A Quasigroup Approach for Conservation Laws in Asymptotically Flat Spacetimes

Abstract

In the framework of the quasigroup approach to conservation laws in general relativity, we show how the infinite-parametric Newman–Unti group of asymptotic symmetries can be reduced to the Poincaré quasigroup. We compute Noether’s charges associated with any element of the Poincaré quasialgebra. The integral conserved quantities of energy momentum and angular momentum, being linear on generators of the Poincaré quasigroup, are identically equal to zero in Minkowski spacetime. We present a definition of the angular momentum free of the supertranslation ambiguity. We provide an appropriate notion of intrinsic angular momentum and a description of the mass reference frame’s center at future null infinity. Finally, in the center of mass reference frame, the momentum and angular momentum are defined by the Komar expression.

1. Introduction

The general covariance of the Einstein equations results in differential conservation laws related to the field equations. From Noether’s theorem, it follows that for an arbitrary diffeomorphism generated by the vector field $\xi$, the invariance of the Lagrangian for the Einstein equations leads to the conservation laws of the form, ${\mathfrak{J}}^{\mu }{\left(\xi \right)}_{,\mu }=0$, where the vector density ${\mathfrak{J}}^{\mu }$ is defined as ${\mathfrak{J}}^{\mu }\left(\xi \right)={\mathfrak{h}}_{,\nu }^{\mu \nu }$, with ${\mathfrak{h}}^{\mu \nu }=-{\mathfrak{h}}^{\nu \mu }$ being the superpotential, which is constructed from the densities of spin, bispin, vector field $\xi$, and its derivatives [1,2,3].

For the Einstein–Hilbert action, we have a simple expression for the Noether current,

$${\mathfrak{J}}^{\mu }=-{\displaystyle \frac{1}{4\pi }}{\left(\sqrt{-g}{\xi }^{[\mu ;\nu ]}\right)}_{,\nu },$$

(1)

which was first obtained by Komar [4]. Komar’s expression provides a fully satisfactory notion of the total mass in stationary, asymptotically flat spacetimes. It is worth noting that the normalization for energy momentum and angular momentum differs by a factor of 2, and it is impossible simply to “renormalize” ${\mathfrak{J}}^{\mu }$. The resolution of this problem was pointed out in [5]. One needs to add to the Einstein–Hilbert action a surface term and apply Noether’s theorem to the total action.

In Minkowski space, the existence of the Poincaré group and corresponding Killing vectors leads to the definitions of total momentum and total angular momentum. However, the situation is more complicated in curved spacetime, even for an isolated system with a vanishing curvature tensor at infinity. While we have a well-defined energy momentum, there is no accordance for the notion of the angular momentum or center of mass [6]. A major difficulty in defining the angular momentum is that the group of asymptotic symmetries is infinite-parametric. Although the asymptotic symmetry group has a unique translation subgroup, there is no canonical Lorentz subgroup [7,8,9,10]. The last one emerges as a factor group of the asymptotic symmetries group by the infinite-dimensional subgroup of supertranslations. Therefore, there does not exist a canonical way of choosing the Poincaré group as a subgroup of the group of asymptotic symmetries. There are too many Poincaré subgroups, one for each supertranslation, which is not translation [11].

These circumstances generate the main difficulties in the numerous attempts to find the correct definition of angular momentum in curved spacetime [12,13,14,15]. Most definitions suffer from the supertranslation ambiguities [7,8,11,12,13,16,17,18,19] (see [20] for review). The origin-dependence of the angular momentum causes additional difficulties. Hitherto, no satisfactory way of resolving these problems has been found.

In flat spacetime, a total angular momentum can be written as $J^{\mu \nu }=M^{\mu \nu }+S^{\mu \nu }$, where the first term, $M^{\mu \nu }=X^{\mu }{P}^{\nu }-P^{\mu }{X}^{\nu }$, is the angular momentum, with $X^{\mu }$ and $P^{\nu }$ being the position vector and momentum of the particle, respectively, and $S^{\mu \nu }$ is the intrinsic angular momentum [18].

In Minkowski spacetime, one can find a particular trajectory, describing the center of mass motion and having the property that the four-momentum is aligned with the observer’s four-velocity [1,2]. Thus, if one wants to generalize this concept to general relativity, then the task would be to find a worldline with similar properties in curved spacetime [21,22,23,24]. An alternative approach is to define a preferred section at ${\mathcal{I}}^{+}$, which can be associated with the rest reference frame. This approach was developed in [25,26], where the so-called "nice sections" were introduced to study the asymptotic fields of an isolated system.

In our paper, the quasigroup approach to the conservation laws developed in [2,27,28,29] is applied to asymptotically flat spacetime. The Poincaré quasigroup at future null infinity (${\mathcal{I}}^{+}$) is introduced and compared with other definitions of asymptotic symmetries that have appeared in the literature. We define the complex Noether charge associated with any element of the Poincaré quasialgebra. It may be regarded as a form of linkages by Tamburino and Winicour [16,17] but with the new gauge conditions for asymptotic symmetries.

We present new definitions of the center of mass and intrinsic angular momentum using the available tools on asymptotically flat spacetimes. Our approach is based on the fact that the intrinsic angular momentum is invariant under the Lorentz boosts [30]. It is shown that the momentum and angular momentum are defined by the Komar expression in the center of the mass reference frame.

The metric’s signature is $(+,-,-,-)$ throughout this paper, with lower case Greek letters ranging and summing from zero to three.

This paper is organized as follows. In Section 2, we review the key ideas and tools that are indispensable for further discussions. In particular, we discuss the non-associative generalization of the transformation groups–quasigroups of transformations and the structure of the group of asymptotic symmetries at future null infinity. In Section 3, we explore the reduction of the Newman–Unti (NU) to the Poincaré quasigroup. In Section 4, we present conserved quantities at future null infinity based on linkages introduced by Tamburino and Winicour. In Section 5, we define a cut of ${\mathcal{I}}^{+}$ associated with the notion center-of-mass reference frame of isolated systems and show that the Komar expression gives the intrinsic angular momentum. Finally, we give our conclusion Section 6, summarizing the obtained results and discuss possible generalizations of our approach. In the appendices, the details of our calculations are presented.

2. Mathematical Background

2.1. Quasigroups of Transformations

The definition of the quasigroup of transformations was first given by Batalin [31]. Below, we outline the main facts from the theory of the smooth quasigroups of transformations.

Let $\mathfrak{M}$ be an n-dimensional manifold and the continuous law of transformation is given by $x^{\prime }=T_{a}x$, $x\in \mathfrak{M}$, where $\left\{a^i\right\}$ is the set of real parameters, $i=1,2,\dots ,r$. The set of transformations $\left\{T_a\right\}$ forms an r-parametric quasigroup of transformations (with right action on $\mathfrak{M}$):

• If there exists a unit element that is common for all $x^{\alpha }$ and corresponds to $a^i=0:$, then $T_a{x|}_{a=0}=x$;
• If the modified composition law holds,

  $$T_a{T}_{b}x=T_{\phi (b,a;x)}x;$$

  (2)
• If the left and right units coincide,

  $$\begin{array}{c}\hfill \phi (a,0;x)=a,\phi (0,b;x)=b;\end{array}$$
• If the modified law of associativity is satisfied,

  $$\begin{array}{c}\hfill \phi (\phi (a,b;x),c;x)=\phi (a,\phi (b,c;T_{a}x);x);\end{array}$$
• If the transformation inverse to $T_a$ exists, $x=T_a^{-1}{x}^{\prime }$.

A group is a set with a binary operation that is closed, associative, has an identity element, and every element has an inverse. In contrast, a quasigroup is a more generalized algebraic structure that only requires the operation to be closed.

The quasigroup structure “falls short” of being a group in one specific way: associativity is modified. In a group the definition is,

$$(T_a·T_b)·T_c=T_a·(T_b·T_c),$$

(3)

and a quasigroup,

$$(T_a·T_b)·T_c=T_a·(T_b·T_{c^{\prime }}),\mathrm{where}{c}^{\prime }\mathrm{depends}\mathrm{on}{T}_a.$$

(4)

A more intuitive understanding is that the sequence in which operations are grouped is significant, as implementing the initial transformation alters the foundational context for subsequent transformations [31,32].

By the other side, the generators of infinitesimal transformations,

$${\mathrm{\Gamma }}_i=(\partial {\left(T_{a}x\right)}^{\alpha }/\partial a^i){|}_{a=0}\partial /\partial x^{\alpha }\equiv R_i^{\alpha }\partial /\partial x^{\alpha },$$

(5)

form quasialgebra and obey commutation relations:

$$[{\mathrm{\Gamma }}_i,{\mathrm{\Gamma }}_j]=C_{ij}^p\left(x\right){\mathrm{\Gamma }}_p.$$

(6)

Here, $C_{ij}^p\left(x\right)$ denotes the structure functions satisfying the modified Jacobi identity:

$$\begin{array}{c}\hfill C_{ij,\alpha }^p{R}_k^{\alpha }+C_{jk,\alpha }^p{R}_i^{\alpha }+C_{ki,\alpha }^p{R}_j^{\alpha }+C_{ij}^l{C}_{kl}^p+C_{jk}^l{C}_{il}^p+C_{ki}^l{C}_{jl}^p=0.\end{array}$$

(7)

Theorem 1.

