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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 [6−8] 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 [6−8]. 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 [11−14], 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. -
We consider the general spherically symmetric background as
ds2=A(r)dt2−B(r)dr2−C(r)r2(dθ2+sin2θdφ2).
(1) By choosing appropriate radial coordinate r, we can set
C(r)=1 , which corresponds to the standard coordinate. We defineD(r)=√A(r)B(r) , and then background metric (1) becomesds2=A(r)dt2−D(r)2A(r)dr2−r2(dθ2+sin2θdφ2).
(2) The null tetrads corresponding to metric (2) are taken as
l=lAμdxμ=dt−D(r)A(r)dr,n=nAμdxμ=A(r)2dt+D(r)2dr,m=mAμdxμ=−r√2(dθ+isinθdφ),ˉm=ˉmAμdxμ=−r√2(dθ−isinθdφ),
(3) which satisfy
ds2=2ln−2mˉm,
(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=A′4D,αA=−βA=−cotθ2√2r,
(6) ΦA01=ΦA10=ΦA02=ΦA20=ΦA12=ΦA21=0,
(7) ΦA00=D′rD3,ΦA22=A2D′4rD3,
(8) ΦA11=18r2D3[2D3−2AD−r2(A′D′−A″D)],
(9) ΛA=−124r2D3[−2D3−r2A′D′+2A(D−2rD′)+rD(4A′+rA″)],
(10) ΨA0=ΨA1=ΨA3=ΨA4=0,
(11) ΨA2=112r2D3[2(AD+rAD′−D3)−rA′(2D+rD′)+r2A″D],
(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 haveΦA00=ΦA22=0,
(13) and the nonvanishing quantities in the background are simplified as
ρA=−1r,μA=−A2r,γA=A′4,αA=−βA=−cotθ2√2r,
(14) ΦA11=18r2(2−2A+r2A″),
(15) ΛA=−124r2(−2+2A+4rA′+r2A″),
(16) ΨA2=112r2(−2+2A−2rA′+r2A″),
(17) We also note that when choosing
A=1−2M/r andD=1 , the background reduces to the Schwarzschild spacetime, where M is the mass of the Schwarzschild black hole. -
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. -
The massless Klein-Gordon equation in the gravitational background is
◻ϕ=∇μ(gμν∂νϕ)=1√−g∂μ(√−ggμν∂νϕ)=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 givesDA=lμA∇μ,ΔA=nμA∇μ,δA=mμA∇μ,ˉδA=ˉmμA∇μ.
(19) By decomposing
gμν in terms of the null tetrads, (18) can be rewritten as[(Δ−2γ+2μ)D+(D−2ρ)Δ−(ˉδ−2α)δ−(δ−2α)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
ΔADA−DAΔA=2γADA,ˉδAδA−δAˉδA=2αA(δ−ˉδ)A.
(22) We also use
ΔAρA=(2γ−μ)AρA−ΨA2−2ΛA,
(23) which is from the background part of that of the Newman-Penrose equation. Then, (20) can be rewritten as
[(Δ−2γ+μ)(D−ρ)−(ˉδ−2α)δ−Ψ2−2Λ]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 toRϕ=24ΛAϕ , whereR is the Ricci scalar (see the appendix). -
The massless Dirac equation can be decomposed into two Weyl equations. For the positive chirality part, the Weyl equation in the gravitational background is
(ˉδ−α)Aχ0−(D−ρ)Aχ1=0,
(25) (Δ−γ+μ)Aχ0−(δ−α)Aχ1=0,
(26) where
χ0 andχ1 are the components of the Weyl spinor. We can eliminateχ0 using the following commutation relation:[Δ+pγ−(q−1)μ]A(ˉδ+pα)A−(ˉδ+pα)A(Δ+pγ−qμ)A=νADA−λAδA+p(αλ+ρν−Ψ3)A+q(−Dν+δλ−4αλ+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 forχ1 in a similar way via the method used to derive the Teukolsky equation [10]. We operate(Δ−γ+2μ)A on (25) and(ˉδ−α)A on (26) and then obtain the difference of them. The terms withχ0 are canceled from (27) withp=q=−1 , and the remaining part is[(Δ−γ+2μ)(D−ρ)−(ˉδ−α)(δ−α)]Aχ1=0,
(28) which gives the wave equation for
s=−1/2 . In a similar way, the wave equation forχ0 (fors=1/2 ) is obtained as[(D−2ρ)(Δ−γ+μ)−(δ−α)(ˉδ−α)]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+ΦA11−ΛA,
(30) (δ+ˉδ)AαA=μAρA+4(αA)2−ΨA2+ΦA11+ΛA,
(31) DAμA=μAρA+ΨA2+2ΛA.
