47,541 research outputs found
Wave Packet Pseudomodes of Variable Coefficient Differential Operators
The pseudospectra of nonselfadjoint linear ordinary differential operators with variable coefficients are considered. It is shown that when a certain winding number or twist condition is satisfied, closely related to Hörmander's commutator condition for partial differential equations, \varepsilon-pseudoeigenfunctions of such operators for exponentially small values of \varepsilon exist in the form of localized wave packets. In contrast to related results of Davies and of Dencker, Sjöstrand, and Zworski, the symbol need not be smooth
Spherically Symmetric Solutions in Higher-Derivative Gravity
Extensions of Einstein gravity with quadratic curvature terms in the action
arise in most effective theories of quantised gravity, including string theory.
This article explores the set of static, spherically symmetric and
asymptotically flat solutions of this class of theories. An important element
in the analysis is the careful treatment of a Lichnerowicz-type `no-hair'
theorem. From a Frobenius analysis of the asymptotic small-radius behaviour,
the solution space is found to split into three asymptotic families, one of
which contains the classic Schwarzschild solution. These three families are
carefully analysed to determine the corresponding numbers of free parameters in
each. One solution family is capable of arising from coupling to a
distributional shell of matter near the origin; this family can then match on
to an asymptotically flat solution at spatial infinity without encountering a
horizon. Another family, with horizons, contains the Schwarzschild solution but
includes also non-Schwarzschild black holes. The third family of solutions
obtained from the Frobenius analysis is nonsingular and corresponds to `vacuum'
solutions. In addition to the three families identified from near-origin
behaviour, there are solutions that may be identified as `wormholes', which can
match symmetrically on to another sheet of spacetime at finite radius.Comment: 57 pages, 6 figures; version appearing in journal; minor corrections
and clarifications to v
Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications
We provide a theory to establish the existence of nonzero solutions of
perturbed Hammerstein integral equations with deviated arguments, being our
main ingredient the theory of fixed point index. Our approach is fairly general
and covers a variety of cases. We apply our results to a periodic boundary
value problem with reflections and to a thermostat problem. In the case of
reflections we also discuss the optimality of some constants that occur in our
theory. Some examples are presented to illustrate the theory.Comment: 3 figures, 23 page
Gravitomagnetic Jets
We present a family of dynamic rotating cylindrically symmetric Ricci-flat
gravitational fields whose geodesic motions have the structure of
gravitomagnetic jets. These correspond to helical motions of free test
particles up and down parallel to the axis of cylindrical symmetry and are
reminiscent of the motion of test charges in a magnetic field. The speed of a
test particle in a gravitomagnetic jet asymptotically approaches the speed of
light. Moreover, numerical evidence suggests that jets are attractors. The
possible implications of our results for the role of gravitomagnetism in the
formation of astrophysical jets are briefly discussed.Comment: 47 pages, 8 figures; v2: minor improvements; v3: paragraph added at
the end of Sec. V and other minor improvements; v4: reference added, typos
corrected, sentence added on p. 24; v5: a few minor improvement
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