16,031 research outputs found
Constructing a Space from the System of Geodesic Equations
Given a space it is easy to obtain the system of geodesic equations on it. In
this paper the inverse problem of reconstructing the space from the geodesic
equations is addressed. A procedure is developed for obtaining the metric
tensor from the Christoffel symbols. The procedure is extended for determining
if a second order quadratically semi-linear system can be expressed as a system
of geodesic equations, provided it has terms only quadratic in the first
derivative apart from the second derivative term. A computer code has been
developed for dealing with larger systems of geodesic equations
Coupling of Linearized Gravity to Nonrelativistic Test Particles: Dynamics in the General Laboratory Frame
The coupling of gravity to matter is explored in the linearized gravity
limit. The usual derivation of gravity-matter couplings within the
quantum-field-theoretic framework is reviewed. A number of inconsistencies
between this derivation of the couplings, and the known results of tidal
effects on test particles according to classical general relativity are pointed
out. As a step towards resolving these inconsistencies, a General Laboratory
Frame fixed on the worldline of an observer is constructed. In this frame, the
dynamics of nonrelativistic test particles in the linearized gravity limit is
studied, and their Hamiltonian dynamics is derived. It is shown that for
stationary metrics this Hamiltonian reduces to the usual Hamiltonian for
nonrelativistic particles undergoing geodesic motion. For nonstationary metrics
with long-wavelength gravitational waves (GWs) present, it reduces to the
Hamiltonian for a nonrelativistic particle undergoing geodesic
\textit{deviation} motion. Arbitrary-wavelength GWs couple to the test particle
through a vector-potential-like field , the net result of the tidal forces
that the GW induces in the system, namely, a local velocity field on the system
induced by tidal effects as seen by an observer in the general laboratory
frame. Effective electric and magnetic fields, which are related to the
electric and magnetic parts of the Weyl tensor, are constructed from that
obey equations of the same form as Maxwell's equations . A gedankin
gravitational Aharonov-Bohm-type experiment using to measure the
interference of quantum test particles is presented.Comment: 38 pages, 7 figures, written in ReVTeX. To appear in Physical Review
D. Galley proofs corrections adde
Eisenhart lift for higher derivative systems
The Eisenhart lift provides an elegant geometric description of a dynamical
system of second order in terms of null geodesics of the Brinkmann-type metric.
In this work, we attempt to generalize the Eisenhart method so as to encompass
higher derivative models. The analysis relies upon Ostrogradsky's Hamiltonian.
A consistent geometric description seems feasible only for a particular class
of potentials. The scheme is exemplified by the Pais-Uhlenbeck oscillator.Comment: V2: 12 pages, minor improvements, references added; the version to
appear in PL
Kinematics and uncertainty relations of a quantum test particle in a curved space-time
A possible model for quantum kinematics of a test particle in a curved
space-time is proposed. Every reasonable neighbourhood V_e of a curved
space-time can be equipped with a nonassociative binary operation called the
geodesic multiplication of space-time points. In the case of the Minkowski
space-time, left and right translations of the geodesic multiplication coincide
and amount to a rigid shift of the space-time x->x+a. In a curved space-time
infinitesimal geodesic right translations can be used to define the (geodesic)
momentum operators. The commutation relations of position and momentum
operators are taken as the quantum kinematic algebra. As an example, detailed
calculations are performed for the space-time of a weak plane gravitational
wave. The uncertainty relations following from the commutation rules are
derived and their physical meaning is discussed.Comment: 6 pages, LaTeX, talk given in the session ``Quantum Fields in Curved
Space'' at the VIII Marcel Grossmann Conference in Jerusalem, Israel, June
199
Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group
We study Sobolev-type metrics of fractional order on the group
\Diff_c(M) of compactly supported diffeomorphisms of a manifold . We show
that for the important special case the geodesic distance on
\Diff_c(S^1) vanishes if and only if . For other manifolds we
obtain a partial characterization: the geodesic distance on \Diff_c(M)
vanishes for and for ,
with being a compact Riemannian manifold. On the other hand the geodesic
distance on \Diff_c(M) is positive for and
.
For we discuss the geodesic equations for these metrics. For
we obtain some well known PDEs of hydrodynamics: Burgers' equation for ,
the modified Constantin-Lax-Majda equation for and the
Camassa-Holm equation for .Comment: 16 pages. Final versio
Eisenhart lift for higher derivative systems
The Eisenhart lift provides an elegant geometric description of a dynamical
system of second order in terms of null geodesics of the Brinkmann-type metric.
In this work, we attempt to generalize the Eisenhart method so as to encompass
higher derivative models. The analysis relies upon Ostrogradsky's Hamiltonian.
A consistent geometric description seems feasible only for a particular class
of potentials. The scheme is exemplified by the Pais-Uhlenbeck oscillator.Comment: V2: 12 pages, minor improvements, references added; the version to
appear in PL
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