(32) Then, (29) can be rewritten as
[(Δ−3γ+μ)(D−2ρ)−(ˉδ−3α)(δ+α)−3Ψ2]Aχ0=0,
(33) -
The Maxwell equation in the gravitational background is
(D−2ρ)Aϕ1−(ˉδ−2α)Aϕ0=Jl,
(34) δAϕ1−(Δ+μ−2γ)Aϕ0=Jm,
(35) (D−ρ)Aϕ2−ˉδAϕ1=Jˉm,
(36) (δ−2α)Aϕ2−(Δ+2μ)Aϕ1=Jn,
(37) where
ϕ0 ,ϕ1 , andϕ2 are complex and constructed from the field strength (the Faraday tensor)Fμν as1 ϕ0=FμνlμAmνA,ϕ1=12Fμν(lμAnνA+ˉmμAmνA),ϕ2=FμνˉmμAnνA.
(38) Jl ,Jn ,Jm , andJˉm are the projections of the currentJμ by the null tetrads asJl=JμlAμ , etc. From (36) and (37), we can construct the wave equation forϕ2 via a similar procedure to that used in the previous subsection. Using the commutation relation (27) withp=0 andq=−2 , we have[(Δ+3μ)(D−ρ)−ˉδ(δ−2α)]Aϕ2=J2,
(39) where
J2 is defined byJ2=(Δ+3μ)AJˉm−ˉδAJn.
(40) In a similar way, we can obtain the wave equation for
ϕ0 from (34) and (35) as[(D−3ρ)(Δ−2γ+μ)−δ(ˉδ−2α)]Aϕ0=J0,
(41) where
J0 is defined byJ0=δAJl−(D−3ρ)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
[(Δ−4γ+μ)(D−3ρ)−(ˉδ−4α)(δ+2α)−6Ψ2]Aϕ0=J0,
(43) -
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) (ˉδ−ˉτ+2α+2ˉβ)BΦA22=0,
(45) (δ+ˉπ−2ˉα−2β)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)(D−ρ)−(ˉδ+2α)(δ−4α)−3Ψ2+2Φ11]AΨB4=T4,
(47) where the source
T4 is defined byT4=(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[(D−5ρ)(Δ−4γ+μ)−(δ+2α)(ˉδ−4α)−3Ψ2+2Φ11]AΨB0=T0,
(49) where the source
T0 is defined byT0=(δ+2α)A[(D−2ρ)AΦB01−δAΦB00]−(D−5ρ)A[(D−ρ)AΦB02−(δ+2α)AΦB01].
(50) For later convenience, we rewrite (49) using (22), (23), and (30)–(32) as
[(Δ−6γ+μ)(D−5ρ)−(ˉδ−6α)(δ+4α)−15Ψ2+2Φ11]AΨB0=T0.
(51) -
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
{[Δ−2(1+s)γ+(1−s+|s|)μ][D−(1+s+|s|)ρ]−[ˉδ−2(1+s)α](δ+2sα)−(1+3s+2s2)Ψ2+13(|s|−3|s|2+2|s|3)Φ11−2δsΛ}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 byδs={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
{[Δ−2(1+s)γ+μ+(|s|−s)(μ−Δf)][D−(1+2s)ρ−(|s|−s)(ρ+Df)]−[ˉδ−2(1+s)α](δ+2sα)−(1+3s+2s2)Ψ2+13(|s|−3|s|2+2|s|3)Φ11−2δsΛ}Aψ(s)=T(s).
