Chandrasekhar waves

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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. The Chandrasekhar waves are different from Einstein–Rosen waves which are also cylindrical waves. The so-called C-energy is conserved in time for Chandrasekhar waves.

Description of the metric

The space-time metric (with $c=1$) of the Chandrasekhar waves is given by

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

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

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

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

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

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

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

satisfying the boundary conditions

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

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

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

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

$C=\nu +\mu .$

The asymptotic behaviours of various functions are given by

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

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

References

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