Teukolsky-like equations with various spins in spherically symmetric spacetime

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Ya Guo, Hiroaki Nakajima and Wenbin Lin. Teukolsky-like equations with various spins in spherically symmetric spacetime[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad2a61
Ya Guo, Hiroaki Nakajima and Wenbin Lin. Teukolsky-like equations with various spins in spherically symmetric spacetime[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad2a61 shu
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Teukolsky-like equations with various spins in spherically symmetric spacetime

    Corresponding author: Wenbin Lin, lwb@usc.edu.cn
  • 1. School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, China
  • 2. School of Mathematics and Physics, University of South China, Hengyang 421001, China

Abstract: We study wave equations with various spins on the background of a general spherically symmetric spacetime. We obtain the unified expression of the Teukolsky-like master equations and the corresponding radial equations with the general spins. We also discuss the gauge dependence in the gravitational-wave equations, which have appeared in previous studies.

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    I.   INTRODUCTION
    • The study of gravitational waves has become a highly important subject in cosmology since the direct observation of gravitational waves by LIGO and VIRGO [1]. Gravitational waves can be described as a solution to the Einstein equation, where the metric is perturbed around a certain background. When the background is chosen as the black hole spacetime, one can, for example, consider the gravitational-wave radiation from a relatively light object rotating around the black hole, which can be studied using the black-hole perturbation theory [2]. However, when the mass of the object is not so light, the contribution to the background spacetime from this object may not be negligible. In such a case, the background must be modified or other methods must be used, such as the post-Newtonian approximation [3]. One systematic method for modifying the background is proposed as effective one-body (EOB) dynamics [4, 5].

      The background of EOB dynamics is deformed by the black hole spacetime and hence may not satisfy the vacuum Einstein equation. Motivated by this, we consider a general spherically symmetric spacetime as a simple example of the background, which is not necessarily a vacuum. A particular form of the background appears in EOB dynamics for the spinless binary system [4, 5]. The spherically symmetric spacetime satisfies the Petrov type D condition, which has played a key role in deriving the gravitational-wave equation in previous studies [68] using the Newman-Penrose formalism [9], similar to the Teukolsky equation [10] in the vacuum case. The advantage of using the Newman-Penrose formalism is that the role of the Einstein equation is restrictive and is merely used to relate the Ricci tensor to the energy-momentum tensor. Therefore, the extension to the non-vacuum case is relatively easier. It has been found that in order to obtain the decoupled wave equation, the gauge condition must be taken such that some of the coupled degrees of freedom vanish [68]. To date, two types of the gravitational-wave equation have been proposed [7, 8] owing to the difference in the gauge conditions.

      In this study, we investigate massless wave equations with different spins, that is, the (massless) Klein-Gordon, Weyl, and Maxwell equations on the same background spacetime. To avoid complexities, we first provide a unified expression for these equations consisting of the Newman-Penrose quantities, as in [1114], and obtain the explicit wave equation and ordinary differential equation for the radial coordinate.

      The remainder of this paper is organized as follows. In Section II, we introduce our parameterization of the background of a spherically symmetric spacetime and present the quantities in the Newman-Penrose formalism. In Section III, we observe the wave equation for spin 0, ±1/2, ±1, and ±2 on this background. Subsequently, we provide the unified expression for these equations with general spin s and obtain the explicit Teukolsky-like master equation and the corresponding radial equation. In Section IV, we discuss the gauge dependence in the gravitational-wave equations proposed in previous studies. Finally, Section V presents a summary and discussion. In Appendix A, we list our notations and conventions.

    II.   BACKGROUND METRIC AND TETRADS
    • We consider the general spherically symmetric background as

      ds2=A(r)dt2B(r)dr2C(r)r2(dθ2+sin2θdφ2).

      (1)

      By choosing appropriate radial coordinate r, we can setC(r)=1, which corresponds to the standard coordinate. We define D(r)=A(r)B(r), and then background metric (1) becomes

      ds2=A(r)dt2D(r)2A(r)dr2r2(dθ2+sin2θdφ2).

