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Gravitational two solitons in Levi-Civita spacetime

Applying the Pomeransky inverse scattering method to the four-dimensional vacuum Einstein equations and using the Levi-Civita solution as a seed, we construct a two-soliton solution with cylindrical symmetry. In our previous work, we constructed the one-soliton solution with a real pole and showed that the singularities that the Levi-Civita background has on an axis can be removed by the choice of certain special parameters, but it still has unavoidable null singularities, as usual one-solitons do. In this work, we show that for the two-soliton solutions, any singularities can be removed by suitable parameter-setting and such solutions describe the propagation of gravitational wave packets. Moreover, in terms of the two-soliton solutions, we mention a time shift phenomenon, the coalescence and the split of solitons as the nonlinear effect of gravitational waves.

that it can generate regular solutions even in higher dimensions. In fact, with regard to vacuum solutions in five dimensions, all currently known asymptotically flat black hole solutions can be discovered or rederived by this method [6,7].
In a cylindrically symmetric case, a diagonal form of a metric makes the vacuum Einstein equations extremely simple structure of a linear wave equation in a flat background. For instance, the Einstein-Rosen metric can be interpreted as superposition of cylindrical gravitational waves with the + mode only [1,10]. However, the existence of off-diagonal nonzero components of a metric drastically changes the structure of the Einstein equations and it yields the × mode together with nonlinearity. Piran et al. [11] numerically studied the nonlinear interaction of cylindrical gravitational waves with both polarization modes such as the gravitational Faraday effect. Tomimatsu [12] studied the gravitational Faraday rotation for the cylindrical gravitational solitons generated by the inverse scattering technique [9]. As one of new attempts to understand strong gravitational effects, one of the present authors has recently constructed the cylindrically symmetric soliton solutions from a Minkowski seed by the Pomeransky inverse scattering method, and has clarified the behavior of the soliton solution including the effect similar to the gravitational Faraday rotation [13,14].
In our previous paper, using the Pomeransky inverse scattering method and regarding the Levi-Cività metric as a seed, we constructed the one-soliton solution that does not admit staticity but cylindrical symmetry [15]. Although Levi-Cività spacetime has singularities on the axis except for Minkowski spacetime, we showed that for the one-soliton solution, such singularities disappear by a certain choice of parameters. However, this solution has singularities on the light cone for any parameters, which is an unavoidable common property for all one-soliton solutions with a real pole and it is well known that such a problem can be resolved by considering two-soliton solutions with two complex-conjugate poles. Therefore, in this paper, we will construct a two-soliton solution with two complex conjugate poles by the Pomeransky method and the Levi-Cività seed. Furthermore, we will show that for certain parameters, the spacetime is entirely free from any singularities even on the axis as well as on the light cone.
In the following section, we will present a two-soliton solution with complex conjugate poles in Levi-Cività spacetime. In Sec. III we will analyze the obtained two-soliton solution by computing the amplitudes and polarization angles for ingoing and outgoing waves and will see the difference from the one-soliton solution. In Sec. IV, we will give the summary and discussion on our results.

