Chandrasekhar waves

In general relativity, Chandrasekhar waves refers to standing cylindrical wave solution of the Einstein's field equations, named after Subrahmanyan Chandrasekhar, who discovered the solution in 1986.^[1] The Chandrasekhar waves are different from Einstein–Rosen waves which are also cylindrical waves.^[2] The so-called C-energy is conserved in time for Chandrasekhar waves.

Description of the metric

The space-time metric (with ${\displaystyle c=1}$) of the Chandrasekhar waves is given by^[1]

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

where (both ${\displaystyle t}$ and ${\displaystyle \rho }$ are measured in units of ${\displaystyle 1/\sigma }$, where ${\displaystyle \sigma }$ is the wave frequency),

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

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

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

and ${\displaystyle F=F(\rho )}$ is the solution of

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

satisfying the boundary conditions

${\displaystyle F=F_{0}\,\,(>0\,\,{\text{and}}\,\,<1)\quad {\text{and}}\quad F_{\rho }=0\quad {\text{for}}\quad \rho =0.}$

The function ${\displaystyle F=F(\rho )}$ has the asymptotic behaviour

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

Since ${\displaystyle (\nu +\mu )_{t}=0}$, we can define the C-energy to be

${\displaystyle C=\nu +\mu .}$

The asymptotic behaviours of various functions are given by

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

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