Remarks on infinite towers of gravitational memories

An infinite tower of gravitational memories was proposed in \cite{Compere:2019odm} by considering the matter-induced vacuum transition in the impulsive limit. We give an alternative realization of the infinite towers of gravitational memories in Newman-Penrose formalism. We also demonstrate that the memories at each order can be associated to the same supertranslation instead of infinite towers of supertranslations or superrotations.


Introduction
Gravitational memory [2][3][4][5] obtained rewed attentions in recent years since its intrinsic connections to BMS supertranslation and soft graviton theorem were revealed [6]. The gravitational memory formula is a Fourier transformation of soft graviton theorem. While the memory effect is a consequence of the fact that the gravitational radiation induces transitions among two different vacua that are connected by BMS supertranslation. The triangle relation [7] has accumulated considerable evidence in its favor in particular in discovering new features of gravitational memory. In [1], it was reported that infinite towers of gravitational memory effects can be generated by matter-induced transitions (see also [8] from the soft theorem side).
In this paper, we show that the infinite towers of gravitational memories can be derived in the Newman-Penrose (NP) formalism [9]. The gravitational wave is characterized by the asymptotic shear σ 0 and σ 0 in the NP formalism. The infinite towers of gravitational memories are derived from the evolutions of the Weyl tensors. The memory effect at each order is encoded in different choice of σ 0 and σ 0 . The infinite towers of gravitational memories can be associated to the unique BMS supertranslation. There are infinite gravitationally-conserved quantities at each order [10] which can be associated to the unique supertranslation charge [11]. The gravitational memories are equivalent to soft graviton theorems at the first two orders [6,12]. While we show that the derivation of equivalence for the first two orders is not valid at the third order. Thus the connection beyond the second order is still obscure. We will demonstrate in linearized gravitational theory. However the nonlinearities can be captured from the contribution of the gravitational wave burst's gravitons [13].
2 Infinite towers of gravitational memories in Newman-Penrose formalism The infinite towers of memory observable in [1] is a displacement memory effect that derived from the geodesic deviation equation at each order in the 1 r expansion. In the NP formalism, σ and σ are the complex shear of the null geodesic generator V " B Br and measure its geodesic deviation. In the linearized theory, they are controlled by the radial NP equation 1 B Hence where Ψ 0 is given as initial data Considering the empty-space case, the evolutions of the Weyl tensors at each order are [10] pn`1qB u Ψ n`1 where Ψ 0 i pi " 1, 2q are at order Opr i´5 q. We use pu, r, z,zq coordinates where z " e iφ cot θ 2 ,z " e´i φ cot θ 2 are the standard stereographic coordinates. The operators ð and ð are defiend in Appendix A.
In the NP formalism, the gravitational wave is characterized by σ 0 and σ 0 . The Weyl 1 Here and in the following, we only list half of the equations while the complex conjugate of those equations should be noticed. tensors in (4)-(7) are completely determined by σ 0 and its time integrations The memories at different orders are completely determined by the same σ 0 . The independence of each memory is encoded in different choice of σ 0 . For instance, σ 0 " Θpuqf pz,zq will induce a permanent change in σ 0 which will lead to a leading memory. 2 Then σ 0 " δpuqf pz,zq will lead to a subleading memory where a permanent change happens in ş du 1 σ 0 instead of σ 0 . Similarly, the nth order memory can be associated to σ 0 " dΘpuq d n´1 u f pz,zq. The memories derived from (10) and (11) are nothing but the higher order terms in (2). While the memory derived from (9) is somewhat different since there is a gap in the expansion in (2) and Ψ 0 1 is not in the expansion. Nevertheless the observational effect of the subleading memory is a time delay [12].

The relation to supertranslation
Memories are associated to asymptotic symmetries. The infinite towers of gravitational memories in [1] are associated to infinite towers of supertranslations and superrotations. However there is only one unique supertranslation [11,14] in the NP formalism in the broadly used Newman-Uniti gauge [15]. Nonetheless the infinite towers of gravitational memories defined in (8)-(11) can be associated to the unique supertranslation. The action of the supertranslation on σ 0 is where T pθ, φq defines the supertranslation. The leading memory is encoded in the transition σ 0`Θ puqδ st σ 0 .
The subleading transition is This transition can be understood as two supertranslations in the following form Then the nth order transition is which can be considered as 2 n´1 supertranslations.

