Formal solutions of any-order mass, angular-momentum, dipole perturbations on the Schwarzschild background spacetime
aa r X i v : . [ g r- q c ] F e b Formal solutions of the any-order mass, angular-momentum, dipoleperturbations on the Schwarzschild background spacetime
Kouji Nakamura ∗ Gravitational-Wave Science Project, National Astronomical Observatory of Japan,Osawa 2-21-1, Mitaka 181-8588, Japan (Dated: February 22, 2021)Formal solutions of the any-order mass, angular-momentum, dipole perturbations onthe Schwarzschild background spacetime are derived. According to the proposal inRef. [K. Nakamura, arXiv:2102.00830 [gr-qc]], the zero-mode problem in the gauge-invariant perturbation theory on general background spacetime was resolved in a specificperturbation theory on the Schwarzschild background spacetime. This resolution impliesthe possibility of the development of the gauge-invariant perturbation theory to any-orderperturbations at least on the Schwarzschild background spacetime. As a result of this reso-lution, we reached to the simple derivation of the above formal solutions of any order.
I. INTRODUCTION
From the pioneer works by the Regge and Wheeler [1], and Zerilli [2], there are many workson the perturbations on the Schwarzschild background spacetime. In all of these researches, theperturbations are decomposed by the spherical harmonics Y lm and classify them into odd-modeand even-mode from their parity, since the Schwarzschild spacetime has the spherical symmetry.In addition, it will be a current consensus that l = , gauge-invariant ” treatments for l = , Conjecture I.1.
If the gauge-transformation rule for a tensor field h ab is given by Y h ab − X h ab = £ ξ ( ) g ab with the background metric g ab , then, there exist a tensor field F ab and a vector field Y a ∗ e-mail:[email protected] ormal solutions of the any-order mass, angular-momentum, dipole perturbations on ... such that h ab is decomposed as h ab = : F ab + £ Y g ab , where F ab and Y a are transformed as Y F ab − X F ab = , Y Y a − X Y a = ξ a ( ) under the gauge transformation. The outline of a proof of Conjecture I.1 is discussed in Refs. [5]. In these arguments, weassumed the existence of the Green functions for some elliptic derivative operators and the kernelmodes of these elliptic derivative operators are ignored. We called these kernel modes as zeromodes and the treatment of these zero modes still unclear. We also called the problem to find thistreatment of these zero modes as zero-mode problem .In the case of the gauge-invariant perturbation theory on the Schwarzschild background space-time, the zero mode is just above l = , l = , l = , l = , l = , l = , l = , II. PROPOSAL OF A STRATEGY TO TREAT PERTURBATIONS ON SPHERICALLYSYMMETRIC BACKGROUND
Here, we use the 2+2 formulation of the perturbation on a spherically symmetric backgroundspacetime which is the direct product M = M × S and the metric on this spacetime is g ab = y ab + r γ ab , y ab = y AB ( dx A ) a ( dx B ) b , γ ab = γ pq ( dx p ) a ( dx q ) b , (1) ormal solutions of the any-order mass, angular-momentum, dipole perturbations on ... x A = ( t , r ) , x p = ( θ , φ ) , and γ pq is the metric on the unit sphere. In the Schwarzschildspacetime, the metric (1) is given by y ab = − f ( dt ) a ( dt ) b + f − ( dr ) a ( dr ) b , f = − Mr , (2) γ ab = ( d θ ) a ( d θ ) b + sin θ ( d φ ) a ( d φ ) b . (3)On this spacetime ( M , g ab ) , we consider the components of the metric perturbation as h ab = h AB ( dx A ) a ( dx B ) b + h Ap ( dx A ) ( a ( dx p ) b ) + h pq ( dx p ) a ( dx q ) b . (4)As discussed in Ref. [7], we decompose these components using a harmonics S = S δ = Y lm : h