Let the given functions $R_i^{\alpha },C_{kj}^p$ obey Equations (6) and (7); then, locally, the quasigroup of transformations is reconstructed as the solution of a set of differential equations:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\tilde{x}}^{\alpha }}{\partial a^i}}=R_j^{\alpha }\left(\tilde{x}\right){\lambda }_i^j(a;x),{\tilde{x}}^{\alpha }\left(0\right)=x^{\alpha },\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\lambda }_j^i}{\partial a^p}}-{\displaystyle \frac{\partial {\lambda }_p^i}{\partial a^j}}+C_{mn}^i\left(\tilde{x}\right){\lambda }_p^m{\lambda }_j^n=0,\hfill \\ \hfill & {\lambda }_j^i(0;x)={\delta }_j^i.\hfill \end{array}$$

(9)

Equation (8) is an analog of the Lie equation, and Equation (9) is the generalized Maurer–Cartan equation [31].

Definition 1.

The Poincaré quasigroup is defined as a semi-direct product of the Lorentz quasigroup and the group of translations.

Poincaré quasialgebra is described by the following set of commutation relations:

$$\begin{array}{c}\hfill [T_a,T_b]=0,[T_a,L_B]=C_{aB}^b\left(x\right)T_b,\left[L_A,L_B\right]=C_{AB}^D\left(x\right)L_D,\end{array}$$

(10)

with $C_{AB}^D,C_{aB}^b$ being the structure functions. The last commutation relations mean that the generators of the Lorentz quasigroup form a closed quasialgebra [29].

2.2. NP Formalism

This section outlines the main facts from the Newman–Penrose (NP) formalism indispensable for future discussions. We follow notations in [32,33,34].

At each point of spacetime, we introduce a null NP basis $\{l,n,m,\overline{m}\}$ with the following non-vanishing scalar products: $l·n=1,m·\overline{m}=-1$. The metric components are given by

$$\begin{array}{c}\hfill g^{\alpha \beta }=l^{\alpha }{n}^{\beta }+l^{\beta }{n}^{\alpha }-m^{\alpha }{\overline{m}}^{\beta }-{\overline{m}}^{\beta }{m}^{\alpha },\end{array}$$

(11)

where bar means complex conjugate.

To introduce the coordinate system, we choose a one-parameter family of two-dimensional space-like cross-sections of future null infinity, ${\mathcal{I}}^{+}$, which are labeled by a coordinate u. We assume that $l=\partial /\partial r$, being a future directed null vector, is tangent to the surface u = const and a null vector n is parallelly propagated along the l congruence. An affine parameter r is normalized by the condition $l^{\alpha }\partial r/\partial x^{\alpha }=1$. Since the topology of ${\mathcal{I}}^{+}$ is $S^2\times R$, “cuts” of ${\mathcal{I}}^{+}$ can be labeled by a complex stereographic coordinate $\zeta =e^{i\phi }cot\theta /2$ [35].

From the null tetrad basis, complex Ricci rotation coefficients are defined as follows:

$$\begin{array}{cc}\hfill \rho & =l_{\alpha ;\beta }{m}^{\alpha }{\overline{m}}^{\beta },\mu =-n_{\alpha ;\beta }{m}^{\alpha }{m}^{\beta },\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \sigma & =l_{\alpha ;\beta }{m}^{\alpha }{m}^{\beta },\lambda =-n_{\alpha ;\beta }{\overline{m}}^{\alpha }{\overline{m}}^{\beta },\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \kappa & =l_{\alpha ;\beta }{m}^{\alpha }{l}^{\beta },\nu =-n_{\alpha ;\beta }{\overline{m}}^{\alpha }{n}^{\beta },\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \tau & =l_{\alpha ;\beta }{m}^{\alpha }{n}^{\beta },\pi =-n_{\alpha ;\beta }{\overline{m}}^{\alpha }{l}^{\beta },\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill ϵ& ={\displaystyle \frac{1}{2}}(l_{\alpha ;\beta }{n}^{\alpha }{l}^{\beta }-m_{\alpha ;\beta }{\overline{m}}^{\alpha }{l}^{\beta }),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \gamma & =-{\displaystyle \frac{1}{2}}(n_{\alpha ;\beta }{l}^{\alpha }{n}^{\beta }-{\overline{m}}_{\alpha ;\beta }{m}^{\alpha }{n}^{\beta }),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \alpha & ={\displaystyle \frac{1}{2}}(l_{\alpha ;\beta }{n}^{\alpha }{\overline{m}}^{\beta }-m_{\alpha ;\beta }{\overline{m}}^{\alpha }{\overline{m}}^{\beta }),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \beta & =-{\displaystyle \frac{1}{2}}(n_{\alpha ;\beta }{l}^{\alpha }{m}^{\beta }-{\overline{m}}_{\alpha ;\beta }{m}^{\alpha }{m}^{\beta }).\hfill \end{array}$$

(19)

The Riemann tensor is decomposed into its irreducible parts in such a way that the corresponding tetrad components are labeled:

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_0=& -C_{\alpha \beta \gamma \delta }{l}^{\alpha }{m}^{\beta }{l}^{\gamma }{m}^{\delta },\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_1=& -C_{\alpha \beta \gamma \delta }{l}^{\alpha }{n}^{\beta }{l}^{\gamma }{m}^{\delta },\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_2=& -{\displaystyle \frac{1}{2}}{C}_{\alpha \beta \gamma \delta }(l^{\alpha }{n}^{\beta }{l}^{\gamma }{n}^{\delta }-l^{\alpha }{n}^{\beta }{m}^{\gamma }{\overline{m}}^{\delta }),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_3=& -C_{\alpha \beta \gamma \delta }{l}^{\alpha }{n}^{\beta }{\overline{m}}^{\gamma }{n}^{\delta },\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_4=& -C_{\alpha \beta \gamma \delta }{n}^{\alpha }{\overline{m}}^{\beta }{n}^{\gamma }{\overline{m}}^{\delta },\hfill \end{array}$$

(24)

where $C_{\alpha \beta \gamma \delta }$ is the Weyl tensor.

Further, it is convenient to introduce the notation

$$\begin{array}{c}\hfill \begin{array}{c}\hfill D=l^{\alpha }{\nabla }_{\alpha },\mathrm{\Delta }=n^{\alpha }{\nabla }_{\alpha },\\ \hfill \delta =m^{\alpha }{\nabla }_{\alpha },\overline{\delta }={\overline{m}}^{\alpha }{\nabla }_{\alpha },\end{array}\end{array}$$

(25)

for the projections of the covariant derivatives onto the null tetrad. Then, one can write

$$\begin{array}{cc}\hfill D=& {\displaystyle \frac{\partial }{\partial r}},\mathrm{\Delta }={\displaystyle \frac{\partial }{\partial u}}+U{\displaystyle \frac{\partial }{\partial r}}+X^A{\displaystyle \frac{\partial }{\partial x^A}},\hfill \\ \hfill \delta =& \omega {\displaystyle \frac{\partial }{\partial r}}+{\xi }^A{\displaystyle \frac{\partial }{\partial x^A}},A=2,3,\hfill \end{array}$$

for some $U,X^A,\omega$, and ${\xi }^A$, where $x^2=\zeta$, $x^3=\overline{\zeta }$ [32,36].

In the following, employing the coordinate freedom, we will use the Bondi coordinates at ${\mathcal{I}}^{+}$. This choice of coordinates implies that a two-dimensional surface $S_{\infty }$, being obtained as a cut $u=\mathrm{const}$, is a two-sphere $S^2$ with the line element written as

$$\begin{array}{c}\hfill d{s}^2={\displaystyle \frac{4d\overline{\zeta }d\zeta }{1+\overline{\zeta }\zeta }}.\end{array}$$

(26)

We choose the NP-basis at ${\mathcal{I}}^{+}$ as follows:

$$\begin{array}{cc}\hfill {\mathrm{\Delta }}^0=& {\displaystyle \frac{\partial }{\partial u}},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\delta }^0=& P{\displaystyle \frac{\partial }{\partial \zeta }}=-{\displaystyle \frac{1}{\sqrt{2}}}\left({\displaystyle \frac{\partial }{\partial \theta }}+{\displaystyle \frac{i}{sin\theta }}{\displaystyle \frac{\partial }{\partial \phi }}\right),\hfill \end{array}$$

(28)

where $P=\zeta (1+\zeta \overline{\zeta })/(\sqrt{2}\left|\zeta \right|)$. Sign "−"in the definition of ${\delta }^0$ is introduced to provide an agreement with the standard form of raising and lowering operators (see Appendix A).

The Weyl tensor components essential for future spin coefficients $\alpha$, $\beta$, $\lambda$, and $\sigma$, asymptotically, are as follows [18,33,36]:

$$\begin{array}{cc}\hfill \alpha =& {\alpha }^0{r}^{-1}+\mathcal{O}\left(r^{-2}\right),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \beta =& {\beta }^0{r}^{-1}+\mathcal{O}\left(r^{-2}\right),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \lambda =& {\lambda }^0{r}^{-1}+\mathcal{O}\left(r^{-2}\right),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill \sigma =& {\sigma }^0{r}^{-2}+\mathcal{O}\left(r^{-3}\right),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_0=& {\mathrm{\Psi }}_0^0{r}^{-5}+\mathcal{O}\left(r^{-6}\right),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_1=& {\mathrm{\Psi }}_1^0{r}^{-4}+\mathcal{O}\left(r^{-5}\right),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_2=& {\mathrm{\Psi }}_2^0{r}^{-3}+\mathcal{O}\left(r^{-4}\right),\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_3=& {\mathrm{\Psi }}_3^0{r}^{-2}+\mathcal{O}\left(r^{-3}\right),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_4=& {\mathrm{\Psi }}_4^0{r}^{-1}+\mathcal{O}\left(r^{-2}\right).\hfill \end{array}$$

(37)

In Bondi coordinates, relationships between functions on ${\mathcal{I}}^{+}$ are as follows [18,22,37]:

$$\begin{array}{cc}\hfill & {\alpha }^0={\displaystyle \frac{\overline{P}}{2}}{\displaystyle \frac{\partial lnP}{\partial \overline{\zeta }}},\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & {\beta }^0=-{\overline{\alpha }}^0,{\lambda }^0={\dot{\overline{\sigma }}}^0,\hfill \end{array}$$

(39)

where ${\lambda }^0={\dot{\overline{\sigma }}}^0$ is the news function; “dot” denotes the derivative $\partial /\partial u$; function ${\lambda }^0$ represents information emerging from the positive null infinity ${\mathcal{I}}^{+}$; and the shear ${\sigma }^0$ corresponds to the shear free parameter. With the shear $h=h_{+}-i{h}_{\times }\propto {\sigma }^0$, the news function is linked to its time-derivative $\dot{h}\propto {\lambda }^0$ gravitational wave polarizations h from gauge-invariant perturbations derived by Thorne in [6,38].

The raising and lowering operators ð and $\overline{ð}$, respectively, are defined by the following expressions:

$$\begin{array}{cc}\hfill ð\eta & =\delta \eta +s({\overline{\alpha }}^0-\beta )\eta ,\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill \overline{ð}\eta & =\overline{\delta }\eta -s({\overline{\alpha }}^0-\beta )\eta ,\hfill \end{array}$$

(41)

where s is the spin weight of function $\eta$ [18,35].

The Weyl tensor is as follows [22]:

$$\begin{array}{cc}\hfill & {\mathrm{\Psi }}_2^0-{\overline{\mathrm{\Psi }}}_2^0={\overline{\sigma }}^0{\overline{\lambda }}^0-{\sigma }^0{\lambda }^0+{\overline{ð}}^2{\sigma }^0-{ð}^2{\overline{\sigma }}^0,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & {\mathrm{\Psi }}_3^0=-ð{\lambda }^0,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & {\mathrm{\Psi }}_4^0=-{\dot{\lambda }}^0.\hfill \end{array}$$

(44)

The mass aspect function [9,26,32],

$$\begin{array}{c}\hfill \mathrm{\Psi }={\mathrm{\Psi }}_2^0+{\sigma }^0{\lambda }^0+{ð}^2{\overline{\sigma }}^0,\end{array}$$

(45)

satisfies the reality condition $\mathrm{\Psi }=\overline{\mathrm{\Psi }}$.

The evolution equations (Bianci identities) have the following asymptotic form:

$$\begin{array}{cc}\hfill {\dot{\mathrm{\Psi }}}_0^0=& ð{\mathrm{\Psi }}_1^0+3{\sigma }^0{\mathrm{\Psi }}_2^0,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\dot{\mathrm{\Psi }}}_1^0=& ð{\mathrm{\Psi }}_2^0+2{\sigma }^0{\mathrm{\Psi }}_3^0,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {\dot{\mathrm{\Psi }}}_2^0=& ð{\mathrm{\Psi }}_3^0+{\sigma }^0{\mathrm{\Psi }}_4^0.\hfill \end{array}$$

(48)

The relations in Equations (43)–(48) have the opposite sign in comparison with similar expressions in Ref. [33]. This is due to the difference in the sign of ${\delta }^0$. In [33], it is defined as ${\delta }^0=-P{\partial }_{\zeta }$.

3. Asymptotic Symmetries and the Poincaré Quasigroup

The asymptotic symmetries of the asymptotically flat spacetime at future null infinity are described by the infinite-parametric Newman–Unti (NU) group [33,36,39,40]. The latter is defined by the transformation ${\mathcal{I}}^{+}\to {\mathcal{I}}^{+}$, with the form

$$\begin{array}{cc}\hfill u\to u^{\prime }=& f(u,\zeta ,\overline{\zeta }),\partial f/\partial u>0,\hfill \\ \hfill \zeta \to {\zeta }^{\prime }=& (\alpha \zeta +\beta )/(\gamma \zeta +\delta ),\alpha \delta -\beta \gamma =1,\hfill \end{array}$$

where $f(u,\zeta ,\overline{\zeta })$ is an arbitrary function.

The infinite-dimensional Bondi–Metzner–Sachs (BMS) group preserving strong conformal geometry is a subgroup of the NU group. The BMS group is defined as follows [9,10,14,21,26]:

$$\begin{array}{cc}\hfill & u\to u^{\prime }=K(\zeta ,\overline{\zeta })(u+a(\zeta ,\overline{\zeta })),\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & \zeta \to {\zeta }^{\prime }=(\alpha \zeta +\beta )/(\gamma \zeta +\delta ),\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill & \alpha \delta -\beta \gamma =1,\hfill \end{array}$$

(51)

where $\alpha$, $\beta$, $\gamma$, and $\delta$ are complex constants, and $a(\zeta ,\overline{\zeta })$ is an arbitrary regular function on $S^2$; moreover,

$$\begin{array}{c}\hfill K(\zeta ,\overline{\zeta })={\displaystyle \frac{(1+\zeta \overline{\zeta })}{\left(\right|\alpha \zeta +\beta {|}^2+|\gamma \zeta +\delta {|}^2)}}.\end{array}$$

(52)

The infinite-parameter normal subgroup of the BMS group,

$$\begin{array}{c}\hfill {\zeta }^{\prime }=\zeta ,u^{\prime }=u+a(\zeta ,\overline{\zeta }),\end{array}$$

(53)

is called the subgroup of supertranslations and contains a four-parameter normal translation subgroup:

$$\begin{array}{c}\hfill a={\displaystyle \frac{p+q\zeta +\overline{q}\overline{\zeta }+g\zeta \overline{\zeta }}{1+\zeta \overline{\zeta }}},\Im p=\Im g=0.\end{array}$$

(54)

The quotient (factor) group of the BMS group by the supertranslations consists of the conformal transformations $S^2\to S^2$, and it is isomorphic to the proper orthochronous Lorentz group.

Since the BMS group is the semi-direct product of the Lorentz group and supertranslation’s group, there is no canonical way to embed the Poincaré group in the BMS group. One has an infinite number of alternatives to extract the Poincaré group. However, at least in Minkowski spacetime, one can elucidate which additional structure on ${\mathcal{I}}^{+}$ the Poincaré group preserves. It turns out to be that the Poincaré group transforms the so-called good cuts [41], cuts with vanishing shear ${\sigma }^0$.

This can be easily seen by considering the transformation of shear under supertranslation Equation (53). We obtain ${\sigma }^0\to {\sigma }^{\prime 0}={\sigma }^0-{ð}^{2}a$, where ð is the “eth” operator on ${\mathcal{I}}^{+}$. Thus, a supertranslation transforms a good cut to a bad cut—a cut with nonvanishing shear. If we impose the condition ${\sigma }^{\prime 0}={\sigma }^0=0$, we obtain ${ð}^{2}a=0$. The solution of this equation with a real a yields a four-parametric subgroup of translations defined by Equation (54).

3.1. Reduction of the NU Group to the Poincaré Quasigroup

The infinitesimal NU group is obtained from the asymptotic Killing equations [39]:

$$\begin{array}{cc}\hfill & {\pounds }_{\xi }{g}_{\mu \nu }={\xi }_{\mu ;\nu }+{\xi }_{\nu ;\mu }=0\left(r^{-n}\right),\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill & l^{\nu }{\pounds }_{\xi }{g}_{\mu \nu }=({\xi }_{\mu ;\nu }+{\xi }_{\nu ;\mu })l^{\nu }=Q(u,\zeta ,\overline{\zeta })l_{\mu },\hfill \end{array}$$

(56)

where $Q(u,\zeta ,\overline{\zeta })$ is an arbitrary function on ${\mathcal{I}}^{+}$.

One can write a general element of NU-algebra as follows:

$$\begin{array}{cc}\hfill & \xi =B(u,\zeta ,\overline{\zeta }){\mathrm{\Delta }}^0+C(u,\zeta ,\overline{\zeta }){\overline{\delta }}^0+\overline{C}(u,\zeta ,\overline{\zeta }){\delta }^0,\hfill \\ \hfill & ðC=0,\hfill \end{array}$$

(57)

where ð, ${\mathrm{\Delta }}^0$, and ${\delta }^0$ are the standard NP operators, and “eth”, $\mathrm{\Delta }$, and $\delta$ are restricted on ${\mathcal{I}}^{+}$.

The generators of the four-parameter translation subgroup are given by

$$\begin{array}{c}\hfill {\xi }_a=B_a(\zeta ,\overline{\zeta }){\mathrm{\Delta }}^0,\end{array}$$

(58)

where the function $B_a$ is assumed to be a real function and is the solution of the following equation:

$$\begin{array}{c}\hfill {ð}^2{B}_a=0,\Im B_a=0(a=0,1,2,3).\end{array}$$

(59)

The generators of the “Lorentz group” are determined as follows:

$$\begin{array}{cc}\hfill & {\xi }_A=B_A(u,\zeta ,\overline{\zeta }){\mathrm{\Delta }}^0+C_A{\overline{\delta }}^0+{\overline{C}}_A{\delta }^0,\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill & ðC_A=0,(A=1,2\dots 6),\hfill \end{array}$$

(61)

where $B_A(u,\zeta ,\overline{\zeta })$ is an arbitrary real function.

The generators of the NU group obey the commutation relations:

$$\begin{array}{cc}\hfill & [{\xi }_a,{\xi }_b]=0,[{\xi }_a,{\xi }_B]=C_{aB}^b(u,\zeta ,\overline{\zeta }){\xi }_b,\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill & \left[{\xi }_A,{\xi }_B\right]=C_{AB}^D(u,\zeta ,\overline{\zeta }){\xi }_D,\hfill \end{array}$$

(63)

where $C_{aB}^b$ and $C_{AB}^D$ are the structure functions. This points out that the NU group is a quasigroup with the closed Lorentz quasialgebra.

