Physics:C-energy

In general relativity, C-energy (short for cylindrical energy) is a quasi-local definition of gravitational energy applicable to space-times with cylindrical symmetry. The concept was introduced by Kip Thorne in 1965 as an attempt to characterize the energy content of infinitely long, cylindrically symmetric systems.

C-energy has been widely used in the analysis of cylindrical gravitational waves, where it provides a useful measure of the gravitational field strength. In standing cylindrical wave solutions, the C-energy may be strictly constant in time (as in Chandrasekhar waves) or constant only on average (as in Einstein–Rosen waves). Although C-energy does not correspond to a globally conserved energy in general relativity, it remains a useful diagnostic tool for studying cylindrically symmetric space-times and gravitational radiation.

Definition

A space-time with cylindrical symmetry about an axis admits two commuting spacelike Killing vector fields, namely

• ${\partial }_{\varphi }$, whose orbits are closed and represent axial symmetry, and
• ${\partial }_z$, whose orbits are open and represent translational symmetry along the axis.

The C-energy is defined geometrically in terms of these Killing vectors 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 $A=[|{\partial }_{\varphi }{|}^2|{\partial }_z{|}^2-({\partial }_{\varphi }\cdot {\partial }_z{)}^2{]}^{\frac{1}{2}}$ is the area (per unit axial length) of the two-dimensional surface spanned by the Killing vectors ${\partial }_{\varphi }$ and ${\partial }_z$.

When the space-time metric is written in the form

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

with $\nu =\nu (t,\rho )$, $\mu =\mu (t,\rho )$ and $q=q(t,\rho )$, the C-energy reduces to the simple form

$C=\nu +\mu .$

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