AB = ∑ l , m ˜ h AB S δ , h Ap = r ∑ l , m (cid:2) ˜ h ( e ) A ˆ D p S δ + ˜ h ( o ) A ε pq ˆ D q S δ (cid:3) , (5) h pq = r ∑ l , m (cid:20) γ pq ˜ h ( e ) S δ + ˜ h ( e ) (cid:18) ˆ D p ˆ D q − γ pq ˆ ∆ (cid:19) S δ + h ( o ) ε r ( p ˆ D q ) ˆ D r S δ (cid:21) , (6)where ˆ D p is the covariant derivative associated with the metric γ pq on S , ˆ ∆ : = ˆ D r ˆ D r ˆ D p : = γ pq ˆ D q , ε pq = ε [ pq ] is the totally antisymmetric tensor on S . The perturbations { ˜ h AB , ˜ h ( e ) A , ˜ h ( e ) , ˜ h ( e ) } are called even-mode perturbations, and the perturbations { ˜ h ( o ) A , ˜ h ( o ) } are called odd-modeperturbations. We choose the harmonics S δ as S δ = Y lm for l ≥ k ( ˆ ∆ + ) m for l = k ( ˆ ∆ ) for l = , (7)where Y lm is the conventional spherical harmonics. We choose the functions k ( ˆ ∆ ) and k ( ˆ ∆ + ) m are k ( ˆ ∆ ) = + δ ln (cid:18) − z + z (cid:19) / δ ∈ R , (8) k ( ˆ ∆ + ) m = = z (cid:26) + δ (cid:18)
12 ln 1 + z − z − z (cid:19)(cid:27) , (9) k ( ˆ ∆ + ) m = ± = ( − z ) / (cid:26) + δ (cid:18)
12 ln 1 + z − z + z − z (cid:19)(cid:27) e ± i φ , (10)where z = cos θ . These functions satisfy the equation ˆ ∆ k ( ˆ ∆ ) = (cid:2) ˆ ∆ + (cid:3) k ( ˆ ∆ + ) m =
0, respec-tively, and are singular when δ =
0, while k ( ˆ ∆ ) ∝ Y and ˆ k ( ˆ ∆ + ) m ∝ Y m when δ = { S δ , ˆ D p S δ , ε pq ˆ D q S δ , γ pq S δ , (cid:0) ˆ D p ˆ D q − γ pq ˆ ∆ (cid:1) S δ ,2 ε r ( p ˆ D q ) ˆ D r S δ } are linearly independent including l = , δ =
0. This linear inde-pendence guarantees the one-to-one correspondence between metric perturbations { h Ap , h pq } and ormal solutions of the any-order mass, angular-momentum, dipole perturbations on ... { ˜ h ( e ) A , ˜ h ( o ) A , ˜ h ( e ) , ˜ h ( e ) , ˜ h ( o ) } . On the other hand, when δ =
0, the linear independence of theharmonics is lost in l = , l = , Proposal II.1.
We decompose the metric perturbations h ab on the background spacetime with themetric (1)–(3) as Eqs. (5)–(6) with the harmonics S δ given by Eq. (7). Then, Eqs. (5)–(6) becomeinvertible including l = , modes. After deriving the field equations using the harmonics S δ , wechoose δ = when we solve these field equations as regularity of solutions. Once we accept Proposal II.1, we can prove Conjecture I.1 including l = , Y h ab − X h ab = £ ξ g ab = ∇ ( a ξ b ) (11)for the linear-order perturbations on spherically symmetric background with the metric (1) play theessential role. Actually, in Ref. [7], we defined gauge-invariant ( ˜ F A ) and gauge-variant variables( ˜ Y ( o ) ) for odd-mode perturbations as˜ F A : = ˜ h ( o ) A + r ¯ D A ˜ h ( o ) , ˜ Y ( o ) : = − r ˜ h ( o ) . (12)For even-mode perturbations, we constructed the gauge-variant variables ˜ Y ( e ) and ˜ Y A by˜ Y ( e ) : = r h ( e ) , ˜ Y A : = r ˜ h ( e ) A − r D A ˜ h ( e ) . (13)The gauge-invariant variables ˜ F AB and ˜ F for even modes are defined by˜ F AB : = ˜ h AB − D ( A ˜ Y B ) , ˜ F : = ˜ h ( e ) − r ˜ Y A ¯ D A r + r ˜ Y ( e ) l ( l + ) . (14)From the above variables ˜ Y ( o ) , ˜ Y ( e ) , and ˜ Y A , we introduce the vector field Y a by Y a : = Y A ( dx A ) a + Y p ( dx p ) a , (15) Y A : = ∑ l , m ˜ Y A S δ , Y p : = ∑ l , m (cid:0) ˜ Y ( e ) ˆ D p S δ + ˜ Y ( o ) ε pr ˆ D r S δ (cid:1) (16)so that its gauge-transformation rule of this vector field Y a is given by Y Y a − X Y a = ξ a . On theother hand, the gauge-invariant variable F ab is defined by F AB : = ∑ l , m ˜ F AB S δ , F Ap : = r ∑ l , m ˜ F Ap ε pq ˆ D q S δ , F pq : = γ pq r ∑ l , m ˜ FS δ . (17) ormal solutions of the any-order mass, angular-momentum, dipole perturbations on ... h ab = F ab + £ Y g ab . (18)Thus, we reached to the following statement: Theorem II.1.