      (2)

      The null tetrads corresponding to metric (2) are taken as

      l=lAμdxμ=dtD(r)A(r)dr,n=nAμdxμ=A(r)2dt+D(r)2dr,m=mAμdxμ=r2(dθ+isinθdφ),ˉm=ˉmAμdxμ=r2(dθisinθdφ),

      (3)

      which satisfy

      ds2=2ln2mˉm,

      (4)

      where the bar denotes the complex conjugate. Superscript (or subscript) A is used as the symbol of the background quantities [10]. Conversely, for the perturbation quantities of the gravitational field, we use superscipt B, which will appear later.

      From the tetrad basis (3), we can compute spin coefficients and the components of the Ricci tensor and Weyl scalars as

      κA=νA=σA=λA=πA=τA=ϵA=0,

      (5)

      ρA=1rD,μA=A2rD,γA=A4D,αA=βA=cotθ22r,

      (6)

      ΦA01=ΦA10=ΦA02=ΦA20=ΦA12=ΦA21=0,

      (7)

      ΦA00=DrD3,ΦA22=A2D4rD3,

      (8)

      ΦA11=18r2D3[2D32ADr2(ADAD)],

      (9)

      ΛA=124r2D3[2D3r2AD+2A(D2rD)+rD(4A+rA)],

      (10)

      ΨA0=ΨA1=ΨA3=ΨA4=0,

      (11)

      ΨA2=112r2D3[2(AD+rADD3)rA(2D+rD)+r2AD],

      (12)

      where we follow the notations of Newman-Penrose [9] and Pirani [15]. The same notation is also used for Teukolsky [10]. We also list the definitions of the above quantities in the appendix. The prime symbol denotes the derivative with respect to r. Equation (11) implies that the background belongs to Petrov type D, which is important to derive the wave equation on this background. As a special case, for D(r)=1, we have

      ΦA00=ΦA22=0,

      (13)

      and the nonvanishing quantities in the background are simplified as

      ρA=1r,μA=A2r,γA=A4,αA=βA=cotθ22r,

      (14)

      ΦA11=18r2(22A+r2A),

      (15)

      ΛA=124r2(2+2A+4rA+r2A),

      (16)

      ΨA2=112r2(2+2A2rA+r2A),

      (17)

      We also note that when choosing A=12M/r and D=1, the background reduces to the Schwarzschild spacetime, where M is the mass of the Schwarzschild black hole.

    III.   WAVE EQUATIONS WITH VARIOUS SPINS
    • Here, we consider the wave equation with various spins s on the background specified by (5)–(12) in the previous section, namely, the Klein-Gordon (s=0), Weyl (s=±1/2), and Maxwell equations (s=±1) and the equation from the Newman-Penrose formalism (s=±2) under the probe (test field) approximation; therefore, the back reactions from matter and electromagnetic fields to the gravitational background are assumed to be negligible. For simplicity, we also assume that the fields are massless and minimally coupled to the background of the gravitational field, unless otherwise specified.

    • A.   spin 0

    • The massless Klein-Gordon equation in the gravitational background is

      ϕ=μ(gμννϕ)=1gμ(ggμννϕ)=T,

      (18)

      where T is the source. μ is the covariant derivative with respect to the curved spacetime (not for the local Lorentz transformation), of which the projection by the null tetrads gives

      DA=lμAμ,ΔA=nμAμ,δA=mμAμ,ˉδA=ˉmμAμ.

      (19)

      By decomposing gμν in terms of the null tetrads, (18) can be rewritten as

      [(Δ2γ+2μ)D+(D2ρ)Δ(ˉδ2α)δ(δ2α)ˉδ]Aϕ=T,

      (20)

      where the superscript A outside the parentheses denotes that all the quantities and operators inside the parentheses are background ones, and we use the relation for the spin coefficients,

      μlμA=2ρA,μnμA=2γA+2μA,μmμA=μˉmμA=2αA.

      (21)

      For later convenience, we rewrite (20) such that the order of the differential operators is rearranged, satisfying the commutation relations

      ΔADADAΔA=2γADA,ˉδAδAδAˉδA=2αA(δˉδ)A.

      (22)

      We also use

      ΔAρA=(2γμ)AρAΨA22ΛA,

      (23)

      which is from the background part of that of the Newman-Penrose equation. Then, (20) can be rewritten as

      [(Δ2γ+μ)(Dρ)(ˉδ2α)δΨ22Λ]Aϕ=12T.

      (24)

      Note that the last term 2ΛAϕ on the right hand side is responsible for the minimal coupling. If we consider curvature coupling, there is a contribution proportional to Rϕ=24ΛAϕ, where R is the Ricci scalar (see the appendix).

