Resonant Scattering of Gravitational Waves with Electromagnetic Waves

Ruodi Yan(闫若帝)$^{1}$ and Yun Kau Lau(刘润球)$^{2\hyperlink{s}{*}}$

doi:10.1088/0256-307X/40/10/100401

⇦Chinese Physics Letters, 2023, Vol. 40, No. 10, Article code 100401⇨ Resonant Scattering of Gravitational Waves with Electromagnetic Waves Ruodi Yan (闫若帝)1 and Yun Kau Lau (刘润球)2* Affiliations 1Department of Physics, Beijing Normal University, Beijing 100875, China 2Institute of Applied Mathematics, Morningside Center of Mathematics, LSSC, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, China Received 2 May 2023; accepted manuscript online 21 September 2023; published online 13 October 2023 ^*Corresponding author. Email: lau@amss.ac.cn Citation Text: Yan R D and Lau Y K 2023 Chin. Phys. Lett. 40 100401 Abstract A certain class of exact solutions of Einstein Maxwell spacetime in general relativity is discussed to demonstrate that at the level of theory, when certain parametric resonance condition is met, the interaction of electromagnetic field with a gravitational wave will display certain Lyapunov instability and lead to exponential amplification of a gravitational wave train described by certain Newman–Penrose component of the Weyl curvature. In some way akin to a free electron laser in electromagnetic theory, by the conversion of electromagnetic energy into gravitational energy in a coherent way, the feasibility of generating a pulsed-laser-like intense beam of gravitational wave is displayed.