To reduce the NU group to the particular Poincaré quasigroup, one needs to impose the constraints on a function $B_A(u,\zeta ,\overline{\zeta })$ and, thus, fix the supertranslational ambiguity in the definition of the Lorentz quasigroup.

In our approach, we use the fact that a group of isometries transforms an arbitrary geodesic to a geodesic one, and the Killing vectors satisfy the geodesic deviation equation for any geodesic [28,29]. In the construction below, only null geodesics passing inward are transformed to geodesics under the transformations of the Poincaré quasigroup. Instead of using the approximate Killing equations, we propagate the asymptotic generators $\xi$ defined on ${\mathcal{I}}^{+}$ inward along the null surface $\mathrm{\Gamma }$ intersecting ${\mathcal{I}}^{+}$ in ${\mathrm{\Sigma }}^{+}$ employing the geodesic deviation equation.

$${\nabla }_l^2\xi +R(\xi ,l)l=0.$$

(64)

Since the geodesic deviation equation is the second-order ordinary differential equation for obtaining the unique solution, we need to impose the initial conditions on the vector $\xi$ and its first derivatives on ${\mathcal{I}}^{+}$. We use the asymptotic Killing equations for determining them.

The key idea behind our approach is to use the geodesic deviation equation only for a null geodesic congruence passing inward ${\mathcal{I}}^{+}$ to define the generators of the Poincare quasigroup. This implies not only that the Poincaré quasigroup transforms an arbitrary geodesic to a geodesic but also that the null geodesics belong to the null congruence defined above.

3.1.1. Minkowski Spacetime

We demonstrate our approach in Minkowski spacetime. The key idea is to reduce the NU group to the ten-parametric Poincaré group, imposing the appropriate conditions on an arbitrary function $B_A$, and thus fixing the supertranslational freedom (see Refs. [27,28,29] for details).

Let us write the Killing vector as

$$\begin{array}{c}\hfill \xi =AD+B\mathrm{\Delta }+\overline{C}\delta +C\overline{\delta }.\end{array}$$

(65)

Using the asymptotic expansion

$$\begin{array}{cc}\hfill A=& A_{1}r+A_0+A_{-1}{r}^{-1}+0\left(r^{-2}\right),\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill B=& B_{1}r+B_0+B_{-1}{r}^{-1}+0\left(r^{-2}\right),\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill C=& C_{1}r+C_0+C_{-1}{r}^{-1}+0\left(r^{-2}\right),\hfill \end{array}$$

(68)

we obtain the solution of the geodesic deviation equation in the following form:

$$\begin{array}{c}\hfill A_{-n}=A_{-n}(u,\zeta ,\overline{\zeta }),C_{-n}=C_{-n}(u,\zeta ,\overline{\zeta }),B_{-n}=0(n\ge 1).\end{array}$$

(69)

The explicit dependence of $A_{-n}(u,\zeta ,\overline{\zeta })$, and $C_{-n}(u,\zeta ,\overline{\zeta })$ is not important to studying the structure of asymptotic symmetries.

To obtain the unique solution of the geodesic deviation equation, we have to impose the conditions on functions $A,B,$ and C and their first derivatives at ${\mathcal{I}}^{+}$. This implies that $A_0,A_1,B_0,B_1,C_0,$ and $C_1$ should be determined. We adapt the asymptotic Killing equations to determine these coefficients:

$$\begin{array}{cc}\hfill & \underset{r\to \infty }{lim}{l}^{\mu }{l}^{\nu }{\pounds }_{\xi }{g}_{\mu \nu }=0,\underset{r\to \infty }{lim}{m}^{\mu }{n}^{\nu }{\pounds }_{\xi }{g}_{\mu \nu }=0,\hfill \\ \hfill & \underset{r\to \infty }{lim}{m}^{\mu }{\overline{m}}^{\nu }{\pounds }_{\xi }{g}_{\mu \nu }=0,\underset{r\to \infty }{lim}r{m}^{\mu }{\overline{m}}^{\nu }{\pounds }_{\xi }{g}_{\mu \nu }=0,\hfill \\ \hfill & \underset{r\to \infty }{lim}r{l}^{\mu }{m}^{\nu }{\pounds }_{\xi }{g}_{\mu \nu }=0.\hfill \end{array}$$

(70)

After some algebra, we obtain

$$\begin{array}{cc}\hfill & A_0=-{\displaystyle \frac{1}{2}}(ð{\overline{C}}_0+\overline{ð}{C}_0)-{\displaystyle \frac{B_0}{2}}({\mu }^0+{\overline{\mu }}^0)+{\displaystyle \frac{1}{2}}({\tau }^0{\overline{C}}_1+{\overline{\tau }}^0{C}_1),\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill & A_1=-{\displaystyle \frac{1}{2}}(ð{\overline{C}}_1+\overline{ð}{C}_1),\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill & B_1=0,\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill & C_0=-ðB_0+{\sigma }^0{\overline{C}}_1,\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill & ðC_0=ð{\sigma }^0{C}_1+{\displaystyle \frac{{\sigma }^0}{2}}(ð{\overline{C}}_1-\overline{ð}{C}_1),\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill & {\dot{C}}_1=0,ðC_1=0,\hfill \end{array}$$

(76)

where “dot” denotes the derivative with respect to the retarded time u. Substituting $C_0$ from Equation (74) in Equation (75), we get

$$\begin{array}{cc}\hfill & {\dot{C}}_1=0,ðC_1=0,\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill & {ð}^2{B}_0-{\displaystyle \frac{{\sigma }^0}{2}}(3ð{\overline{C}}_1-\overline{ð}{C}_1)-{\overline{C}}_1ð{\sigma }^0-C_1\overline{ð}{\sigma }^0=0.\hfill \end{array}$$

(78)

A general solution of this system can be written as

$$\begin{array}{cc}\hfill & B_0=B_t+ð\eta {\overline{C}}_1+\overline{ð}\eta C_1+{\displaystyle \frac{u-\eta }{2}}(ð{\overline{C}}_1+\overline{ð}{C}_1),\hfill \\ \hfill & {ð}^2{B}_t=0,ðC_1=0,\hfill \end{array}$$

(79)

where ${\sigma }^0={ð}^2\eta$.

The system of differential constraints, Equations (77) and (78), is a unique one that determines functions $B_0$ and $C_1$ and restricts the NU group to a particular Poincaré group. Thus, in Minkowski spacetime, one can reduce the NU group to the Poincaré group even for “bad” cuts (${\sigma }^0\ne 0$).

Now, an arbitrary Killing vector at ${\mathcal{I}}^{+}$, ${\xi }^0{=\xi |}_{{\mathcal{I}}^{+}}$, can be written as

$${\xi }^0=B_0(u,\zeta ,\overline{\zeta }){\mathrm{\Delta }}^0+C_1(\zeta ,\overline{\zeta }){\overline{\delta }}^0+{\overline{C}}_1(\zeta ,\overline{\zeta }){\delta }^0.$$

(80)

To specify the generators of the Poincaré group, one should impose the additional conditions on functions $B_0$ and $C_1$.

The generators of translations are,

$$\begin{array}{c}\hfill l_a=B_a{\mathrm{\Delta }}^0,a=0,1,2,3,\end{array}$$

(81)

with $\Im B_a=0$ and $B_a$ denoting solutions of the differential equation ${ð}^2{B}_a=0$. With real $B_a$, we obtain four independent solutions of this equation, yielding

$$\begin{array}{c}\hfill l_a=(1,sin\theta cos\phi ,sin\theta sin\phi ,cos\theta ).\end{array}$$

(82)

The generators of boosts and rotations are given by

$${\xi }_A=B_A{\mathrm{\Delta }}^0+C_A{\overline{\delta }}^0+{\overline{C}}_A{\delta }^0,$$

(83)

where $ðC_A=0$ and

$$\begin{array}{c}\hfill B_A=ð\eta {\overline{C}}_A+\overline{ð}\eta C_A+{\displaystyle \frac{u-\eta }{2}}(ð{\overline{C}}_A+\overline{ð}{C}_A),\end{array}$$

(84)

with $\Im B_A=\Im \eta =0$ and ${\sigma }^0={ð}^2\eta$. Imposing additional conditions,

$$\begin{array}{cc}\hfill & \overline{ð}{C}_A+ð{\overline{C}}_A=0\left(\mathrm{rotations}\right),\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill & \overline{ð}{C}_A-ð{\overline{C}}_A=0\left(\mathrm{boosts}\right),\hfill \end{array}$$

(86)

we obtain six independent solutions of equation $ðC_A=0$. It is convenient to divide them into two groups, writing $C_A=\{L_i,R_i\}$, where $L_i$ and $R_i$ describe the rotations and boosts, respectively. We denote the generators of the complex Lorentz group as ${\mathrm{\Gamma }}_A={\overline{C}}_A{\delta }^0={\xi }_A{\partial }_{\zeta }$. Using the results of Ref. [40], we obtain the generators of the boost and rotations. They are shown in

Table 1. Generators of the complex Lorentz group. (Note that $L_1$, $L_2$, and $L_3$ are defined from $L_x$, $L_y$, and $L_z$ (see Appendix B)).

