Einstein-Maxwell-dilaton theory in Newman-Penrose formalism
aa r X i v : . [ g r- q c ] A ug Einstein-Maxwell-dilaton theory in Newman-Penroseformalism
Wen-Di Tan
Center for Joint Quantum Studies and Department of Physics,School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China A BSTRACT . In this paper, we study the four dimensional Einstein-Maxwell-dilaton (EMD) theories in the Newman-Penrose (NP) formalism. We adapt theequations of motion into the NP formalism, and obtain the solution space thatis asymptotic to the flat space-time. We then investigate the gravitational andelectromagnetic memory effects. We find that the dilaton does not contributeto the displacement nor the kick memory effects, but it does contribute to thetime-delayed memory effect.
In 1960s, to understand the gravitational radiation in full Einstein theory, Bondi and hiscollaborators established an elegant framework for axisymmetric isolated systems anddemonstrated that gravitational waves exist in the full Einstein theory rather than an ar-tifact of linearization [1]. In this framework, they chose a suitable coordinates systemand expanded the metric fields in inverse powers of the radial coordinate r . Imposingthe proper boundary conditions, the equations of motion can be solved order by orderin /r expansions. In this framework, the gravitational radiation is characterized by thenews functions and the mass of the system decreases whenever the news function exists.Shortly, this framework was extended to asymptotically flat space-times by Sachs [2].Meanwhile, Newman and Penrose [3] developed a new approach to understand gravita-tional radiation by means of a tetrad or spinor formalism. They derived a compact setof first order differential equations involving linear combinations of the equations for theRiemann tensor, expressed in the Ricci rotation coefficients or the spinor affine connec-tion. These equations are equivalent to the empty space Einstein equations. From theseequations, one can investigate the asymptotic behavior of the fields systematically, underthe condition that the space-time should approach to flatness at infinity. The asymptoticflatness condition is imposed on the Riemann tensor rather than the metric. W. T AN This formalism is motivated by the strong belief that the essential element of a space-time is its light-cone structure and it is the most effective way for grasping the inherentsymmetries of the space-times such as the black-hole solutions of general relativities. Inthis formalism, the geometrical property of the space-times is more transparent and it isthe most satisfactory way to study the fermion-coupled theories. The asymptotically flatsolutions of the empty Newman-Penrose equations were later derived by Newman andUnti [4], and the News functions and the mass-loss formula were successfully recovered.When matter fields are coupled, one expects that the matter equations of motion are alsoadaptable into the NP formalism. This was indeed done for Einstein-Maxwell gravity[5, 6]. However other theories are less studied in NP formalism.In this paper, we study the four dimensional Einstein-Maxwell-dilaton (EMD) the-ories in Newman-Penrose formalism. Including the Kaluza-Klein theory which arisesfrom five dimensional Einstein gravity reduced on a circle, the four-dimensional EMDtheories are a class of theories that can be embedded into various supergravities whichoriginate from string theories or M-theory. In these theories, the matter sector includesthe Maxwell field A and a dilatonic scalar ϕ , both of which are massless and minimallycoupled to gravity. The dilaton is non-minimally coupled to the Maxwell kinetic term inthe form of an exponential function e aϕ where a is the dilaton coupling constant. In [7],the authors used Newman-Penrose formalism to analyze the perturbations of the Kerr-Newman dilatonic black hole background. However, EMD theories have not been fullystudied in NP formalism elsewhere. In this paper, we adapt the equations of motion ofEMD theories into NP formalism and obtain the solution space that approaches the flat-ness asymptotically. We then examine the mass-loss formula and the charge conservation.As a direct application, we also study the memory effects.First reported by Zel’dovich and Polnarev [8] in linearized gravity and further studiedby Christodoulou in full Einstein gravity [9], gravitational memory effects are a largegroup of observational effects for gravitational radiation which is characterized by thechange of the asymptotic shear ∆ σ [10] (see also [11–17] for relevant developments).The memory effects also exist in Maxwell theory named electromagnetic memory effects[18, 19]. We should notice that the memory effects can be classified by the observationaleffects.In the recent years, there have been renewed interests in memory effects. Stromingerand Zhiboedov [20] discovered an intriguing triangular relation of three ingredients: theBMS supertranslation symmetry, the leading soft graviton theorem and a displacementgravitational memory effect. This memory effect is a displacement of two parallel iner-tial detectors caused by the radiative energy flux and it turns out to be mathematicallyequivalent to Weinberg’s soft graviton theorem [21] by the Fourier or the inverse Fouriertransformations.Pasterski, Strominger and Zhiboedov [22] later discovered a spin memory effect which INSTEIN -M AXWELL - DILATON THEORY IN N EWMAN -P ENROSE FORMALISM r = r hypersurface near the null infinity. In [25], theauthors considered the motion of a charged observer and investigated both the gravita-tional and electromagnetic memory effects in a unified manner in the Einstein-Maxwelltheory. It is thus of great interest to study the memory effect in EMD theories to uncoverthe possible observational effects due to the non-minimal coupling between the scalar andthe electromagnetic field. The fully understood memory effects in EMD theories may alsohelp to understand the memory effects in string theories [26] and M-theory.The paper is organized as follows. In section 2, we will give a brief introduction ofNP formalism and derive the NP equations of the four-dimensional Einstein-Maxwell-dilaton theories. The asymptotically-flat solution space of these theories will be derivedin section 3. We also examine the charge conservation and the mass-loss formula. Insection 4, we will obtain the memory effects based on the investigation of [25] in EMDtheories. We conclude the paper in section 5. The Newman-Penrose formalism is a special tetrad formalism with two real null basisvectors e = l , e = n , and two complex null basis vectors e = m , e = ¯ m . These basisvectors have the orthogonality relations l · m = l · ¯ m = n · m = n · ¯ m = 0 , (2.1)and are normalized as l · n = 1 , m · ¯ m = − . (2.2)The metric is obtained from the basis vectors as g µν = η ab ( e aµ )( e bν ) = n µ l ν + l µ n ν − m µ ¯ m ν − m ν ¯ m µ . (2.3)where e aµ represents the basis vector l, n, m, ¯ m , µ is the coordinate index, while a isthe tetrad index, η ab is the metric component under the tetrad form. The connectioncoefficients, called spin coefficients in the NP formalism with special Greek symbols (wewill follow the convention of [27]), are presented as follows κ = Γ = l ν m µ ∇ ν l µ , π = − Γ − l ν ¯ m µ ∇ ν n µ ,ǫ = 12 (Γ − Γ ) = 12 ( l ν n µ ∇ ν l µ − l ν ¯ m µ ∇ ν m µ ) . (2.4) W. T AN τ = Γ = n ν m µ ∇ ν l µ , ν = − Γ = − n ν ¯ m µ ∇ ν n µ ,γ = 12 (Γ − Γ ) = 12 ( n ν n µ ∇ ν l µ − n ν ¯ m µ ∇ ν m µ ) . (2.5) σ = Γ = m ν m µ ∇ ν l µ , µ = − Γ = − m ν ¯ m µ ∇ ν n µ ,β = 12 (Γ − Γ ) = 12 ( m ν n µ ∇ ν l µ − m ν ¯ m µ ∇ ν m µ ) . (2.6) ρ = Γ = ¯ m ν m µ ∇ ν l µ , λ = − Γ = − ¯ m ν ¯ m µ ∇ ν n µ ,α = 12 (Γ − Γ ) = 12 ( ¯ m ν n µ ∇ ν l µ − ¯ m ν ¯ m µ ∇ ν m µ ) . (2.7)We use five complex scalars to represent ten independent components of the Weyl tensors Ψ = − C , Ψ = − C , Ψ = − C , Ψ = − C , Ψ = − C . (2.8)Ricci tensors are defined by four real and three complex scalars as follows Φ = − R , Φ = − R , Φ = − R , Φ = − R , Φ = −
14 ( R + R ) , Φ = − R , Φ = − R , Λ = 124 R = 112 ( R − R ) , Φ = − R , Φ = − R , (2.9)where Λ is the cosmological constant. Considered as directional derivatives, the basisvectors are represented by special symbols: D = l µ ∂ µ , ∆ = n µ ∂ µ , δ = m µ ∂ µ . (2.10)The equations that describing NP formalism include three classes:(1)The commutation relations of the basis vectors and the structure constants [ e a , e b ] = (Γ cba − Γ cab ) e c = C cab e c , (2.11)where e a is the basis vector, and C cab is the structure constant. The general tensorialformalism does not consider these relations since the coordinate basis is commutative.An example is as follows [∆ , D ] = [ n, l ] = [ e , e ] = (Γ c − Γ c ) e c = − Γ ∆ + Γ D − (Γ − Γ ) δ − (Γ − Γ ) δ. (2.12)Giving the spin coefficients their symbols, we get ∆ D − D ∆ = ( γ + γ ) D + ( ǫ + ǫ )∆ − ( τ + π ) δ − ( τ + π ) δ. (2.13) INSTEIN -M AXWELL - DILATON THEORY IN N EWMAN -P ENROSE FORMALISM − Ψ = C = R = Γ , − Γ , + Γ (Γ + Γ − Γ + Γ + Γ ) − Γ (Γ + Γ − Γ + Γ + Γ ) . (2.14)Substituting for the directional derivatives and the spin coefficients their designated sym-bols, we obtain Dσ − δκ = σ (3 ǫ − ǫ + ρ + ρ ) + κ ( π − τ − β − α ) + Ψ . (2.15)(3)The Bianchi identities. It is similar to the Bianchi identities in the tensorial form,i.e. R | + R | + R | = 0 , (2.16)where “ | ” represents the covariant derivative in tetrad form. It can be rewritten in thefollowing form − δ Ψ + D Ψ + (4 α − π )Ψ − ρ + + ǫ )Ψ + 3 κ Ψ + [ Ricci ] = 0 . (2.17)Here [ Ricci ] = − D Φ + δ Φ + 2( ǫ + ρ )Φ + 2 σ Φ − κ Φ − κ Φ + ( π − α − β )Φ . (2.18)As for Maxwell theory, in NP formalism the antisymmetric Maxwell-tensor F µν isreplaced by the three complex scalars