    • B.   spin ±1/2

    • The massless Dirac equation can be decomposed into two Weyl equations. For the positive chirality part, the Weyl equation in the gravitational background is

      (ˉδα)Aχ0(Dρ)Aχ1=0,

      (25)

      (Δγ+μ)Aχ0(δα)Aχ1=0,

      (26)

      where χ0 and χ1 are the components of the Weyl spinor. We can eliminate χ0 using the following commutation relation:

      [Δ+pγ(q1)μ]A(ˉδ+pα)A(ˉδ+pα)A(Δ+pγqμ)A=νADAλAδA+p(αλ+ρνΨ3)A+q(Dν+δλ4αλ+2Ψ3)A=0,

      (27)

      where p and q are arbitrary constants, and we just use νA=λA=ΨA3=0 for the last equality. Hence, (27) holds not only in the vacuum, but also for the background specified by (5)–(12). We obtain the wave equation for χ1 in a similar way via the method used to derive the Teukolsky equation [10]. We operate (Δγ+2μ)A on (25) and (ˉδα)A on (26) and then obtain the difference of them. The terms with χ0 are canceled from (27) with p=q=1, and the remaining part is

      [(Δγ+2μ)(Dρ)(ˉδα)(δα)]Aχ1=0,

      (28)

      which gives the wave equation for s=1/2. In a similar way, the wave equation for χ0 (for s=1/2) is obtained as

      [(D2ρ)(Δγ+μ)(δα)(ˉδα)]Aχ0=0.

      (29)

      We note that (28) and (29) take the same forms as in the vacuum case [10]. For later convenience, we rewrite (29) as in the previous subsection. We use (22), (23), and

      DAγA=ΨA2+ΦA11ΛA,

      (30)

      (δ+ˉδ)AαA=μAρA+4(αA)2ΨA2+ΦA11+ΛA,

      (31)

      DAμA=μAρA+ΨA2+2ΛA.

      (32)

      Then, (29) can be rewritten as

      [(Δ3γ+μ)(D2ρ)(ˉδ3α)(δ+α)3Ψ2]Aχ0=0,

      (33)
    • C.   spin ±1

    • The Maxwell equation in the gravitational background is

      (D2ρ)Aϕ1(ˉδ2α)Aϕ0=Jl,

      (34)

      δAϕ1(Δ+μ2γ)Aϕ0=Jm,

      (35)

      (Dρ)Aϕ2ˉδAϕ1=Jˉm,

      (36)

      (δ2α)Aϕ2(Δ+2μ)Aϕ1=Jn,

      (37)

      where ϕ0, ϕ1 , and ϕ2 are complex and constructed from the field strength (the Faraday tensor) Fμν as 1

      ϕ0=FμνlμAmνA,ϕ1=12Fμν(lμAnνA+ˉmμAmνA),ϕ2=FμνˉmμAnνA.

      (38)

      Jl, Jn, Jm , and Jˉm are the projections of the current Jμ by the null tetrads as Jl=JμlAμ, etc. From (36) and (37), we can construct the wave equation for ϕ2 via a similar procedure to that used in the previous subsection. Using the commutation relation (27) with p=0 and q=2, we have

      [(Δ+3μ)(Dρ)ˉδ(δ2α)]Aϕ2=J2,

      (39)

      where J2 is defined by

      J2=(Δ+3μ)AJˉmˉδAJn.

      (40)

      In a similar way, we can obtain the wave equation for ϕ0 from (34) and (35) as

      [(D3ρ)(Δ2γ+μ)δ(ˉδ2α)]Aϕ0=J0,

      (41)

      where J0 is defined by

      J0=δAJl(D3ρ)AJm.

      (42)

      Note that (39) and (41) take the same forms as in the vacuum case [10]. For later convenience, we rewrite (41) using (22), (23), and (30)–(32) as

      [(Δ4γ+μ)(D3ρ)(ˉδ4α)(δ+2α)6Ψ2]Aϕ0=J0,

      (43)
    • D.   spin ±2

    • The wave equations for the spin ±2 are obtained from the perturbed Einstein equation or the perturbed Newman-Penrose equation. In a previous paper [8], we studied and obtained the wave equations, and in the next section, we revisit the derivation to discuss the gauge dependence. Then, we provide the result of the equations.

