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.^[1] In standing cylindrical gravitational waves, the C-energy may be constant in time (Chandrasekhar waves) or constant in time on average (Einstein–Rosen waves).^[2]

Definition

Space-times with cylindrical symmetry about an axis has two commuting space-like Killing vectors, namely ${\displaystyle \partial _{\phi }}$ and ${\displaystyle \partial _{z}}$, in which the orbit of ${\displaystyle \partial _{\phi }}$ is closed and the orbit of ${\displaystyle \partial _{z}}$ is open. The definition of the C-energy in terms of these Killing vectors is given by^[3][4]

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

where ${\displaystyle g_{ij}}$ is the metric tensor and ${\displaystyle A=[|\partial _{\phi }|^{2}|\partial _{z}|^{2}-(\partial _{\phi }\cdot \partial _{z})^{2}]^{\frac {1}{2}}}$ is the two-dimensional surface (per unit axial length), spanned by ${\displaystyle \partial _{\phi }}$ and ${\displaystyle \partial _{z}}$.

If the space-time metric is of the form

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

with ${\displaystyle \nu =\nu (t,\rho )}$, ${\displaystyle \mu =\mu (t,\rho )}$ and ${\displaystyle q=q(t,\rho )}$, then the C-energy may be defined as^[3]

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

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