φ = F = F µν l µ m ν ,φ = 12 ( F + F ) = 12 F µν ( l µ n ν + m µ m ν ) ,φ = F = F µν m µ n ν . (2.19)Correspondingly, the Maxwell equations in tetrad form F [ ab | c ] = 0 , η nm F an | m = 0 , (2.20)can be replaced by those equations φ | − φ | = 0 , φ | − φ | = 0 ,φ | − φ | = 0 , φ | − φ | = 0 . (2.21)Expanding these equations in the terms of the ordinary derivatives and spinor coefficients,then expressing them in the symbols above, we can get the Maxwell equations in NP W. T AN formalism. Similar disposition can be used to deal with the Klein-Gordon equation forscalar field, where we define Ω = Dϕ , Ω = ∆ ϕ , Ω = δϕ , ¯Ω = δϕ .The freedom of the rotation of the basis vectors, see e.g. in [3], will allow us to set π = κ = ǫ = 0 , ρ = ¯ ρ, τ = ¯ α + β. (2.22)From those conditions, one can find that l is tangent to a null geodesic with an affineparameter. Also, the congruence of the null geodesic is hypersurface orthogonal, that is, l is proportional to the gradient of a scalar field. So it is convenient to choose the scalarfield as coordinate u = x , and the affine parameter as r = x . Thus, the basis vectorsand the co-tetrad must have the form n µ ∂ µ = ∂∂u + U ∂∂r + X A ∂∂x A , l µ ∂ µ = ∂∂r , m µ ∂ µ = ω ∂∂r + L A ∂∂x A ,n µ dx µ = (cid:2) − U − X A ( ωL A + ω ¯ L A ) (cid:3) du + dr + ( ω ¯ L A + ωL A ) dx A ,l µ dx µ = du, m µ dx µ = − X A L A du + L A dx A . (2.23)where L A L A = 0 , L A ¯ L A = − . We will use the standard stereographic coordinates z = e iφ cot θ and ¯ z = e − iφ cot θ in this work.The Lagrangian of four-dimensional Einstein-Maxwell-dilaton theories are L = √− g (cid:20) R − e aϕ F + 12 ( ∂ϕ ) (cid:21) , F = dA. (2.24)This class of theories is generalized from the Einstein-Maxwell theory to include a realdilatonic scalar. When the dilaton coupling constant a takes the following specific values a = 0 , √ , , √ , the EMD theories can be embedded into N = 2 D = 4 STU super-gravity, with global SL (2 , R ) × SL (2 , R ) × SL (2 , R ) triality symmetries that correspondto the string/weak, T and U dualities of string theory []. Einstein-Maxwell theory, whichis the bosonic sector of N = 2 supergravity, can be reduced from the a = 0 case, whilethe a = √ case can be Kaluza-Klein theory. Now we suppose that a is an arbitrary realconstant.The dilaton, Maxwell and Einstein equations can be derived from the Lagrangian(2.24) ∂ µ ( √− gg µν ∂ ν ϕ ) + a √− ge aϕ F = 0 . (2.25) ∂ ν ( √− ge aϕ F µν ) = 0 . (2.26) R µν = 12 e aϕ F µρ F ρν − g µν e aϕ F − ∂ µ ϕ∂ ν ϕ. (2.27)According to these equations of motion in the tensorial form, we can easily recast theminto the NP formalism. We divide these equations into three groups [4]: Note that we use the signature (+ , − , − , − ) . Hence the third term in the Lagrangian is ( ∂ϕ ) ratherthan most used convention − ( ∂ϕ ) , see e.g. in [28] INSTEIN -M AXWELL - DILATON THEORY IN N EWMAN -P ENROSE FORMALISM σ ( u, z, ¯ z ) and P ( u, z, ¯ z ) .III. The u-derivative equationsThis group of equations characterizes the propagation of the components of Weyltensor, the dilatonic scalar field and the Maxwell fields off the hypersurface in the u-direction (time direction), from null surface to null surface. Radial equations Dρ = ρ + σσ + 12 e aϕ φ φ + 14 (Ω ) , (2.28) Dσ = 2 ρσ + Ψ , (2.29) Dτ = τ ρ + τ σ + Ψ + 12 e aϕ φ φ + 14 Ω Ω , (2.30) Dα = ρα + βσ + 12 e aϕ φ φ + 14 Ω ¯Ω , (2.31) Dβ = ασ + ρβ + Ψ , (2.32) Dγ = τ α + τ β + Ψ + 12 e aϕ φ φ + 16 Ω Ω + 112 Ω ¯Ω , (2.33) Dλ = ρλ + σµ + 12 e aϕ φ φ + 14 ( ¯Ω) , (2.34) Dµ = ρµ + σλ + Ψ + 112 (Ω ¯Ω − Ω Ω ) , (2.35) Dν = τ µ + τ λ + Ψ + 12 e aϕ φ φ + 14 Ω ¯Ω , (2.36) DU = τ ω + τ ω − ( γ + γ ) , (2.37) DX A = τ L A + τ ¯ L A , (2.38) Dω = ρω + σω − τ, (2.39) DL A = ρL A + σ ¯ L A , (2.40) D Ψ − δ Ψ = 4 ρ Ψ − α Ψ + 12 e aϕ [( aφ − a e aϕ φ − a e aϕ φ ) φ Ω − ( aφ − a e aϕ φ − a e aϕ φ ) φ Ω + φ Dφ − φ δφ − σφ φ +2 βφ φ ] −
12 Ω δ Ω + 14 Ω D Ω + 14 Ω D Ω − ρ Ω Ω − σ Ω ¯Ω+ 12 ( α + β )(Ω ) , (2.41) D Ψ − δ Ψ = 3 ρ Ψ − α Ψ − λ Ψ + 12 e aϕ [( aφ − ae aϕ φ + 12 ae aϕ φ ) φ ¯Ω − ( aφ − ae aϕ φ ) φ Ω W. T AN − ae aϕ φ φ Ω + φ δφ − φ ∆ φ − αφ φ + 2 ρφ φ + 2 γφ φ − τ φ φ ] + 14 Ω δ Ω + 14 Ω δ Ω −
12 Ω ∆Ω −
12 ( α + τ )Ω Ω+ 14 ρ (Ω Ω + Ω ¯Ω) + 14 σ (Ω) −
14 ( µ − γ − γ )(Ω ) − τ Ω ¯Ω+ 112 (Ω D Ω + Ω D Ω − ¯Ω D Ω − Ω D ¯Ω) , (2.42) D Ψ − δ Ψ = 2 ρ Ψ − λ Ψ + 12 e aϕ [( aφ − ae aϕ φ − ae aϕ φ ) φ Ω − ( aφ − ae aϕ φ − ae aϕ φ ) φ Ω + φ Dφ − φ δφ + 2 µφ φ − βφ φ ] + 14 ( ¯Ω D Ω + Ω D ¯Ω) −
12 ¯Ω δ ¯Ω − ρ Ω ¯Ω + 12 µ Ω ¯Ω+ 12 ( α − β )( ¯Ω) −
112 (Ω δ Ω + Ω δ Ω − ¯Ω δ Ω − Ω δ ¯Ω) , (2.43) D Ψ − δ Ψ = ρ Ψ + 2 α Ψ − λ Ψ + 12 e aϕ [( 12 ae aϕ φ − aφ ) φ Ω − ae aϕ φ φ Ω − ( 12 ae aϕ φ − ae aϕ φ − aφ ) φ ¯Ω − φ ∆ φ + φ δφ + 2 αφ φ + 2 νφ φ − γφ φ − λφ φ ]+ 12 ¯Ω∆ ¯Ω −
14 ¯Ω δ Ω −
14 Ω δ ¯Ω −