(56) We find that
μA−ΔAf andρA+DAf can simultaneously vanish by choosingf=lnr , namely,2 ψ(s)=r|s|−s˜ψ(s),T(s)=r|s|−s˜T(s). (57) Then, (56) is simplified as
{[Δ−2(1+s)γ+μ][D−(1+2s)ρ]−[ˉδ−2(1+s)α](δ+2sα)−(1+3s+2s2)Ψ2+13(|s|−3|s|2+2|s|3)Φ11−2δsΛ}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 [11−14]; 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 ofΦA11 andΛA to the wave equation.By substituting the background, the explicit form of the unified wave equation (58) is
r2A∂2ψ(s)∂t2−1D(r2A)s∂∂r[(r2A)s+1D∂ψ(s)∂r]−1sinθ∂∂θ(sinθ∂ψ(s)∂θ)−1sin2θ∂2ψ(s)∂φ2+(2srD−sr2A′AD)∂ψ(s)∂t−2iscotθsinθ∂ψ(s)∂φ+[s2cot2θ−s−s(2s+1)rAD′D3+13(1−δs+3s+2s2)(1−AD2−2rA′D2+2rAD′D3+r2A′D′2D3−r2A″2D2)+16(|s|−3|s|2+2|s|3)(1−AD2−r2A′D′2D3+r2A″2D2)]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 ofA=1−2M/r andD=1 , (59) reduces to the Teukolsky master equation with spin s on the background of the Schwarzschild spacetime. In the case ofD=1 , the above is simplified asr2A∂2ψ(s)∂t2−1(r2A)s∂∂r[(r2A)s+1∂ψ(s)∂r]−1sinθ∂∂θ(sinθ∂ψ(s)∂θ)−1sin2θ∂2ψ(s)∂φ2+(2sr−sr2A′A)∂ψ(s)∂t−2iscotθsinθ∂ψ(s)∂φ+[s2cot2θ−s+13(1−δs+3s+2s2)(1−A−2rA′−12r2A″)+16(|s|−3|s|2+2|s|3)(1−A+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)=e−iωteimφR(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(2srD−sr2A′AD)+s(2s+1)rAD′D3−13(1−δs+3s+2s2)(1−AD2−2rA′D2+2rAD′D3+r2A′D′2D3−r2A″2D2)−16(|s|−3|s|2+2|s|3)(1−AD2−r2A′D′2D3+r2A″2D2)−λ(s)]R=0,
(62) 1sinθddθ(sinθdSdθ)+(−m2sin2θ−2smcotθsinθ−s2cot2+s+λ(s))S=0,
(63) where
λ(s) is the separation constant. From (63), we can find thatS(θ)eimφ coincides with the spin-weighted spherical harmonicssYlm(θ,φ) with spin s, where l and m take the values ofl=|s|, |s|+1, |s|+2, …,m=−l, −l+1, …, l−1, l,
(64) respectively.
λ(s) becomes the eigenvalue ofsYlm(θ,φ) , which is given byλ(s)=(l−s)(l+s+1).
(65) For the nonhomogeneous case, we expand
ψ(s) andT(s) in terms ofsYlm(θ,φ) asψ(s)=∫dω∑l,mR(s)lmω(r)sYlm(θ,φ)e−iωt,
(66) −2r2T(s)=∫dω∑l,mG(s)lmω(r)sYlm(θ,φ)e−iωt.
(67) Then,
R(s)lmω(r) satisfies1D(r2A)sddr[(r2A)s+1DdR(s)lmωdr]+[r2ω2A+iω(2srD−sr2A′AD)+s(2s+1)rAD′D3−13(1−δs+3s+2s2)(1−AD2−2rA′D2+2rAD′D3+r2A′D′2D3−r2A″2D2)−16(|s|−3|s|2+2|s|3)(1−AD2−r2A′D′2D3+r2A″2D2)−λ(s)]R(s)lmω=G(s)lmω.