Generator	${\mathbf{\Gamma }}_{\mathit{A}}={\mathit{\xi }}_{\mathit{A}}\frac{\mathit{\partial }}{\mathit{\partial }\mathit{\zeta }}$
${\mathrm{\Gamma }}_1=L_1$	${\displaystyle \frac{i}{2}}(1-{\zeta }^2){\displaystyle \frac{\partial }{\partial \zeta }}$
${\mathrm{\Gamma }}_2=L_2$	$-{\displaystyle \frac{1}{2}}(1+{\zeta }^2){\displaystyle \frac{\partial }{\partial \zeta }}$
${\mathrm{\Gamma }}_3=L_3$	$i\zeta {\displaystyle \frac{\partial }{\partial \zeta }}$
${\mathrm{\Gamma }}_4=R_1$	${\displaystyle \frac{1}{2}}(1-{\zeta }^2){\displaystyle \frac{\partial }{\partial \zeta }}$
${\mathrm{\Gamma }}_5=R_2$	${\displaystyle \frac{i}{2}}(1+{\zeta }^2){\displaystyle \frac{\partial }{\partial \zeta }}$
${\mathrm{\Gamma }}_6=R_3$	$\zeta {\displaystyle \frac{\partial }{\partial \zeta }}$

.

Let us introduce a complex vector ${\xi }_c$ at ${\mathcal{I}}^{+}$ such that an arbitrary element of the infinitesimal NU group is written as $\xi ={\xi }_c+{\overline{\xi }}_c$, where

$$\begin{array}{c}\hfill {\xi }_c={\xi }_s{\mathrm{\Delta }}^0+{\xi }_1{\delta }^0.\end{array}$$

(87)

(Hereafter, we omit the index “0” in ${\xi }^0$.) We specify the vector ${\xi }_s$ as follows:

$$\begin{array}{cc}\hfill & {\xi }_s={\xi }_t+ð\eta {\xi }_1+{\displaystyle \frac{u-\eta }{2}}ð{\xi }_1,\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill & {ð}^2{\xi }_t^0=0,\overline{ð}{\xi }_1=0,\hfill \end{array}$$

(89)

where $\Im {\xi }_t^0=\Im \eta =0$ and ${ð}^2\eta ={\sigma }^0$. Comparing this expression with Equation (80), we find $B_0={\xi }_s+{\overline{\xi }}_s$ and $C_1={\overline{\xi }}_1$.

A straightforward computation shows that ${\xi }_s$ obeys the following differential equation:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {ð}^2{\xi }_s=& {\displaystyle \frac{3}{2}}{\sigma }^0ð{\xi }_1+ð{\sigma }^0{\xi }_1,\hfill \\ \hfill \overline{ð}{\xi }_1=& 0.\hfill \end{array}\end{array}$$

(90)

Thus, instead of employing Equations (77) and (78) to reduce the NU group to the Poincaré group in Minkowski spacetime, one can consider the equivalent differential constraints with Equation (90).

3.1.2. General Case: Asymptotically Flat Spacetime with Radiation

As was mentioned above, the NU group is an infinite-dimensional group, and therefore, there is no existing unique way to reduce the NU group to the finite-dimensional group even in Minkowski spacetime. All attempts suffer in supertranslational ambiguity. To overcome this issue, we impose on ${\xi }_c$ the differential constraints restricting the NU group to a particular Poincaré quasigroup [29]:

$$\begin{array}{cc}\hfill {ð}^2{\xi }_s=& {\displaystyle \frac{3}{2}}{\sigma }^0ð{\xi }_1+ð{\sigma }^0{\xi }_1,\hfill \end{array}$$

(91)

$$\begin{array}{cc}\hfill \overline{ð}{\xi }_1=& 0.\hfill \end{array}$$

(92)

In our research, rather than using the complete infinite-dimensional BMS [42,43,44,45,46,47] or Weyl BMS group, we constrain the NU group to a finite-dimensional Poincaré quasigroup by imposing geometric constraints sourced from the geodesic deviation equation. A principal advantage of the quasigroup formalism lies in its intrinsic capability to manage time-dependent structure functions inherent in radiating spacetimes. Concurrently, the constraints enforced by Equations (91) and (92) serve to remove supertranslation ambiguities.

Since the spin weight of the asymptotic shear ${\sigma }^0$ is two, one can write ${\sigma }^0={ð}^2\eta$, where $\eta =\eta (u,\zeta ,\overline{\zeta })$ is a complex function. Then, a general solution of Equation (91) can be written as

$$\begin{array}{c}\hfill {\xi }_s={\xi }_t+ð\eta {\xi }_1+{\displaystyle \frac{u-\eta }{2}}ð{\xi }_1,\end{array}$$

(93)

where ${ð}^2{\xi }_t=0$.

We consider Equation (91) the differential constraint restricting the NU group to a particular Poincaré quasigroup [29]. In the absence of radiation, the differential constraint Equation (91) is compatible with the Killing equations, and the Poincaré quasigroup becomes the Poincaré group. Note that the same constraint was obtained in [15,48] in the twistor theory framework.

The structure of the Poincaré quasialgebra is as follows:

The generators of translations are given by

$${\xi }_a={\xi }_{ta}+{\overline{\xi }}_{ta}=B_a{\mathrm{\Delta }}^0,$$

(94)

where $B_a={\overline{B}}_a$, with $B_a$ being the solution of the following equation:

$$\begin{array}{c}\hfill {ð}^2{B}_a=0.\end{array}$$

(95)

There are four independent solutions of this equation if we assume that $\Im B_a$ = 0.

The generators of boosts and rotations are given by

$$\begin{array}{c}\hfill {\xi }_A={\xi }_{cA}+{\overline{\xi }}_{cA},\end{array}$$

(96)

where

$${\xi }_{cA}=(ð\eta {\overline{C}}_A+{\displaystyle \frac{u-\eta }{2}}ð{\overline{C}}_A){\mathrm{\Delta }}^0+{\overline{C}}_A{\delta }^0,$$

(97)

and $ðC_A=0$. There are six independent solutions of the equation $ðC_A=0$ if we impose additional constraints:

$$\begin{array}{cc}\hfill & \overline{ð}{C}_A+ð{\overline{C}}_A=0\left(\mathrm{rotations}\right),\hfill \end{array}$$

(98)

$$\begin{array}{cc}\hfill & \overline{ð}{C}_A-ð{\overline{C}}_A=0\left(\mathrm{boosts}\right),\hfill \end{array}$$

(99)

The straightforward computation shows that the generators of the Poincaré quasigroup obey at commutation relations ${\mathcal{I}}^{+}$:

$$\begin{array}{cc}\hfill & [{\xi }_a,{\xi }_b]=0,[{\xi }_a,{\xi }_B]=C_{aB}^b(u,\zeta ,\overline{\zeta }){\xi }_b,\hfill \end{array}$$

(100)

$$\begin{array}{cc}\hfill & \left[{\xi }_A,{\xi }_B\right]=C_{AB}^D(u,\zeta ,\overline{\zeta }){\xi }_D,\hfill \end{array}$$

(101)

where $C_{AB}^D$ and $C_{aB}^b$ are the structure functions. The last commutation relations mean that the generators of the Lorentz quasigroup form a closed algebra.

4. Energy Momentum and Angular Momentum at ${\mathcal{I}}^{+}$

As is well known, the general covariance of the Einstein equations results in differential conservation laws related to the field equations. From Noether’s theorem, it follows that for an arbitrary diffeomorphism generated by the vector field $\xi$, the invariance of the Lagrangian for the Einstein equations leads to the conservation laws of the form

$${\mathfrak{J}}^{\mu }{\left(\xi \right)}_{,\mu }=0,$$

(102)

where the vector density ${\mathfrak{J}}^{\mu }$ is defined as ${\mathfrak{J}}^{\mu }\left(\xi \right)={\mathfrak{h}}_{,\nu }^{\mu \nu }$, with ${\mathfrak{h}}^{\mu \nu }=-{\mathfrak{h}}^{\nu \mu }$ being the superpotential, which is constructed from the densities of the spin, bispin, vector field $\xi$, and its derivatives [1,2,3].

For the Einstein–Hilbert action, one obtains a simple expression of up to a factor of 2,

$${\mathfrak{J}}^{\mu }=-{\displaystyle \frac{1}{4\pi }}{\left(\sqrt{-g}{\xi }^{[\mu ;\nu ]}\right)}_{,\nu }$$

(103)

that was first given by Komar [4]. It is impossible to simply “renormalize” ${\mathfrak{J}}^{\mu }$ by a factor of 2 because the normalization for energy momentum and angular momentum differs by a factor of 2. The resolution of this problem is known as pointed out in [5]. One needs to add to the Einstein–Hilbert action I a surface term $I_s$ and apply Noether’s theorem to $I+I_s$. Komar’s expression provides a fully satisfactory notion of the total mass in stationary, asymptotically flat spacetimes.

As is known, the Komar integral is not invariant under a change in the choice of generators of time-like translations in the equivalence class associated with a given BMS translation. Moreover, the resulting energy would not be the Bondi energy, but instead the Newman–Unti energy [39]. The Bondi and Newman–Unti masses evaluated at null infinity are crucial for comprehending conserved quantities. The Bondi mass is associated with the standard BMS time-like translation and shows a monotonic decrease. From the other side, the Newman–Unti mass uses an arbitrary generator that is neither unique nor necessarily monotonic, since there are infinitely many such functions (related by supertranslations).

Our approach ensures that ${\mathcal{L}}_{\xi }$ reduces to the Bondi mass linked with pure time translations, rigorously formulated within the Newman–Penrose framework, thereby ensuring adherence to pertinent flux balance laws. This is distinguished by its gauge independence attributable to the use of a null tetrad rather than coordinate systems. However, when addressing coordinate-dependent quantities such as the Bondi or NU mass, it is imperative to choose a specific gauge. The constraints in Equations (91) and (92) perform a role akin to establishing the Bondi gauge, identifying a specific “radiation-adapted” reference frame that facilitates the interpretation of conserved quantities. In the absence of these constraints, computation of the Komar-type integral (Equation (104)) using an arbitrary BMS generator would yield a quantity similar to the NU energy, accompanied by its intrinsic gauge ambiguities.

