2}_{\mu\nu}\mathbf{t}_{\alpha}{}^{\beta}\simeq{}R_{....}R_{....}$).
Normally, the expansion of a pseudotensor expression up to second order can
only be some linear combination of three tensors $\\{B,S,K\\}$ or
$\\{B,V,S\\}$ [6, 8, 9] which are each certain quadratic expressions in the
curvature.

According to a review article (4.2.2 in [7]): “Therefore, in vacuum in the
leading $r^{5}$ order any coordinate and Lorentz-covariant quasilocal energy-
momentum expression which is non-spacelike and future pointing must be
proportional to the Bel-Robinson ‘momentum’
$B_{\mu\lambda\xi\kappa}t^{\lambda}t^{\xi}t^{\kappa}$.” Note that here
$t^{\alpha}$ is timelike unit vector and ‘momentum’ means 4-momentum (see
(28)). This is a strong test. The Bel-Robinson tensor $B$ has many nice
properties such as fully symmetric, traceless and divergence free [10]. It is
known that $B$ contributes positivity in a small sphere region and perhaps it
may thought that it is the only one. However, we recently proposed an
alternative $V$ (see (18)) which has the identical ‘Bel-Robinson momentum’ at
the same limit, i.e.,
$(B_{\mu\lambda\xi\kappa}-V_{\mu\lambda\xi\kappa})t^{\lambda}t^{\xi}t^{\kappa}\equiv
0$. Confined to a small spherical or cubical regions [11], $B$ and $V$ cannot
be distinguished. One may suspect that $V$ is redundant because $B$ can manage
all the jobs. But we claim not.

As the basic requirement for the quasilocal energy is any closed 2-surface, we
examined the energy-momentum and angular momentum in other regions (see Table
1). We find for the energy in a small half-cylinder when $h\neq\sqrt{3}a$ give
different values if substituting $\mathbf{t}$ by $B$ and $V$, which means that
they are distinguishable. Only for one particular ratio $h=\sqrt{3}a$, $B$ and
$V$ both give the same ‘Bel-Robinson momentum’ value, however we lose the
distinction between them again. Therefore we turn to examining the angular
momentum in a small half-cylinder, and show that when replacing $\mathbf{t}$
by $B$ and $V$ in the angular momentum expression they contribute different
values, thereby clarifying that the two tensors are really distinguishable.

Here we remark that some components of the angular momentum in a hemi-sphere
show that $B$ contributes a null result while $V$ gives non-zero values (see
section 3.2). The reason comes from the fully symmetric property of $B$, while
$V$ only has some certain symmetry property (see (19)). Consequently, $V$ is
non-replaceable.

##  2 Technical background

Using a Taylor series expansion, the metric tensor can be written as

$\displaystyle
g_{\alpha\beta}(x^{\lambda})=g_{\alpha\beta}|_{x^{\lambda}_{0}}+\partial_{\mu}g_{\alpha\beta}|_{x^{\lambda}_{0}}(x^{\mu}-x^{\mu}_{0})+\frac{1}{2}\partial^{2}_{\mu\nu}g_{\alpha\beta}|_{x^{\lambda}_{0}}(x^{\mu}-x^{\mu}_{0})(x^{\nu}-x^{\nu}_{0})+...,$
(4)

where the metric signature is $+2$. For simplicity, let $x^{\lambda}_{0}=0$
and at the origin in RNC

$\displaystyle g_{\alpha\beta}|_{0}$ $\displaystyle=$
$\displaystyle\eta_{\alpha\beta},\quad\quad\partial_{\mu}g_{\alpha\beta}|_{0}=0,$
(5) $\displaystyle-3\partial^{2}_{\mu\nu}g_{\alpha\beta}|_{0}$
$\displaystyle=$ $\displaystyle
R_{\alpha\mu\beta\nu}+R_{\alpha\nu\beta\mu},\quad\quad-3\partial_{\nu}\Gamma^{\mu}{}_{\alpha\beta}|_{0}=R^{\mu}{}_{\alpha\beta\nu}+R^{\mu}{}_{\beta\alpha\nu}.$
(6)

Three basic tensors [6, 8, 9] that commonly occurred in pseudotensors are:

$\displaystyle
B_{\alpha\beta\mu\nu}\equiv{}B_{(\alpha\beta\mu\nu)}:=R_{\alpha\lambda\mu\sigma}R_{\beta}{}^{\lambda}{}_{\nu}{}^{\sigma}+R_{\alpha\lambda\nu\sigma}R_{\beta}{}^{\lambda}{}_{\mu}{}^{\sigma}-\frac{1}{8}g_{\alpha\beta}g_{\mu\nu}\mathbf{R}^{2},$
(7) $\displaystyle
S_{\alpha\beta\mu\nu}\equiv{}S_{(\alpha\beta)(\mu\nu)}\equiv{}S_{\mu\nu\alpha\beta}:=R_{\alpha\mu\lambda\sigma}R_{\beta\nu}{}^{\lambda\sigma}+R_{\alpha\nu\lambda\sigma}R_{\beta\mu}{}^{\lambda\sigma}+\frac{1}{4}g_{\alpha\beta}g_{\mu\nu}\mathbf{R}^{2},$
(8) $\displaystyle
K_{\alpha\beta\mu\nu}\equiv{}K_{(\alpha\beta)(\mu\nu)}\equiv{}K_{\mu\nu\alpha\beta}:=R_{\alpha\lambda\beta\sigma}R_{\mu}{}^{\lambda}{}_{\nu}{}^{\sigma}+R_{\alpha\lambda\beta\sigma}R_{\nu}{}^{\lambda}{}_{\mu}{}^{\sigma}-\frac{3}{8}g_{\alpha\beta}g_{\mu\nu}\mathbf{R}^{2},$
(9)

where $\mathbf{R}^{2}=R_{\rho\tau\xi\kappa}R^{\rho\tau\xi\kappa}$.

It may be worthwhile to mention that $B$ has a very good analog with the
electromagnetic energy-momentum tensor $\mathbf{T}^{\mu\nu}$. In Minkowski
coordinates $(t,x,y,z)$:

$\displaystyle\mathbf{T}^{00}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(E_{a}E^{a}+B_{a}B^{a}),$ (10)
$\displaystyle\mathbf{T}^{0i}$ $\displaystyle=$
$\displaystyle\delta^{ij}\epsilon_{jab}E^{a}B^{b},$ (11)
$\displaystyle\mathbf{T}^{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\delta^{ij}(E_{a}E^{a}+B_{a}B^{a})-2(E^{i}E^{j}+B^{i}B^{j})\right].$
(12)

where $\vec{E}$ and $\vec{B}$ refer to the electric and magnetic field
density. In order to appreciate the nice properties of $B$, we compare the
energy density with $S$ and $K$

$\displaystyle
B_{0000}=E^{2}_{ab}+H^{2}_{ab},\quad{}S_{0000}=2(E^{2}_{ab}-H^{2}_{ab}),\quad{}K_{0000}=-E^{2}_{ab}+3H^{2}_{ab},$
(13)

where the evaluation has used the electric part $E_{ab}$ and magnetic part
$H_{ab}$, defined in terms of the Weyl tensor [12]: $E_{ab}:=C_{a0b0}$ and
$H_{ab}:=*C_{a0b0}$, where $*C_{\alpha\beta\mu\nu}$ means its dual. Likewise
for the linear momentum density (i.e., Poynting vector)

$B_{000i}=2\epsilon_{ijk}E^{jd}H^{k}{}_{d},\quad{}S_{000i}=0,\quad{}K_{000i}=2\epsilon_{ijk}E^{jd}H^{k}{}_{d}.$
(14)

Finally, the stress,

$\displaystyle B_{00ij}$ $\displaystyle=$
$\displaystyle\delta_{ij}(E_{ab}E^{ab}+H_{ab}H^{ab})-2(E_{id}E_{j}{}^{d}+H_{id}H_{j}{}^{d}),$
(15) $\displaystyle S_{00ij}$ $\displaystyle=$
$\displaystyle-2\left[\delta_{ij}(E_{ab}E^{ab}-H_{ab}H^{ab})+2(E_{id}E_{j}{}^{d}-H_{id}H_{j}{}^{d})\right],$
(16) $\displaystyle K_{00ij}$ $\displaystyle=$
$\displaystyle\delta_{ij}(5E_{ab}E^{ab}-3H_{ab}H^{ab})-4E_{id}E_{j}{}^{d}.$
(17)

We observe that summing up $S$ and $K$ has exactly the same energy as $B$:
$(B_{0000}-S_{0000}-K_{0000})\equiv{}0\equiv(B_{00ij}-S_{00ij}-K_{00ij})\delta^{ij}$.
It is natural to define the alternative 4th rank tensor [9] as follows