      Here, we take the gauge such that the following quantities vanish [8]:

      λB=σB=0,

      (44)

      (ˉδˉτ+2α+2ˉβ)BΦA22=0,

      (45)

      (δ+ˉπ2ˉα2β)BΦA00=0,

      (46)

      where the superscript B denotes the perturbation part. Under the above gauge, the wave equation for the perturbation part of the Weyl scalar ΨB4 is

      [(Δ+2γ+5μ)(Dρ)(ˉδ+2α)(δ4α)3Ψ2+2Φ11]AΨB4=T4,

      (47)

      where the source T4 is defined by

      T4=(Δ+2γ+5μ)A[(ˉδ+2α)AΦB21(Δ+μ)AΦB20](ˉδ+2α)A[ˉδAΦB22(Δ+2γ+2μ)AΦB21)].

      (48)

      In a similar way, we can also obtain the wave equation for ΨB0 as

      [(D5ρ)(Δ4γ+μ)(δ+2α)(ˉδ4α)3Ψ2+2Φ11]AΨB0=T0,

      (49)

      where the source T0 is defined by

      T0=(δ+2α)A[(D2ρ)AΦB01δAΦB00](D5ρ)A[(Dρ)AΦB02(δ+2α)AΦB01].

      (50)

      For later convenience, we rewrite (49) using (22), (23), and (30)–(32) as

      [(Δ6γ+μ)(D5ρ)(ˉδ6α)(δ+4α)15Ψ2+2Φ11]AΨB0=T0.

      (51)
    • E.   Unified wave equation for general spin s

    • We can unify the above wave equations (24), (28), (33), (39), (43), (47), and (51) and express the unified equation for general spin s as

      {[Δ2(1+s)γ+(1s+|s|)μ][D(1+s+|s|)ρ][ˉδ2(1+s)α](δ+2sα)(1+3s+2s2)Ψ2+13(|s|3|s|2+2|s|3)Φ112δsΛ}A˜ψ(s)=˜T(s),

      (52)

      where we collectively denote the fields and sources as

      ˜ψ(s)={ΨB4for s=2ϕ2for s=1χ1fors=1/2ϕfors=0χ0fors=1/2ϕ0for s=1ΨB0for s=2,˜T(s)={T4for s=2J2for s=10fors=1/2T/2fors=00fors=1/2J0for s=1T0for s=2.

      (53)

      δs is defined by

      δs={1fors=00otherwise.

      (54)

      We rewrite (52) in a slightly simpler form using the following redefinitions:

      ψ(s)=exp[(|s|s)f]˜ψ(s),T(s)=exp[(|s|s)f]˜T(s),

      (55)

      where f is a function of r, which will be determined soon. By substituting (55) into (52), we have

      {[Δ2(1+s)γ+μ+(|s|s)(μΔf)][D(1+2s)ρ(|s|s)(ρ+Df)][ˉδ2(1+s)α](δ+2sα)(1+3s+2s2)Ψ2+13(|s|3|s|2+2|s|3)Φ112δsΛ}Aψ(s)=T(s).

      (56)

      We find that μAΔAf and ρA+DAf can simultaneously vanish by choosing f=lnr, namely, 2

      ψ(s)=r|s|s˜ψ(s),T(s)=r|s|s˜T(s).

      (57)

      Then, (56) is simplified as

      {[Δ2(1+s)γ+μ][D(1+2s)ρ][ˉδ2(1+s)α](δ+2sα)(1+3s+2s2)Ψ2+13(|s|3|s|2+2|s|3)Φ112δsΛ}Aψ(s)=T(s).

      (58)

      The advantage of the above form is that in the vacuum ΦA11=ΛA=0, the equation (58) depends on s but not |s|. The same transformation (57) has been performed in the vacuum case [10]. Similar equations to (52) and (58) were studied in [1114]; however, these were considered for positive and negative s separately, or restricted to positive s. Here, we obtain the completely unified expression for both positive and negative s. Moreover, we find the contributions of ΦA11 and ΛA to the wave equation.

