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.Thorne, K. S. (1965). Energy of infinitely long, cylindrically symmetric systems in general relativity. Physical Review, 138(1B), B251.
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).Bini, D., Geralico, A., & Plastino, W. (2019). Cylindrical gravitational waves: C-energy, super-energy and associated dynamical effects. Classical and Quantum Gravity, 36(9), 095012.Nikiel, K., & Szybka, S. J. (2025). Halilsoy and Chandrasekhar standing gravitational waves in the linear approximation. Physical Review D, 111(10), 104015. 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.Bondi, H. (1990). The mass of cylindrical systems in general relativity. Proceedings of the Royal Society of London A, 427(1873), 259–264.
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 byChandrasekhar, S. (1986). Cylindrical waves in general relativity. Proceedings of the Royal Society of London A, 408(1835), 209–232.Chandrasekhar, S., & Ferrari, V. (1987). On the dispersion of cylindrical impulsive gravitational waves. Proceedings of the Royal Society of London A, 412(1842), 75–91.
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.
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