Draft:Chandrasekhar waves

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In general relativity, Chandrasekhar waves are a class of standing cylindrical gravitational-wave solutions of Einstein’s field equations. They describe vacuum space-times with cylindrical symmetry in which gravitational waves oscillate in time without net energy transport along the radial direction. Chandrasekhar waves form one of the best-studied examples of exact standing-wave solutions in general relativity and are closely related to, but distinct from, Einstein–Rosen waves.

The solution was first derived by Subrahmanyan Chandrasekhar in 1986, who showed that nontrivial standing cylindrical waves can exist without violating regularity conditions on the symmetry axis. Subsequent work clarified the physical interpretation of the solution, its relation to other cylindrical wave models, and its behavior under linear perturbations and backreaction effects.

A key distinguishing feature of Chandrasekhar waves is that the associated C-energy—a quasi-local measure of gravitational energy adapted to cylindrical symmetry—remains constant in time, in contrast to Einstein–Rosen waves, for which the C-energy oscillates periodically.

Description of the metric

In units where $c=1$, the space-time metric of Chandrasekhar waves can be written as

$d{s}^2=e^{2\nu }[(dt{)}^2-(d\rho {)}^2]-e^{-2\mu }(\rho d\phi {)}^2-e^{2\mu }(dz-qd\phi {)}^2$

where the metric functions depend on the time coordinate $t$ and the radial coordinate $\rho$; both $t$ and $\rho$ are measured in units of $1/\sigma$, with $\sigma$ denoting the wave frequency. The functions $\mu$, $q$, and $\nu$ are given by

$\mu =\frac{1}{2}\ln \frac{1-F^2}{1+F^2-2F\cos t},$

$q\sigma =-\frac{2\rho F_{\rho }}{1-F^2}\cos t-4\underset{0}{\overset{\rho }{\int }}\frac{\rho F^2}{(1-F^2{)}^2}d\rho ,$

$(\nu +\mu {)}_t=0,(\nu +\mu {)}_{\rho }=\frac{\rho }{(1-F^2{)}^2}(F^2+F_{\rho }^2),$

where $F=F(\rho )$ satisfies the nonlinear ordinary differential equation

$F(1+F^2)+\frac{1-F^2}{\rho }(\rho F_{\rho }{)}_{\rho }+2F{F}_{\rho }^2=0.$

Regularity on the symmetry axis requires the boundary conditions

$F=F_0(>0\text{and}<1)\text{and}{F}_{\rho }=0\text{for}\rho =0.$

Because $(\nu +\mu {)}_t=0$, the associated C-energy,

$C=\nu +\mu ,$

is conserved in time, reflecting the standing-wave character of the solution.

Asymptotic behaviour

For large radial distances, the function $F(\rho )$ exhibits the asymptotic behavior

$F\sim \frac{\text{const.}}{\sqrt{\rho }}\cos (\rho +\frac{1}{16}\ln \rho +b)+O\left({\rho }^{-\frac{3}{2}}\right)\text{as}\rho \to \mathrm{\infty }.$

Near the axis and at infinity, the metric functions and the C-energy behave as

$e^{2\mu }\to \frac{1-F_0^2}{1+F_0^2-2F_0\cos t}\sim O(1),q=O({\rho }^2),C=O({\rho }^2)\text{as}\rho \to 0,$

$e^{2\mu }\to 1,q\to -\text{const.}\sqrt{\rho }\cos t-\text{const.}\rho ,C=O(\rho )\text{as}\rho \to \mathrm{\infty }.$