Cylindrical gravitational waves: radiation and resonance
aa r X i v : . [ phy s i c s . g e n - ph ] J un Cylindrical gravitational waves: radiation andresonance
Yu-Zhu Chen, a, Yu-Jie Chen, b Shi-Lin Li, b and Wu-Sheng Dai b, a Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin, 300071,P. R. China b Department of Physics, Tianjin University, Tianjin 300350, P.R. China
Abstract:
In the weak field approximation the gravitational wave is approximated as alinear wave, which ignores the nonlinear effect. In this paper, we present an exact generalsolution of the cylindrical gravitational wave. The exact solution of the cylindrical gravi-tational wave is far different from the weak field approximation. This solution implies thefollowing conclusions. (1) There exist gravitational monopole radiations in the cylindricalgravitational radiation. (2) The gravitational radiation may generate the resonance in thespacetime. (3) The nonlinearity of the gravity source vanishes after time averaging, so theobserved result of a long-time measurement may be linear. [email protected] [email protected]. ontents General relativity is a nonlinear theory. The quadrupole radiation, the polarization, thespin of the graviton, however, are mostly obtained under the linear approximation [1, 2].In the linear approximation, it is supposed that the energy and the momentum of thesource are invariable. As a result, the monopole and the dipole of the source are invariableand make no contribution to the gravitational radiation [3]. Besides, when the spacetime isregarded as a Minkowski spacetime, the dipole vanishes in the center-of-mass coordinate [2].Therefore, the leading contribution of the gravitational radiation is made by the variationof the quadrupole [1, 2], i.e., the gravitational quadrupole radiation.In this paper, we present an exact cylindrical solution which implies a gravitationalmonopole radiation. The exact nonlinear solution is far different from the linear approx-imation. By the Birkhoff theorem we know that there is no gravitational wave in thespherical vacuum solution of the Einstein equation [1–3]. The simplest metric involving thegravitational radiation is the Einstein-Rosen metric. The source of an infinite length mayexist in the Einstein-Rosen metric. It is convenient to discuss the gravitational radiationwith the Einstein-Rosen metric, such as the energy loss of the source by the gravitationalradiation [4], the C-energy, the super-energy, and the associated dynamical effect of thecylindrical gravitational wave [5], the energy-momentum of the gravitational wave [6], thesuperenergy flux of the Einstein-Rosen wave [7], the nonlinear effect such as the Faradayrotation and the time-shift phenomenon of the cylindrical gravitational soliton solution [8],the nonlinear evolution of cylindrical gravitational waves [9], the twisted gravitational wave[10], the scattering of the gravitational waves [11], the gravitational collapse of the energy of– 1 –ravitational waves [12] , the asymptotic structure of the radiation spacetime [13], the inter-action between the gravitational wave and the cosmic string [14, 15], the cosmic censorshiphypothesis [16], and the midisuperspace quantization [17–19].We also discuss the resonance of the gravitational radiation in the spacetime. In