II. TWO-SOLITON SOLUTION
The most general cylindrically symmetric solution to the four-dimensional vacuum Einstein equations can be described in the Kompaneets-Jordan-Ehlers form [16] where the functions ψ, ω, and γ depend on the time coordinate t and radial coordinate ρ only. Let us define a 2 × 2 metric g and a metric function f by respectively.
As shown by Belinsky and Sakharov [9], the vacuum Einstein equation with cylindrical symmetry (in general, with two commuting Killing vectors) is completely integrable, and it admits a pair of linear equations, which is called Lax pair where λ is a spectral parameter (complex parameter), D ρ and D t are the commuting differential operators defined by and U and V are the 2 × 2 matrices defined by U = ρ g ,ρ g −1 and V = ρ g ,t g −1 , respectively, and the generating matrix Ψ(λ, ρ, t) is a 2 × 2 matrix such that Ψ(λ = 0, ρ, t) = g. Therefore, one can obtain the 2 × 2 metric g from the generating matrix Ψ(λ, ρ, t) (more precisely, by normalizing the metric such that det g = ρ 2 ).
To generate new cylindrically symmetric solutions, we start from one solution of the Lax pair such that Ψ 0 (λ = 0, ρ, t) = G 0 (G 0 : a seed metric) and then must dress it as where in particular, for a two-soliton solution, the dressing matrix χ is given by with where, following the notation in [2], we use the successive Latin subscripts to denote the summation of the z and φ components, and µ i (i = 1, 2) are defined by The constant vectors m   (k: a nonzero constant)). After dressing, taking the limit of λ → 0, one can obtain the metric g = Ψ(λ = 0, ρ, t). In the Belinsky-Zakarov method, one must normalize the obtained metric. Furthermore, for reality of the metric, one must put a 1 =ā 2 .
In this paper, following the Pomeransky method [8], in which one need not normalize the metric obtained from the dressed generating matrix. For the Levi-Cività metric, which we choose as a seed, the 2 × 2 metric g 0 and the metric function f 0 are written as respectively, where the parameters b and d are independent, and are assumed to be positive without loss of generality. Let us remove trivial solitons at t = a 1 and t = a 2 with a vector (1, 0) (this corresponds to the inverse transformation g 0 = χ(λ = 0, ρ, t) −1 g 0 with m (k) 0 = (1, 0) (k = 1, 2) in the Belinski-Zakharov method), and then we have the metric and dress the generating matrix Ψ 0 corresponding to the metric g 0 : Next, add back nontrivial solitons with BZ vectors m (1) 0 = (1, a) and m (2) 0 = (1,ā), and then we obtain a two-soliton solution as where Γ 0kl is Γ kl evaluated at a = 0.
This is how we can obtain a two-soliton solution, whose metric can be written in the Kompaneets-Jordan-Ehlers form (1), where the functions ψ, ω and γ are explicitly written as Here, [ ] denotes the real part of [ ]. When d = 1, we recover the metric in [14], in which case the seed is a Minkowski metric, and when c = 0, this metric reduces to the Levi-Cività metric. 1 The function ω can be rewritten in terms of X and Y as A shift of the time coordinate allows us to fix the parameters a i as a pure imaginary where i denotes the imaginary unit and q is assumed to be positive without loss of generality.

A. Asymptotic behaviors of gravitational waves
In this subsection, we analyze the asymptotic behaviors of gravitational waves at spacetime boundaries for the obtained two-soliton solution by computing the asymptotic behaviors of the metric and ingoing and outgoing wave amplitudes, where we basically follow the definitions given in Refs. [11,12] (see Appendix A). First, let us consider the symmetric axis ρ = 0. In order to focus on the physical propagation process of cylindrically symmetric gravitational soliton waves, we investigate the condition for the parameters c and d such that there do not exist any gravitational sources on the axis, i.e., the C -energy density could become finite on the axis.
Note that the C -energy density is proportional to γ ,ρ . Near ρ 0, the metric function e 2γ behaves asymptotically as where [ ] denotes the imaginary part of [ ].
For d = 1, 3, 5, the C -energy density diverges on ρ = 0 as but for d = 1, 3, 5, it vanishes as where we assume c I = 0 for d = 3 because if not, γ ,ρ diverges on ρ = 0. The case of d = 1 (two-soliton solution obtained from Minkowski metric) have been previously analyzed by one of the authors [14]. This is why we focus on d = 3 (c I = 0) and d = 5 only in what follows.
Near ρ = 0, the metric behaves as deficit angle ∆ on the axis is defined as where φ is replaced byφ for d = 3, and ∆φ is the periodicity of the angular coordinate. Then we By choosing a suitable value of ∆φ to be ∆ = 0, we get rid of conical singularities on the axis.
Near ρ = 0, the ingoing and outgoing total amplitudes, A and B, behave as respectively. The polarization angles, θ A and θ B , take the following asymptotic forms tan 2θA − tan 2θB tan 2θA − tan 2θB At late time t → ∞ or at early time t → −∞, the metric behaves as The total amplitudes and polarization angles behave as and At spacelike infinity ρ → ∞, the metric behaves as The total amplitudes and polarization angles behave as and At past null infinity u → −∞, the metric behaves as where ω p and f p are defined by The asymptotic forms of the amplitudes and the polarization angles are where and At future null infinity v → ∞, the metric behaves as where ω f and f f are defined by The asymptotic forms of the amplitudes and the polarization angles are where N f and D f are defined by and and v = ∞ for d = 3, 5 is exactly the same as for d = 1 discussed in [14]. In Fig. 3, the blue- colored graphs and the red-colored graphs denote lim u→−∞ A √ −u and lim v→∞ B √ v for (k, θ) = (1000, nπ/6), respectively, where we take n = 1, 3, 4 to see the typical behaviors of waves. The dashed and dotted blue-colored (red-colored) graphs denote at null infinity u = −∞ (v = ∞), respectively. It is worth noting that for large value of k, both amplitudes near past and future null infinity are composed of × mode waves.
To see that a time shift phenomenon happens, as was already explained in [14], let us consider the incoming massless test particle which starts from past null infinity, propagates along the null geodesic v = 0, is reflected on the axis ρ = 0 and then propagates to future null infinity along the null geodesic u = 0. An incident wave packet has a peak near v < 0, while a reflectional wave packet has a peak at u > 0. This means that an observer at past null infinity sees an ingoing wave packet earlier than an incoming radial photon, but at future null infinity he sees the outgoing wave packet after the outgoing photon. We may consider that a gravitational wave packet can propagate at slower speed than the light velocity.