According to the framework developed by Nester et al. [49], the gravitational energy momentum is linked to the Hamiltonian by applying specific boundary conditions. Pitts [50] showed that utilizing Noether’s first theorem leads to an infinite set of conserved currents.

Therefore, we do not claim to identify the definitive energy and angular momentum among the infinitely many alternatives. Instead, we propose a geometric criterion that is particularly suitable for the analysis of radiating isolated systems at future null infinity and for resolving the supertranslation ambiguity in angular momentum.

Our results should be situated within the contemporary framework of gravitational energy momentum in terms of the function ${\lambda }^0$, which represents information emerging from the positive null infinity ${\mathcal{I}}^{+}$ and ${\sigma }^0$ corresponds to the shear-free parameter. With shear $h=h_{+}-i{h}_{\times }\propto {\sigma }^0$, the news function is linked to its time-derivative $\dot{h}\propto {\lambda }^0$ gravitational wave polarizations h from gauge-invariant perturbations derived by Thorne and Ruiz [6] and Thorne [38].

For an asymptotically flat at future null infinity spacetime, the modified “gauge-invariant” Komar integral (linkage) was introduced by Tamburino and Winicour [16,17]. The computation leads to the following coordinate-independent expression [8,12,13]:

$$\begin{array}{c}\hfill {\mathcal{L}}_{\xi }=-{\displaystyle \frac{1}{4\pi }}\Re \oint \left[B\left({\mathrm{\Psi }}_2^0+{\sigma }^0{\lambda }^0-{ð}^2{\overline{\sigma }}^0\right)+\overline{C}\left({\mathrm{\Psi }}_1^0-{\sigma }^0{ð}^2{\overline{\sigma }}^0-(1/2)ð\left({\sigma }^0{\overline{\sigma }}^0\right)\right)\right]d\mathrm{\Omega },\end{array}$$

(104)

where ${\lambda }^0={\dot{\overline{\sigma }}}^0$, and $\xi$ is an arbitrary generator of the NU group at ${\mathcal{I}}^{+}$,

$$\begin{array}{c}\hfill \xi =B{\mathrm{\Delta }}^0+\overline{C}{\delta }^0+C{\overline{\delta }}^0,ðC=0.\end{array}$$

(105)

A complex generator, ${\xi }_c={\xi }_s{\mathrm{\Delta }}^0+{\xi }_1{\delta }^0$, of the Poincaré quasigroup yields the complex Noether charge that can be written as [11]

$$\begin{array}{c}\hfill Q_{{\xi }_c}=-{\displaystyle \frac{1}{8\pi }}\oint \left\{{\xi }_s\left({\mathrm{\Psi }}_2^0-{\sigma }^0{\lambda }^0-{ð}^2{\overline{\sigma }}^0\right)+{\xi }_1\left({\mathrm{\Psi }}_1^0-{\sigma }^0ð{\overline{\sigma }}^0-(1/2)ð\left({\sigma }^0{\overline{\sigma }}^0\right)\right)\right\}d\mathrm{\Omega }.\end{array}$$

(106)

The complex Noether charge $Q_{{\xi }_c}$ encodes both the real conserved quantity and phase information. The physical (real) conserved quantity is defined by ${\mathcal{L}}_{\xi }$.

Setting $B=({\xi }_s+{\overline{\xi }}_s)/2$ and ${\overline{C}}_1={\xi }_1$, we obtain

$${\mathcal{L}}_{\xi }=Q_{{\xi }_c}+{\overline{Q}}_{{\xi }_c}=Q_{{\xi }_c}+Q_{{\overline{\xi }}_c}.$$

(107)

The factor $1/8\pi$ in Equation (106) is determined by calculating the intrinsic angular momentum, $J=ma$, for the Kerr metric with the rotational Killing vector, $L_z$, yielding ${\xi }_1=-L_z·\overline{m}=i/\sqrt{2}sin\theta$. The Weyl scalar, ${\psi }_1^0=\sigma /\sqrt{2}$ is computed with respect to a Bondi frame with shear-free cross-sections [11,51].

As in the previous section, to reduce the NU group to the Poincaré quasigroup, we impose the following differential constraints:

$$\begin{array}{cc}\hfill {ð}^2{\xi }_s=& {\displaystyle \frac{3}{2}}{\sigma }^0ð{\xi }_1+ð{\sigma }^0{\xi }_1,\hfill \end{array}$$

(108)

$$\begin{array}{cc}\hfill \overline{ð}{\xi }_1=& 0.\hfill \end{array}$$

(109)

One can show that the following integral identity is valid:

$$\begin{array}{c}\hfill \oint {ð}^2{\xi }_s{\overline{\sigma }}^{0}d\mathrm{\Omega }=\oint {\xi }_1\left(ð{\sigma }^0{\overline{\sigma }}^0-{\displaystyle \frac{3}{2}}ð\left({\sigma }^0{\overline{\sigma }}^0\right)\right)d\mathrm{\Omega }.\end{array}$$

(110)

Using this identity in Equation (106), one can rewrite it in the equivalent form introduced in [15]:

$$\begin{array}{c}\hfill Q_{{\xi }_c}=-{\displaystyle \frac{1}{8\pi }}\oint \left\{{\xi }_s\left({\mathrm{\Psi }}_2^0+{\sigma }^0{\lambda }^0+{ð}^2{\overline{\sigma }}^0\right)+{\xi }_1\left({\mathrm{\Psi }}_1^0+{\sigma }^0ð{\overline{\sigma }}^0+(1/2)ð\left({\sigma }^0{\overline{\sigma }}^0\right)\right)\right\}d\mathrm{\Omega }.\end{array}$$

(111)

Now, employing Equations (108) and (109) and performing integration by parts, we find that Equation (111) can be recast as

$$\begin{array}{c}\hfill Q_{{\xi }_c}=-{\displaystyle \frac{1}{8\pi }}\oint \left({\xi }_s({\mathrm{\Psi }}_2^0+{\sigma }^0{\lambda }^0)+{\xi }_1{\mathrm{\Psi }}_1^0\right)d\mathrm{\Omega }.\end{array}$$

(112)

We adopt this as the definition of the conserved quantities on ${\mathcal{I}}^{+}$ associated with the generators of the Poincaré quasigroup, writing [11,15,29,52]

$$\begin{array}{c}\hfill {\mathcal{L}}_{\xi }=Q_{{\xi }_c}+{\overline{Q}}_{{\xi }_c}.\end{array}$$

(113)

The integral four-momentum is given by

$$P_a=-{\displaystyle \frac{1}{4\pi }}\Re \oint l_a({\mathrm{\Psi }}_2^0+{\sigma }^0{\lambda }^0)d\mathrm{\Omega }.$$

(114)

where $l^a$ denotes the four independent solutions of ${ð}^2{l}^a=0$ representing pure translations, ${\mathrm{\Psi }}_2^1$ is the leading-order Weyl scalar (mass aspect), ${\sigma }^0$ is the asymptotic shear (encodes gravitational radiation), ${\lambda }^0={\dot{\overline{\sigma }}}^0$ is the news functions (rate of change of shear), and some of the physical components are denoted by $P^0=M$ (Bondi mass). The total energy/mass of the system and $P^1$, $P^2$, and $P^3$ are the components of linear momentum in three spatial directions.

Using the Bianchi identities, we compute the loss of energy momentum and obtain the standard expression for the flux balance law (see, e.g., [8]):

$${\dot{P}}_a=-\left({\displaystyle \frac{1}{4\pi }}\right)\oint l_a{\left|{\lambda }^0\right|}^{2}d\mathrm{\Omega }.$$

(115)

This shows that energy/momentum decreases monotonically during gravitational wave emission.

The angular momentum is given by

$$M_i=-{\displaystyle \frac{1}{4\pi }}\Re \oint \left({\xi }_i({\mathrm{\Psi }}_2^0+{\sigma }^0{\lambda }^0)+{\overline{L}}_i{\mathrm{\Psi }}_1^0\right)d\mathrm{\Omega },$$

(116)

where $L_i$ denotes the three rotation generators on $S^2$, ${\mathrm{\Psi }}_0^1$ is the Weyl scalar component related to angular momentum, and ${\xi }_i$ is the solution of Equation (108) (adapted to radiation),

$$\begin{array}{c}\hfill {\xi }_i=ð\eta {\overline{L}}_i+{\displaystyle \frac{u-\eta }{2}}ð{\overline{L}}_i.\end{array}$$

(117)

Here, $L_i$ is the solution of the equation $ðL_i=0$ such that $\overline{ð}{L}_i+ð{\overline{L}}_i=0$ and ${\sigma }^0={ð}^2\eta$.