lit-erature, the resonance between the gravitational radiation and the matter (especially thegravitational radiation detectors) are considered [20–23]. In this paper, we focus on theresonance of gravitational waves.In the linear approximation, two gravitational radiations do not interact with eachother. In this paper, we show the interaction between two cylindrical gravitational radia-tions. When two cylindrical gravitational radiations exist simultaneously, the interactionterms arise both in the metric and the energy-momentum tensor. Especially, we show that,though the interaction always exists in nonlinear gravity waves, the nonlinearity of thegravity source vanishes when taking a time average. This implies that the observed resultof a long-time measurement may be linear, though the gravity is nonlinear.In section 2, we present an exact general cylindrical gravitational wave solution. Insection 3, we show the existence of the cylindrical gravitational monopole radiation. Insection 4, we discuss the resonance of the gravitational radiation. In section 5, we showthe interaction of two cylindrical gravitational radiations. The conclusion and outlooks aregiven in section 6. In this section, we present a general vacuum solution of the cylindrical gravitational wave.With the cylindrical gravitational wave solution, we discuss the gravitational monopoleradiation, the resonance, and the interaction of the cylindrical gravitational radiations inthe latter sections.The cylindrical gravitational wave is described by the dimensional Einstein-Rosenmetric [24], ds = e γ ( t,ρ ) − ψ ( t,ρ ) (cid:0) − dt + dρ (cid:1) + e − ψ ( t,ρ ) ρ dφ + e ψ ( t,ρ ) dz . (2.1)The Einstein tensor G µν = R µν − η µν R in the orthogonal frame [25] with η µν = diag ( − , , , is G = G = e ψ − γ " ρ ∂γ∂ρ − (cid:18) ∂ψ∂ρ (cid:19) − (cid:18) ∂ψ∂t (cid:19) , (2.2) G = G = e ψ − γ (cid:18) ρ ∂γ∂t − ∂ψ∂t ∂ψ∂ρ (cid:19) , (2.3) G = e ψ − γ "(cid:18) ∂ψ∂ρ (cid:19) − (cid:18) ∂ψ∂t (cid:19) + ∂ γ∂ρ − ∂ γ∂t , (2.4) G = e ψ − γ "(cid:18) ∂ψ∂ρ (cid:19) − (cid:18) ∂ψ∂t (cid:19) + ∂ γ∂ρ − ∂ γ∂t + 2 ∂ ψ∂t − ∂ ψ∂ρ − ρ ∂ψ∂ρ . (2.5)– 2 –e consider the gravitational field outside the source, i.e., ρ = 0 , which satisfies G µν =0 . For G µν = 0 , eqs. (2.2), (2.3), (2.4), and (2.5) can be simplified as [24] ∂γ∂ρ = ρ (cid:18) ∂ψ∂t (cid:19) + ρ (cid:18) ∂ψ∂ρ (cid:19) , (2.6) ∂γ∂t = 2 ρ ∂ψ∂t ∂ψ∂ρ . (2.7)For gravitational wave solutions, we require ∂ψ∂t = 0 . Then we obtain the equation of ψ from eqs. (2.6) and (2.7) [24], − ∂ ψ∂t + ∂ ψ∂ρ + 1 ρ ∂ψ∂ρ = 0 . (2.8)The equation of ψ (2.8) is a linear equation. The general solution of eq. (2.8) is ψ = Z ∞−∞ dτ Z ∞ dλA ( τ, λ ) J ( λρ ) cos ( λ ( t − τ ) + α ( τ, λ ))+ Z ∞−∞ dτ Z ∞ dλB ( τ, λ ) Y ( λρ ) cos ( λ ( t − τ ) + β ( τ, λ ))+ κ t + κ ln ρ + κ , (2.9)where α ( τ, λ ) , β ( τ, λ ) , A ( τ, λ ) , and B ( τ, λ ) are arbitrary functions of τ and λ , J is theBessel function of first kind, Y is the Bessel function of second kind, and κ , κ , and κ are constants. From eqs. (2.6) and (2.7) we can see that the equations of γ are also linearequations. Substituting eq. (2.9) into eqs. (2.6) and (2.7) gives the general solution of γ : γ = − ρ Z ∞−∞ dτ Z ∞−∞ dτ Z ∞ dλ Z ∞ dλ ( F + + F − ) − κ ρ Z ∞−∞ dτ Z ∞ dλ [ A ( τ, λ ) J ( λρ ) sin ( λ ( t − τ ) + α ( τ, λ )) + B ( τ, λ ) Y ( λρ ) sin ( λ ( t − τ ) + β ( τ, λ ))] − κ Z ∞−∞ dτ Z ∞ dλ [ A ( τ, λ ) J ( λρ ) cos ( λ ( t − τ ) + α ( τ, λ )) + B ( τ, λ ) Y ( λρ ) cos ( λ ( t − τ ) + β ( τ, λ ))]+ 2 κ κ t + 12 κ ρ + κ ln ρ + κ (2.10)with F + = λ λ λ + λ { A ( τ , λ ) A ( τ , λ ) [ J ( λ ρ ) J ( λ ρ ) + J ( λ ρ ) J ( λ ρ )] × cos ( λ t + λ t − λ τ − λ τ + α ( τ , λ ) + α ( τ , λ ))+ A ( τ , λ ) B ( τ , λ ) [ J ( λ ρ ) Y ( λ ρ ) + Y ( λ ρ ) J ( λ ρ )] × cos ( λ t + λ t − λ τ − λ τ + α ( τ , λ ) + β ( τ , λ ))+ B ( τ , λ ) A ( τ , λ ) [ Y ( λ ρ ) J ( λ ρ ) + J ( λ ρ ) Y ( λ ρ )] × cos ( λ t + λ t − λ τ − λ τ + β ( τ , λ ) + α ( τ , λ ))+ B ( τ , λ ) B ( τ , λ ) [ Y ( λ ρ ) Y ( λ ρ ) + Y ( λ ρ ) Y ( λ ρ )] × cos ( λ t + λ t − λ τ − λ τ + β ( τ , λ ) + β ( τ , λ ))] (2.11)– 3 –nd F − = λ λ λ − λ { A ( τ , λ ) A ( τ , λ ) [ J ( λ ρ ) J ( λ ρ ) − J ( λ ρ ) J ( λ ρ )] × cos ( λ t − λ t − λ τ + λ τ + α ( τ , λ ) − α ( τ , λ ))+ A ( τ , λ ) B ( τ , λ ) [ J ( λ ρ ) Y ( λ ρ ) − Y ( λ ρ ) J ( λ ρ )] × cos ( λ t − λ t − λ τ + λ τ + α ( τ , λ ) − β ( τ , λ ))+ B ( τ , λ ) A ( τ , λ ) [ Y ( λ ρ ) J ( λ ρ ) − J ( λ ρ ) Y ( λ ρ )] × cos ( λ t − λ t − λ τ + λ τ + β ( τ , λ ) − α ( τ , λ ))+ B ( τ , λ ) B ( τ , λ ) [ Y ( λ ρ ) Y ( λ ρ ) − Y ( λ ρ ) Y ( λ ρ )] × cos ( λ t − λ t − λ τ + λ τ + β ( τ , λ ) − β ( τ , λ )) } . (2.12)Here κ in eq. (2.10) can be eliminated by a coordinate transformation and, without lossof generality, we set κ = 0 .The solutions (2.9) and (2.10) are the general solutions of the cylindrical gravitationalwave. The solutions given in ref. [24] and the solutions mentioned after are particular casesof the solutions (2.9) and (2.10). In this section, we give an exact cylindrical gravitational monopole radiation solution. Theradiation is a wave produced by a source. For example, the plane electromagnetic wave isnot an electromagnetic radiation and thee electromagnetic wave produced by the antennais an electromagnetic radiation. The gravitational radiation can be recognized by observingif a wave solution has a time-varying source.Next (1) we first choose a particular solution and show that the solution is a gravita-tional wave solution. (2) We separate the monopole radiation solution from this gravita-tional wave solution.
In this section, we present an exact solution, and we show that the solution is a gravi-tational wave by investigating the Weinberg energy-momentum pseudotensor [1] and theLaudau-Lifscitz energy-momentum pseudotensor [26]. The energy-momentum pseudoten-sors are defined to describe the energy-momentum of the gravitational field. If the energy-momentum pseudotensor of the gravitational field is time-varying, we say that the solutionrepresents a gravitational wave [26].Taking A ( τ, λ ) = Aδ ( ω − λ ) δ ( τ ) ,B ( τ, λ ) = Bδ ( ω − λ ) δ ( τ ) ,κ = κ = κ = 0 (3.1)– 4 –n eqs. ( ) and ( ) , where ω = 0 is a constant, we obtain a particular solution ψ = AJ ( ωρ ) cos ( ωt + α ) + BY ( ωρ ) cos ( ωt + β ) , (3.2) γ = f ( ρ ) − ABπ ωt sin ( α − β ) − A ωρJ ( ωρ ) J ( ωρ ) cos (2 ωt + 2 α ) − B ωρY ( ωρ ) Y ( ωρ ) cos (2 ωt + 2 β ) − AB ωρ [ J ( ωρ ) Y ( ωρ ) + J ( ωρ ) Y ( ωρ )] cos (2 ωt + α + β ) (3.3)with α ≡ α (0 , ω ) , β ≡ β (0 , ω ) , and f ( ρ ) = A ω ρ (cid:2) J ( ωρ ) + 2 J ( ωρ ) − J ( ωρ ) J ( ωρ ) (cid:3) + B ω ρ (cid:2) Y ( ωρ ) + 2 Y ( ωρ ) − Y ( ωρ ) Y ( ωρ ) (cid:3) + AB ω ρ [2 J ( ωρ ) Y ( ωρ ) + 4 J ( ωρ ) Y ( ωρ ) − J ( ωρ ) Y ( ωρ ) − J ( ωρ ) Y ( ωρ )] cos ( α − β ) . (3.4)Note here that ψ given by eq. (3.2) has been given in ref. [24]; γ given by eq. (3.3) isobtained in the present paper.In the Cartesian coordinates x = ρ cos φ,y = ρ sin φ, (3.5)the Einstein-Rosen metric reads ds = − e γ − ψ dt + e − ψ ρ (cid:0) x e γ + y (cid:1) dx + e − ψ ρ (cid:0) y e γ + x (cid:1) dy + 2 xyρ e − ψ (cid:0) e γ − (cid:1) dxdy + e ψ dz . (3.6)It can be checked by the numerical method that the metric (3.6) with ψ given by eq. (3.2)and γ given by eq. (3.3) has a time-varying Weinberg energy-momentum pseudotensor [1]and a time-varying Laudau-Lifscitz energy-momentum pseudotensor [26].Besides, there is a special case of the solutions (3.2) and (3.3) satisfying the out-goingwave condition: when t − r = const , ψ and γ remain unchanged at ρ → ∞ . When A = B,β = α − π , (3.7)the approximation of ψ and γ at ρ → ∞ are ψ = A r πωρ cos (cid:18) ωt − ωρ + α + 14 π (cid:19) ,γ = A π [cos (2 ωt − ωρ + 2 α ) − ωt − ωρ )] . (3.8)In addition, it can be found that when the out-going wave condition is satisfied, the curva-ture of the spacetime vanishes at ρ → ∞ . That is, the spacetime is asymptotically flat inspite of that the metric is not asymptotic to the Minkowski metric.– 5 – .2 Gravitational radiation In this section, we show that the solutions (3.2) and (3.3) contain a gravitational radiation.As mentioned above, the gravitational radiation is a gravitational wave solution with asource.The solutions (3.2) and (3.3) have two special cases. A = 0 : ψ rad = BY ( ωρ ) cos ( ωt + β ) , (3.9) γ rad = B ω ρ (cid:2) Y ( ωρ ) + 2 Y ( ωρ ) − Y ( ωρ ) Y ( ωρ ) (cid:3) − B ωρY ( ωρ ) Y ( ωρ ) cos (2 ωt + 2 β ) , (3.10) B = 0 : ψ nrad = AJ ( ωρ ) cos ( ωt + α ) , (3.11) γ nrad = A ω ρ (cid:2) J ( ωρ ) + 2 J ( ωρ ) − J ( ωρ ) J ( ωρ ) (cid:3) − A ωρJ ( ωρ ) J ( ωρ ) cos (2 ωt + 2 α ) . (3.12)These two cases, eqs. (3.9), (3.10), (3.11), and (3.12), have been given in ref. [24].In this paper, we point out that ψ rad and γ rad reprensent radiations, but ψ nrad and γ nrad do not reprensent radiations. Here we use the subscripts "rad" to denote the gravitationalradiation and use "nrad" to denote the nonradiation gravitational wave.Next we show that ψ rad and γ rad describe a gravitational monopole radiation, whichhas a time-varying energy density or a monopole, and ψ nrad and γ nrad are pure gravitationalwaves without sources.The solutions (3.2) and (3.3) represent a gravitational wave in the vacuum for ρ > .Nevertheless, when ρ = 0 , there may exist a source. The singularity in the solutions (3.2)and (3.3), according to ref. [24], might be interpreted as a matter presented along the z axis. Just as the Coulomb potential /r , when r = 0 , the solution is a vacuum solution;nevertheless, there is a point charge at r = 0 . We use a standard mathematical analysismethod to calculate the energy-momentum tensor at ρ = 0 . More details of this methodcan be found in our previous work [27]. Replacing ρ by p ρ + ǫ in eqs. (3.2) and (3.3) andsubstituting the metric (2.1) into eqs. (2.2), (2.3), (2.4) and (2.5), by the Einstein equation G µν = 8 πT µν , we arrive at T ( ω, ǫ ) = T ( ω, ǫ )= e ψ − γ π ω ǫ ρ + ǫ h AJ (cid:16) ω p ρ + ǫ (cid:17) cos ( ωt + α ) + BY (cid:16) ω p ρ + ǫ (cid:17) cos ( ωt + β ) i , (3.13) T ( ω, ǫ ) = e ψ − γ π ω ǫ ρ p ρ + ǫ h AJ (cid:16) ω p ρ + ǫ (cid:17) sin ( ωt + α ) + BY (cid:16) ω p ρ + ǫ (cid:17) sin ( ωt + β ) i × h AJ (cid:16) ω p ρ + ǫ (cid:17) cos ( ωt + α ) + BY (cid:16) ω p ρ + ǫ (cid:17) cos ( ωt + β ) i , (3.14)– 6 – ( ω, ǫ ) = e ψ − γ π ω ǫ ρ + ǫ h AJ (cid:16) ω p ρ + ǫ (cid:17) cos ( ωt + α ) + BY (cid:16) ω p ρ + ǫ (cid:17) cos ( ωt + β ) i − e ψ − γ π ω ǫ p ρ + ǫ [ A J (cid:16) ω p ρ + ǫ (cid:17) J (cid:16) ω p ρ + ǫ (cid:17) cos (2 ωt + 2 α )+ B Y (cid:16) ω p ρ + ǫ (cid:17) Y (cid:16) ω p ρ + ǫ (cid:17) cos (2 ωt + 2 β )] − e ψ − γ π ω ǫ p ρ + ǫ AB [ J (cid:16) ω p ρ + ǫ (cid:17) Y (cid:16) ω p ρ + ǫ (cid:17) + J (cid:16) ω p ρ + ǫ (cid:17) Y (cid:16) ω p ρ + ǫ (cid:17) ] cos (2 ωt + α + β ) , (3.15) T ( ω, ǫ ) = e ψ − γ π ω ǫ ρ + ǫ h AJ (cid:16) ω p ρ + ǫ (cid:17) cos ( ωt + α ) + BY (cid:16) ω p ρ + ǫ (cid:17) cos ( ωt + β ) i + e ψ − γ π ω ǫ ρ + ǫ h AJ (cid:16) ω p ρ + ǫ (cid:17) cos ( ωt + α ) + BY (cid:16) ω p ρ + ǫ (cid:17) cos ( ωt + β ) i − e ψ − γ π ω ǫ p ρ + ǫ [ A J (cid:16) ω p ρ + ǫ (cid:17) J (cid:16) ω p ρ + ǫ (cid:17) cos (2 ωt + 2 α )+ B Y (cid:16) ω p ρ + ǫ (cid:17) Y (cid:16) ω p ρ + ǫ (cid:17) cos (2 ωt + 2 β )] − e ψ − γ π ω ǫ p ρ + ǫ AB [ J (cid:16) ω p ρ + ǫ (cid:17) Y (cid:16) ω p ρ + ǫ (cid:17) + J (cid:16) ω p ρ + ǫ (cid:17) Y (cid:16) ω p ρ + ǫ (cid:17) ] cos (2 ωt + α + β ) . (3.16)The energy-momentum tensor is given by T µν ( ω ) = lim ǫ → T µν ( ω, ǫ ) . (3.17)When ρ = 0 , we have T µν ( ω ) = lim ǫ → T µν ( ω, ǫ ) = 0 . (3.18)When ρ = 0 , using lim z → J ( z ) = 1 , lim z → Y ( z ) = 2 π ln z, lim z → J ν ( z ) = 1Γ ( ν + 1) (cid:16) z (cid:17) ν , ν = 0 , lim z → Y ν ( z ) = − Γ ( ν ) π (cid:16) z (cid:17) − ν , ν = 0 , (3.19)– 7 –e have T ( ω ) = lim ǫ → T ( ω, ǫ ) = T ( ω, ǫ ) = e ψ − γ B π cos ( ωt + β ) lim ǫ → ǫ ( ρ + ǫ ) ,T ( ω ) = lim ǫ → T ( ω, ǫ ) = − e ψ − γ B π sin (2 ωt + 2 β ) lim ǫ → ωǫ ρ ( ρ + ǫ ) ln (cid:16) ω p ρ + ǫ (cid:17) ,T ( ω ) = lim ǫ → T ( ω, ǫ ) = e ψ − γ B π cos ( ωt + β ) lim ǫ → ǫ ( ρ + ǫ ) ,T ( ω ) = lim ǫ → T ( ω, ǫ ) = e ψ − γ B π (cid:20) cos ( ωt + β ) − πB cos ( ωt + β ) (cid:21) lim ǫ → ǫ ( ρ + ǫ ) . (3.20)By use of [28] lim ǫ → ǫ ( x + y + ǫ ) = πδ ( x ) δ ( y ) , we have T ( ω ) = T ( ω ) = lim ǫ → T ( ω, ǫ ) = e ψ − γ B π δ ( x ) δ ( y ) cos ( ωt + β ) , (3.21) T ( ω ) = lim ǫ → T ( ω, ǫ ) = e ψ − γ B π δ ( x ) δ ( y ) cos ( ωt + β ) , (3.22) T ( ω ) = lim ǫ → T ( ω, ǫ ) = e ψ − γ B π δ ( x ) δ ( y ) (cid:20) cos ( ωt + β ) − πB cos ( ωt + β ) (cid:21) . (3.23)Because lim ǫ → T ( ω, ǫ ) T ( ω, ǫ ) = 0 ,T ( ω ) should be regraded as zero: T ( ω ) = 0 . (3.24) T ( ω ) = 0 can be explained from a different way which we will show below.Integrating T ( ω, ǫ ) over the whole space E ( ω, ǫ ) ≡ Z √ gdρdθdφT ( ω, ǫ )= ω ǫ L z A (cid:0) J ( ωǫ ) + J ( ωǫ ) (cid:1) cos ( ωt + α ) + B (cid:0) Y ( ωǫ ) + Y ( ωǫ ) (cid:1) cos ( ωt + β )+ 2 AB ( J ( ωǫ ) Y ( ωǫ ) + J ( ωǫ ) Y ( ωǫ )) cos ( ωt + α ) cos ( ωt + β )] (3.25)and taking the limit ǫ → give E ( ω ) ≡ lim ǫ → E ( ω, ǫ ) = B L z π cos ( ωt + β ) (3.26)– 8 –ith L z ≡ R ∞−∞ dz . Again, integrating T ( ǫ ) over the whole space P ρ ( ω, ǫ ) = Z √ gdρdθdφT ( ω, ǫ )= ω ǫ L z A (cid:0) ωǫJ ( ωǫ ) + ωǫJ ( ωǫ ) − J ( ωǫ ) J ( ωǫ ) (cid:1) sin ( ωt + α ) cos ( ωt + α )+ B (cid:0) ωǫY ( ωǫ ) + ωǫY ( ωǫ ) − Y ( ωǫ ) Y ( ωǫ ) (cid:1) sin ( ωt + β ) cos ( ωt + β )+ AB ( ωǫJ ( ωǫ ) Y ( ωǫ ) + ωǫJ ( ωǫ ) Y ( ωǫ ) − J ( ωǫ ) Y ( ωǫ )) cos ( ωt + α ) sin ( ωt + β )+ AB ( ωǫJ ( ωǫ ) Y ( ωǫ ) + ωǫJ ( ωǫ ) Y ( ωǫ ) − J ( ωǫ ) Y ( ωǫ )) sin ( ωt + α ) cos ( ωt + β )] (3.27)and taking the limit ǫ → give P ρ ( ω ) ≡ lim ǫ → P ρ ( ω, ǫ ) = lim ǫ → B L z ω π ǫ ln ǫ = 0 . (3.28)Then the energy-momentum tensor of the metric (2.1) with ψ given by eq. (3.2) and γ given by eq. (3.3) is T µν = e ψ − γ B π δ ( x ) δ ( y ) × diag (cid:18) cos ( ωt + β ) , cos ( ωt + β ) , cos ( ωt + β ) , cos ( ωt + β ) − πB cos ( ωt + β ) (cid:19) (3.29)with x = ρ cos φ and y = ρ sin φ .The energy-momentum tensor corresponding to ψ nrad and γ nrad in eqs. (3.11) and(3.12) is zero and the energy-momentum tensor corresponding to ψ rad and γ rad , eq. (3.29),is not zero. In other words, the solutions with ψ rad and γ rad in eqs. (3.9) and (3.10)have a time-varying energy density. In general relativity, the total energy itself is themonopole of the source. The solution with ψ rad and γ rad describes a gravitational monopoleradiation with the time-varying total energy of the source (3.29). The solution with ψ nrad and γ nrad is a wave solution without sources, which has no singularity and is just like aplane electromagnetic wave.There is a problem in the gravitational quadrupole radiation. If insisting that theenergy of the source is invariable in a process of gravitational radiations, where the energyof the gravitational radiation comes form. If the energy of the gravitational radiation comesfrom the source, then the total energy of the source should decrease with time instead ofbeing invariable. In general relativity, the charge of the gravitational filed is the energydensity, just like the charge of the electromagnetic field is the electric charge density. Themonopole in general relativity is the integration of the energy density, i.e., the total energy.The contribution of the gravitational radiation should be made by the variation of themonopole. In the following we show that the solution given above contains resonance structures.– 9 –o see this, we take A ( τ, λ ) = Aδ ( ω − λ ) δ ( τ ) ,B ( τ, λ ) = Bδ ( ω − λ ) δ ( τ ) ,κ = κ = κ = 0 (4.1)in the general solution (2.9) and (2.10): ψ = A J ( ω ρ ) cos ( ω t + α ) + B Y ( ω ρ ) cos ( ω t + β ) , (4.2) γ = − A ω ρJ ( ω ρ ) J ( ω ρ ) cos (2 ω t + 2 α ) − B ω ρY ( ω ρ ) Y ( ω ρ ) cos (2 ω t + 2 β ) − A B ω ω ω + ω ρ [ J ( ω ρ ) Y ( ω ρ ) + Y ( ω ρ ) J ( ω ρ )] cos ( ω t + ω t + α + β )+ A ω ρ (cid:2) J ( ω ρ ) + 2 J ( ω ρ ) − J ( ω ρ ) J ( ω ρ ) (cid:3) + B ω ρ (cid:2) Y ( ω ρ ) + 2 Y ( ω ρ ) − Y ( ω ρ ) Y ( ω ρ ) (cid:3) − A B ω ω ω − ω ρ [ J ( ω ρ ) Y ( ω ρ ) − Y ( ω ρ ) J ( ω ρ )] cos ( ω t − ω t + α − β ) , (4.3)where A = A (0 , ω ) , B = B (0 , ω ) , α = α (0 , ω ) , and β = β (0 , ω ) .The resonance occurs when ω = ω . When ω → ω , γ diverges due to the existenceof the factor ω − ω . Taking ǫ = ω − ω → , we have lim ω → ω γ = lim ǫ → γ ∼ − A B π ω cos ( α − β ) lim ǫ → ǫ . (4.4)The resonance also appears in eq. (3.3). In eqs. (3.2) and (3.3), the radiation part andthe nonradiation part have the same frequency ω . When A = 0 and B = 0 , γ in eq. (3.3)has an aperiodic term being proportional to t . This is also a resonance which means theinpact stores up and increase with the time t .The solutions (4.2) and (4.3) show that the radiation part (with the frequency ω )resonates with the nonradiation part (with the frequency ω ). In this section, we consider the interaction between the radiations. The theory of gravityis a nonlinear theory, so gravitational radiations in principle interact with each other.