D. Coalescence and split of solitons
In addition to the time shift phenomena, when k q, physically and mathematically interesting phenomenon such as coalescence and split of solitons happens, as pointed out in [14]. As seen in the blue-colored and red-colored graphs show the ingoing amplitude A √ −u at past null infinity u = −∞ and outgoing amplitudes B √ v at future null infinity v = ∞, respectively, and the dashed and dotted graphs show the + mode wave amplitudes and the × mode wave amplitudes, respectively. Here, (a) and (b) in Fig. 4 are plotted for the parameters (k, θ, q) = (5, nπ/6, q) and (1/1000, nπ/6, q), respectively, where we take n = 1, 3, 4. From these graphs, we can see that at least, either of ingoing and outgoing waves can have two peaks.
For n = 3, there are two incident wave packets, one with a small peak and another with a large peak near past null infinity, and after reflection we have two reflectional wave packets, one with a small peak and one with a large peak near future null infinity. This obviously shows that two gravitational solitons collide with each other, which occurs near the axis ρ 0, and then the larger one of two solitons overtakes the smaller one (see the case n = 3 in Fig. 1).
For n = 4, the incident waves incoming from past null infinity u = −∞ (the blue-colored graphs) have two peaks and the reflectional waves outgoing to future null infinity v = ∞ (the red-colored graphs) have one peak only. As seen from the case of n = 4 in Fig. 1, two wave packets seem to change into one wave packet after the reflection at the axis. Therefore, this can be interpreted as the coalescence of two solitons.
In contrast to n = 4, for n = 1, the incident waves (the blue-colored graphs) have one peak near past null infinity, and the reflectional waves (the red-colored graphs) have two peaks near future null infinity. From the case n = 1 in Fig. 1, one wave packet seems to change into two wave packets after the reflection at the axis. This phenomenon can be interpreted as the split of soliton waves.
The phenomena of the coalescence and split do not happen for other solitons than ones in general relativity.
Finally, we comment on which polarization mode contributes to the total ingoing and outgoing amplitudes at null infinity. As seen from the behaviors of the dashed and dotted graphs in Fig. 4, for the small values of k the × mode only mainly contributes to both ingoing and outgoing amplitudes, while for the large values of k both the + and × modes contribute to them to the same order.

IV. CONCLUSIONS
In this paper, applying the Pomeransky method to a cylindrically symmetric spacetime and starting from the Levi-Cività background, we have constructed the two-soliton solutions with two complex conjugate poles to the vacuum Einstein equations with cylindrical symmetry. As shown in the previous work, although the Levi-Cività spacetime generally includes singularities on its axis of symmetry, for the one-soliton solution with d = 3, such singularities can be removed [15]. In this work, we have analytically shown that as for the two-soliton solution with d = 3, 5, singularities on an axis entirely disappear in addition to null singularities which one solitonic solution with a real pole has in common. The regular solutions with d = 3, 5 describe the propagation of gravitational wave packets that come into the region near the symmetric axis from past null infinity, then leave for future null infinity after reflection at the axis. Moreover, we have studied nonlinear effect of solitons such as a time shift phenomenon and the gravitational Faraday effect. We have seen that these effects are essentially similar to the case of d = 1, which was investigated in [14].
Finally, we point out the essential differences of two-soliton solutions with d = 3, 5 from ones with d = 1. (iii) Null infinity v → ∞ or u → −∞: Even though the asymptotic forms of the metric at null infinity entirely differ for each of d, the asymptotic forms of wave packet are exactly same. Therefore, as happens for d = 1, for d = 3, 5, two gravitational solitons can coalesce into a single soliton, and also that a single soliton can split into two via the nonlinear effect of gravitational waves. Such phenomena cannot be seen for solitons of other integrable equations such as solitons of the KdV equation.
The amplitudes of ingoing and outgoing waves with the + mode are defined as, respectively, and the amplitudes of ingoing and outgoing waves with the × mode are defined as, respectively, where the advanced ingoing and outgoing null coordinates u and v are defined by u = (t − ρ)/2 and v = (t + ρ)/2, respectively. The total amplitudes of ingoing and outgoing waves are defined by respectively, and moreover the total amplitude of cylindrical gravitational waves is written as The polarization angles θ A and θ B for the respective wave amplitudes are defined as The vacuum Einstein equations can be written in terms of these quantities as follows: and