$V:=S+K\equiv{}B+W,$ (18)

where
$W_{\alpha\beta\mu\nu}:=\frac{3}{2}S_{\alpha\beta\mu\nu}-\frac{1}{8}(5g_{\alpha\beta}g_{\mu\nu}-g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu})\mathbf{R}^{2}$.
For a comparison of $B$ and $V$, we find that it is more convenient to use
$(B+W)$ instead of $(S+K)$ for the representation of $V$. Both $V$ and $W$
satisfy the following properties:

$\displaystyle
X_{\alpha\beta\mu\nu}\equiv{}X_{(\alpha\beta)(\mu\nu)}\equiv{}X_{\mu\nu\alpha\beta},\quad{}X_{\alpha\beta\mu}{}^{\mu}\equiv{}0\equiv{}X_{\alpha\mu\beta}{}^{\mu}.$
(19)

However, unlike $B$ (see (7)), they are not fully symmetric. Intuitively,
referring to (18), $V$ may contain more non-trivial independent components
than $B$ and indeed it is the case (see section 3.3).

In our work, we are mainly dealing with expression of the 4th rank which are
quadratic in the curvature tensor. There are four tensors which form a basis
with appropriate symmetries [13], we use

$\displaystyle\tilde{B}_{\alpha\beta\mu\nu}$ $\displaystyle:=$ $\displaystyle
R_{\alpha\lambda\mu\sigma}R_{\beta}{}^{\lambda}{}_{\nu}{}^{\sigma}+R_{\alpha\lambda\nu\sigma}R_{\beta}{}^{\lambda}{}_{\mu}{}^{\sigma},\quad{}\tilde{S}_{\alpha\beta\mu\nu}:=R_{\alpha\mu\lambda\sigma}R_{\beta\nu}{}^{\lambda\sigma}+R_{\alpha\nu\lambda\sigma}R_{\beta\mu}{}^{\lambda\sigma},\quad$
(20) $\displaystyle\tilde{K}_{\alpha\beta\mu\nu}$ $\displaystyle:=$
$\displaystyle
R_{\alpha\lambda\beta\sigma}R_{\mu}{}^{\lambda}{}_{\nu}{}^{\sigma}+R_{\alpha\lambda\beta\sigma}R_{\nu}{}^{\lambda}{}_{\mu}{}^{\sigma},\quad\tilde{T}_{\alpha\beta\mu\nu}:=-\frac{1}{8}g_{\alpha\beta}g_{\mu\nu}\mathbf{R}^{2}.$
(21)

They are designed to describe the gravitational energy expression based on the
pseudotensor (see (22)) and are manifestly symmetric in the last two indices,
i.e., $\tilde{M}_{\alpha\beta\mu\nu}=\tilde{M}_{\alpha\beta(\mu\nu)}$. Then
$\tilde{M}_{\alpha\beta\mu\nu}=\tilde{M}_{(\alpha\beta)\mu\nu}$ and it also
naturally turns out
$\tilde{M}_{\alpha\beta\mu\nu}=\tilde{M}_{\mu\nu\alpha\beta}$.

##  3 Energy-momentum tensors of $B$ and $V$

###  3.1 Alternative gravitational energy-momentum tensor $V$

Let $x^{\mu}=(t,x,y,z)$ and using a RNC Taylor expansion around any point,
consider all the possible combinations of the small region in vacuum. The
total energy-momentum density pseudotensor is in general expressed as

${\cal{T}}_{\alpha}{}^{\beta}=\kappa^{-1}G_{\alpha}{}^{\beta}+(a_{1}\tilde{B}_{\alpha}{}^{\beta}{}_{\xi\kappa}+a_{2}\tilde{S}_{\alpha}{}^{\beta}{}_{\xi\kappa}+a_{3}\tilde{K}_{\alpha}{}^{\beta}{}_{\xi\kappa}+a_{4}\tilde{T}_{\alpha}{}^{\beta}{}_{\xi\kappa})x^{\xi}x^{\kappa}+{\cal{}O}(\mbox{Ricci},x)+{\cal{}O}(x^{3}),$
(22)

where $a_{1}$ to $a_{4}$ are constants. Since our concern is the vacuum case,
so $G_{\alpha\beta}=0=T_{\alpha\beta}$. Then the first order linear in Ricci
terms ${\cal{}O}({\mbox{Ricci}},x)$ vanish. The lowest order non-vanishing
term is of second order, and compared to this in the small region limit we
ignore the third order terms ${\cal{}O}(x^{3})$. It should be noted that
${\cal{T}}_{\alpha}{}^{\beta}$ in (2) or (22) is a pseudotensor, but in the
Taylor expansion on the right hand side in (22) the coefficients of the
various powers of $x$ are tensors. As argued in [13],
$\partial^{2}_{\mu\nu}{\cal{T}}_{\alpha}{}^{\beta}(0)$ must be some linear
combination of 4 tensors, here we use {$\tilde{B}$, $\tilde{S}$, $\tilde{K}$,
$\tilde{T}$}. From now on, we only keep the second order term and drop the
others. There are two physical conditions which can constrain the unlimited
combinations between {$\tilde{B}$, $\tilde{S}$, $\tilde{K}$, $\tilde{T}$}:
4-momentum conservation and positivity, both considered in the small region
vacuum limit (i.e., not restricted to a 2-sphere).

First condition: energy-momentum conservation. Consider (2) and (22) in vacuum

$\displaystyle
0=4\,\partial_{\beta}\mathbf{t}_{\alpha}{}^{\beta}=(a_{1}-2a_{2}+3a_{3}-a_{4})g_{\alpha\beta}x^{\beta}\mathbf{R}^{2}.$
(23)

Therefore, the constraint for the conservation of the energy-momentum density
is

$a_{4}=a_{1}-2a_{2}+3a_{3}.$ (24)

No single element from {$\tilde{B}$, $\tilde{S}$, $\tilde{K}$, $\tilde{T}$}
can satisfy (23), however certain linear combinations of them can. Eliminate
$\tilde{T}$ which is absorbed by $\tilde{B}$, $\tilde{S}$ or $\tilde{K}$,
comparing (2) and using (24), rewrite (22)

$\displaystyle\mathbf{t}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\left[a_{1}(\tilde{B}_{\alpha\beta\xi\kappa}+\tilde{T}_{\alpha\beta\xi\kappa})+a_{2}(\tilde{S}_{\alpha\beta\xi\kappa}-2\tilde{T}_{\alpha\beta\xi\kappa})+a_{3}(\tilde{K}_{\alpha\beta\xi\kappa}+3\tilde{T}_{\alpha\beta\xi\kappa})\right]x^{\xi}x^{\kappa}$
(25) $\displaystyle=$
$\displaystyle(a_{1}B_{\alpha\beta\xi\kappa}+a_{2}S_{\alpha\beta\xi\kappa}+a_{3}K_{\alpha\beta\xi\kappa})x^{\xi}x^{\kappa}$
$\displaystyle=$
$\displaystyle\left[a_{1}B_{\alpha\beta\xi\kappa}+a_{3}V_{\alpha\beta\xi\kappa}+(a_{2}-a_{3})S_{\alpha\beta\xi\kappa}\right]x^{\xi}x^{\kappa}.$

Consider all the possible expressions for the pseudotensors (some of which
explicitly included the flat metric), there indeed does appear linear
combinations of these three tensors [6, 8, 9]. Explicitly one can use either
$\\{B,S,K\\}$ or $\\{B,V,S\\}$.

Second condition: non-negative gravitational energy. For simplicity, we use a
small sphere. For any quantity at $t=t_{0}$ we consider the limiting value for
the radius $r:=\sqrt{x^{2}+y^{2}+z^{2}}$. The 4-momentum at time $t=0$ is

$\displaystyle
2\kappa{}P_{\mu}=\int{}\mathbf{t}^{\rho}{}_{\mu\xi\kappa}x^{\xi}x^{\kappa}d\Sigma_{\rho}=\mathbf{t}^{0}{}_{\mu{}ij}\int{}x^{i}x^{j}d^{3}x=\mathbf{t}^{0}{}_{\mu{}ij}\delta^{ij}\,\frac{4\pi{}r^{5}}{15}.$
(26)

Thus, from (25)

$P_{\mu}=(-E,\vec{P})=-\frac{r^{5}}{60G}\left[a_{1}B_{\mu{}0ij}+a_{3}V_{\mu{}0ij}+(a_{2}-a_{3})S_{\mu{}0ij}\right]\delta^{ij}.$
(27)

