Energy of Infinitely Long, Cylindrically Symmetric Systems in General Relativity

Abstract

A definition of energy is proposed for systems invariant under rotations about, and translations along, a symmetry axis. This energy (which is called "cylindrical energy" or "$C$ energy") takes the form of a covariant vector ${P}^{i}$, which obeys the conservation law ${P}^{i}$; $i=0$. $C$ energy is localizable and locally measurable: The component of ${P}^{i}$ along the world line of an observer is the $C$-energy density he measures. Near the symmetry axis of a static system, where strong gravitational fields are absent, $C$-energy density reduces to proper mass density ${{T}_{0}}^{0}$. $C$ energy is propagated by Einstein-Rosen gravitational waves and by cylindrical electromagnetic waves. In vacuo and in the presence of electromagnetic fields the $C$ energy on a space-like hyper-surface is minimized when the system is static; and the difference between the "potential" part and the "kinetic" part is a Lagrangian for the Einstein-Maxwell field equations. $C$ energy can be a powerful tool in the analysis of finite as well as infinite cylindrically symmetric systems. Here it is used to elucidate the nature of Einstein-Rosen gravitational radiation, and to suggest and support the conjecture of flux resistance to gravitational collapse: In any configuration of electromagnetic fields collapsing toward a singularity, each electric and magnetic-field line is either entirely ejected from the collapsing region or entirely swallowed by it as collapse proceeds; there can be no flux threading a collapsed region.