Dispersion of an initially impulsive cylindrical gravitational wave in the general two-polarization (non-diagonal) metric: collision singularities and C-energy transport

Chandrasekhar and Ferrari (1987) constructed the exact solution describing the dispersion of an initially impulsive cylindrical gravitational wave - the spacetime is flat at t=0 except at a single radius, where the time-derivative of the Weyl scalars has delta-function singularities - and found that after reflection at the symmetry axis the wave collides with the still-ingoing waves, developing singularities in the metric function nu and in the Weyl scalar Psi_2. The construction lives entirely in the diagonal cylindrical metric, where a single gravitational polarization (+ mode) is active and the governing equation for ln Psi is linear, so solutions superpose. The general non-diagonal (two-polarization) metric is governed by the nonlinear cylindrical Ernst equation, and three linked questions remain open: (1) the dispersion of impulsive initial data in the non-diagonal class, including whether such data admit closed-form or special-function representations at all; (2) the nature of the axis-reflection and collision singularities when both modes are active, in particular whether the 1987 collision divergences persist, weaken, or disappear in the two-polarization setting; and (3) the C-energy account of the two-mode impulsive solution, whose flux quadratures are nonlinear in the field derivatives.

Background

Cylindrical gravitational waves - spacetimes with two commuting spacelike Killing fields, generating translations along and rotations about a fixed symmetry axis, whose metric depends only on time $t$ and a radial coordinate $\varpi$ - are the simplest family of exact solutions of Einstein's vacuum equations that describe propagating radiation; they were introduced by Einstein and Rosen (1937). Because exact radiative solutions are rare, this family serves as the standard analytic laboratory for nonlinear wave effects in general relativity, and two variants matter here. In the diagonal (Einstein-Rosen) form only one gravitational polarization is present. In the general non-diagonal form - often called the Jordan-Ehlers metric, written in the notation of Chandrasekhar (1986) as

ds^2=e^{2\nu}(dt^2-d\varpi^2)-\varpi\left[\chi\,d\phi^2+\chi^{-1}(dz-q_2\,d\phi)^2\right],

both polarization states are active: the $+$ mode (the function $\ln\Psi$ , where $\Psi=\varpi/\chi$ is, up to sign, the coefficient of $dz^2$ in the metric) and the $\times$ mode (the twist function $q_2$ ; its potential $\omega$ , defined by $\omega_{,t}=(\varpi/\chi^{2})\,q_{2,\varpi}$ and $\omega_{,\varpi}=(\varpi/\chi^{2})\,q_{2,t}$ , is the form in which the twist enters the field equations). Gravitational waves, like electromagnetic waves, carry two polarization states; cylindrical symmetry admits exactly these two.

The two cases differ sharply in their governing equations. In the diagonal case ( $q_2=0$ ) the vacuum equations reduce to a single linear cylindrical wave equation for $\ln\Psi$ ,

(\ln\Psi)_{,tt}-\frac{1}{\varpi}\left[\varpi(\ln\Psi)_{,\varpi}\right]_{,\varpi}=0,

so superpositions of solutions are again solutions, exactly as for the scalar wave equation in cylindrical coordinates. In the general case the two modes couple: they combine into a complex Ernst potential $Z=\Psi+\mathrm{i}\omega$ (real part the metric function $\Psi$ itself, imaginary part the twist potential) satisfying the nonlinear cylindrical Ernst equation

\mathrm{Re}\,Z\left[Z_{,tt}-\frac{1}{\varpi}\left(\varpi Z_{,\varpi}\right)_{,\varpi}\right]=(Z_{,t})^2-(Z_{,\varpi})^2,

(the Ernst equation is the compact complex-potential form of the vacuum equations that underlies most modern solution-generating techniques), and no superposition principle survives.

Chandrasekhar and Ferrari (1987) exploited the linearity of the diagonal case to construct, in closed form, the dispersion of a wave packet with impulsive initial data. As the published abstract states, the spacetime is flat at $t=0$ except at a single radius $\varpi=\varpi_1$ (taken as $\varpi_1=1$ ), with the time-derivative of the Weyl scalars (the components $\Psi_0,\dots,\Psi_4$ of the Weyl curvature tensor in a suitable null tetrad) exhibiting delta-function singularities there - an initially impulsive wave. The solution is given by a Bessel integral and, piecewise, by Legendre functions of degree $-1/2$ ; the packet disperses, reflects at the symmetry axis at $t=\varpi_1$ , and - the paper's principal finding - a singularity of the metric function $\nu$ and of the Weyl scalar $\Psi_2$ develops where the reflected wave collides with the still-ingoing waves (at $\varpi=t-\varpi_1$ for $t>\varpi_1$ ), with $\ln\Psi$ diverging logarithmically at the collision locus and the divergence propagating into $\nu$ , $\Psi_2$ , and the C-energy. On the axis after the reflection the Weyl tensor is algebraically special of Petrov type D (the Petrov classification is the algebraic classification of the Weyl tensor by its principal null directions; type D is the Schwarzschild-like, doubly degenerate case).