      By substituting the background, the explicit form of the unified wave equation (58) is

      r2A2ψ(s)t21D(r2A)sr[(r2A)s+1Dψ(s)r]1sinθθ(sinθψ(s)θ)1sin2θ2ψ(s)φ2+(2srDsr2AAD)ψ(s)t2iscotθsinθψ(s)φ+[s2cot2θss(2s+1)rADD3+13(1δs+3s+2s2)(1AD22rAD2+2rADD3+r2AD2D3r2A2D2)+16(|s|3|s|2+2|s|3)(1AD2r2AD2D3+r2A2D2)]ψ(s)=2r2T(s).

      (59)

      Note that (59) with s=±2 is different from the equation obtained in our previous study [8]. However, this is simply because of the difference in the transformation (57), and they are equivalent. We also note that in the case of A=12M/r and D=1, (59) reduces to the Teukolsky master equation with spin s on the background of the Schwarzschild spacetime. In the case of D=1, the above is simplified as

      r2A2ψ(s)t21(r2A)sr[(r2A)s+1ψ(s)r]1sinθθ(sinθψ(s)θ)1sin2θ2ψ(s)φ2+(2srsr2AA)ψ(s)t2iscotθsinθψ(s)φ+[s2cot2θs+13(1δs+3s+2s2)(1A2rA12r2A)+16(|s|3|s|2+2|s|3)(1A+12r2A)]ψ(s)=2r2T(s).

      (60)

      First, we consider the homogeneous case. The equation allows the separation of the variables, and we assume the product form of the solution to be

      ψ(s)=eiωteimφR(r)S(θ),

      (61)

      where ω is the frequency of the waves, and m is constant. Then, the separated equations are

      1D(r2A)sddr[(r2A)s+1DdRdr]+[r2ω2A+iω(2srDsr2AAD)+s(2s+1)rADD313(1δs+3s+2s2)(1AD22rAD2+2rADD3+r2AD2D3r2A2D2)16(|s|3|s|2+2|s|3)(1AD2r2AD2D3+r2A2D2)λ(s)]R=0,

      (62)

      1sinθddθ(sinθdSdθ)+(m2sin2θ2smcotθsinθs2cot2+s+λ(s))S=0,

      (63)

      where λ(s) is the separation constant. From (63), we can find that S(θ)eimφ coincides with the spin-weighted spherical harmonics sYlm(θ,φ) with spin s, where l and m take the values of

      l=|s|, |s|+1, |s|+2, ,m=l, l+1, , l1, l,

      (64)

      respectively. λ(s) becomes the eigenvalue of sYlm(θ,φ), which is given by

      λ(s)=(ls)(l+s+1).

      (65)

      For the nonhomogeneous case, we expand ψ(s) and T(s) in terms of sYlm(θ,φ) as

      ψ(s)=dωl,mR(s)lmω(r)sYlm(θ,φ)eiωt,

      (66)

      2r2T(s)=dωl,mG(s)lmω(r)sYlm(θ,φ)eiωt.

      (67)

      Then, R(s)lmω(r) satisfies

      1D(r2A)sddr[(r2A)s+1DdR(s)lmωdr]+[r2ω2A+iω(2srDsr2AAD)+s(2s+1)rADD313(1δs+3s+2s2)(1AD22rAD2+2rADD3+r2AD2D3r2A2D2)16(|s|3|s|2+2|s|3)(1AD2r2AD2D3+r2A2D2)λ(s)]R(s)lmω=G(s)lmω.

      (68)

      We again note that in the case of A=12M/r and D=1, (68) reduces to the Teukolsky radial equation with spin s on the background of the Schwarzschild spacetime. In the case of D=1, (68) reduces to

      1(r2A)sddr[(r2A)s+1dR(s)lmωdr]+[r2ω2A+iω(2srsr2AA)13(1δs+3s+2s2)(1A2rA12r2A)16(|s|3|s|2+2|s|3)(1A+12r2A)λ(s)]R(s)lmω=G(s)lmω,

      (69)

      which reduces to [11] for positive s.

    IV.   GAUGE DEPENDENCE IN GRAVITATIONAL-WAVE EQUATIONS
    • In previous studies, two types of the gravitational-wave equations have appeared. One can be found in [6, 8], and the other in [7]. Because these equations are obtained from the same set of coupled equations in the Newman-Penrose formalism but with different gauges, both equations should describe the gravitational wave correctly. Here, one question can be raised: although the unknown variables ΨB4 and ΨB0 are gauge-invariant quantities, why can we have two (or more, in principle) forms of the wave equations for each variable? To answer this question, we revisit the derivation of the gravitational-wave equation on the background, with emphasis on the gauge dependence.