DOI:10.1088/0256-307X/40/10/100401 © 2023 Chinese Physics Society Article Text In electromagnetic theory, an intense light beam, known to have a diverse number of practical applications in science and engineering, usually is generated by a laser whose operating principle depends on the quantum stimulated emission of photons by molecules in a medium. A shortcoming of this kind of laser sources is that frequency of the light beam emitted depends heavily on the energy levels of the kind of molecules employed and as a result the tunable frequency range is severely limited. One way to overcome this spectral limitation is to employ free electron laser whose operating principle is classical in nature. The underlying physics hinges on the resonant interaction between the electromagnetic field generated by a highly relativistic electron beam and that of an oscillating magnetic field. When frequency of electromagnetic field of an electron beam and that generated by an oscillating magnetic field come close to each other, resonant interaction occurs between the two electric fields and an intense beam of light is generated.[1] The aim of the present work is to show that, within the context of Einstein's general theory of relativity, when certain resonance condition is met, a spacetime describing the interaction between gravitational wave and electromagnetic field will display certain Lyapunov instability in a sense to be described in what follows. The pumping of electromagnetic energy into a gravitational field will also generate an intense beam of gravitational waves. It is known that the ergosphere of a rotating black hole is able to amplify gravitational wave by means of an electromagnetic field.[2] Our work presents another way of amplification without necessarily the mediation of a black hole. Though the underlying operational principles behind a free electron laser and that presented in this work are different, both share the common feature that certain resonance condition is required to generate an intense beam. On the basis of this loose analogy and, as we shall see later, the coherent conversion between gravitational and electromagnetic energy, the beam of intense gravitational wave may be termed as “gravilaser”. It should be admitted that the class of spacetimes considered is highly idealistic and quite remote from experimental realisation and practical applications in the near future, still it demonstrates that at the level of theory that such a mechanism of generating an intense beam of gravitational wave train in laboratory is allowed by general relativity. This is in contrast to the general belief that gravitational wave generated in laboratory is too weak to be considered (see, e.g., Ref. [3]). In this Letter, the class of spacetimes to be considered in the present work is first described. Then we present a theoretical discussion on the parametric resonance (or Lyapunov instability) phenomena originated from the Einstein Maxwell field equations and the nature of spacetime singularity of the class of spacetime concerned. Numerical simulation is presented to illustrate in details how an intense beam of pulse gravitational wave train is generated by the coherent conversion of electromagnetic energy into gravitational energy and the waveform of the intense beam. Finally, we provide some brief remarks to conclude our work. Background and Notations. We give a brief description of the class of spacetime to be considered in what follows, therefrom we develop our work. Consider a spacetime whose metric is given as[4,5] \begin{align} ds^2=\,&2e^{2\zeta} du dv - e^{2\eta} \cosh{\omega} (dx^2 + e^{2\chi} dy^2)\notag\\ &- e^{2\eta + \chi} \sinh{\omega}\, dx dy, \tag {1} \end{align} where $\zeta$, $\eta$, $\omega$, and $\chi$ are functions of null coordinates $u, v$ to be determined by the Einstein Maxwell equations; $(\partial/\partial x)^a$ and $(\partial / \partial y)^a$ are spatial Killing vectors that describe the translation invariance of spacetime in the $x,y$ directions. A Newman–Penrose (NP) tetrad[6] pertained to the spacetime metric may be defined as[5] \begin{align} l^a=\,&e^{-\zeta} \frac{\partial}{\partial u}, \tag {2} \\ n^a=\,&e^{-\zeta} \frac{\partial}{\partial v}, \tag {3} \\ m^a=\,&\frac{1}{\sqrt{2}} e^{-\eta} \sqrt{\cosh{\omega}}\Big( e^{i\phi} \frac{\partial}{\partial x} - ie^{-\chi} \frac{\partial}{\partial y} \Big), \tag {4} \\ \bar{m}^a=\,&\frac{1}{\sqrt{2}} e^{-\eta} \sqrt{\cosh{\omega}}\Big(e^{-i\phi} \frac{\partial}{\partial x} + ie^{-\chi} \frac{\partial}{\partial y}\Big), \tag {5} \end{align} where $\sin{\phi} = \tanh{\omega}$. Some relevant spin coefficients of the NP tetrad may be worked out to be[5] \begin{align} \kappa=\,&\kappa' = \tau = \tau' = \alpha = \alpha' = 0, \notag\\ \rho=\,&-e^{-\zeta} \Big(\frac{\partial \eta}{\partial u} + \frac{1}{2} \frac{\partial \chi}{\partial u}\Big), \notag\\ \rho'=\,&-e^{-\zeta} \Big(\frac{\partial \eta}{\partial v} + \frac{1}{2} \frac{\partial \chi}{\partial v}\Big), \notag\\ \sigma=\,&\frac{1}{2} e^{-\zeta}(1+i\sinh{\omega}) \frac{\partial \chi}{\partial u} \notag\\ &+\frac{i}{2} e^{-\zeta} \Big(\frac{1}{\cosh{\omega}} + i\tanh{\omega}\Big) \frac{\partial \omega}{\partial u}, \notag\\ \sigma'=\,&\frac{1}{2} e^{-\zeta} (1 + i\sinh{\omega}) \frac{\partial \chi}{\partial v} \notag\\ &+\frac{i}{2} e^{-\zeta} \Big(\frac{1}{\cosh{\omega}} + i\tanh{\omega}\Big) \frac{\partial \omega}{\partial v}. \tag {6} \end{align} Provided that the null congruence defined by ${\partial}/{\partial v}$ is shear-free so that $\sigma'=0$, the NP structure equations may be substantially simplified and the metric