11,321 research outputs found

### Results on the spectrum of R-Modes of slowly rotating relativistic stars

The paper considers the spectrum of axial perturbations of slowly uniformly
rotating general relativistic stars in the framework of Y. Kojima. In a first
step towards a full analysis only the evolution equations are treated but not
the constraint. Then it is found that the system is unstable due to a continuum
of non real eigenvalues. In addition the resolvent of the associated generator
of time evolution is found to have a special structure which was discussed in a
previous paper. From this structure it follows the occurrence of a continuous
part in the spectrum of oscillations at least if the system is restricted to a
finite space as is done in most numerical investigations. Finally, it can be
seen that higher order corrections in the rotation frequency can qualitatively
influence the spectrum of the oscillations. As a consequence different
descriptions of the star which are equivalent to first order could lead to
different results with respect to the stability of the star

### On the Completeness of the Quasinormal Modes of the Poeschl-Teller Potential

The completeness of the quasinormal modes of the wave equation with
Poeschl-Teller potential is investigated. A main result is that after a large
enough time $t_0$, the solutions of this equation corresponding to
$C^{\infty}$-data with compact support can be expanded uniformly in time with
respect to the quasinormal modes, thereby leading to absolutely convergent
series. Explicit estimates for $t_0$ depending on both the support of the data
and the point of observation are given. For the particular case of an ``early''
time and zero distance between the support of the data and observational point,
it is shown that the corresponding series is not absolutely convergent, and
hence that there is no associated sum which is independent of the order of
summation.Comment: 22 pages, 2 figures, submitted to Comm. Math. Phy

### A new result on the Klein-Gordon equation in the background of a rotating black hole

This short paper should serve as basis for further analysis of a previously
found new symmetry of the solutions of the wave equation in the gravitational
field of a Kerr black hole. Its main new result is the proof of essential
self-adjointness of the spatial part of a reduced normalized wave operator of
the Kerr metric in a weighted L^2-space. As a consequence, it leads to a purely
operator theoretic proof of the well-posedness of the initial value problem of
the reduced Klein-Gordon equation in that field in that L^2-space and in this
way generalizes a corresponding result of Kay (1985) in the case of the
Schwarzschild black hole. It is believed that the employed methods are
applicable to other separable wave equations

### On a new symmetry of the solutions of the wave equation in the background of a Kerr black hole

This short paper derives the constant of motion of a scalar field in the
gravitational field of a Kerr black hole which is associated to a Killing
tensor of that space-time. In addition, there is found a related new symmetry
operator S for the solutions of the wave equation in that background. That
operator is a partial differential operator with a leading order time
derivative of the first order that commutes with a normal form of the wave
operator. That form is obtained by multiplication of the wave operator from the
left with the reciprocal of the coefficient function of its second order time
derivative. It is shown that S induces an operator that commutes with the
generator of time evolution in a formulation of the initial value problem for
the wave equation in the setting of strongly continuous semigroups

### On the r-mode spectrum of relativistic stars

We present a mathematically rigorous proof that the r-mode spectrum of
relativistic stars to the rotational lowest order has a continuous part. A
rigorous definition of this spectrum is given in terms of the spectrum of a
continuous linear operator. This study verifies earlier results by Kojima
(1998) about the nature of the r-mode spectrum.Comment: 6 pages, no figure

### Galileo early cruise, including Venus, first Earth, and Gaspra encounters

This article documents Deep Space Network (DSN) support for the Galileo cruise to Jupiter. The unique trajectory affords multiple encounters during this cruise phase. Each encounter had or will have unique requirements for data acquisition and DSN support configurations. An overview of the cruise and encounters through the asteroid Gaspra encounter is provided

### Quantifying excitations of quasinormal mode systems

Computations of the strong field generation of gravitational waves by black
hole processes produce waveforms that are dominated by quasinormal (QN)
ringing, a damped oscillation characteristic of the black hole. We describe
here the mathematical problem of quantifying the QN content of the waveforms
generated. This is done in several steps: (i) We develop the mathematics of QN
systems that are complete (in a sense to be defined) and show that there is a
quantity, the ``excitation coefficient,'' that appears to have the properties
needed to quantify QN content. (ii) We show that incomplete systems can (at
least sometimes) be converted to physically equivalent complete systems. Most
notably, we give a rigorous proof of completeness for a specific modified model
problem. (iii) We evaluate the excitation coefficient for the model problem,
and demonstrate that the excitation coefficient is of limited utility. We
finish by discussing the general question of quantification of QN excitations,
and offer a few speculations about unavoidable differences between normal mode
and QN systems.Comment: 27 pages, 14 figures. To be published in: J. Math. Phys. (1999

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