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

• , whose orbits are closed and represent axial symmetry, and
• , whose orbits are open and represent translational symmetry along the axis.

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

where is the metric tensor and is the area (per unit axial length) 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.

References