coefficients in Eq. (1) are specified by a pair of functions $[f(u), g(v)]$, \begin{align} g(v) = -|c|^2 v^2, \tag {7} \end{align} and $c$ is an arbitrary complex constant; $f(u)$ is governed by a Sturm–Liouville-type ordinary differential equation (ODE) given by \begin{align} \frac{{d}^2 f}{{d}u^2} + 2|\tilde{\sigma}|^2 f = 0, \tag {8} \end{align} where \begin{align} \tilde{\sigma}=e^\zeta \sigma=\,&\frac{1}{2} (1 + i\sinh{\omega}) \frac{\partial \chi}{\partial u}\notag\\ & \frac{1}{2}\Big(\frac{1}{\cosh{\omega}} + i\tanh{\omega}\Big)\frac{\partial \omega}{\partial u}. \tag {9} \end{align} The ordered pair $[f(u),\tilde\sigma]$ together with Eq. (7) then generates an exact solution of the Einstein Maxwell equation. The parameters $(\eta, \chi, \zeta)$ are further subject to \begin{align} \tag {10} &2\eta + \chi = \ln{[f(u) + g(v)]},\\ & \tag {11} \zeta = -\frac{1}{4} \ln{[f(u) + g(v)]}. \end{align} The NP equations only require $\omega$ and $\chi$ to be a function of $u$[5] and may be regarded as a free parameter in the spacetime metric. Here, $\omega\ne 0$ means that the two Killing vectors are not orthogonal to each other and this generates non-trivial polarization for the gravitational wave; $\chi$ describes the freedom in specifying the norm of $(\partial / \partial y)^a$ and is governed by Eq. (10). The spacetime metric in Eq. (1) takes the form \begin{align} {d}s^2=\,& \frac{2}{(f - |c|^2 v^2)} {d}u {d}v - (f - |c|^2 v^2)\notag\\ &\cosh{\omega} (e^{- \chi} {d}x^2 + e^\chi {d}y^2) \notag\\ &- 2(f - |c|^2 v^2) \sinh{\omega} \,{d}x {d}y. \tag {12} \end{align} The non-zero Weyl curvature components are given by \begin{align} \varPsi_0=\,& \frac{1}{(f - |c|^2 v^2)^{1/2}} \Big[2\frac{{d} f}{{d} u} + i(f - |c|^2 v^2)\notag\\ &\cdot\Big(\sinh{\omega} \frac{\partial \chi}{\partial u} - \frac{1}{\cosh{\omega}} \frac{\partial w}{\partial u}\Big)\Big] \tilde{\sigma} \notag\\ &+(f-|c|^2 v^2)^{1/2} \frac{\partial \tilde{\sigma}}{\partial u}, \tag {13} \\ \varPsi_2=\,& \frac{|c|^2 v}{2 (f - |c|^2 v^2)^{3/2}} \frac{{d} f}{{d} u}. \tag {14} \end{align} The NP components of the Maxwell field are given as \begin{align} \phi_0 = \frac{c v}{(f - |c|^2 v^2)^{1/4}} \tilde{\sigma} e^{-i F},~~ \phi_2 = \frac{c}{(f - |c|^2 v^2)^{1/4}} e^{-i F}, \tag {15} \end{align} where \begin{align} F = \int \Big(\sinh{\omega} \frac{\partial \chi}{\partial u} + \frac{1}{\cosh{\omega}} \frac{\partial \omega}{\partial u}\Big) {d}u. \tag {16} \end{align} Theoretical Analysis. We undertake further analysis of the structure of spacetime whose metric is given in Eq. (1). This will provide a basis on which we build up the numerical waveform. In what follows, we shall consider the simple case in which $\omega=0$. Parametric Resonance and Lyapunov Instability. Among the solutions which satisfy Eq. (8), there is a class of solutions involving the scattering of gravitational waves and electromagnetic waves which are of particularly physical interests to us. Consider \begin{align} 2|\tilde{\sigma}|^2 =\sigma_0^2 [1 + h \cos{(\gamma u)}], \tag {17} \end{align} which is a simple harmonic potential of frequency $\sigma_0$ subject to a sufficiently small periodic perturbation. Here $0\le h \ll 1$ is a constant, $\gamma = 2\sigma_0 + \varepsilon$, (see, e.g., chapter V section 27 in Ref. [7]) where $\varepsilon$ is a sufficiently small detuned parameter so that the periodic perturbation becomes a sideband deviated slightly from $2\sigma_0$. Written as a first order system of ODE, Eq. (8) may be given as \begin{align} \frac{{d}}{{d} u} \begin{pmatrix} f(u) \\ \xi(u) \end{pmatrix} = \boldsymbol{A} \begin{pmatrix} f(u) \\ \xi(u) \end{pmatrix}, \tag {18} \end{align} where $\xi = {d} f / {d} u$ and \begin{align*} \boldsymbol{A} = \begin{pmatrix} 0 & 1 \\ -\tilde{\sigma}^2 & 0 \end{pmatrix}. \end{align*} Provided $|\tilde{\sigma}| > 1$, $|A^n| = \tilde{\sigma}^{2n} \rightarrow \infty$ when $n \rightarrow \infty$, which means that $f(u)$ is divergent as $u\rightarrow \infty$.[8] It is well known that this is a simple case of Lyapunov instability when Eq. (8) is looked on from a dynamical system perspective. The aim of the present work is to work out the implications of this Lyapunov instability on the propagation of gravitational wave characterized by the Weyl curvature component $\varPsi_0$. To this end, define $q = |\sigma_0|^2h/16$, Eq. (8) together with Eq. (17) may be written as the standard Mathieu equation[9] \begin{align} \frac{{d}^2 f}{{d} u^2} + [|\sigma_0|^2 + 16q \cos{(\gamma u)}] f = 0. \tag {19} \end{align} A periodic solution with period $2\pi / \gamma$ may be given as \begin{align} f(u) = ae^{su} ce_1(u,q) + be^{su} se_1(u,q). \tag {20} \end{align} Here, $ce_1(u,q)$ and $se_1(u,q)$ are two periodic functions known as the Mathieu function given as \begin{align} \begin{split} ce_1 (u,q) = \cos{\Big(\frac{\gamma u}{2}\Big)} + q \cos{\Big(\frac{3\gamma u}{2}\Big)} + \mathcal{O}(q^2), \\ se_1 (u,q) = \sin{\Big(\frac{\gamma u}{2}\Big)} + q \sin{\Big(\frac{3\gamma u}{2}\Big)} + \mathcal{O}(q^2), \end{split} \tag {21} \end{align} where $s$ is a positive constant defined by \begin{align} s^2 = \frac{1}{4} \Big[\Big(\frac{1}{2}h \sigma_0\Big)^2 - \varepsilon^2\Big]. \tag {22} \end{align} We further assume $-\sigma_0 h / 2 < \varepsilon < \sigma_0 h / 2$ so that the system described in Eq. (20) is Lyapunov unstable. In what follows, we shall work only with the zeroth-order term in the perturbative expansion in Eq. (21), with terms involving the small parameter $q$ left out. Figure 1 displays the variation of $f(u)$ with respect to $u$. We shall seek to justify numerically this step in the following.