The factor ψ in the metric (2.1) satisfies a linear equation, eq. (2.8), so ψ = ψ + ψ (5.1)is also a solution when ψ and ψ are the solutions of eq. (2.8). For convenience, we take ψ and ψ given by eq. (3.9) as an example, ψ = B Y ( ω ρ ) cos ( ω t + β ) ,ψ = B Y ( ω ρ ) cos ( ω t + β ) . (5.2)– 10 –hese two radiations have the frequencies ω and ω , respectively. Substituting eq. (5.1)into eqs. (2.6) and (2.7), we have γ = γ + γ − γ int , (5.3)where γ and γ are given by eq. (3.10) with frequencies ω and ω and γ int = B B ω ω ω + ω ρ [ Y ( ω ρ ) Y ( ω ρ ) + Y ( ω ρ ) Y ( ω ρ )] cos ( ω t + ω t + β + β )+ B B ω ω ω − ω ρ [ Y ( ω ρ ) Y ( ω ρ ) − Y ( ω ρ ) Y ( ω ρ )] cos ( ω t − ω t + β − β ) . (5.4) γ int can be understood as an interaction between two gravitational radiations.It should be emphasized that in this case when ω = ω = ωγ int = B B ωρY ( ωρ ) Y ( ωρ ) cos (2 ωt + β + β ) − B B ω ρ (cid:2) Y ( ωρ ) + 2 Y ( ωρ ) − Y ( ωρ ) Y ( ωρ ) (cid:3) cos ( β − β ) (5.5)does not diverge. Two radiations do not resonate. The resonance occurs only between theradiation part and the non-radiation part. Now we calculate the energy-momentum tensor of the radiations (5.1) and (5.3).By the same procedure in section 2, by eqs. (3.21), (3.22), (3.23), we have T ( ω , ω ) = T ( ω , ω ) = T ( ω ) + T ( ω ) + T int ( ω , ω ) ,T ( ω , ω ) = T ( ω , ω ) = 0 ,T ( ω , ω ) = T ( ω ) + T ( ω ) + T int ( ω , ω ) ,T ( ω , ω ) = T ( ω ) + T ( ω ) + T int ( ω , ω ) (5.6)with the interaction term T int ( ω , ω ) = e ψ − γ B B π δ ( x ) δ ( y ) cos ( ω t + β ) cos ( ω t + β ) . (5.7)It can be seen that T µν ( ω , ω ) which involves both ω and ω can be written in threeparts: T µν ( ω ) which involves only ω , T µν ( ω ) which involves only ω , and T int ( ω , ω ) which involves both ω and ω . Here T int ( ω , ω ) is an interaction term due to the nonlin-earity of the gravity.Nevertheless, though the nonlinearity of the gravity leads to the existence of the inter-action term T int ( ω , ω ) , the time average of T int vanishes: h T int i = Z ∞ dtT int ( ω , ω ) = 0 . (5.8)The vanishing of the time average of interaction term implies that the result of a long-timemeasurement may be linear. That is, when the typical time of a detector is more larger– 11 –han the period of a gravitational wave, the source of radiations observed may be a linearone though the gravitational wave is nonlinear.As an analogy, we consider a simple example of an energy superposition of two planeelectromagnetic waves. When two plane electromagnetic waves E = A cos ( ω t + α ) and E = A cos ( ω t + β ) superpose together, the electric field is E = E + E = A cos ( ω t + α ) + A cos ( ω t + β ) , where E is the electric field and A is the amplitude. The energy density of the electric fieldis ε = E = ( E + E ) = ε + ε + ε int , (5.9)where ε = A cos ( ω t + α ) , ε = A cos ( ω t + β ) , and the