Substituting ${\xi }_i$ in Equation (116) and performing integration by parts, we find

$$M_i=-{\displaystyle \frac{1}{4\pi }}\Re \oint {\overline{L}}_i\mathcal{J}dS\mathrm{\Omega },$$

(118)

where

$$\begin{array}{cc}\hfill \mathcal{J}=& {\mathrm{\Psi }}_1^0+{\displaystyle \frac{3}{2}}ð\eta ({\mathrm{\Psi }}_2^0+{\sigma }^0{\lambda }^0)+{\displaystyle \frac{\eta }{2}}\left(ð{\mathrm{\Psi }}_2^0+ð\left({\sigma }^0{\lambda }^0\right)\right).\hfill \end{array}$$

(119)

It yields the following expression for the angular momentum loss:

$$\begin{array}{c}\hfill {\dot{M}}_i=-{\displaystyle \frac{1}{4\pi }}\Re \oint {\overline{L}}_i{\displaystyle \frac{\partial \mathcal{J}}{\partial u}}d\mathrm{\Omega }.\end{array}$$

(120)

Geroch and Winicour have given a list of properties that conserved quantities $P(\xi ,\mathrm{\Sigma })$ defined at ${\mathcal{I}}^{+}$ and should have the following [12]:

• $P(\xi ,\mathrm{\Sigma })$ should be linear in the generators of the asymptotic symmetry group.
• $P(\xi ,\mathrm{\Sigma })$ should be invariant with respect to Weyl rescalings of the form ${\tilde{g}}_{\mu \nu }={\tilde{\mathrm{\Omega }}}^2{g}_{\mu \nu }$, where $\tilde{\mathrm{\Omega }}$ constitutes a smooth function that is non-vanishing across the conformal transformation.
• The expression $P(\xi ,\mathrm{\Sigma })$ should depend on the geometry of ${\mathcal{I}}^{+}$ and behavior of generators in the neighborhood of ${\mathcal{I}}^{+}$.
• $P(\xi ,\mathrm{\Sigma })$ should be proportional to the corresponding Komar integral for the exact symmetries and coincide with the Bondi four-momentum when $\xi$ is a BMS translation.
• $P(\xi ,\mathrm{\Sigma })$ should be defined also for the system with radiation on ${\mathcal{I}}^{+}$.
• There should exist a flux integral $\mathcal{I}$ that is linear in $\xi$ and that gives the difference $P(\xi ,{\mathrm{\Sigma }}^{\prime })-P(\xi ,\mathrm{\Sigma })$ for ${\mathrm{\Sigma }}^{\prime }$ and ${\mathrm{\Sigma }}^{\prime }$, two closed surfaces on ${\mathcal{I}}^{+}$.
• In Minkowski spacetime, $P(\xi ,\mathrm{\Sigma })$ should vanish identically.

Our definition of the conserved quantities, ${\mathcal{L}}_{\xi }=Q_{{\xi }_c}+{\overline{Q}}_{{\xi }_c}$, where the complex Noether charge is given by Equation (112), is free from the supertranslation ambiguity and satisfies all these conditions.

5. Center of Mass and Intrinsic Angular Momentum

In special relativity, total angular momentum is given by $J^{\mu \nu }=X^{\mu }{P}^{\nu }-P^{\mu }{X}^{\nu }+S^{\mu \nu }$, where the two first terms are the angular momentum, and $S^{\mu \nu }$ denotes intrinsic angular momentum, with $X^{\mu }$ and $P^{\nu }$ being the position vector and momentum of the particle, respectively. Using space-like translation freedom, one can transform the given reference frame to the center-of-mass reference frame (CMRF), where $P^{\nu }$ is aligned with the observer’s four-velocity. In the CMRF, one has $J^{\mu \nu }{P}_{\nu }=0$. Applying the transformation $X^{\mu }=P^{\mu }/P_{\nu }{P}^{\nu }$, we obtain the Dixon condition on the intrinsic angular momentum, $S^{\mu \nu }{P}_{\nu }=0$ [53].

The Dixon condition implies that the boost generators do not contribute to the internal part of the total angular momentum [30]. At future null infinity, this condition reduces to the requirement that the linkage $L_{\xi }=0$ for the boost generators. We adopt this requirement for the general case by imposing the condition that the Noether charge $K_i=0$ for the boost generators:

$$\begin{array}{c}\hfill K_i=-{\displaystyle \frac{1}{4\pi }}\Re \oint \left({\xi }_i^R({\mathrm{\Psi }}_2^0+{\sigma }^0{\lambda }^0)+{\overline{R}}_i{\mathrm{\Psi }}_1^0\right)d\mathrm{\Omega },\end{array}$$

(121)

where

$$\begin{array}{c}\hfill {\xi }_i^R=ð\eta {\overline{R}}_i+{\displaystyle \frac{u-\eta }{2}}ð{\overline{R}}_i,\end{array}$$

(122)

and $R_i$ is the solution of the equation $ðR_i=0$ such that $\overline{ð}{R}_i-ð{\overline{R}}_i=0$.

As one can see, quantities $R_i$ and $L_i$ are related by relations $R_i=\pm i{L}_i$ (see Table 1 in Section 3). Then, the condition $K_i=0$, or Dixon condition, can be written as

$$\begin{array}{c}\hfill \Im \oint \left({\xi }_i({\mathrm{\Psi }}_2^0+{\sigma }^0{\lambda }^0)+{\overline{L}}_i{\mathrm{\Psi }}_1^0\right)d\mathrm{\Omega }=0,\end{array}$$

(123)

where ${\xi }_i$ is defined by Equation (117). In terms of the complex Noether charge, Equation (123) can be recast as $\Im Q_{{\xi }_c}=0$. This condition defines the center-of-mass frame at null infinity by requiring that boost charges vanish. In the framework of special relativity, the center-of-mass reference frame is characterized by the vanishing of total spatial momentum: $\overline{P}=0$. Our condition represents a generalization applicable to asymptotically flat spacetimes in the presence of gravitational radiation.

To transform the given reference frame at null infinity to the CMRF frame, we use supertranslation freedom to align the total momentum with the observer’s four-velocity. To explain in detail, let us consider the Noether charge associated with the mass aspect,

$$\begin{array}{c}\hfill {\mathcal{Q}}_a=\oint {\xi }_a\mathrm{\Psi }d\mathrm{\Omega }=\oint {\xi }_a({\mathrm{\Psi }}_2^0+{\sigma }^0{\lambda }^0+{ð}^2{\overline{\sigma }}^0)d\mathrm{\Omega }.\end{array}$$

(124)

In the CMRF, only component ${\mathcal{Q}}_0$ is different from zero. Thus, to align the total momentum with the observer’s four-velocity, one should require ${\mathcal{Q}}_i=0$ for $i=1,2,3$. Using this condition in Equation (123), we obtain

$$\begin{array}{c}\hfill \Im \oint \left({\overline{L}}_i{\mathrm{\Psi }}_1^0-{\xi }_i{ð}^2{\overline{\sigma }}^0\right)d\mathrm{\Omega }=0,\end{array}$$

(125)

where the integration is performed over the section defined by equation $u-{\eta }_0=0$. Further simplification can be performed by employing Equation (108). The computation yields

$$\begin{array}{c}\hfill \Im \oint {\overline{L}}_i\left(2{\mathrm{\Psi }}_1^0-ð{\sigma }^0{\overline{\sigma }}^0-3{\sigma }^0ð{\overline{\sigma }}^0\right)d\mathrm{\Omega }=0.\end{array}$$

(126)

The obtained results are an integral expression of the condition defining the center of mass deduced in [22]. This corresponds to the Dixon condition imposed on the intrinsic angular momentum, $S^{\mu \nu }{P}_{\nu }=0$.

To proceed further, we expand ${\xi }_i$, ${\sigma }^0$, and $\eta$ in terms of spin-weighted spherical harmonics:

$$\begin{array}{cc}\hfill {\xi }_i=& \sum _{l=0}^{\infty }\sum _{m=-l}^l{\xi }_{lm}^i{Y}_{lm},\hfill \end{array}$$

(127)

$$\begin{array}{cc}\hfill {\sigma }^0=& \sum _{l=2}^{\infty }\sum _{m=-l}^l{h}_{lm}\left(u\right){}_{2}Y_{lm},\hfill \end{array}$$

(128)

$$\begin{array}{cc}\hfill \eta =& {\eta }_0+\sum _{l=1}^{\infty }\sum _{m=-l}^l{\eta }_{lm}{Y}_{lm}.\hfill \end{array}$$

(129)

From the relation ${\sigma }^0={ð}^2\eta$, it follows that

$$\begin{array}{c}\hfill {\eta }_{lm}={\displaystyle \frac{2h_{lm}}{\sqrt{l(l^2-1)(l+2)}}},l\ge 2.\end{array}$$

(130)

We use the freedom in the choice of $\eta$ to eliminate in Equation (127) the contribution of the term with $l=0$, choosing the cut as $u-{\eta }_0=0$. We obtain

$$\begin{array}{cc}\hfill {\xi }_i=& \sum _{l=1}^{\infty }\sum _{m=-l}^l{\xi }_{lm}^i{Y}_{lm}.\hfill \end{array}$$

(131)

Using Equation (131) in Equation (124), we have

$$\begin{array}{c}\hfill {\mathcal{J}}_i=\sum _{l=1}^{\infty }\sum _{m=-l}^l{\xi }_{lm}^i\oint Y_{lm}({\mathrm{\Psi }}_2^0+{\sigma }^0{\lambda }^0+{ð}^2{\overline{\sigma }}^0)d\mathrm{\Omega }.\end{array}$$

(132)

To proceed further, we consider the supermomenta $P_{lm}\left(\mathrm{\Sigma }\right)$ introduced in [25],

$$\begin{array}{c}\hfill P_{lm}\left(\mathrm{\Sigma }\right)=-{\displaystyle \frac{1}{\sqrt{4\pi }}}{\int }_{\mathrm{\Sigma }}{Y}_{lm}\mathrm{\Psi }d\mathrm{\Omega },\end{array}$$

(133)

where $\mathrm{\Sigma }$ is an arbitrary section of ${\mathcal{I}}^{+}$. A nice section is defined by the requirement that the supermomentum $P_{lm}\left(\mathrm{\Sigma }\right)=0$ for $l\ge 1$. Thus, only the component $P_0$ of the total momentum is non-vanishing, providing us with a geometric notion of a reference frame at rest. This, in general, involves the need to make a Lorentz boost, which keeps $\mathrm{\Sigma }$ fixed and aligns the generator of time translations with the total momentum [26]. As one can see, for the nice sections, the quantity ${\mathcal{J}}_i=0$.