The energy-momentum values associated with $\\{B,V,S\\}$ are

$\displaystyle
B_{\mu{}0ij}\delta^{ij}\equiv{}V_{\mu{}0ij}\delta^{ij}=(E^{2}_{ab}+H^{2}_{ab},2\epsilon_{cab}E^{ad}H^{b}{}_{d}),~{}{}S_{\mu{}0ij}\delta^{ij}=-10(E^{2}_{ab}-H^{2}_{ab},0).$
(28)

Here we emphasize that in a small sphere region, the energy-momentum of $B$ or
$V$ is inside the light cone, $-P_{0}\geq|\vec{P}|\geq 0$. Observing (27),
basically we are considering positive energy, $B$ and $V$ already satisfy this
condition and the remaining job is to find $\\{a_{2},a_{3}\\}$. Equation (28)
shows that $S_{\mu{}0ij}\delta^{ij}$ cannot ensure positivity, since we should
allow for any magnitude of $|E_{ab}|$ and $|H_{ab}|$. The only possibility for
(27) to guarantee positivity is to require $a_{1}+a_{3}\geq{}10|a_{2}-a_{3}|$.
However, if we insist on the pure ‘Bel-Robinson momentum’ [7], obviously, we
only have one choice $a_{2}=a_{3}$.

###  3.2 Computing energy-momentum and angular momentum

The Papapetrou pseudotensor [9] gives a certain linear combination of $B$ and
$V$:
$2\kappa{}P^{\alpha\beta}=\frac{1}{9}(4B^{\alpha\beta}{}_{\xi\kappa}-V^{\alpha\beta}{}_{\xi\kappa})x^{\xi}x^{\kappa}$.
The energy using (26) in a small sphere is

$P_{0}=-\frac{r^{5}}{540G}(4B_{00ij}-V_{00ij})\delta^{ij}\equiv-\frac{r^{5}}{180G}B_{00ij}\delta^{ij},$
(29)

where $(B_{00ij}-V_{00ij})\delta^{ij}\equiv 0$. Before we proceed, one might
question that perhaps $V$ is superfluous since $B$ and $V$ have so far shown
no distinction. We claim that $B$ and $V$ are distinct because they are
constructed from different basic quadratic curvatures
$\\{\tilde{B},\tilde{S},\tilde{K},\tilde{T}\\}$: $B=\tilde{B}+\tilde{T}$ and
$V=\tilde{S}+\tilde{K}+\tilde{T}$. Strictly speaking, we claim $B$ and $V$ are
fundamentally different [9]. But this raises a question regarding how to see
the distinction clearly. We realize that it is impossible to distinguish $B$
and $V$ if we consider 4-momentum or angular momentum in a small sphere. So we
change our strategy to evaluating these physical quantities in other
quasilocal volume elements (see Table 1).

We claim $B$ and $V$ can have different energy values, for instance, in a
small box with different dimensions. Here we give a concrete example: let
$a=b$, $c=a+\Delta$ and $|\Delta|<<a$. The energy for substituting
$\mathbf{t}$ by $B$ is
$P^{B}_{0}\simeq\frac{a^{5}}{12}(B^{0}{}_{0ij}\delta^{ij}+\frac{2\Delta}{a}B^{0}{}_{033})$.
Similarly for $V$,
$P^{V}_{0}\simeq\frac{a^{5}}{12}(V^{0}{}_{0ij}\delta^{ij}+\frac{2\Delta}{a}V^{0}{}_{033})$.
Thus, generally, $B$ and $V$ are separable:
$P^{V}_{0}-P^{B}_{0}\simeq\frac{a^{4}\Delta}{6}W^{0}{}_{033}\neq 0$. Following
the restriction that the quasilocal energy-momentum must be a multiple of
‘Bel-Robinson momentum’ [7]. We can fulfill this requirement using either $B$
or $V$ in a small region for a perfect sphere or a box with
$a\equiv{}b\equiv{}c$, i.e., a cube [11], for a cylinder or half-cylinder we
need $h\equiv\sqrt{3}a$. These are desirable results, but unfortunately, we
lose the distinction between $B$ and $V$ again.

