2,568 research outputs found
Projectiles, pendula, and special relativity
The kind of flat-earth gravity used in introductory physics appears in an
accelerated reference system in special relativity. From this viewpoint, we
work out the special relativistic description of a ballistic projectile and a
simple pendulum, two examples of simple motion driven by earth-surface gravity.
The analysis uses only the basic mathematical tools of special relativity
typical of a first-year university course.Comment: 9 pages, 5 figures; to appear in American Journal of Physic
Analytic approximations to the spacetime of a critical gravitational collapse
We present analytic expressions that approximate the behavior of the
spacetime of a collapsing spherically symmetric scalar field in the critical
regime first discovered by Choptuik. We find that the critical region of
spacetime can usefully be divided into a ``quiescent'' region and an
``oscillatory'' region, and a moving ``transition edge'' that separates the two
regions. We find that in each region the critical solution can be well
approximated by a flat spacetime scalar field solution. A qualitative nonlinear
matching of the solutions across the edge yields the right order of magnitude
for the oscillations of the discretely self-similar critical solution found by
Choptuik.Comment: 12 pages, Revtex, 9 figures included with eps
Ballistic trajectory: parabola, ellipse, or what?
Mechanics texts tell us that a particle in a bound orbit under gravitational
central force moves on an ellipse, while introductory physics texts approximate
the earth as flat, and tell us that the particle moves in a parabola. The
uniform-gravity, flat-earth parabola is clearly meant to be an approximation to
a small segment of the true central-force/ellipse orbit. To look more deeply
into this connection we convert earth-centered polar coordinates to
``flat-earth coordinates'' by treating radial lines as vertical, and by
treating lines of constant radial distance as horizontal. With the exact
trajectory and dynamics in this system, we consider such questions as whether
gravity is purely vertical in this picture, and whether the central force
nature of gravity is important only when the height or range of a ballistic
trajectory is comparable to the earth radius. Somewhat surprisingly, the
answers to both questions is ``no,'' and therein lie some interesting lessons.Comment: 7 pages, 3 figure
The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method
The periodic standing wave (PSW) method for the binary inspiral of black
holes and neutron stars computes exact numerical solutions for periodic
standing wave spacetimes and then extracts approximate solutions of the
physical problem, with outgoing waves. The method requires solution of a
boundary value problem with a mixed (hyperbolic and elliptic) character.
We present here a new numerical method for such problems, based on three
innovations: (i) a coordinate system adapted to the geometry of the problem,
(ii) an expansion in multipole moments of these coordinates and a filtering out
of higher moments, and (iii) the replacement of the continuum multipole moments
with their analogs for a discrete grid. We illustrate the efficiency and
accuracy of this method with nonlinear scalar model problems. Finally, we take
advantage of the ability of this method to handle highly nonlinear models to
demonstrate that the outgoing approximations extracted from the standing wave
solutions are highly accurate even in the presence of strong nonlinearities.Comment: RevTex, 32 pages, 13 figures, 6 table
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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