(68) We again note that in the case of
A=1−2M/r andD=1 , (68) reduces to the Teukolsky radial equation with spin s on the background of the Schwarzschild spacetime. In the case ofD=1 , (68) reduces to1(r2A)sddr[(r2A)s+1dR(s)lmωdr]+[r2ω2A+iω(2sr−sr2A′A)−13(1−δs+3s+2s2)(1−A−2rA′−12r2A″)−16(|s|−3|s|2+2|s|3)(1−A+12r2A″)−λ(s)]R(s)lmω=G(s)lmω,
(69) which reduces to [11] for positive s.
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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 andΨB0 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 ofD=1 because in [7], only this case is considered. Here, our gauge conditions (44) and (45) reduce toλB=0 [8]. We begin with the following three equations in the Newman-Penrose formalism:(δ+4β−τ)Ψ4−(Δ+4μ+2γ)Ψ3+3νΨ2=(ˉδ−ˉτ+2ˉβ+2α)Φ22−(Δ+2γ+2ˉμ)Φ21−2λΦ12+2νΦ11+ˉνΦ20,
(70) (D+4ϵ−ρ)Ψ4−(ˉδ+4π+2α)Ψ3+3λΨ2=(ˉδ−2ˉτ+2α)Φ21−(Δ+2γ−2ˉγ+ˉμ)Φ20+ˉσΦ22−2λΦ11+2νΦ10,
(71) (Δ+μ+ˉμ+3γ−ˉγ)λ−(ˉδ+π−ˉτ+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(δ−4α)AΨB4−(Δ+2γ+4μ)AΨB3+3νBΨA2=ˉδAΦB22−(Δ+2γ+2μ)AΦB21+2νBΦA11,
(73) (D−ρ)AΨB4−(ˉδ+2α)AΨB3+3λBΨA2=(ˉδ+2α)AΦB21−(Δ+μ)AΦB20−2λBΦA11,
(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 withΨB3 are canceled by (27) withp=2 ,q=−4 , and the remainder becomes[(Δ+2γ+5μ)(D−ρ)−(ˉδ+2α)(δ−4α)]AΨB4+(3Ψ2+2Φ11)A(Δ+2γ+5μ)AλB−(3Ψ2−2Φ11)A(ˉδ+2α)AνB=T4−λBΔA(3Ψ2+2Φ11)A+νBˉδA(3Ψ2−2Φ11)A,
(76) where
T4 is defined byT4=(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(3Ψ2−2Φ11)A=−3μA(3Ψ2+2Φ11)A+8μAΦA11,
(78) ˉδA(3Ψ2+2Φ11)A=−3πA(3Ψ2−2Φ11)A,
(79) and then substituting the above into (76), we obtain
[(Δ+2γ+5μ)(D−ρ)−(ˉδ+2α)(δ−4α)−3Ψ2]AΨB4+2ΦA11[(Δ+2γ+2μ)AλB+(ˉδ+2α)AνB]=T4−4λB[(Δ+2μ)Φ11]A,
(80) where we use (75) and
ˉδAΦA11=0 from the spherical symmetry. By eliminating the terms withνB using (75) again, we have[(Δ+2γ+5μ)(D−ρ)−(ˉδ+2α)(δ−4α)−3Ψ2+2Φ11]AΨB4=T4−4(Δ+2γ+4μ)A(ΦA11λB).
(81) Moreover, using (78) and (79), the above can be rewritten as
[(Δ+2γ+5μ)(D−ρ)−(ˉδ+2α)(δ−4α)−3Ψ2+2Φ11]AΨB4=T4+(3Ψ2−2Φ11)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)(D−ρ)−(ˉδ+2α)(δ−4α)−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 ofΨB3 asλB=1(3Ψ2+2Φ11)A[−(D−ρ)AΨB4+(ˉδ+2α)A(Ψ3+Φ21)B−(Δ+μ)AΦB20].
