C-energy

In general relativity, C-energy describes a definition of energy that may be applicable to space-times with cylindrical symmetry. The definition was first introduced by Kip Thorne in 1965. In standing cylindrical gravitational waves, the C-energy may be constant in time (Chandrasekhar waves) or constant in time on average (Einstein–Rosen waves).

Definition

Space-times with cylindrical symmetry about an axis have two commuting space-like Killing vectors, namely ${\partial }_{\varphi }$ and ${\partial }_z$, in which the orbit of ${\partial }_{\varphi }$ is closed and the orbit of ${\partial }_z$ is open. The definition of the C-energy in terms of these Killing vectors is given by

$C=-\frac{1}{2}\ln \left(\frac{g^{ij}{A}_{,i}{A}_{,j}}{|{\partial }_z{|}^2}\right),$

where $g_{ij}$ is the metric tensor and Failed to parse (syntax error): {\displaystyle A = \left|\partial_\phi\right|^2\left|\partial_z\right|^2 - \left(\partial_\phi \cdot \partial_z\right)^2\right|^{\frac{1}{2}}} is the two-dimensional surface (per unit axial length), spanned by ${\partial }_{\varphi }$ and ${\partial }_z$.

If the space-time metric is of the form

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

with $\nu =\nu (t,\rho )$, $\mu =\mu (t,\rho )$ and $q=q(t,\rho )$, then the C-energy may be defined as

$C=\nu +\mu .$

In Chandrasekhar waves for which $q\ne 0$, $C$ is constant in time, whereas in Einstein–Rosen waves for which $q=0$, $C$ varies periodically in time.

References

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