If the gauge-transformation rule for a tensor field h ab is given by Y h ab − X h ab = £ ξ ( ) g ab . Here, g ab is the background metric with the spherical symmetry. Then, there exist a tensorfield F ab and a vector field Y a such that h ab is decomposed as h ab = : F ab + £ Y g ab , where F ab andY a are transformed as Y F ab − X F ab = , Y Y a − X Y a = ξ a ( ) under the gauge transformation. III. l = , SOLUTIONS TO THE LINEARIZED EINSTEIN EQUATIONS
As shown in Refs. [3, 4], the linearized Einstein equation ( ) G ba = π ( ) T ba for the linear metricperturbation (18) with the vacuum background Einstein equation G ba = π T ba = ( ) G ba = π ( ) T ba , (19)where ( ) G ba and ( ) T ba are the gauge-invariant part of the first-order perturbation of the Einsteintensor and the energy-momentum tensor. Since we only consider the perturbations on the vacuumbackground solution based on the conventional general relativity, the linear metric perturbation h ab is not included in ( ) T ac nor ( ) T ca . Here, we also note that the first order perturbation of thiscontinuity equation is given by ∇ a ( ) T ba = . (20)We decompose the components of the linear perturbation of ( ) T ac as ( ) T ac = ∑ l , m ˜ T AC S δ ( dx A ) a ( dx C ) c + r ∑ l , m (cid:8) ˜ T ( e ) A ˆ D p S δ + ˜ T ( o ) A ε pq ˆ D q S δ (cid:9) ( dx A ) ( a ( dx p ) c ) + r ∑ l , m (cid:26) ˜ T ( e ) γ pq S δ + ˜ T ( e ) (cid:18) ˆ D p ˆ D q − γ pq ˆ D r ˆ D r (cid:19) S δ + ˜ T ( o ) ε s ( p ˆ D q ) ˆ D s S δ (cid:9) ( dx p ) ( a ( dx q ) c ) . (21)Now, we consider the solutions to the Einstein equation for l = , S δ through δ =
0. For this reason, as discussed in Ref. [7], we may choose˜ T ( e ) = ˜ T ( o ) = l = , T ( e ) A = T ( o ) A = l = T ( e ) = l = ormal solutions of the any-order mass, angular-momentum, dipole perturbations on ... A. l = odd mode perturbations If we impose the regularity on the harmonics S δ by choosing δ =
0, there is no l = l = m = F Ap ( dx A ) ( a ( dx p ) b ) = (cid:18) Mr Z dr r a ( t , r ) (cid:19) sin θ ( dt ) ( a ( d φ ) b ) + £ V ( o ) g ab , (22) V ( o ) a = (cid:0) β ( t ) + W ( o ) ( t , r ) (cid:1) r sin θ ( d φ ) a , (23)where β ( t ) is an arbitrary function of t . The function a ( t , r ) , which corresponds to the angular-momentum perturbation, is given by a ( t , r ) = − π M r f Z dt ˜ T ( o ) r + a = − π M Z drr f ˜ T ( o ) t + a . (24)Furthermore r f ∂ r W ( o ) of the variable W ( o ) in Eq. (23) is determined the evolution equation ∂ t ( r f ∂ r W ( o ) ) − f ∂ r ( f ∂ r ( r f ∂ r W ( o ) ) + r f [ − ( − f )] ( r f ∂ r W ( o ) ) = π f ˜ T ( o ) r . (25) B. l = , even mode perturbations Since the component ˜ T ( e ) vanishes in both l = , F AB is traceless and we introduce the components of ˜ F AB by˜ F AB ( dx A ) a ( dx B ) b = : X ( e ) (cid:8) − f ( dt ) a ( dt ) b − f − ( dr ) a ( dr ) b (cid:9) + Y ( e ) ( dt ) ( a ( dr ) b ) . (26)To evaluate the evolution equations of X ( e ) , Y ( e ) , and ˜ F , it is convenient to introduce Moncriefvariable Φ ( e ) by Φ ( e ) : = r Λ (cid:20) f X ( e ) − Λ ˜ F + r f ∂ r ˜ F (cid:21) , Λ : = ( l − )( l + ) + ( − f ) . (27)