12 ( α − τ )Ω ¯Ω − ν Ω ¯Ω − σ (Ω ) + 14 λ (Ω Ω + Ω ¯Ω) + 14 ( µ + 2 γ − γ )( ¯Ω) , (2.44) Dφ − δφ = 2 ρφ − αφ − ae aϕ ( φ + φ )Ω + 12 ae aϕ φ ¯Ω+ 12 ae aϕ φ Ω , (2.45) Dφ − δφ = ρφ − λφ + 12 ae aϕ φ Ω − ae aϕ φ Ω + 12 ae aϕ ( φ − φ ) ¯Ω . (2.46) Non-radial equations ∆ λ = δν − ( µ + µ ) λ − (3 γ − γ ) λ + 2 αν − Ψ , (2.47) ∆ ρ = δτ − ρµ − σλ − ατ + ( γ + γ ) ρ − Ψ + 112 Ω Ω −
112 Ω ¯Ω , (2.48) ∆ α = δγ + ρν − ( τ + β ) λ + ( γ − γ − µ ) α − Ψ , (2.49) ∆ µ = δν − µ − λλ − ( γ + γ ) µ + 2 βν − e aϕ φ φ −
14 (Ω ) , (2.50) ∆ β = δγ − µτ + σν + β ( γ − γ − µ ) − αλ − e aϕ φ φ −
14 Ω Ω , (2.51) ∆ σ = δτ − σµ − ρλ − βτ + (3 γ − γ ) σ − e aϕ φ φ −
14 (Ω) , (2.52) ∆ ω = δU + ν − λω + ( γ − γ − µ ) ω, (2.53) ∆ L A = δX A − λ ¯ L A + ( γ − γ − µ ) L A , (2.54) δρ − δσ = ρτ − σ (3 α − β ) − Ψ + 12 e aϕ φ φ + 14 Ω Ω , (2.55) INSTEIN -M AXWELL - DILATON THEORY IN N EWMAN -P ENROSE FORMALISM δα − δβ = µρ − λσ + αα + ββ − αβ − Ψ + 12 e aϕ φ φ + 112 Ω Ω + 16 Ω ¯Ω , (2.56) δλ − δµ = µτ + λ ( α − β ) − Ψ + 12 e aϕ φ φ + 14 Ω ¯Ω , (2.57) δω − ¯ δω = µ − µ − ( α − β ) ω + ( α − β ) ω, (2.58) δ ¯ L A − ¯ δL A = ( α − β ) ¯ L A − ( α − β ) L A . (2.59) The u-derivative equations ∆Ψ − δ Ψ = (4 γ − µ )Ψ − (4 τ + 2 β )Ψ + 3 σ Ψ − e aϕ [( aφ − ae aϕ φ ) φ Ω + 12 ae aϕ φ φ Ω +( 12 ae aϕ φ − ae aϕ φ − aφ ) φ Ω + φ Dφ − φ δφ + 2 βφ φ − σφ φ ] −
12 Ω D Ω + 14 Ω δ Ω + 14 Ω δ Ω − β Ω Ω − λ (Ω ) + 14 σ (Ω Ω + Ω ¯Ω) + 14 ρ (Ω) . (2.60) ∆Ψ − δ Ψ = ν Ψ + (2 γ − µ )Ψ − τ Ψ + 2 σ Ψ + 12 e aϕ [( aφ − ae aϕ φ − ae aϕ φ ) φ Ω + ( 12 ae aϕ φ + 12 ae aϕ φ ) φ Ω − aφ φ ¯Ω + φ ∆ φ − φ δφ − γφ φ − ρφ φ + 2 αφ φ + 2 τ φ φ ]+ 14 Ω∆Ω + 14 Ω ∆Ω −
12 Ω δ Ω + 12 ( µ − γ )Ω Ω − ρ Ω Ω − ν (Ω ) + 14 τ (Ω Ω + Ω ¯Ω) + 14 ( τ − β + 2 α )(Ω) −
112 (Ω δ Ω + Ω δ Ω − Ω δ ¯Ω − ¯Ω δ Ω) . (2.61) ∆Ψ − δ Ψ = 2 ν Ψ − µ Ψ + (2 β − τ )Ψ + σ Ψ − e aϕ [( aφ − ae aϕ φ ) φ Ω + 12 ae aϕ φ φ Ω +( 12 ae aϕ φ − ae aϕ φ − aφ ) φ Ω + φ Dφ − φ δφ − βφ φ +2 µφ φ ] −
12 Ω D Ω + 14 ¯Ω δ Ω + 14 Ω δ ¯Ω + 12 β Ω ¯Ω − µ (Ω Ω + Ω ¯Ω) − λ ( ¯Ω) + 14 ρ (Ω ) + 112 (Ω ∆Ω + Ω ∆Ω − ¯Ω∆Ω − Ω∆ ¯Ω) . (2.62) ∆Ψ − δ Ψ = 3 ν Ψ − (2 γ + 4 µ )Ψ + (4 β − τ )Ψ + 12 e aϕ [( aφ − ae aϕ φ − ae aϕ φ ) φ Ω − ( aφ − ae aϕ φ ) φ ¯Ω+ 12 aφ φ e aϕ Ω + φ ∆ φ − φ δφ + 2 γφ φ − νφ φ + 2 λφ φ − αφ φ ] + 14 ¯Ω∆Ω + 14 Ω ∆ ¯Ω −
12 Ω δ Ω + 12 ( µ + γ )Ω ¯Ω W. T AN − ν (Ω Ω + Ω ¯Ω) − ν ( ¯Ω) + 12 λ Ω Ω −
14 ( α + β )(Ω ) . (2.63) ∆ φ − δφ = (2 γ − µ ) φ − τ φ + σφ − ae aϕ φ Ω + 12 ae aϕ φ Ω + 12 ae aϕ φ Ω − ae aϕ φ Ω . (2.64) ∆ φ − δφ = νφ − µφ − ( α − β ) φ − ae aϕ φ Ω − ae aϕ φ Ω + 12 ae aϕ φ ¯Ω + 12 ae aϕ φ Ω . (2.65) ∆Ω + D Ω = δ Ω + δ ¯Ω + ( γ + γ − µ − µ )Ω + 2 ρ Ω − α ¯Ω − α Ω + ae aϕ ( φ + φ − φ φ − φ φ ) . (2.66) The main condition of approaching flatness at infinity is Ψ = Ψ r + O ( r − ) . Newmanand Unti [4] listed the fall-off conditions of the rest quantities by solving the empty spaceNewman-Penrose equations ρ = − r − + O ( r − ) , σ = O ( r − ) , α = O ( r − ) , β = O ( r − ) ,τ = O ( r − ) , λ = O ( r − ) , µ = O ( r − ) , γ = O (1) ,ν = O (1) , U = O ( r ) , X z = O ( r − ) , ω = O ( r − ) ,L z = O ( r − ) , L ¯ z = O ( r − ) , Ψ = O ( r − ) , Ψ = O ( r − ) , Ψ = O ( r − ) , Ψ = O ( r − ) . (3.1)The fall-off of the matter fields can not violate the asymptotic conditions above, so wechoose φ = φ r + O ( r − ) , (3.2) ϕ = ϕ r + ϕ r + O ( r − ) , (3.3)and φ = φ r + O ( r − ) , φ = φ r + O ( r − ) . (3.4)Using the conditions above, we work out the asymptotically flat solution space of Newman-Penrose equations in Einstein-Maxwell-dilaton theories. The solutions of the radial equa-tions are as follows. One should notice that here the boundary topology is an arbitrary 2surface but not S . Here the “ ð ” operator is defined as ð η s = P ¯ P − s ∂ ¯ z ( ¯ P s η s ) = P ∂ ¯ z η s + 2 sα η s , ð η s = ¯ P P s ∂ z ( P − s η s ) = ¯ P ∂ z η s − sα η s . (3.5)where s is the spin weight of the field η . The spin weights of relevant fields are listed inTable 1. INSTEIN -M AXWELL - DILATON THEORY IN N EWMAN -P ENROSE FORMALISM ð ∂ u γ ν µ σ λ Ψ Ψ Ψ Ψ Ψ φ φ φ s − − − − − = Ψ ( u, z, ¯ z ) r + Ψ ( u, z, ¯ z ) r + O ( r − ) ,φ = φ ( u, z, ¯ z ) r + φ ( u, z, ¯ z ) r + O ( r − ) ,ϕ = ϕ r + ϕ r + O ( r − ) ,ρ = − r + − ϕ − σ σ r − ϕ ϕ r + 148 r ( − φ φ − ϕ − ϕ − ϕ ϕ + 8 σ Ψ + 8 σ Ψ − σ σ ϕ − σσ ) ) + O ( r − ) ,σ = σ ( u, z, z ) r + − + σ ϕ + 4 σ σ σ r + 13 r ( − Ψ + σ ϕ ϕ ) + O ( r − ) ,L z = − σ ¯ P ( u, z, ¯ z ) r + 124 r (4 ¯ P Ψ − P σ ϕ −
24 ¯
