C-energy
In general relativity, C-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 or constant only on average. 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- , whose orbits are closed and represent axial symmetry, and
- , whose orbits are open and represent translational symmetry along the axis.
where is the metric tensor and is the area of the two-dimensional surface spanned by the Killing vectors and.
When the space-time metric is written in the form
with, and, the C-energy reduces to the simple form
In Chandrasekhar waves, for which, the C-energy is constant in time, whereas in Einstein–Rosen waves, where, the C-energy varies periodically with time.