1. l = mode solutions In this case, it is convenient to introduce the variable m ( t , r ) : = − ( − f ) Φ ( e ) , (28)which corresponds to the mass perturbation. The constraint equations in the l = t -component of Eq. (20) and these yields m ( t , r ) = π Z dr (cid:20) r f ˜ T tt (cid:21) + M = π Z dt (cid:2) r f ˜ T rt (cid:3) + M , M ∈ R . (29) ormal solutions of the any-order mass, angular-momentum, dipole perturbations on ... F should satisfy its evolution equation − f ∂ t ˜ F + ∂ r ( f ∂ r ˜ F ) + r ( − f ) ˜ F = − r m ( t , r ) + π (cid:20) − f ˜ T tt + f ˜ T rr (cid:21) . (30)We denote the solution to this equation by ˜ F = : ∂ t ϒ . Using the solution (29) and ϒ , we obtain F ab = r (cid:18) M + π Z dr (cid:20) r f ˜ T tt (cid:21)(cid:19) (cid:18) ( dt ) a ( dt ) b + f ( dr ) a ( dr ) b (cid:19) + (cid:20) π r Z dt (cid:18) f ˜ T tt + f ˜ T rr (cid:19)(cid:21) ( dt ) ( a ( dr ) b ) + £ V ( e ) g ab , (31) V ( e ) a : = (cid:18) + f ϒ + r f ∂ r ϒ + γ ( r ) (cid:19) ( dt ) a + f r ∂ t ϒ ( dr ) a , (32)where γ ( r ) is an arbitrary function of r .
2. l = mode solutions In this mode, we can obtain the variables ˜ F , Y ( e ) , and X ( e ) , if we obtain the variable Φ ( e ) as asolution to the linearized Einstein equations. The l = m = F ab = − π r f ( − f ) (cid:20) + f T rr + r f ∂ r ˜ T rr − ˜ T ( e ) − T ( e ) r (cid:21) cos θ ( dt ) a ( dt ) b + π r (cid:26) ˜ T tr − r f ( − f ) ∂ t ˜ T tt (cid:27) cos θ ( dt ) ( a ( dr ) b ) + π r ( − f ) f ( − f ) (cid:20) ˜ T tt − r f ( − f ) ∂ r ˜ T tt (cid:21) cos θ ( dr ) a ( dr ) b − π r ( − f ) ˜ T tt cos θγ ab + £ V ( e ) g ab , (33) V ( e ) a : = − r ∂ t Φ ( e ) cos θ ( dt ) a + (cid:0) Φ ( e ) − r ∂ r Φ ( e ) (cid:1) cos θ ( dr ) a − r Φ ( e ) sin θ ( d θ ) a . (34) IV. EXTENSION TO THE HIGHER-ORDER PERTURBATIONS
In the higher-order perturbation theory, the metric perturbation of each order is given by theexpansion of the full metric ¯ g ab under an appropriate gauge choice X as X ¯ g ab = X ∗ ε ¯ g ab = n ∑ k = ε k k ! ( k ) X g ab + o ( ε n ) , ( ) g ab = g ab . (35)Following the procedure proposed in Ref. [6], the n -th order metric perturbation ( n ) g ab is decom-posed into the gauge-invariant and gauge-variant parts as ( k ) g ab = : ( n ) F ab − n ∑ k = n ! ( n − l ) ! ∑ { j i }∈ J k C k ( { j i } ) £ j − ( ) Y £ j − ( ) Y · · · £ j k − ( k ) Y ( n − k ) g ab , (36) ormal solutions of the any-order mass, angular-momentum, dipole perturbations on ... C k ( { j i } ) : = k ∏ i = ( i ! ) j i j i ! , J k : = ( ( j , ..., j k ) ∈ N k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ∑ i = i j i = k ) . (37)Here, we note that Eq. (36) is not only the definition of the gauge-invariant variable ( n ) F ab but alsothe definition of the gauge-variant variable ( n ) Y a and these variables are recursively defined. If wehave the set of the gauge-variant variables { ( ) Y a , ..., ( n ) Y a } , we can define the gauge-invariantvariable ( n ) Q of the perturbation of an arbitrary tensor field Q = ∑ nk = ε k k ! ( k ) Q + o ( ε n ) as ( n ) Q = : ( n ) Q − n ∑ k = n ! ( n − l ) ! ∑ { j i }∈ J k C k ( { j i } ) £ j − ( ) Y £ j − ( ) Y · · · £ j k − ( k ) Y ( n − k ) Q . (38)This decomposition formula (38) implies that the n -th order perturbation of the Einstein tensorand the energy-momentum tensor are also decomposed as ( n ) G ba = : ( n ) G ba − n ∑ k = n ! ( n − l ) ! ∑ { j i }∈ J k C k ( { j i } ) £ j − ( ) Y £ j − ( ) Y · · · £ j k − ( k ) Y ( n − k ) G ba , (39) ( n ) T ba = : ( n ) T ba − n ∑ k = n ! ( n − l ) ! ∑ { j i }∈ J k C k ( { j i } ) £ j − ( ) Y £ j − ( ) Y · · · £ j k − ( k ) Y ( n − k ) T ba . (40)Then, the n -th order perturbation of the Einstein equation is given in the gauge-invariant form as ( n ) G ba = π ( n ) T ba . (41)This due to the lower-order Einstein equations. Note that ( n ) G ba should have the form ( n ) G ba = ( ) G ba h ( n ) F cd i + ( NL ) G ba hn ( i ) F cd (cid:12)(cid:12)(cid:12) i < n oi . (42)Here, the first term in Eq. (42) is the collection of the linear term of ( n ) F ab and its second termis the non-linear term consists of the lower-order metric perturbation ( i ) F ab with i < n . ThroughEq. (42), the n -th order perturbation (41) of the Einstein equation is given in the form as ( n ) G ba h ( n ) F cd i = − ( NL ) G ba hn ( i ) F cd (cid:12)(cid:12)(cid:12) i < n oi + π ( n ) T ba = : 8 π ( n ) T ba . (43)The right-hand side 8 π ( n ) T ba of Eq. (43) may be regarded an effective energy-momentum tensorfor the n -th order metric perturbation ( n ) F ab .The vacuum background condition G ba = ∇ a ( ) G ba = ∇ a ( n ) T ba =
0. Furthermore, it is trivial that the term ormal solutions of the any-order mass, angular-momentum, dipole perturbations on ... ( NL ) G ba hn ( i ) F cd (cid:12)(cid:12)(cid:12) i < n oi does not contain the n -th order metric perturbation ( n ) F ab by its def-inition, and the n -th order perturbation ( n ) T ba does not include ( n ) F ab , neither, because our back-ground spacetime is vacuum. Then, ( n ) T ba does not include ( n ) F ab .We have to emphasize that this situation is same as that we used when we derive the solutions(22), (31), and (33) from the linearized Einstein equation ( ) G ba = π ( ) T ba under the condition ∇ a ( ) T ba =
