C-energy

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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

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

The C-energy is defined geometrically in terms of these Killing vectors by

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

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 area (per unit axial length) of the two-dimensional surface spanned by the Killing vectors ${\displaystyle \partial _{\phi }}$ and ${\displaystyle \partial _{z}}$.

When the space-time metric is written in 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 ${\displaystyle q=q(t,\rho )}$, the C-energy reduces to the simple form

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

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

References