Gravitational waves, black holes and cosmic strings in cylindrical symmetry

The emission of gravitational waves by black holes is under intense investigation, in preparation for the expected observational study of black holes and other astrophysical objects by upcoming gravitational-wave detectors. Most of this work is either numerical or in an approximation, since we lack a full physical understanding of the dynamics of black holes and gravitational waves. The accepted theory of black holes mainly concerns statics and asymptotics; for instance, there are standard asymptotic definitions of mass, angular momentum and the energy flux of gravitational waves, with surface gravity defined for stationary black holes. One might expect such physical quantities to play a more local role in dynamical processes. A framework for addressing black-hole dynamics was introduced a few years ago [1]. Black holes were given a local, dynamical definition in terms of trapping horizons, which are essentially the locations where light waves are marginally trapped by the gravitational field. Some basic properties of black holes were established, such as that the horizon is achronal and therefore locally one-way traversible, and a second law expressing the increase of area of the trapping horizon. A comprehensive picture of the spherically symmetric case has since been developed, involving local, dynamical definitions of physical quantities such as energy [2] and surface gravity [3], related by local, dynamical equations including a first law. This article introduces the corresponding physical, geometrical quantities and equations in cylindrical symmetry, which has the additional complexity of gravitational waves. In vacuo, these are known as Einstein-Rosen gravitational waves [4]. Of course, cylindrical black holes are rather unphysical, but their study is a precursor to that of axially symmetric black holes, in particular their interaction with gravitational waves. The main conceptual issue is whether the energy flux of gravitational waves admits a local, dynamical definition. This turns out to be so, indeed quite natural, in that there is a covariant conservation law for the combined energy-momentum of the gravitational waves and matter. Due to some peculiar properties of black holes in cylindrical symmetry, naked singularities may also be a feature of gravitational collapse. Consequently the framework is extended to include axial singularities, particularly dynamic cosmic strings. The article is organised as follows. Section II reviews cylindrical symmetry in the sense of Melvin [5,6] and discusses regular axes and axial singularities. Section III defines black holes and related ideas of gravitational trapping. Section IV reviews the modified Thorne energy [7], which turns out to be the appropriate definition of gravitational energy. Section V introduces a preferred flow of time and the corresponding energy-momentum vector of the matter. Section VI introduces the energy-momentum vector of the gravitational waves, states the conservation law, and introduces the gravitational potential and effective energy tensor of the gravitational waves. Section VII states the first law in terms of the energy flux of the gravitational waves, plus energy-flux and work terms for the matter. Section VIII defines the surface gravity. Section IX collects basic laws of black-hole dynamics, including the projection of the first law along a trapping horizon, which involves area and surface gravity as expected. Section X discusses some static examples including black holes and cosmic strings. Section XI defines dynamic cosmic strings and discusses issues of cosmic censorship and predictability. Section XII concludes. An Appendix lists various expressions in a standard coordinate system. Einstein gravity is assumed, though the application of the Einstein equation is stated as appropriate.