The energy bookkeeping for cylindrical waves uses Thorne's C-energy, a conserved energy per unit cylindrical length; in the above variables $C=\nu+\frac{1}{2}\ln\Psi$ , with time and radial fluxes given by quadratures quadratic in the first derivatives of $\ln\Psi$ ,

C_{,t}=\tfrac{1}{2}\,\varpi\,(\ln\Psi)_{,t}(\ln\Psi)_{,\varpi},\qquad C_{,\varpi}=\tfrac{1}{4}\,\varpi\left\{\left[(\ln\Psi)_{,t}\right]^2+\left[(\ln\Psi)_{,\varpi}\right]^2\right\}.

This quadratic structure means that the C-energy of a superposition of pulses is not the sum of the individual C-energies: in the diagonal class the quadratures contain cross (interference) terms that the linear superposition of $\ln\Psi$ does not determine.

For the general non-diagonal metric the C-energy convention is fixed by the same references (Thorne 1965; the Hamiltonian-density formulation of Chandrasekhar 1986; Piran, Safier, and Stark 1985): the energy per unit length remains $C=\nu+\frac{1}{2}\ln\Psi$ , but the fluxes acquire cross-polarization contributions from the twist potential $\omega$ . In the normalization of Piran et al. (1985), writing $\ln\Psi=2\psi$ and denoting this energy function by $\gamma$ (so that $C=\gamma$ in the notation above),

\gamma_{,t}=2\varpi\,\psi_{,t}\psi_{,\varpi}+\frac{e^{4\psi}}{2\varpi}\,\omega_{,t}\omega_{,\varpi},\qquad \gamma_{,\varpi}=\varpi\left(\psi_{,t}^{2}+\psi_{,\varpi}^{2}\right)+\frac{e^{4\psi}}{4\varpi}\left(\omega_{,t}^{2}+\omega_{,\varpi}^{2}\right);

these reduce exactly to the diagonal quadratures quoted above when $\omega$ (equivalently $q_2$ ) vanishes.

The entire 1987 construction, moreover, lives in the diagonal world: for the non-diagonal metric the Ernst equation is nonlinear, the Bessel-superposition route is unavailable, and the behavior of impulsive data - including the fate of the collision singularity found in the diagonal solution - is unknown.

Problem Statement

Chandrasekhar and Ferrari (1987) analyzed the dispersion of an initially impulsive cylindrical gravitational wave packet - initial data at $t=0$ consisting of flat spacetime except at a single radius $\varpi=\varpi_1$ , with delta-function singularities in the time-derivative of the Weyl scalars - but only in the diagonal cylindrical metric, where a single gravitational polarization is active and the governing equation for $\ln\Psi$ is linear. The problem is the completion of this dispersion analysis in the general non-diagonal (two-polarization) cylindrical vacuum spacetime, together with the singularity and energy-transport questions that its diagonal construction leaves open.

Common objective: determine how initially impulsive cylindrical gravitational disturbances disperse, reflect, and collide in the general cylindrical vacuum geometry, and what the associated singularity structure and energy transport are. In the general cylindrical metric of Chandrasekhar (1986),

ds^2=e^{2\nu}(dt^2-d\varpi^2)-\varpi\left[\chi\,d\phi^2+\chi^{-1}(dz-q_2\,d\phi)^2\right],

with the two gravitational degrees of freedom - the $+$ mode $\ln\Psi$ ( $\Psi=\varpi/\chi$ ) and the $\times$ mode $q_2$ (twist potential $\omega$ ) - coupled through the nonlinear cylindrical Ernst equation for the complex potential $Z$ quoted in the background, the problem asks:

(1) Representation. Construct, or establish the non-existence of, solutions describing the dispersion of initially impulsive data in this non-diagonal metric - the two-mode analog of the 1987 diagonal construction - whether in closed form, special functions, or quadratures; determine whether delta-type initial data can be prescribed at all in this class, and what replaces the 1987 identification of the initial delta-function through a Bessel integral.

(2) Singularity structure. Characterize the singularities of the two-mode dispersion: the behavior at reflection on the symmetry axis $\varpi=0$ at $t=\varpi_1$ ; the locus and strength of the collision singularities that in the diagonal solution develop where the reflected wave meets the still-ingoing waves (at $\varpi=t-\varpi_1$ for $t>\varpi_1$ ) - the logarithmic divergence of $\ln\Psi$ and the divergence of $\nu$ , the Weyl scalar $\Psi_2$ , and the C-energy; and whether these divergences persist, weaken, or disappear when both modes are active.

(3) Energy transport. Compute the C-energy and its time and radial fluxes for the two-mode impulsive solution, in the two-mode convention stated in the background.

All three parts are required for a complete resolution, and no single part alone resolves the task. Parts (2) and (3) presuppose a solution of part (1); if part (1) is settled negatively - no delta-type impulsive data can be prescribed in this class - then parts (2) and (3) are to be addressed for the most general impulsive-type data that can be prescribed.

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