      We focus on the wave equation for ΨB4 and consider the case of D=1 because in [7], only this case is considered. Here, our gauge conditions (44) and (45) reduce to λB=0 [8]. We begin with the following three equations in the Newman-Penrose formalism:

      (δ+4βτ)Ψ4(Δ+4μ+2γ)Ψ3+3νΨ2=(ˉδˉτ+2ˉβ+2α)Φ22(Δ+2γ+2ˉμ)Φ212λΦ12+2νΦ11+ˉνΦ20,

      (70)

      (D+4ϵρ)Ψ4(ˉδ+4π+2α)Ψ3+3λΨ2=(ˉδ2ˉτ+2α)Φ21(Δ+2γ2ˉγ+ˉμ)Φ20+ˉσΦ222λΦ11+2νΦ10,

      (71)

      (Δ+μ+ˉμ+3γˉγ)λ(ˉδ+πˉτ+ˉβ+3α)ν+Ψ4=0.

      (72)

      We split all the quantities in the above into the background (A) and perturbation parts (B); for instance, Ψ4=ΨA4+ΨB4, etc. We keep the first order of the perturbation only. The background part of the above equations is satisfied, and the perturbation part becomes

      (δ4α)AΨB4(Δ+2γ+4μ)AΨB3+3νBΨA2=ˉδAΦB22(Δ+2γ+2μ)AΦB21+2νBΦA11,

      (73)

      (Dρ)AΨB4(ˉδ+2α)AΨB3+3λBΨA2=(ˉδ+2α)AΦB21(Δ+μ)AΦB202λBΦA11,

      (74)

      (Δ+2γ+2μ)AλB(ˉδ+2α)AνB+ΨB4=0.

      (75)

      Now, we obtain the wave equation for ΨB4 using the same procedure as in the previous section. We operate (Δ+2γ+5μ)A to (74) and (ˉδ+2α)A to (73) and then find the difference of them. The terms with ΨB3 are canceled by (27) with p=2, q=4 , and the remainder becomes

      [(Δ+2γ+5μ)(Dρ)(ˉδ+2α)(δ4α)]AΨB4+(3Ψ2+2Φ11)A(Δ+2γ+5μ)AλB(3Ψ22Φ11)A(ˉδ+2α)AνB=T4λBΔA(3Ψ2+2Φ11)A+νBˉδA(3Ψ22Φ11)A,

      (76)

      where T4 is defined by

      T4=(Δ+2γ+5μ)A[(ˉδ+2α)AΦB21(Δ+μ)AΦB20](ˉδ+2α)A[ˉδAΦB22(Δ+2γ+2μ)AΦB21].

      (77)

      For the third line in (76), we have

      ΔA(3Ψ22Φ11)A=3μA(3Ψ2+2Φ11)A+8μAΦA11,

      (78)

      ˉδA(3Ψ2+2Φ11)A=3πA(3Ψ22Φ11)A,

      (79)

      and then substituting the above into (76), we obtain

      [(Δ+2γ+5μ)(Dρ)(ˉδ+2α)(δ4α)3Ψ2]AΨB4+2ΦA11[(Δ+2γ+2μ)AλB+(ˉδ+2α)AνB]=T44λB[(Δ+2μ)Φ11]A,

      (80)

      where we use (75) and ˉδAΦA11=0 from the spherical symmetry. By eliminating the terms with νB using (75) again, we have

      [(Δ+2γ+5μ)(Dρ)(ˉδ+2α)(δ4α)3Ψ2+2Φ11]AΨB4=T44(Δ+2γ+4μ)A(ΦA11λB).

      (81)

      Moreover, using (78) and (79), the above can be rewritten as

      [(Δ+2γ+5μ)(Dρ)(ˉδ+2α)(δ4α)3Ψ2+2Φ11]AΨB4=T4+(3Ψ22Φ11)A(Δ+2γ+2μ)AλB(Δ+2γ+5μ)A[(3Ψ2+2Φ11)AλB],

      (82)

      So far, we have not used a gauge condition. If we take the gauge condition as λB=0, Eqs. (80) and (82) are reduced to the wave equation (47) in [6, 8] as

      [(Δ+2γ+5μ)(Dρ)(ˉδ+2α)(δ4α)3Ψ2+2Φ11]AΨB4=T4,

      (83)

      which has a similar form to the vacuum case ΦA11=ΛA=0. However, from (74), λB can be expressed in terms of ΨB3 as

      λB=1(3Ψ2+2Φ11)A[(Dρ)AΨB4+(ˉδ+2α)A(Ψ3+Φ21)B(Δ+μ)AΦB20].