Is it possible to keep a multiple of ‘Bel-Robinson momentum’ and still able to
tell the difference between $B$ and $V$ naturally? Yes, it is possible: we
turn to examining the angular momentum (see, e.g., $\S$20.3 in [8]) which can
be defined as follows

$\displaystyle J^{\mu\nu}:=\int(x^{\mu}\mathbf{t}^{\nu
0}{}_{\xi\kappa}-x^{\nu}\mathbf{t}^{\mu
0}{}_{\xi\kappa})x^{\xi}x^{\kappa}d^{3}x,$ (30)

where $\mathbf{t}$ can be $B$ or $V$. According to Table 1, we observe that
the angular momentum vanishes for a perfect sphere, ellipsoid, box or
cylinder. Conversely, both hemi-sphere and half-cylinder ($h\equiv\sqrt{3}a$)
have non-vanishing angular momentum. In these regions, the angular momentum
values for $B$ and $V$ are distinguishable, i.e., $V$ is no longer
superfluous. Moreover, we remark that for a hemi-sphere, if we substitute
$\mathbf{t}$ by the completely symmetric $B$,
$J^{12}_{B}=\frac{\pi}{12}(B_{1023}-B_{2013})a^{6}\equiv{}0$. However, if
consider $V$, $J^{12}_{V}=\frac{\pi}{12}(V_{1023}-V_{2013})a^{6}\neq 0$
generally. Thus, the difference between $B$ and $V$ becomes sharply manifest,
showing that in this case $V$ is essential, not redundant.

Perfect- |  $P_{\mu}=\frac{4\pi}{15}\mathbf{t}^{0}{}_{\mu{}ij}\delta^{ij}a^{5}$, $r\in[0,a],~{}\theta\in[0,\pi],~{}\phi\in[0,2\pi]$
---|---
sphere |  $J^{0m}=(0,0,0)$, $(J^{12},J^{13},J^{23})=(0,0,0)$
Ellipsoid |  $P_{\mu}=\frac{4\pi}{15}(\mathbf{t}^{0}{}_{\mu{}11}a^{2}+\mathbf{t}^{0}{}_{\mu{}22}b^{2}+\mathbf{t}^{0}{}_{\mu{}33}c^{2})abc$, $x\in[-a,a],~{}y\in[-b,b],~{}z\in[-c,c]$
|  $J^{0m}=(0,0,0)$, $(J^{12},J^{13},J^{23})=(0,0,0)$
Hemi- |  $P_{\mu}=\frac{2\pi}{15}\mathbf{t}^{0}{}_{\mu{}ij}\delta^{ij}a^{5}$, $r\in[0,a],~{}\theta\in[0,\pi/2],~{}\phi\in[0,2\pi]$
sphere | $J^{0m}=\frac{\pi}{24}(2\mathbf{t}^{0}{}_{013},2\mathbf{t}^{0}{}_{023},\mathbf{t}^{0}{}_{0ij}\delta^{ij}+\mathbf{t}^{0}{}_{033})a^{6}$
|
$J^{12}=\frac{\pi}{12}(\mathbf{t}^{1}{}_{023}-\mathbf{t}^{2}{}_{013})a^{6}$,
$J^{13}=\frac{\pi}{24}(\mathbf{t}^{1}{}_{0ij}\delta^{ij}+\mathbf{t}^{1}{}_{033}-2\mathbf{t}^{3}{}_{013})a^{6}$,
|
$J^{23}=\frac{\pi}{24}(\mathbf{t}^{2}{}_{0ij}\delta^{ij}+\mathbf{t}^{2}{}_{033}-2\mathbf{t}^{3}{}_{023})a^{6}$
Box |  $P_{\mu}=\frac{1}{12}(\mathbf{t}^{0}{}_{\mu 11}a^{2}+\mathbf{t}^{0}{}_{\mu 22}b^{2}+\mathbf{t}^{0}{}_{\mu 33}c^{2})abc$ , $x\in[-\frac{a}{2},\frac{a}{2}],~{}y\in[-\frac{b}{2},\frac{b}{2}],~{}z\in[-\frac{c}{2},\frac{c}{2}]$
|  $J^{0m}=(0,0,0)$, $(J^{12},J^{13},J^{23})=(0,0,0)$
Cylinder |  $P_{\mu}=\frac{\pi}{4}\mathbf{t}^{0}{}_{\mu{}ij}\delta^{ij}a^{4}h+\frac{\pi}{12}\mathbf{t}^{0}{}_{\mu 33}(h^{2}-3a^{2})a^{2}h$, $\rho\in[0,a],~{}\varphi\in[0,2\pi],~{}z\in[-\frac{h}{2},\frac{h}{2}]$
|  $J^{0m}=(0,0,0)$, $(J^{12},J^{13},J^{23})=(0,0,0)$
Half- |  $P_{\mu}=\frac{\pi}{8}\mathbf{t}^{0}{}_{\mu{}ij}\delta^{ij}a^{4}h+\frac{\pi}{24}\mathbf{t}^{0}{}_{\mu 33}(h^{2}-3a^{2})a^{2}h$, $\rho\in[0,a],~{}\varphi\in[0,\pi],~{}z\in[-\frac{h}{2},\frac{h}{2}]$
cylinder |  $J^{01}=\frac{4}{15}\mathbf{t}^{0}{}_{012}a^{5}h$, $J^{02}=\frac{1}{18}\mathbf{t}^{0}{}_{033}a^{3}h^{3}+\frac{2}{15}(\mathbf{t}^{0}{}_{011}+2\mathbf{t}^{0}{}_{022})a^{5}h$, $J^{03}=\frac{1}{9}\mathbf{t}^{0}{}_{023}a^{3}h^{3}$
|
$J^{12}=\frac{1}{18}\mathbf{t}^{1}{}_{033}a^{3}h^{3}+\frac{2}{15}(\mathbf{t}^{1}{}_{011}+2\mathbf{t}^{1}{}_{022}-2\mathbf{t}^{2}{}_{012})a^{5}h$,
$J^{13}=\frac{1}{9}\mathbf{t}^{1}{}_{023}a^{3}h^{3}-\frac{4}{15}\mathbf{t}^{3}{}_{012}a^{5}h$
|
$J^{23}=\frac{1}{18}(2\mathbf{t}^{2}{}_{023}-\mathbf{t}^{3}{}_{033})a^{3}h^{3}-\frac{2}{15}(\mathbf{t}^{3}{}_{011}+2\mathbf{t}^{3}{}_{022})a^{5}h$
Table 1: Energy-momentum and angular momentum in different small regions,
$\mathbf{t}$ can be $B$ or $V$