interference term ε int = 2 A A cos ( ω t + α ) cos ( ω t + β ) . (5.10)The time average of ε int vanishes: h ε int i = Z ∞ dtε int = 0 . (5.11)The interaction behavior of the energy-momentum tensor (5.6) is similar to the interferencebehavior of the energy density (5.9). In linear approximation, the gravitational radiation is a quadrupole radiation, two radia-tions superpose linearly and do not interact with each other. Only the resonance betweenthe radiation and the detector is considered as the probing scheme of the radiation.In this paper, we discuss the gravitational radiation based on the exact cylindrical grav-itational wave solutions rather than the linear approximation. (1) We present a cylindricalgravitational monopole radiation solution which indicates that the leading contribution ofthe gravitational radiation may be the monopole rather than a quadrupole. (2) We con-sider a new kind of the resonance between gravitational radiations. Expect the gravity, theradiation and the spacetime are two separate concepts. Nevertheless, in general relativ-ity, the concepts of the radiation and the spacetime are mixed up. The radiation and thespacetime are both described by the metric. We attempt to separate the radiation and thespacetime in this paper. Following this idea, we regard the metric without the radiation asa spacetime background. That is, ψ nrad and γ nrad in eqs. (3.11) and (3.12) are regarded asthe spacetime background. This idea works well. The spacetime is also a kind of matter. ψ nrad and γ nrad are periodic with the intrinsic frequency ω . When the gravitational radi-ation ψ rad ( γ rad ) with the same frequency act on the periodic spacetime background, theresonance occurs. That makes the resonance in the spacetime consistent with the resonancein Newton mechanics or other physical theories expect the gravity. We suppose that the– 12 –esonance between the gravitational radiation and the spacetime background exists in spiteof the symmetry of the system. In recent years, the gravitational wave detection makesrapid progress. It can be expected that the resonance between the gravitational radiationand the spacetime background can be found. (3) We investigate the interaction betweenthe cylindrical gravitational radiations. The interaction arises both in the metric and theenergy-momentum tensor. Nevertheless, the time average of the interaction term in theenergy-momentum tensor vanishes, which indicates that the energy-momentum tensor donot interact with each other directly in the time-averaging level.In future works, based on gravitational monopole radiation solutions obtained in thispaper, we may figure out if the gravitational dipole radiation solution exists or not. Besides,the gravitational radiation will lead to the energy loss of the source. With the conservationlaw of the energy, we may define the energy of the cylindrical gravitational radiation in ourframework. We can also consider that the matter wave resonates with the gravitationalradiation based on the preceding work on scattering [29–31]. Acknowledgments
We are very indebted to Dr G. Zeitrauman for his encouragement. This work is supportedin part by Nankai Zhide foundation and NSF of China under Grant No. 11575125 and No.11675119.
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