25 research outputs found

### Spacetime Encodings IV - The Relationship between Weyl Curvature and Killing Tensors in Stationary Axisymmetric Vacuum Spacetimes

The problem of obtaining an explicit representation for the fourth invariant
of geodesic motion (generalized Carter constant) of an arbitrary stationary
axisymmetric vacuum spacetime generated from an Ernst Potential is considered.
The coupling between the non-local curvature content of the spacetime as
encoded in the Weyl tensor, and the existence of a Killing tensor is explored
and a constructive, algebraic test for a fourth order Killing tensor suggested.
The approach used exploits the variables defined for the B\"{a}ckland
transformations to clarify the relationship between Weyl curvature, constants
of geodesic motion, expressed as Killing tensors, and the solution generation
techniques. A new symmetric non-covariant formulation of the Killing equations
is given. This formulation transforms the problem of looking for fourth-order
Killing tensors in 4D into one of looking for four interlocking two-manifolds
admitting fourth-order Killing tensors in 2D.Comment: 15 page

### Spacetime Encodings III - Second Order Killing Tensors

This paper explores the Petrov type D, stationary axisymmetric vacuum (SAV)
spacetimes that were found by Carter to have separable Hamilton-Jacobi
equations, and thus admit a second-order Killing tensor. The derivation of the
spacetimes presented in this paper borrows from ideas about dynamical systems,
and illustrates concepts that can be generalized to higher- order Killing
tensors. The relationship between the components of the Killing equations and
metric functions are given explicitly. The origin of the four separable
coordinate systems found by Carter is explained and classified in terms of the
analytic structure associated with the Killing equations. A geometric picture
of what the orbital invariants may represent is built. Requiring that a SAV
spacetime admits a second-order Killing tensor is very restrictive, selecting
very few candidates from the group of all possible SAV spacetimes. This
restriction arises due to the fact that the consistency conditions associated
with the Killing equations require that the field variables obey a second-order
differential equation, as opposed to a fourth-order differential equation that
imposes the weaker condition that the spacetime be SAV. This paper introduces
ideas that could lead to the explicit computation of more general orbital
invariants in the form of higher-order Killing Tensors.Comment: 9 page

### Spacetime Encodings I- A Spacetime Reconstruction Problem

This paper explores features of an idealized mathematical machine (algorithm)
that would be capable of reconstructing the gravitational nature (the
multipolar structure or spacetime metric) of a compact object, by observing
gravitational radiation emitted by a small object that orbits and spirals into
it. An outline is given of the mathematical developments that must be carried
out in order to construct such a machine.Comment: 5 pages, 2 figure

### A Nonlinear Coupling Network to Simulate the Development of the r-mode Instablility in Neutron Stars I. Construction

R-modes of a rotating neutron star are unstable because of the emission of
gravitational radiation. We explore the saturation amplitudes of these modes
determined by nonlinear mode-mode coupling. Modelling the star as
incompressible allows the analytic computation of the coupling coefficients.
All couplings up to n=30 are obtained, and analytic values for the shear
damping and mode normalization are presented. In a subsequent paper we perform
numerical simulations of a large set of coupled modes.Comment: 15 pages 3 figure

### Nonlinear Couplings of R-modes: Energy Transfer and Saturation Amplitudes at Realistic Timescales

Non-linear interactions among the inertial modes of a rotating fluid can be
described by a network of coupled oscillators. We use such a description for an
incompressible fluid to study the development of the r-mode instability of
rotating neutron stars. A previous hydrodynamical simulation of the r-mode
reported the catastrophic decay of large amplitude r-modes. We explain the
dynamics and timescale of this decay analytically by means of a single three
mode coupling. We argue that at realistic driving and damping rates such large
amplitudes will never actually be reached. By numerically integrating a network
of nearly 5000 coupled modes, we find that the linear growth of the r-mode
ceases before it reaches an amplitude of around 10^(-4). The lowest parametric
instability thresholds for the r-mode are calculated and it is found that the
r-mode becomes unstable to modes with 13<n<15 if modes up to n=30 are included.
Using the network of coupled oscillators, integration times of 10^6 rotational
periods are attainable for realistic values of driving and damping rates.
Complicated dynamics of the modal amplitudes are observed. The initial
development is governed by the three mode coupling with the lowest parametric
instability. Subsequently a large number of modes are excited, which greatly
decreases the linear growth rate of the r-mode.Comment: 3 figures 4 pages Submitted to PR