      (84)

      Substituting the above into (82) gives

      [F12(Δ+2γ+2μF1)(Dρ)(ˉδ+2α)(δ4α)3Ψ2+2Φ11]AΨB4=F12(Δ+2γ+2μF1)A×[(ˉδ+2α)A(Ψ3+Φ21)B(Δ+μ)AΦB20](ˉδ+2α)A[ˉδAΦB22(Δ+2γ+2μ)AΦB21],

      (85)

      or

      [(Δ+2γ+2μF1)(Dρ)F2(ˉδ+2α)(δ4α)3Ψ22Φ11]AΨB4=(Δ+2γ+2μF1)A[(ˉδ+2α)A(Ψ3+Φ21)B(Δ+μ)AΦB20]F2(ˉδ+2α)A[ˉδAΦB22(Δ+2γ+2μ)AΦB21],

      (86)

      where F1 and F2 are defined by

      F1=Δ[ln(3Ψ2+2Φ11)A],F2=(3Ψ2+2Φ113Ψ22Φ11)A.

      (87)

      Thus, if we take the gauge condition as ΨB3=0, (86) is reduced to the wave equation in [7] as

      [(Δ+2γ+2μF1)(Dρ)F2(ˉδ+2α)(δ4α)3Ψ22Φ11]AΨB4=˜T4,

      (88)

      where ˜T4 is defined by

      ˜T4=(Δ+2γ+2μF1)A[(ˉδ+2α)AΦB21(Δ+μ)AΦB20]F2(ˉδ+2α)A[ˉδAΦB22(Δ+2γ+2μ)AΦB21].

      (89)

      Next, we consider the gauge transformation in the wave equation, for which we take the following tetrad rotations 3 [16]:

      lμlμ,mμmμ+alμ,ˉmμˉmμ+ˉalμ,nμnμ+ˉamμ+aˉmμ+aˉalμ,

      (90)

      nμnμ,mμmμ+bnμ,ˉmμˉmμ+ˉbnμ,lμlμ+ˉbmμ+bˉmμ+bˉbnμ,

      (91)

      lμeclμ,nμecnμ,mμeiϑmμ,ˉmμeiϑˉmμ.

      (92)

      To avoid changing the background, we assume that parameters a, b, c, and ϑ are in the first order of the perturbation, and hence for the perturbation quantities, the transformation of the first order is sufficient. In the wave equation (80), the left hand side is gauge-invariant becauseΨB4 is so, which imples that the right hand side must also be invariant. Using

      λBλB+(ˉδ+2α)Aˉa,ΦB21ΦB21+2ΦA11ˉa,ΦB20ΦB20,ΦB22ΦB22,

      (93)

      source termT4 transforms as

      T4T4+4[(ˉδ+2α)(Δ+2γ+3μ)]A(ΦA11ˉa).

      (94)

      We can find that this transformation is cancelled by that of other terms on the right hand side of (80), where we use (27) and ˉδAΦA11=0. We can also show that ˜T4 , defined by (89), has a nontrivial gauge transformation under (90)–(92), which is cancelled by that of ΨB3 as

      ΨB3ΨB3+3ΨA2ˉa.

      (95)

      Thus, the origin of the gauge dependence of the gravitational-wave equation is due to that of the source term, particularly ΦB21. Note that in the vacuum case, this dependence does not appear because ΦA11=0. We can also show that the two gravitational-wave equations (83) and (88) coincide in the vacuum background because of

      F1=3μA,F2=1,

      (96)

      in the vacuum.

    V.   SUMMARY AND DISCUSSION
    • In this study, we investigated the wave equations with various spins on the background of a general spherically symmetric spacetime. By introducing spin variable s, we unified these equations using s itself,|s| , and δs. The transformation (57) to simplify the equation was possible, and the form of (57), in turn, became the same as in the vacuum case although the background, particularly spin coefficientρA, was deformed.