Infinite towers of conservation laws
Memories have their intrinsic connections to conserved quantities. For instance, the leading memory is related to Bondi mass aspect [6] while the subleading memory is related to angular momentum flux [12]. The infinite towers of gravitational memories in [1] are related to the Noether charges of the infinite towers of supertranslations and superrotations.
In the NP formalism, there are infinite gravitationally-conserved quantities derived from the evolution equations of the Weyl tensors at each order [10]. Equation (4) leads to the conservation of mass where the relations in (34) have been used. Equation (7) yields the infinite amount of subleading conservation laws as when applying the relations in (34). Equations (5) and (6) only induce identically vanishing quantities The gravitationally-conserved conserved quantities can be reorganized as a unique charge in expansion (see, for instance, [16][17][18][19] for relevant developments). The supertranslation charge at each order proposed in [11] is given by 3 All the conserved quantities in equations (17)- (21) can be recovered from certain order of the supertranslation charge (22).

Comment on the relation to soft theorems
Memories and soft theorems are mathematically equivalent in the context of the triangle relation [7]. However such relation is not easy to verify beyond the subsubleading order since the soft graviton theorems beyond the subsubleading order do not have an universal factorization property [8,20]. Correspondingly, the higher order memory formulas (11) are in more complicated forms. The first two orders of memories are shown to be equivalent to the leading soft graviton theorems [6] and subleading soft graviton theorem [12].
The key observation of the equivalence is the fact that the soft factors, after Fourier transform and projected on the null infinity, can be considered as classical fields that satisfy classical equations of motion, namely the Einstein equation. Hence the equivalence between the first two orders of memories and soft theorems can be interpreted as [6,12] S p0q zz " γ zz σ 0 , S p1q zz " Including local stress-energy tensor, the classical equations of motion 4 that the leading factor and the subleading soft factor satisfy are which is the real (electric) part of 3 A factor 1 2 is missing on the righthand side of both eq.(2.26) and eq.(2.27) in [11]. To compare the results in [11] to those in [10], one should notice that, in [11], the signature is p´,`,`,`q and a unit sphere boundary is chosen. Moreover one needs to do the rescaling Ψ pmq 0 Ñ 2 pm`3qpm`2q Ψ pmq 0 pm ě 1q. 4 Here we follow the results in [11] in the signature p´,`,`,`q in order to compare to the computation in [6,12].
which is the imaginary (magnetic) part of 5 Regarding to the third memory, it is supposed to be equivalent to the subsubleading soft factor discovered in [20]. After Fourier transform and projected on the null infinity, the subsubleading soft factor should be interpreted as classical field and the classical equation of motion for the subsubleading soft factor should be related to Equation (29) can be re-organized as to connect to the expression ş du 1 du 2 γ zz σ 0 . However, considering stress-energy tensor from massless particles or localized wave packets which puncture the null infinity at points pu k , z k ,z k q as in [12], direct computation shows that equation (30) does not fulfill when ş du 1 du 2 γ zz σ 0 is replaced by the subsubleading soft factor. The connection at the third order seems to be still of a mystery.

Conclusion
We demonstrated that there are infinite towers of gravitational memories which can be associated to the same supertranslation in the NP formalism. An infinite number of gravitationally-conserved quantities can be obtained from each order of the supertranslation charge. We also commented on the relation between the infinite towers of gravitational memories and the infinite towers of soft graviton theorems. The equivalence of those two subjects is not yet clear beyond the second order. 5 The relation Ψ 0 2´Ψ 0 2 " ð 2 σ 0´ð 2 σ 0 needs to be utilized.
As a final remark, it is worthwhile to point out that the analysis performed in gravitational theory can be easily extended to electromagnetism by applying the results in [10] and [21].
where η and ζ are with spin weight´l´1 and l`1 respectively, and the expression AðB has spin weight zero.