P σ ( σ ) ) + 112 r ( ¯ P Ψ − P ϕ ϕ σ ) + O ( r − ) ,L ¯ z = P ( u, z, ¯ z ) r + 18 r ( P ϕ + 8 P σ σ ) + 16 r P ϕ ϕ + 1384 r (16 P φ φ + 5 P ϕ + 32 P ϕ + 48 P ϕ ϕ − P σ Ψ − P σ Ψ + 112 P σ σ ϕ + 384 P ( σ ) ( σ ) ) + O ( r − ) ,L z = − r ¯ P + ϕ P r + ϕ ϕ P r + 1384 ¯ P r (16 φ φ − ϕ + 32 ϕ + 48 ϕ ϕ + 32 σ Ψ )+ O ( r − ) ,L ¯ z = − σ P + 124 P r (4Ψ + σ ϕ ) + 112 P r (Ψ + σ ϕ ϕ ) + O ( r − ) ,α = α r + σ α r + 18 r ( α ϕ + 8 α σ σ + ϕ ð ϕ )+ 124 r (24 α σ ( σ ) − α Ψ + 4 σ Ψ − φ φ − ω ϕ + 4 α ϕ ϕ + 5 α σ ( ϕ ) − σ ϕ ð ϕ + 4 ϕ ð ϕ + 2 ϕ ð ϕ ) + O ( r − ) ,β = − α r − α σ r + 18 r ( − α ϕ − − σ σ α )+ 124 r (8 ð Ψ + 4 α Ψ − α ( σ ) σ − φ φ + 2 ω ϕ − α ϕ ϕ − α σ ϕ + 4 ϕ ð ϕ − ϕ ð ϕ + σ ϕ ð ϕ ) + O ( r − ) , W. T AN τ = 18 r ( − + ϕ ð ϕ ) + 124 r (8 ð Ψ + 4 σ Ψ − φ φ − σ ϕ ð ϕ + 8 ϕ ð ϕ ) + O ( r − ) ,ω = ω r + 18 r ( − − ω σ + ϕ ð ϕ ) + 124 r (4 ð Ψ + 24 σ σ ω + 8 σ Ψ − φ φ + 3 ω ϕ − σ ϕ ð ϕ + 4 ϕ ð ϕ ) + O ( r − ) , Ψ = Ψ ( u, z, ¯ z ) r + 14 r (6 φ φ − ð Ψ − ω ϕ − ϕ ð ϕ + ϕ ð ϕ − σ ϕ ð ϕ ) + O ( r − ) ,X z = 124 r (4 ¯ P Ψ − ¯ P ϕ ð ϕ ) + 124 r ( − P ð Ψ − P σ Ψ + 4 ¯ P φ φ + ¯ P σ ϕ ð ϕ − P ϕ ð ϕ ) + O ( r − ) ,γ = γ + 112 r ( γ ϕ + γ ϕ − + ϕ ∂ u ϕ ) + 124 r (8 ð Ψ + 4 α Ψ − α Ψ − φ φ − µ ϕ + 2 U ϕ + 4 ϕ ∂ u ϕ − ϕ ∂ u ϕ − ð ϕ ð ϕ − ϕ ðð ϕ − α ϕ ð ϕ + α ϕ ð ϕ ) + O ( r − ) ,λ = λ r − σ µ r + 124 r (24 σ σ λ + 12 σ Ψ − φ φ + 3 λ ϕ + ( γ + γ ) σ ϕ − ð ϕ ) + σ ϕ ∂ u ϕ ) + O ( r − ) ,µ = µ r + 112 r ( − − σ λ + U ϕ − ϕ ∂ u ϕ ) + 18 r (8 σ σ µ + 4 ð Ψ − φ φ − γ + γ ) ϕ ϕ + 2 U ϕ − ð ϕ ð ϕ − ϕ ðð ϕ − ϕ ∂ u ϕ ) + O ( r − ) ,U = − r ( γ + γ ) + U + 16 r ( γ ϕ + γ ϕ − − + ϕ ∂ u ϕ )+ 124 r ( − φ φ + 4 ð Ψ + 4 ð Ψ − µ ϕ + 2 ϕ U + 4 ϕ ∂ u ϕ − ϕ ∂ u ϕ − ð ϕ ð ϕ − ϕ ðð ϕ − ϕ ðð ϕ ) + O ( r − ) , Ψ = Ψ ( u, z, ¯ z ) r + 112 r (12 φ φ + 3 µ ϕ + 8( γ + γ ) ϕ ϕ − ð Ψ − ϕ U − ϕ ∂ u ϕ + 5 ϕ ∂ u ϕ + 2 ð ϕ ð ϕ + 3 ϕ ðð ϕ ) + O ( r − ) ,ν = ν − Ψ r + 124 r ( − φ φ + 12 ð Ψ + ð γ ϕ + ð γ ϕ − γ ϕ ð ϕ − γ ϕ ð ϕ − ð ϕ ∂ u ϕ + ϕ ð ∂ u ϕ ) + O ( r − ) , Ψ = Ψ r + 112 r (6 φ φ − ð Ψ − ( ð γ + ð γ ) ϕ + ( γ + γ ) ϕ ð ϕ + 2 ð ϕ ∂ u ϕ − ϕ ð ∂ u ϕ ) + O ( r − ) ,φ = φ ( u, z, ¯ z ) r + 12 r ( − aφ ϕ − ð φ ) + 18 r (4 λ φ + 8 ω φ + 4 σ σ φ + 4 σ ð φ + 4 ð φ − a ( γ + γ ) φ ϕ + φ ϕ − a φ ϕ − aφ ϕ + 2 a ð φ ϕ + 4 a ð φ ϕ + 4 aφ ð ϕ − aφ ∂ u ϕ ) + O ( r − ) , INSTEIN -M AXWELL - DILATON THEORY IN N EWMAN -P ENROSE FORMALISM φ = φ ( u, z, ¯ z ) r + 12 r ( − ð φ − aφ ϕ − aφ ϕ ) + O ( r − ) , Ψ = Ψ r − ð Ψ r + O ( r − ) . (3.6)The solutions of non-radial equations are as follows Ψ = ð ν − ∂ u λ − γ λ ,U = µ , γ = − ∂ u ln ¯ P , ω = ð σ ,ν = ð ( γ + γ ) , α = 12 ¯ P ∂ z ln P,λ = ∂ u σ + σ (3 γ − γ ) , Ψ = ð µ − ð λ ,µ = − P ¯ P ∂ z ∂ ¯ z ln P ¯ P , Ψ − Ψ = ð σ − ð σ + σ λ − σ λ . (3.7)And we obtain the solutions of the u-derivative equations which determine the propaga-tion of the fields off the null hypersurface ∂ u Ψ = 32 φ φ − λ ϕ − ( γ + 5 γ )Ψ + ð Ψ −
14 ( γ + γ ) σ ϕ + 3 σ Ψ − σ ϕ ∂ u ϕ − ϕ ð ϕ + 12 ( ð ϕ ) . (3.8) ∂ u Ψ = φ φ − ν ϕ − γ + 2 γ )Ψ + 2 σ Ψ + 112 ( ð γ + ð γ ) ϕ − γ ϕ ð ϕ + 16 γ ϕ ð ϕ + ð Ψ − ϕ ∂ u ð ϕ + 13 ð ϕ ∂ u ϕ + 112 ϕ ð ∂ u ϕ . (3.9) ∂ u Ψ = 12 φ φ − γ + γ )Ψ + σ Ψ + ð Ψ −
112 ( ∂ u γ + ∂ u γ ) ϕ + 112 ( γ + γ ) ϕ ∂ u ϕ + 16 ( ∂ u ϕ ) − ϕ ∂ u ϕ . (3.10) ∂ u Ψ = ð Ψ − γ + γ )Ψ . (3.11) ∂ u φ = − ( γ + 3 γ ) φ + σ φ + ð φ − aφ ϕ . (3.12) ∂ u φ = ð φ − γ + γ ) φ . (3.13) ∂ u ϕ = − γ + γ ) ϕ − ðð ϕ . (3.14)From the solutions above, we find that there is no constraint at the order O ( r ) of φ and ϕ , and at the order O ( r ) of σ . So σ , φ and ϕ are related to the news functions inthe system which indicate gravitational, electromagnetic and scalar radiations. σ has aspecial geometric meaning that it represents the asymptotic shear of l (see [3] and [24]),4 W. T AN the change of which at early time u i and late time u f is equivalent to the time integrationof the asymptotic shear of n , i.e. λ when we set the boundary topology to be S × R ,i.e. P = ¯ P = P s = z ¯ z √ (see [10] and [24]). The memory effects [10, 24], which will bediscussed in the following section, are controlled by the time integration of the asymptoticshear of n , i.e. λ , thus the change of the asymptotic shear at early time and late time ∆ σ is a very important quantity that characterizes gravitational memory effects.According to eq(3.12), we can find that this time evolution equation involves the cou-pling constant a which represents the non-minimal coupling of the electromagnetic fieldand the scalar field. φ is related to the electric dipole [29]. Our result is consistent withthe result in [28]. The phenomenon that the coupling constant a do not appear in the timeevolution functions of four tetrad components of Weyl tensor reflects that the scalar fieldis minimally coupled to gravity.From eq(3.10) and eq(3.13) we can consider the conservation laws and the loss ofmass in EMD theory. Here we work in the unit 2-sphere case. From eq(3.10), we can find ∂ u Ψ = 12 φ φ − σ ∂ u σ − ð ∂ u σ + ð Ψ + 16 ( ∂ u ϕ ) − ϕ ∂ u ϕ . (3.15)Define the mass density M = 12 (Ψ + Ψ ) + 12 ( σ ∂ u σ + σ ∂ u σ ) + 12 ( ð σ + ð σ ) + 112 ϕ ∂ u ϕ . (3.16)We can obtain ∂ u M = 12 φ φ + ∂ u σ ∂ u σ + 14 ( ∂ u ϕ ) . (3.17)Considering the signature convention in NP formalism, we get the mass loss theorem inEMD theories: The mass density at any angle of the system can never increase. It is aconstant if and only if there is no news. Our mass-loss formula generalizes the one in [28]by removing the constraint of axisymmetry.As for Maxwell part, we work in retarded radial gauge A r = 0 . The Maxwell-tensoris constructed as F µν = ( φ + φ )( n µ l ν − l µ n ν ) + ( φ − φ )( m µ ¯ m ν − ¯ m µ m ν ) + φ ( l µ m ν − m µ l ν )+ φ ( l µ ¯ m ν − ¯ m µ l ν ) + φ ( ¯ m µ n ν − n µ ¯ m ν ) + φ ( m µ n ν − n µ m ν ) . (3.18)We represent the Newman-Penrose variables of Maxwell parts in terms of the gauge fields A µ , A u = − ( φ + φ ) , ∂ u A z = − φ ¯ P , A z = − φ ¯ P , ( ∂ z A z − ∂ ¯ z A z ) = φ − φ P ¯ P . (3.19) ∂ u (cid:18) A u P ¯ P (cid:19) = ∂ u ( ∂ z A z + ∂ ¯ z A z ) , (3.20) INSTEIN -M AXWELL - DILATON THEORY IN N EWMAN -P ENROSE FORMALISM A u = A u ( u, z, ¯ z ) r + O ( r − ) , A z = A z ( u, z, ¯ z ) + A z ( u, z, ¯ z ) r + O ( r − ) . (3.21)From eq(3.13), we find ∂ u φ = ð φ = P s ∂ ¯ z φ − ∂ ¯ z P s φ = P s ∂ ¯ z ( φ /P s ) . (3.22)Taking the real part of eq(3.22), we get ∂ u ( φ + φ P s ∂ ¯ z ( φ /P s ) + ∂ z ( φ /P s )) . (3.23)Substituting eq(3.19) into eq(3.23), we find ∂ u A u = ∂ u ( P s ∂ ¯ z A z + P s ∂ z A z ) . (3.24)Defining the flux, Φ = A u − P s ( ∂ ¯ z A z + ∂ z A z ) , (3.25)we find ∂ u Φ = 0 . (3.26)This means that the flux does not change with time. According to Gauss’s Law, wecan conclude that the charge is conserved, which again generalizes the result in [28] toasymptotically flat case. According to the solution space in the previous section, we can derive the memory effects.We will examine the motion of a charged time-like particle to specify the observationaleffects in a unified expression [25]. That is to say, we consider the contributions of thegravitational radiation, electromagnetic radiation and scalar radiation at the same timeby considering the effects of the motion of a charged particle caused by the radiations inEMD theories. Gravitational memory effects, characterized by the non-linear contributionto the overall change in the shear of outgoing null surfaces at the future null infinity [10],have a large group of observational effects of the gravitational radiation. Displacementmemory effects are observational effects about a location displacement of the observers,i.e. [20] describes a distance shift of two parallel inertial detectors near the null infinitycaused by the radiative energy flux. Spin memory effects [22, 24] are memory effectscharacterized by the observational phenomena that the radiation causes the observers torotate. i.e. [22] is about a relative time delay of two beams of light on clockwise and6
W. T AN counterclockwise orbits induced by the radiative angular momentum flux. [24] discoversa kind of spin memory effect characterized by the time delay of a free-falling massiveparticle constrained on a time-like, r = r hypersurface. The memory effects also existin Maxwell theory called electromagnetic memory effects, i.e. [18, 19, 30, 31]. [19] is achange of the velocity (a“kick”) of a charged particle. Here we will consider all thesethree kinds of memory effects. The charged particle will be constrained on a time-like, r = r hypersurface. r = r is a fixed radial distance which is very large, means that theparticle is very far from the gravitational and electromagnetic source. The induced metricof this hypersurface can be derived by inserting the solution space in the previous sectioninto eq(2.3) and eq(2.23), which in series expansions is given by ds = [1 + 1 r (Ψ + Ψ − ϕ ∂ u ϕ ) + 112 r (12 φ φ − ð Ψ − ð Ψ + 2 ð ϕ ð ϕ + 2 ϕ ðð ϕ − ϕ ∂ u ϕ + 2 ϕ ∂ u ϕ ) + O ( r − )] du + 2[ − ð σ P s + 16 P s r (4Ψ − ϕ ð ϕ )+ O ( r − )] dudz + 2[ − ð σ P s + 16 P s r (4Ψ − ϕ ð ϕ ) + O ( r − )] dud ¯ z + [ − σ r P s + 13 P s r (Ψ + σ ϕ ) + O ( r − )] dz + [ − σ r P s + 13 P s r (Ψ + σ ϕ )+ O ( r − )] d ¯ z − r P s + − ϕ + 4 σ σ P s + − ϕ ϕ P s r + O ( r − )] dzd ¯ z. (4.1)One should notice that here we fix the topology of the 2-surface, i.e. P = ¯ P = P s = z ¯ z √ . The induced Maxwell field on the r = r hypersurface is F uz = − φ P s + 12 P s r ( aφ ϕ − σ φ + 2 ð φ ) + O ( r − ) ,F u ¯ z = − φ P s + 12 P s r ( aφ ϕ − σ φ + 2 ð φ ) + O ( r − ) ,F z ¯ z = φ − φ P s + ð φ − ð φ P s r + O ( r − ) . (4.2)The dilaton field on the hypersurface is ϕ ( u, r , z, ¯ z ) = ϕ ( u, z, ¯ z ) r + ϕ ( u, z, ¯ z ) r + O ( r − ) , (4.3)where ∂ u ϕ + ðð ϕ = 0 . (4.4)The equation of motion of free falling charged particle on this hypersurface is V ν ( ∇ ν V µ + qF νµ ) = 0 . (4.5) INSTEIN -M AXWELL - DILATON THEORY IN N EWMAN -P ENROSE FORMALISM ∇ is the covariant derivative onthis 3 dimensional hypersurface and q is the charge of the particle.According to [24], we impose that V has the following asymptotic expansion V u = 1 + ∞ X a =1 V ua r a , V z = ∞ X a =2 V za r a . (4.6)Then we solve (4.5) order by order. The solution up to relevant order is V u = −
12 (Ψ + Ψ ) + 16 ϕ ∂ u ϕ ,V z = − P s ð σ + qP s A z ,V u = q P s A z A z + 16 ( ð Ψ + ð Ψ ) + 38 (Ψ + Ψ ) − ð σ ð σ − φ φ −
14 (Ψ + Ψ ) ϕ ∂ u ϕ − ϕ ∂ u ϕ + 16 ϕ ∂ u ϕ + 124 ( ϕ ∂ u ϕ ) −