      We also discussed the gauge dependence in the form of the gravitational-wave equations from a previous study. The gauge dependence of the wave equation originates from that of the source term, and hence it cannot be avoided, except in the vacuum case. If we take another gauge, the form of the gravitational-wave equation changes, which also affects the unified expressions (52) and (58).

      A similar analysis using the metric perturbation was performed in [1719], where the wave equations resembled the Regge-Wheeler and the Zerilli equations [20, 21], and its generalization to general spin s has also been studied [22]. In the vacuum case, there is a transformation between the Teukolsky and Regge-Wheeler equations, known as the Chandrasekhar transformation [23]. Moreover, for s0, the relationship between R(|s|)lmω(r) and R(|s|)lmω(r)can be regarded as a special case of the Chandrasekhar transformation [10, 24]. It would be interesting to investigate whether similar relations hold in the current case as in [25].

      A possible generalization would be to extend the results to the axisymmetric background, which contains the Kerr black hole as an example. In the vacuum case, the backgrounds satisfying the type D condition are fully classified as the so-called the Kinnersley metric [26]. It would be interesting to find the non-vacuum extension of the Kinnersley metric and the wave equation on that background with general spin. The gravitational-wave equation on a certain non-vacuum axisymmetric background was proposed in [27]. However, this equation does not allow for the separation of the variables, hence, further modification is needed [28].

      The study of the (gravitational) wave equation using the Newman-Penrose formalism is applicable to not only EOB dynamics but also modified gravitational theories, as long as the gravitational degrees of freedom are described by the metric tensor, such as the scalar-tensor, Horndeski [29], and degenerated higher order scalar-tensor theories [30]. In these theories, the equations of motion become complicated. However, in the Newman-Penrose formalism, the role of the Einstein equation is rather restrictive. In particular, the original Newman-Penrose equations (70)–(72) for the wave equation originate from the Bianchi identity, which also takes the same form in these theories. Then, one can perform a similar analysis to ours. It would be interesting to obtain wave equations in these theories, especially the equations for other degrees of freedom, for example, the perturbed scalar field in the above theories.

    APPENDIX A NOTATIONS AND CONVENTIONS
    • In this paper, we follow the notations and conventions of Newman-Penrose [9] and Pirani [15]. Here, we list some of the definitions for convenience. The metric has the sign (+) , and the Riemann curvature is decomposed as

      Rμναβ=Cμναβ12(gμαRνβgμβRνα+gνβRμαgναRμβ)+16R(gμαgνβgμβgνα),

      (A1)

      where Cμναβ is the Weyl tensor. Ricci tensor Rμν and Ricci scalar R are defined by

      Rμν=Rρμνρ,R=gμνRμν.

      (A2)

      The twelve spin coefficients are defined by

      κ=mμlννlμ,τ=mμnννlμ,ϵ=12(nμlννlμˉmμlννmμ),σ=mμmννlμ,ρ=mμˉmννlμ,γ=12(nμnννlμˉmμnννmμ),ν=ˉmμnννnμ,μ=ˉmμmννnμ,β=12(nμmννlμˉmμmννmμ),λ=ˉmμˉmννnμ,π=ˉmμlννnμ,α=12(nμˉmννlμˉmμˉmννmμ),

      (A3)

      where μ is the covariant derivative with respect to the curved spacetime. The Ricci tensor is decomposed into the following components:

      Φ00=12Rμνlμlν,Φ01=12Rμνlμmν,Φ02=12Rμνmμmν,Φ10=12Rμνlμˉmν,Φ11=14Rμν(lμnν+mμˉmν),Φ12=12Rμνnμmν,Φ20=12Rμνˉmμˉmν,Φ21=12Rμνnμˉmν,Φ22=12Rμνnμnν,

      Λ=124R=112Rμν(lμnνmμˉmν).

      (A4)

      Finally, the Weyl scalars are defined by

      Ψ0=Cμνλρlμmνlλmρ,Ψ1=Cμνλρlμnνlλmρ,Ψ2=12Cμνλρ(lμnνlλnρlμnνmλˉmρ),Ψ3=Cμνλρlμnνˉmλnρ,Ψ4=Cμνλρnμˉmνnλˉmρ.

      (A5)
    ACKNOWLEDGEMENTS
    • The authors thank Prof. Ruffini for useful discussions.

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