0. Then, the solutions to Eq. (43) should be given in the form2 ( n ) F Ap ( dx A ) ( a ( dx p ) b ) = (cid:18) Mr Z dr r a n ( t , r ) (cid:19) sin θ ( dt ) ( a ( d φ ) b ) + £ V ( n , o ) g ab , (44) V ( n , o ) a = (cid:0) β n ( t ) + W ( n , o ) ( t , r ) (cid:1) r sin θ ( d φ ) a (45)with a n ( t , r ) = − π M r f Z dt ( n ) ˜ T ( o ) r + a n = − π M Z drr f ( n ) ˜ T ( o ) t + a n (46)for the l = ( n ) F ab = r (cid:18) M n + π Z dr (cid:20) r f ( n ) ˜ T tt (cid:21)(cid:19) (cid:18) ( dt ) a ( dt ) b + f ( dr ) a ( dr ) b (cid:19) + (cid:20) π r Z dt (cid:18) f ( n ) ˜ T tt + f ( n ) ˜ T rr (cid:19)(cid:21) ( dt ) ( a ( dr ) b ) + £ V ( n , e ) g ab , (47) V ( n , e ) a : = (cid:18) + f ϒ n + r f ∂ r ϒ n + γ n ( r ) (cid:19) ( dt ) a + f r ∂ t ϒ n ( dr ) a (48)with arbitrary function γ n ( r ) for the l = ( n ) F ab = − π r f ( − f ) (cid:20) + f ( n ) ˜ T rr + r f ∂ r ( n ) ˜ T rr − ( n ) ˜ T ( e ) − ( n ) ˜ T ( e ) r (cid:21) cos θ ( dt ) a ( dt ) b + π r (cid:26) ( n ) ˜ T tr − r f ( − f ) ∂ t ( n ) ˜ T tt (cid:27) cos θ ( dt ) ( a ( dr ) b ) + π r ( − f ) f ( − f ) (cid:20) ( n ) ˜ T tt − r f ( − f ) ∂ r ( n ) ˜ T tt (cid:21) cos θ ( dr ) a ( dr ) b − π r ( − f ) ( n ) ˜ T tt cos θγ ab + £ V ( n , e ) g ab , (49) V ( n , e ) a : = − r ∂ t Φ ( n , e ) cos θ ( dt ) a + (cid:0) Φ ( n , e ) − r ∂ r Φ ( n , e ) (cid:1) cos θ ( dr ) a − r Φ ( n , e ) sin θ ( d θ ) a (50)for the l = ormal solutions of the any-order mass, angular-momentum, dipole perturbations on ... V. SUMMARY
In summary, we extended the linear-order solution of the mass perturbation ( l = l = l = l = , S δ defined by Eq. (7) instead of the conventional harmonic func-tion Y lm . The parameter δ in the harmonics S δ is chosen so that the linear independence of thetensor harmonics is guaranteed when δ =
0, and S δ is realize Y lm when δ =
0. Then, we pro-posed Proposal II.1 as a strategy of a gauge-invariant treatment of the l = , l = , ( n ) T ba , i.e., ( NL ) G ba hn ( i ) F cd (cid:12)(cid:12)(cid:12) i < n oi and ( n ) T ba .This evaluation will depend on the situations which we want to clarify. Therefore, we leave furtherdevelopments as future works. [1] T. Regge and J. A. Wheeler, Phys. Rev. (1957), 1063.[2] F. Zerilli, Phys. Rev. Lett. (1970), 737; F. Zerilli, Phys. Rev. D (1970), 2141.[3] K. Nakamura, Prog. Theor. Phys. , (2003), 723;[4] K. Nakamura, Prog. Theor. Phys. (2005), 481;[5] K. Nakamura, Class. Quantum Grav. (2011), 122001; K. Nakamura, Prog. Theor. Exp. Phys. (2013), 043E02.[6] K. Nakamura, Class. Quantum Grav.31