112 ( ð ϕ ð ϕ + ϕ ðð ϕ ) ,V z = P s [2 ð σ σ + 23 Ψ + 12 ð σ (Ψ + Ψ )] − P s Z dv
12 ( ð Ψ + ð Ψ + 2 q ð A u ) , − qP s σ A z + qP s A z − P ð σ ϕ ∂ u ϕ , (4.7)where we have set all integration constants of u to zero since we require that the chargedparticle is static initially. At r − order, we can see that V has angular components. In otherwords, gravitational and electromagnetic radiations characterized by σ and A z cause freefalling charged particle to rotate over some tiny angle about the “center” of the spacetime r = 0 . The leading memory effect is the velocity kick of the charged particle ∆ V z = − r ( P s ð ∆ σ − qP s ∆ A z ) + O ( r − ) . (4.8)The leading memory effect consists of two parts, namely the gravitational part − P s ð ∆ σ and electromagnetic part qP s ∆ A z . They are mathematically equivalent to leading softgraviton theorem [20] and leading soft photon theorem [31] respectively by a Fouriertransformation. That is why we call this a unified expression of leading gravitationalmemory effect and leading electromagnetic memory effect. It is the same as the resultin Einstein-Maxwell theory [25], which means that the scalar field has no contribution tothe leading memory effect. Besides this, we can not see the coupling effect in the leadingmemory effect. We should notice that the change of the velocity of the charged particle(the velocity kick) is considered as distinct effect from the displacement memory effect.The gravitational memory is a property of gravitational wave characterized by the changeof the asymptotic shear ∆ σ [10]. The velocity kick we discuss in this paper and therelative displacement of nearby observers(e.g. in [20]) are different observational effectsof the gravitational wave with memory.8 W. T AN According to the treatment in electromagnetism [30], the sub-leading memory effectis a position displacement of the charged particle . ∆ z = Z V z du = − r Z du ( P s ð σ − qP s A z ) + O ( r − ) . (4.9)The gravitational contribution − R ( P s ð σ ) du has a relevance to sub-leading soft gravitontheorem(see [22] for specific discussion), and the electromagnetic contribution R ( qP s A z ) du has a relevance to subleading soft photon theorem (see [30] for further discussion). So wecall this a unified expression of subleading gravitational memory effect and subleadingelectromagnetic memory effect. The result is the same as [25], we do not see the couplingeffect in this sub-leading memory effect either.Another sub-leading observational memory effect is a time delay of the observe [24,32]. It is a kind of spin memory effect. The time delay of a charge particle will alsohave contributions from the electromagnetic radiation and the scalar radiation. Since V istime-like, the infinitesimal change of the proper time can be derived from the co-vector dχ = { r (3Ψ + 3Ψ − ϕ ∂ u ϕ ) + 1 r [ −
18 (Ψ + Ψ ) −
16 ( ð Ψ + ð Ψ ) + ð σ ð σ + 12 φ φ − q P s A z A z + 112 ( ð ϕ ð ϕ + ϕ ðð ϕ ) − ϕ ∂ u ϕ + 112 ϕ ∂ u ϕ + 112 (Ψ + Ψ ) ϕ ∂ u ϕ −
172 ( ϕ ∂ u ϕ ) ] } du + O ( r − ) . (4.10)The electromagnetic contribution ( φ φ − q P s A z A z ) comes one order higher than thegravitational contribution (Ψ + Ψ ) in the r expansion, but the scalar contribution − ϕ ∂ u ϕ appears the same order as the gravitational contribution, which means thatthe scalar effect is stronger than the electromagnetic effect and it is of the same orderas the gravitational effect. The coupling constant a does not show in eq(4.10) whichmeans that we can not find the effect of the non-minimal coupling of the scalar field andthe electromagnetic field at this order. We can find a scalar-gravitational coupled term (Ψ + Ψ ) ϕ ∂ u ϕ which is the same order as the electromagnetic contribution. Exceptthis, we can not find any other term about the coupling of the gravitation and matter fieldsat this order in this spin memory effect. In this work, we studied the EMD theory in NP formalism. We derived the NP equationsof the EMD theory and obtained the asymptotically-flat solution space. The solution We have used the fact that du = dχ + O ( r − ) , where χ is the proper time. We have used the fact that dz = V z r du + O ( r − ) . INSTEIN -M AXWELL - DILATON THEORY IN N EWMAN -P ENROSE FORMALISM r expansion as the the gravitational contribution, and it is one order lower thanthe electromagnetic contribution. Furthermore. we also discovered a scalar-gravitationalcoupled term in the same order as the electromagnetic contribution in the time-delaymemory effect. However, we found that there was no observable effect associated withthe non-minimal coupling between the Maxwell and the dilaton field in all of these threememory effects. Acknowledgements
The author thanks Pujian Mao for useful discussion and H. Lu for useful suggestions.This work is supported in part by the NSFC (National Natural Science Foundation ofChina) under the Grant Nos. 11905156 and 11935009.
