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Published April 12, 1965 | public
Journal Article Open

Energy of infinitely long, cylindrically symmetric systems in general relativity

Thorne, Kip S.

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 Pi, which obeys the conservation law Pi; i=0. C energy is localizable and locally measurable: The component of Pi 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 T00. 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.

Additional Information

©1965 The American Physical Society. Received 30 September 1964; revised 7 December 1964. The author is indebted to Professor J. A. Wheeler for many discussions in which he emphasized the role that an energy-like quantity could play in the stability analysis of Melvin's magnetic universe. It was out of these discussions that the work presented here grew. Professor Wheeler and Professor M. A. Melvin both provided valuable criticism of this presentation of C energy.

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