Fig. 1. Graph of $f(u)$ at $\varepsilon = 0.2$, $\sigma_0 = 1$, $h = 0.5$. It is a periodic function with the exponential function as envelope.

Scattering of Gravitational Waves with Electromagnetic Waves. Given $(f,g)$ defined in Eqs. (20) and (7), we shall seek to construct a spacetime that describes the scattering of gravitational and electromagnetic waves. To this end, further refine the definition of $(f,g)$ as \begin{align} g(v) = -|c|^2v^2 \theta(v), \tag {23} \end{align} where $\theta(v)$ is the step function that can be regarded as influencing $|c|$ by making it such that \begin{align*} |c|=\begin{cases}|c|, ~~{\rm for}~~ v > 0, \\ 0,~ ~{\rm for}~~ v \leqslant 0, \end{cases} \end{align*} \begin{align} \tilde{f}(u)=\begin{cases} f(0)~~{\rm for}~~u < 0, \\ f(u),~~{\rm for}~~u \geqslant 0. \end{cases} \tag {24} \end{align} Putting them into the metric (12) then we have

Fig. 2. Scattering picture of gravitational and electromagnetic waves. In region $u \leqslant 0, v \leqslant 0$, spacetime is Minkowskian; in region $u \leqslant 0, v \geqslant 0$, spacetime contains electromagnetic field with only $\phi_2$ component; $u \geqslant 0, v\leqslant 0$, spacetime only contains gravitational wave. Gravitational wave and full coupling between gravitational and electromagnetic waves takes place at $u, v \geqslant 0$.