###  3.3 Counting the independent components of $B$, $V$ and $W$

Basically $B$, $V$ and $W$ are fourth rank tensor and could have 256
components. However, by symmetry, they only have a relatively small number of
independent components. The counting of the number of independent components
of $B$ has already been done, here we claim there is no common term between
$B$ and $W$, i.e., $\\{B\\}\bigcap\,\\{W\\}=\\{\emptyset\\}$. We verify this
statement as follows:

First, we count the components of $B$. In principle, $B$ is fully symmetric,
by explicit examination it reduces to 35. There is a formula that directly
gives this number. A $k$th rank totally symmetric tensor in $n$ dimensional
space has $C^{n+k-1}_{k}$ components. For our case $C^{4+4-1}_{4}=35$. Since
$B$ is completely tracefreeness, there are 10 additional constraints which
reduce the number of components. Therefore, we have left only 25 for $B$ (for
another argument see [14]).

Next we count the number of independent components of $V$. $V$ does not have
the totally symmetric property, but as mentioned in (19) that
$V_{\alpha\beta\mu\nu}\equiv{}V_{(\alpha\beta)(\mu\nu)}\equiv{}V_{\mu\nu\alpha\beta}$.
This reduces $V$ to 55 components. However, the completely traceless condition
gives two extra constraints indicated in (19) again:
$V^{\alpha}{}_{\alpha\mu\nu}\equiv{}0\equiv{}V^{\alpha}{}_{\mu\alpha\nu}$.
Consequently, we have $55-10-10=35$ for $V$.

Finally, we count the number of independent components of $W$. Observing that
$V$ and $W$ are similar. Referring to (19), there should thus be at most 35
components. However, take care an extra condition
$W_{\alpha(\beta\mu\nu)}\equiv{}0$ which gives 25 more constraints. Hence we
find $35-25=10$ for $W$.

###  3.4 Physical meaning of the fully tracefreeness property

It is easy to check that $B$ and $V$ are fully trace free. We are going to
verify that this mathematical property and the physical conservation laws are
in a 1-1 correspondence in the quasilocal limit. Consider a linear combination
between $\\{\tilde{B},\tilde{S},\tilde{K},\tilde{T}\\}$, let

$A:=a_{1}\tilde{B}+a_{2}\tilde{S}+a_{3}\tilde{K}+a_{4}\tilde{T}.$ (31)

We observe that there are only two distinct traces because of the symmetry:

$\displaystyle
8A^{\alpha}{}_{\mu\alpha\nu}\equiv(a_{1}-2a_{2}+3a_{3}-a_{4})g_{\mu\nu}\mathbf{R}^{2},\quad{}2A^{\alpha}{}_{\alpha\mu\nu}\equiv(a_{1}+a_{2}-a_{4})g_{\mu\nu}\mathbf{R}^{2}.$
(32)

The totally traceless condition requires that the above two equations vanish
simultaneously:

$\displaystyle 0=a_{1}-2a_{2}+3a_{3}-a_{4},\quad{}0=a_{1}+a_{2}-a_{4}.$ (33)

The first equation in (33) is the same as (24), which indicates one of the
mathematical conditions identical to the energy-momentum conservation
criterion: solving the equations in (33), we obtain $a_{2}=a_{3}$, and this is
proportional to the ‘Bel-Robinson momentum’ requirement found from (27); we
have noted that the fully tracefreeness property is related to some physical
conditions.

##  4 Conclusion

For describing positivity, the Bel-Robinson tensor is the best, and perhaps
has been thought to be the only possibility. We recently proposed an
alternative $V$ in such a way that it shares the same energy-momentum as $B$
does in the small sphere limit. One might think that $B$ and $V$ cannot be
distinguished, but we claim they can. After examining the energy found from
other 2-surfaces such as in ellipsoid, box, cylinder and half-cylinder
($h\neq\sqrt{3}a$), we demonstrate that $V$ is not redundant because $B$ and
$V$ are distinguishable. However, if we insist to achieve a multiple of pure
‘Bel-Robinson momentum’ from Szabados’s argument in Living Review, the
distinction between $B$ and $V$ will be lost once more. For a shape such that
both $B$ and $V$ give a multiple of the pure ‘Bel-Robinson momentum’ we can
turn to investigate the angular momentum. Thus when replacing $\mathbf{t}$ by
either $B$ or $V$, indeed they do lead to different angular momentum values
for a hemi-sphere or half-cylinder with $h=\sqrt{3}a$. Moreover, we emphasize
that some of the components of the angular momentum give a null result for $B$
and a non-vanishing result for $V$. The reason is based on the elegant
completely symmetric property of $B$, while $V$ is not fully symmetric. Thus
$V$ can play an essential irreplaceable role.

The tensors $B$ and $V$ are constructed from different fundamental quadratic
curvatures $\\{\tilde{B},\tilde{S},\tilde{K},\tilde{T}\\}$. As a double check,
we counted the independent components of $B$ and $V$ and find that they are
not the same. Finally, we discover the necessary and sufficient conditions for
$B$ and $V$: fully tracefreeness and conservation of future pointing non-
spacelike pure ‘Bel-Robinson momentum’ in the small region limit.

## Acknowledgment

The author would like to thank Dr. Peter Dobson, Professor Emeritus, HKUST,
for reading the manuscript and providing some helpful comments. This work was
supported by NSC 95-2811-M-032-008, NSC 96-2811-M-032-001, NSC
97-2811-M-032-007 and NSC 98-2811-M-008-078.

## References

  * [1] Trautman A 1962 in An introduction to Current Research, ed L Witten (New York: Wiley) p169-198
  * [2] Landau L D and Lifshitz E M 1962 The classical theory of fields, 2nd edition (Reading, MA: Addison-Wesley) (Oxford: Pergamon, 1975)
  * [3] Bergmann P G and Thomson R 1953 Phys. Rev. 89 400
  * [4] Papapetrou A 1948 Proc. Roy. Irish. Acad. A52 11-23
  * [5] Weinberg S 1972 Gravitation and Cosmology, (New York: Wiley) p371
  * [6] So L L and Nester J M 2009 Phys. Rev. D 79 084028
  * [7] Szabados L B 2009 Living Rev. Relativity 12 4
  * [8] Misner C W, Thorne K S and Wheeler J A 1973 Gravitation (San Francisco, CA: Freeman)
  * [9] So L L 2009 Class. and Quantum Grav. 26 185004
  * [10] Senovilla J M M 2000 Class. Quantum Grav. 17 2799
  * [11] Garecki J 1977 Acta Phys. Pol. B8 159
  * [12] Carmeli M 1982 Classical Fields General relativuty and Gauge Theory (John Wiley $\&$ Sons)
  * [13] Deser S, Franklin J S and Seminaea D 1999 Class. Quantum Grav. 16 2815
  * [14] Gomez-Lobo A G P 2008 Class. Quantum. Grav. 25 015006