References [1] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, “Gravitational waves ingeneral relativity. 7. Waves from axisymmetric isolated systems,”
Proc. Roy. Soc. Lond.
A269 (1962) 21–25.[2] R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptoticallyflat space-time,”
Proc. Roy. Soc. Lond.
A270 (1962) 103–126.[3] E. Newman and R. Penrose, “An approach to gravitational radiation by a method ofspin coefficients,”
J. Math. Phys. (1962) 566–278.[4] E. Newman and T. W. J. Unti, “Behavior of asymptotically flat empty spaces,” J. Math. Phys. (1962) 891–901.[5] B. Kozarzewski, “Asymptotic properties of the electromagnetic and gravitationalfields,” Acta Phys. Polon (1965) 775.[6] A. R. Exton, E. Newman, and R. Penrose, “Conserved quantities in theEinstein-Maxwell Theory,” J. Math. Phys (1969) 1566–1570.[7] R. Casadio, B. Harms, Y. Leblanc, and P. H. Cox, “Perturbations in theKerr-Newman dilatonic black hole background: 1. Maxwell waves,” Phys. Rev. D (1997) 4948–4961.0 W. T AN [8] Y. B. Zel’dovich and A. G. Polnarev, “Radiation of gravitational waves by a clusterof superdense stars,” Soviet. Astronomy (Aug. 1974) 17.[9] D. Christodoulou, “Nonlinear nature of gravitation and gravitational waveexperiments,” Phys. Rev. Lett (1991) 1486–1489.[10] J. Frauendiener, “Note on the memory effect,” Class. Quant. Grav (1992)1639–1641.[11] V. B. Braginsky and L. P. Grishchuk, “Kinematic resonance and memory effect infree mass gravitational antennas,” Sov. Phys. JETP (1985) 427–430.[12] V. B. Braginsky and K. S. Thorne, “Gravitational-wave bursts with memory andexperimental prospects,” Nature (May, 1987) 123–125.[13] A. G. Wiseman and C. M. Will, “Christodoulou’s nonlinear gravitational wavememory: Evaluation in the quadrupole approximation,”
Phys. Rev. D (1991)R2945–R2949.[14] K. S. Thorne, “Gravitational-wave bursts with memory: The Christodoulou effect,” Phys. Rev D (1992) 520–524.[15] P. D. Lasky, E. Thrane, Y. Levin, J. Blackman, and Y. Chen, “Detectinggravitational-wave memory with LIGO: implications of GW150914,” Phys. Rev.Lett (2016) 061102.[16] D. A. Nichols, “Spin memory effect for compact binaries in the post Newtonianapproximation,”
Phys. Rev. D (2017) 084048.[17] H. Yang and D. Martynov, “Testing gravitational memory generation with compactbinary mergers,” Phys. Rev. Lett (2018) 071102.[18] L. Susskind, “Electromagnetic memory,” arXiv:1507.02584 [hep-th] .[19] L. Bieri and D. Garfinkle, “An electromagnetic analogue of gravitational wavememory,”
Class. Quant. Grav (2013) 195009.[20] A. Strominger and A. Zhiboedov, “Gravitational memory, BMS supertranslationand soft theorems,” JHEP (2016) 086.[21] S. Weinberg, “Infrared photons and gravitons,” Phys. Rev (1965) B516–B524.[22] S. Pasterski, A. Strominger, and A. Zhiboedov, “New gravitational memories,”
JHEP (2016) 053. INSTEIN -M AXWELL - DILATON THEORY IN N EWMAN -P ENROSE FORMALISM arXiv:1404.4091 [hep-th] .[24] P. Mao and X. Wu, “More on gravitational memory,”
JHEP (2019) 058.[25] P. Mao and W. Tan, “On gravitational and electromagnetic memory,” arXiv:1912.01840 [gr-qc] .[26] H. Afshar, E. Esmaeili, and M. M. Sheikh-Jabbari, “String memory effect,” JHEP (2019) 053.[27] Chandrasekhar, “The Newman-Penrose formalism,” The mathematical theory ofblack holes pp.40–55.[28] H. Lu, P. Mao, and J. Wu, “Asymptotic structure of Einstein-Maxwell-dilatontheory and its five dimensional origin,”
JHEP (2019) 005.[29] A. I. Janis and E. T. Newman, “Structure of gravitational sources,” J. Math. Phys (1965) 902–914.[30] P. Mao, H. Ouyang, J. B. Wu, and X. Wu, “New electromagnetic memories andsoft photon theorems,” Phys. Rev. D (2017) 125011.[31] S. Pasterski, “Asymptotic symmetries and electromagnetic memory,” JHEP (2017) 154.[32] E. E. Flanagan, A. M. Grant, A. I. Harte, and D. A. Nichols, “Persistentgravitational wave observables: general framework,” Phys. Rev. D.99