Handling of Spacetime Singularity. From Eq. (13), it may be seen that there is a curvature singularity whenever $f - |c|^2 v^2=0$. The periodic nature of $f(u)$ [see Eq. (20)] means that singularities will also appear periodically at constant $v$. Further investigation reveals that this is not the spacetime singularity considered in the Hawking–Penrose singularity theorem.[10,11] The affine parameter $u$ of the null geodesic congruence defined by $\partial/\partial u$ is extendible beyond the singularity, in contrast to the existence of incomplete null geodesics ending abruptly considered in the Hawking–Penrose singularity theorem. The occurrence of curvature singularity in the present context originates from the divergent behavior of null geodesics, as may be seen from the divergent behavior of the spin coefficients $\rho,\rho'$ in Eq. (6). To avoid further complication due to the ill-behavior of null geodesics and without compromising the physics we are interested in, we shall introduce a cutoff function $\theta(u) = \theta_1(u) + \theta_2(u)$, with \begin{align} \theta_1(u) = \begin{cases} 1,~ u < u_i - \delta, \\ 0,~ u \geqslant u_i - \delta, \end{cases} ~~~ \theta_2(u)=\begin{cases} 0,~u < u_i + \delta, \\ 1,~u \geqslant u_i + \delta, \end{cases} \tag {25} \end{align} where $\delta$ is a sufficiently small real number, $u_i$ ($i=1, \dots, n\dots $) represents the solutions of $f(u) = g(v)$, and $n$ is a natural number. The gravitational waveform to be simulated numerically is essentially given by $\theta(u) |\varPsi_0(u)|$. The introduction of cutoff function means that the propagation of $\varPsi_0$ resembles a pulsed wave in electromagnetism. In electromagnetism, such an operation corresponds to a temporal filter to alter or attenuate the shape of a waveform. It is unclear how to implement this physically in the context of gravitational wave physics. We will take it only as a permissible mathematical operation at present. Numerical Simulation of Gravitational Waveform. In general, $\varPsi_0$ is complex and in what follows we shall numerically plot the modulus of these complex quantities and see how they propagate in the outgoing null direction described by the affine parameter $u$. This will enable us to see how intense beam of gravitational waves characterized by $\varPsi_0$ is generated. If we take the modulus of $\varPsi_0$ and $\phi_0, \phi_2$ as a measure of gravitational and electromagnetic energy respectively, then we may see that an inter-exchange of energy between gravitational and electromagnetic energy takes place. Consider $f(u)$ as defined in Eq. (8) and set \begin{align} a = b = 1,~~ \sigma_0 = 1,~~ c = 2,~~ g(v) = -4 v^2, \tag {26} \end{align} and $\varepsilon = 0.2$, $h = 0.5$ as perturbation. Figure 3 describes the global variation of $|\varPsi_0|$, $|\varPsi_2|$, $|\phi_2|$, $|\phi_0|$ in the $(u,v)$ plane. As may be seen from the figure, the sharp increase in $|\varPsi_0|$ is accompanied by the corresponding attenuation in $|\varPsi_2|$, $|\phi_2|$, and $|\phi_0|$. To better understand the increase of $|\varPsi_0|$ in relation to $|\varPsi_2|$, $|\phi_2|$, and $|\phi_0|$, we further take snapshot of $|\varPsi_0|$ at constant $v$.

Fig. 4. Waveform of $|\varPsi_0|$ at $\varepsilon = 0.2$, $\sigma_0 = 1$, $h = 0.5$ in the interval $u \in (0,40)$ with singularities cutoff in different $v = {\rm const}$ slice. The vertical dotted lines represent the solutions of $f(u) = g(v)$, where the spacetime singularities are located. The dashed lines represent the envelope. Correlations may be observed between $\varPsi_0$ and $\phi_2$. The larger the $g(v)$ is, the later the parametric resonance appears. However, the shape of the waveforms remains unchanged. (a) Waveform at $v = 0.05$ slice, where $g(v) = -0.01$. Such a slice is very close to the $v = 0$ boundary slice. (b) Waveform at the $v = 0.5$ slice, where $g(v) = -1$. (c) Waveform at the $v = 1$ slice, where $g(v) = -4$.