(84) Substituting the above into (82) gives
[F−12(Δ+2γ+2μ−F1)(D−ρ)−(ˉδ+2α)(δ−4α)−3Ψ2+2Φ11]AΨB4=F−12(Δ+2γ+2μ−F1)A×[(ˉδ+2α)A(Ψ3+Φ21)B−(Δ+μ)AΦB20]−(ˉδ+2α)A[ˉδAΦB22−(Δ+2γ+2μ)AΦB21],
(85) or
[(Δ+2γ+2μ−F1)(D−ρ)−F2(ˉδ+2α)(δ−4α)−3Ψ2−2Φ11]AΨB4=(Δ+2γ+2μ−F1)A[(ˉδ+2α)A(Ψ3+Φ21)B−(Δ+μ)AΦB20]−F2(ˉδ+2α)A[ˉδAΦB22−(Δ+2γ+2μ)AΦB21],
(86) where
F1 andF2 are defined byF1=Δ[ln(3Ψ2+2Φ11)A],F2=(3Ψ2+2Φ113Ψ2−2Φ11)A.
(87) Thus, if we take the gauge condition as
ΨB3=0 , (86) is reduced to the wave equation in [7] as[(Δ+2γ+2μ−F1)(D−ρ)−F2(ˉδ+2α)(δ−4α)−3Ψ2−2Φ11]AΨB4=˜T4,
(88) where
˜T4 is defined by˜T4=(Δ+2γ+2μ−F1)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]:lμ→lμ,mμ→mμ+alμ,ˉmμ→ˉmμ+ˉalμ,nμ→nμ+ˉamμ+aˉmμ+aˉalμ,
(90) nμ→nμ,mμ→mμ+bnμ,ˉmμ→ˉmμ+ˉbnμ,lμ→lμ+ˉbmμ+bˉmμ+bˉbnμ,
(91) lμ→e−clμ,nμ→ecnμ,mμ→eiϑmμ,ˉmμ→e−iϑˉmμ.
(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 because
ΨB4 is so, which imples that the right hand side must also be invariant. UsingλB→λB+(ˉδ+2α)Aˉa,ΦB21→ΦB21+2ΦA11ˉa,ΦB20→ΦB20,ΦB22→ΦB22,
(93) source term
T4 transforms asT4→T4+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 ofΨB3 asΨB3→ΨB3+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 becauseΦA11=0 . We can also show that the two gravitational-wave equations (83) and (88) coincide in the vacuum background because ofF1=−3μA,F2=1,
(96) in the vacuum.
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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| , andδs . 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 [17–19], 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
s≠0 , the relationship betweenR(−|s|)lmω(r) andR(|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.
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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 asRμναβ=Cμναβ−12(gμαRνβ−gμβRνα+gνβRμα−gναRμβ)+16R(gμαgνβ−gμβgνα),
(A1) where
Cμναβ is the Weyl tensor. Ricci tensorRμν and Ricci scalarR are defined byRμν=Rρμνρ,R=gμνRμν.
(A2) The twelve spin coefficients are defined by
κ=mμlν∇νlμ,τ=mμnν∇νlμ,ϵ=12(nμlν∇νlμ−ˉmμlν∇νmμ),σ=mμmν∇νlμ,ρ=mμˉmν∇νlμ,γ=12(nμnν∇νlμ−ˉmμnν∇νmμ),ν=−ˉmμnν∇νnμ,μ=−ˉmμmν∇νnμ,β=12(nμmν∇νlμ−ˉmμmν∇νmμ),λ=−ˉmμˉmν∇νnμ,π=−ˉmμlν∇νnμ,α=12(nμˉmν∇νlμ−ˉmμˉmν∇νmμ),
(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(lμnν−mμˉmν).
(A4) Finally, the Weyl scalars are defined by
Ψ0=−Cμνλρlμmνlλmρ,Ψ1=−Cμνλρlμnνlλmρ,Ψ2=−12Cμνλρ(lμnνlλnρ−lμnνmλˉmρ),Ψ3=−Cμνλρlμnνˉmλnρ,Ψ4=−Cμνλρnμˉmνnλˉmρ.
(A5) -
The authors thank Prof. Ruffini for useful discussions.
Teukolsky-like equations with various spins in spherically symmetric spacetime
- Received Date: 2024-01-15
- Available Online: 2024-08-15
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.