### Avenues for analytic exploration in axisymmetric spacetimes: Foundations and the triad formalism

Axially symmetric spacetimes are the only vacuum models for isolated systems with continuous symmetries that also include dynamics. For such systems, we review the reduction of the vacuum Einstein field equations to their most concise form by dimensionally reducing to the three-dimensional space of orbits of the Killing vector, followed by a conformal rescaling. The resulting field equations can be written as a problem in three-dimensional gravity with a complex scalar field as source. This scalar field, the Ernst potential, is constructed from the norm and twist of the spacelike Killing field. In the case where the axial Killing vector is twist-free, we discuss the properties of the axis and simplify the field equations using a triad formalism. We study two physically motivated triad choices that further reduce the complexity of the equations and exhibit their hierarchical structure. The first choice is adapted to a harmonic coordinate that asymptotes to a cylindrical radius and leads to a simplification of the three-dimensional Ricci tensor and the boundary conditions on the axis. We illustrate its properties by explicitly solving the field equations in the case of static axisymmetric spacetimes. The other choice of triad is based on geodesic null coordinates adapted to null infinity as in the Bondi formalism. We then explore the solution space of the twist-free axisymmetric vacuum field equations, identifying the known (unphysical) solutions together with the assumptions made in each case. This singles out the necessary conditions for obtaining physical solutions to the equations

### Formal Solution of the Fourth Order Killing equations for Stationary Axisymmetric Vacuum Spacetimes

An analytic understanding of the geodesic structure around non-Kerr
spacetimes will result in a powerful tool that could make the mapping of
spacetime around massive quiescent compact objects possible. To this end, I
present an analytic closed form expression for the components of a the fourth
order Killing tensor for Stationary Axisymmetric Vacuum (SAV) Spacetimes. It is
as yet unclear what subset of SAV spacetimes admit this solution. The solution
is written in terms of an integral expression involving the metric functions
and two specific Greens functions. A second integral expression has to vanish
in order for the solution to be exact. In the event that the second integral
does not vanish it is likely that the best fourth order approximation to the
invariant has been found. This solution can be viewed as a generalized Carter
constant providing an explicit expression for the fourth invariant, in addition
to the energy, azimuthal angular momentum and rest mass, associated with
geodesic motion in SAV spacetimes, be it exact or approximate. I further
comment on the application of this result for the founding of a general
algorithm for mapping the spacetime around compact objects using gravitational
wave observatories.Comment: 5 Page

### Spacetime Encodings II - Pictures of Integrability

I visually explore the features of geodesic orbits in arbitrary stationary
axisymmetric vacuum (SAV) spacetimes that are constructed from a complex Ernst
potential. Some of the geometric features of integrable and chaotic orbits are
highlighted. The geodesic problem for these SAV spacetimes is rewritten as a
two degree of freedom problem and the connection between current ideas in
dynamical systems and the study of two manifolds sought. The relationship
between the Hamilton-Jacobi equations, canonical transformations, constants of
motion and Killing tensors are commented on. Wherever possible I illustrate the
concepts by means of examples from general relativity. This investigation is
designed to build the readers' intuition about how integrability arises, and to
summarize some of the known facts about two degree of freedom systems. Evidence
is given, in the form of orbit-crossing structure, that geodesics in SAV
spacetimes might admit, a fourth constant of motion that is quartic in momentum
(by contrast with Kerr spacetime, where Carter's fourth constant is quadratic).Comment: 11 pages, 10 figure