From Fig. 4, it may be seen that the exponential increase of the amplitude of $\varPsi_0$ is at the expense of the corresponding decrease in amplitudes of $\phi_2$. $\varPsi_0$ draws on energy not just from the electromagnetic field but also from the $\varPsi_2$. When certain resonance condition is met, the pumping of electromagnetic energy is transferred to the large amplitude of $\varPsi_0$. Further, the energy transfer is coherent in the sense that the increase of $\varPsi_0$ and the decrease of $\phi_2$ are at the same frequency. Like the free electron laser, the frequency of $\varPsi_0$ is tunable by adjusting the periodicity of $\phi_0$ and $\phi_2$. In the vacuum limit when the Maxwell field is absent, the parametric resonance phenomenon persists, however this corresponds only to the exchange of energy between the Weyl curvature components $\varPsi_0$ and $\varPsi_2$ of the gravity field. As the numerical resolution in Fig. 5 is not good enough to see the pulse nature of $\varPsi_0$ due to the cutoff of singularities. In Fig. 5, single period of the wave propagation is displayed to illustrate the pulsed nature of the gravitational wave propagation.

Fig. 5. Pulsed nature of gravitational wave propagation.

Fig. 6. Propagation of $\varPsi_0$ and $\varPsi_2$ at the $v = 1$ slice where $g(v) = -4$ is a delay function which determines the onset of instability at constant $v$.

Fig. 7. Propagation of $\varPsi_0$ and $\phi_0$ at the $v = 1$ slice where $g(v) = -4$.

Next, we plot at the $v = 1$ slice the exponential increase in $|\varPsi_0|$ in relation to $\varPsi_2$, $\phi_0$ in Figs. 6 and 7 along the outgoing null direction $u$. From Fig. 4(c) (which displays the exponential increase of $|\varPsi_0|$ in relation to $\phi_2$) and Figs. 6 and 7, we may see that the function $g(v)$ determines the time $u$ at which instability is switched on. So far we have considered only an approximate solution to the Mathieu equation with the small parameter $q$ totally left out. To understand the possible contribution from terms involving $q$ in Eq. (21), we repeat the numerical waveform simulations with terms up to second order in $q$ included in Eq. (21). No noticeable difference in the waveform from that displayed in Fig. 4 is observed. The exponential amplification of gravitational wave amplitude is dominated by the zeroth-order term given in Eq. (19). Concluding Remarks. The present work serves to point out, at least at the theoretical level, that an intense pulse gravitational wave train may be generated by the conversion of electromagnetic energy into gravitational energy, provided that certain resonance condition is met, in some way similar to that for a free electron laser. The class of examples considered is admittedly highly idealistic and more work remains to be carried out to explore this feasibility at a more practical level. It offers hope, however remote it may seem for the time being, that in future an intense beam of gravitational wave akin to a laser in electromagnetic theory may be generated in a laboratory, unlike the current situation when we only count on astrophysical sources for gravitational wave detection. As far as detection of gravitational waves from an astrophysical source is concerned, it is conceivable that, provided we have in advance knowledge of the frequency of the wave to be detected, electromagnetic energy may be pumped in to enhance the amplitude of the gravitational waves so that the signal-to-noise ratio may be substantially enhanced.[12] This remains a feasibility to be explored in future. Acknowledgement. The work was supported by the National Key Research and Development Program of China (Grant No. 2021YFC2202501). References Fundamentals of Interferometric Gravitational Wave DetectorsA family of cylindrically symmetric solutions to Einstein-Maxwell equationsEinstein-Maxwell Spacetime with Two Commuting Spacelike Killing Vector Fields and Newman-Penrose FormalismThe singularities of gravitational collapse and cosmologyGravitational-electromagnetic resonance

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