Full Nonlinear Theory Research Articles

12.—Spectral Theory of Rotating Chains

SynopsisTwo eigenvalue problems associated with steady rotations of a chain are considered. To compare the spectra of these two problems, let σK(n) denote the set of all angular velocities with which a chain of unit length with one end fixed and the other free can rotate in a vertical plane so as to have exactly n nodes on thevertical axis (including the fixed end). In a linearised theory σK(n) is a single point, i.e.In the full non-linear theory σK(n) is an infinite interval lying to the right of ⍵n. Indeed,This is established in [1].Next, let σM(β, n) denote the set of all angular velocities with which a chain, having ends fixed at unit distance apart on the vertical axis, can rotate in a vertical plane so as to have exactly n+l nodes on the vertical axis (including the ends) and so that the tension takes the value β at the lower end. This problem, in which the length of the chain is not prescribed is a model for a spinning process in which the ‘chain’ is continuously created in a rotating configuration. For β>0, we again have in the linearised theory that σM(β, n) is a singleton, i.e.In the full non-linear theory σM(β, n) lies to the left of λn(β). Although unable to determine exactly σM(β, n) for β>0, we havewhere ⍵n λn(β), and are all characterised as the nth zeros of known combinations of Bessel functions.

Asymptotically simple space-time manifolds

Asymptotic simplicity is shown to be k-stable (k≥3) at any Minkowski metric on R4 in both the Whitney fine Ck topology and a coarser topology (in which the Ck twice-convariant symmetric tensors form a Banach manifold whose connected components consist of tensor field asymptotic to one another at null infinity). This result, together with a sequential method for solving the field equations previously proposed by the authors, allows a fairly straightforward proof that a well-known result in the linearized theory holds in the full nonlinear theory as well: There are no nontrivial (i.e., non-Minkowskian) asymptotically simple vacuum metrics on R4 whose conformal curvature tensors result from prescribing zero initial data on past null infinity.

Note on the Dynamics of Gravitational Sources

This paper is concerned with multipole structure for the sources of the gravitational field in the full (axially symmetric) nonlinear gravitational theory. Definitions proposed recently by Janis and Newman for the mass of the source, its linear momentum, dipole moment, and its spin or angular momentum are examined. These definitions are contrasted with those given by other authors. The advantages of the new definitions are discussed and the manner in which they enter into the dynamics of the system is studied. An appendix considers some transformation properties of these multiple moments.