Employing the condition ${\mathcal{J}}_i=0$ in Equation (116), after some computations, we find that the intrinsic angular momentum, $J_i$, can be written as the Komar angular momentum:

$$J_i=-{\displaystyle \frac{1}{4\pi }}\Re \oint {\overline{L}}_i\left({\mathrm{\Psi }}_1^0+{\sigma }^0ð{\overline{\sigma }}^0\right)d\mathrm{\Omega }.$$

(134)

This is the angular momentum in the center-of-mass frame where orbital angular momentum vanishes. An independent calculation by Gallo and Moreschi [26] confirms Equation (134).

Hence, the angular momentum loss is given by

$${\dot{J}}_i=-{\displaystyle \frac{1}{4\pi }}\Re \oint {\overline{L}}_i\left({\dot{\mathrm{\Psi }}}_1^0+{\overline{\lambda }}^0ð{\overline{\sigma }}^0+{\sigma }^0ð{\lambda }^0\right)d\mathrm{\Omega },$$

(135)

Using the Bianci identities [34,37],

$$\begin{array}{cc}\hfill & {\mathrm{\Psi }}_2^0-{\overline{\mathrm{\Psi }}}_2^0={\overline{\sigma }}^0{\overline{\lambda }}^0-{\sigma }^0{\lambda }^0+{\overline{ð}}^2{\sigma }^0-{ð}^2{\overline{\sigma }}^0,\hfill \end{array}$$

(136)

$$\begin{array}{cc}\hfill & {\dot{\mathrm{\Psi }}}_1^0=ð{\mathrm{\Psi }}_2^0-2{\sigma }^0ð{\lambda }^0,\hfill \end{array}$$

(137)

Equation (135) reduces to

$$\begin{array}{c}\hfill {\dot{J}}_i=-{\displaystyle \frac{1}{8\pi }}\Re \oint {\overline{L}}_i\left(3{\overline{\lambda }}^0ð{\overline{\sigma }}^0-3{\sigma }^0ð{\lambda }^0+{\overline{\sigma }}^0ð{\overline{\lambda }}^0-{\lambda }^0ð{\sigma }^0\right)d\mathrm{\Omega }.\end{array}$$

(138)

This expression agrees with the results obtained in [54].

To compute the angular momentum loss, we express ${\sigma }^0$ and ${\overline{L}}_i$ in terms of spin-weighted spherical harmonics:

$$\begin{array}{cc}\hfill {\sigma }^0=& \sum _{l=2}^{\infty }\sum _{m=-l}^l{h}_{lm}\left(u\right){}_{2}Y_{lm},\hfill \end{array}$$

(139)

$$\begin{array}{cc}\hfill {\overline{L}}_i=& \sum _{m=-1}^1{L}_{im}{}_{-1}Y_{lm}.\hfill \end{array}$$

(140)

The computation yields (see Appendix B for details)

$$\begin{array}{cc}\hfill {\dot{J}}_i=& {\displaystyle \frac{1}{8\pi }}\Re \sum _{l,m,l^{\prime },m^{\prime }}\left(h_{lm}{\dot{\overline{h}}}_{l^{\prime }{m}^{\prime }}-{\dot{h}}_{lm}{\overline{h}}_{l^{\prime }{m}^{\prime }}\right)c_{lm{l}^{\prime }{m}^{\prime }}^i\hfill \\ \hfill =& {\displaystyle \frac{1}{8\pi }}\Re \sum _{l,m,l^{\prime },m^{\prime }}{h}_{lm}{\dot{\overline{h}}}_{l^{\prime }{m}^{\prime }}\left(c_{lm{l}^{\prime }{m}^{\prime }}^i-{\overline{c^i}}_{l^{\prime }{m}^{\prime }lm}\right),\hfill \end{array}$$

(141)

where the coefficients $c_{lm{l}^{\prime }{m}^{\prime }}^i$ are given in terms of the Wigner 3-j symbols:

$$\begin{array}{cc}\hfill c_{lm{l}^{\prime }{m}^{\prime }}^i=& {(-1)}^{m^{\prime }}\sqrt{{\displaystyle \frac{3(2l+1)(2l^{\prime }+1)}{8\pi }}}\{3\sqrt{(l^{\prime }+2)(l^{\prime }-1)}\left(\begin{array}{ccc}l& l^{\prime }& 1\\ -2& 1& 1\end{array}\right)\hfill \\ \hfill & +\sqrt{(l+3)(l-2)}\left(\begin{array}{ccc}l& l^{\prime }& 1\\ -3& 2& 1\end{array}\right)\}\sum _{m^{″}=-1}^1{L}_{i{m}^{″}}\left(\begin{array}{ccc}l& l^{\prime }& 1\\ m& -m^{\prime }& m^{″}\end{array}\right).\hfill \end{array}$$

(142)

In Appendix B, it is demonstrated that, for the specific case $l=2$, the loss of the z-component of angular momentum is represented by the equation

$$\begin{array}{cc}\hfill {\dot{J}}_z=& {\displaystyle \frac{1}{4\pi }}\Im \sum _{m=-2}^{m=2}m{h}_{2m}{\dot{\overline{h}}}_{2m}.\hfill \end{array}$$

(143)

where $h_{lm}$ denotes the spherical harmonic decomposition coefficients of the shear.

Finally, in the last equation, we obtain the momenta carried away by the terms of the gravitational wave polarizations h from gauge-invariant perturbations derived by Thorne in [54] and Lousto [38].

The angular momentum, as expressed in Equation (143), represents a classical conserved quantity that has been extensively studied in relation to various astrophysical entities, including black holes, accretion disks, the collapse of black holes, supernovae, and gravastars, among others. In the context of quantum gravity applications, the concept of gravitational memory ${\sigma }^0$ emerges as a compelling subject for investigation, particularly concerning gravitational wave bursts with memory, higher-dimensional frameworks, nonlinear characteristics of gravitation, and gravitational wave experiments. Our goal in this paper was to demonstrate that the Poincaré quasigroup algebra in a different way to obtain the classical angular moment and that allows us to obtain the energy momentum, angular momentum, and gravitational memory, which can be used for several physical examples in gravitational waves and other topics. These quantities have been demonstrated using various formalisms. For instance, Milton et al. [16] employed the decomposition of the Weyl scalar ${\psi }_4$ into spin-weighted spherical harmonics. Similarly, Lousto et al. [52] derived the outgoing radiation, represented by the Weyl scalar ${\psi }_4$ within the Kinnersley tetrad, as discussed by Thorne, K. [53] in terms of gauge-invariant perturbations. In Appendix A, we present a derivation of the spin-weighted spherical harmonics, and Appendix B addresses the angular momentum analysis within the context of the Poincaré quasigroup, drawing upon the different wave treatment of Milton, Lousto, and Thorne. This observation presents an excellent opportunity for further exploration in future studies utilizing our formalism, but nevertheless, the main focus of this research was to attain angular momentum in a nonlinear representation.

6. Discussion and Conclusions

In this paper, we address the task of identifying conserved quantities, such as energy momentum, angular momentum, and linear moment radiation with an isolated system, within asymptotically flat spacetimes as governed by general relativity. Due to the complex nature of gravitational radiation, gauge choices, or field-dependent transformations being too restrictive with traditional group structures, quasigroup use offers a better, more transparent way to describe the algebra of asymptotic symmetries. Our investigation demonstrates that the application of quasigroup symmetries facilitates a systematic and unified representation of the conserved quantities.

By employing the quasigroup methodology, we developed a framework that effectively resolves the conventional challenges presented by supertranslation ambiguity at null infinity. Our method facilitates the establishment of a geometric and algebraic structure invariant under supertranslations; in particular, we derive the intrinsic angular momentum and delineate it with the center-of-mass frame reference. In conjunction with methodologies for modulating supertranslation freedom, this highlights the pivotal importance of selecting suitable sections of null infinity for rigorous physical analysis.

Utilizing methodologies such as the spin coefficient formalism and Newman–Penrose scalar quantities, in conjunction with an in-depth analysis of the Bondi–Metzner–Sachs group, we derived explicit formulae that describe the loss of angular momentum and energy conveyed by gravitational radiation with the Weyl tensor ${\mathrm{\Psi }}_4$ in terms of the spin-weighted spherical harmonics. It is important to remark that our results for conservative quantities are well known and in concordance with established results using other methodologies.

Furthermore, by employing Poincaré quasigroups, we identified conserved quantities in terms of polarizations $h_{+}$ and $h_{\times }$, which are directly related to the physics of gravitational waves in astronomy, specifically for estimating parameters of black holes, neutron stars, and supernovae. In future work, these quantities could be obtained using data gathered from the detectors of the LIGO/Virgo/KAGRA international collaboration. Using tools such as numerical simulations in relativity allows for standardizing waveform extraction methods; in the field of black hole physics, for verifying the Kerr no-hair theorem and quantifying spin parameters; and in the domain of quantum gravity, for interconnections with BMS symmetries, soft graviton theorems, and the black hole information paradox.

This investigation enhances our understanding of the asymptotic structure of spacetime and offers a dependable methodology for distinctly identifying conserved quantities within gravitational systems. Subsequent research endeavors might extend these concepts to encompass more generalized frameworks, such as spacetimes affected by gravitational forces and intricate matter configurations, thereby augmenting both theoretical insights and their astrophysical significance.