60 research outputs found
Non-uniqueness of the Dirac theory in a curved spacetime
We summarize a recent work on the subject title. The Dirac equation in a
curved spacetime depends on a field of coefficients (essentially the Dirac
matrices), for which a continuum of different choices are possible. We study
the conditions under which a change of the coefficient fields leads to an
equivalent Hamiltonian operator H, or to an equivalent energy operator E. In
this paper, we focus on the standard version of the gravitational Dirac
equation, but the non-uniqueness applies also to our alternative versions. We
find that the changes which lead to an equivalent operator H, or respectively
to an equivalent operator E, are determined by initial data, or respectively
have to make some point-dependent antihermitian matrix vanish. Thus, the vast
majority of the possible coefficient changes lead neither to an equivalent
operator H, nor to an equivalent operator E, whence a lack of uniqueness. We
show that even the Dirac energy spectrum is not unique.Comment: 13 pages (standard 12pt article format). Text of a talk given at the
1st Mediterranean Conference on Classical and Quantum Gravity, Kolymbari
(Greece), Sept. 14-18, 200
Gravitational Energy Loss and Binary Pulsars in the Scalar Ether-Theory of Gravitation
Motivation is given for trying a theory of gravity with a preferred reference
frame (``ether'' for short). One such theory is summarized, that is a scalar
bimetric theory. Dynamics is governed by an extension of Newton's second law.
In the static case, geodesic motion is recovered together with Newton's
attraction field. In the static spherical case, Schwarzschild's metric is got.
An asymptotic scheme of post-Minkowskian (PM) approximation is built by
associating a conceptual family of systems with the given weakly-gravitating
system. It is more general than the post-Newtonian scheme in that the velocity
may be comparable with . This allows to justify why the 0PM approximation of
the energy rate may be equated to the rate of the Newtonian energy, as is
usually done. At the 0PM approximation of this theory, an isolated system loses
energy by quadrupole radiation, without any monopole or dipole term. It seems
plausible that the observations on binary pulsars (the pulse data) could be
nicely fitted with a timing model based on this theory.Comment: Text of a talk given at the 4th Conf. on Physics Beyond the Standard
Model, Tegernsee, June 2003, submitted to the Proceedings (H. V.
Klapdor-Kleingrothaus, ed.
Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations
One may ask whether the relations between energy and frequency and between
momentum and wave vector, introduced for matter waves by de Broglie, are
rigorously valid in the presence of gravity. In this paper, we show this to be
true for Dirac equations in a background of gravitational and electromagnetic
fields. We first transform any Dirac equation into an equivalent canonical
form, sometimes used in particular cases to solve Dirac equations in a curved
spacetime. This canonical form is needed to apply the Whitham Lagrangian
method. The latter method, unlike the WKB method, places no restriction on the
magnitude of Planck's constant to obtain wave packets, and furthermore
preserves the symmetries of the Dirac Lagrangian. We show using canonical Dirac
fields in a curved spacetime, that the probability current has a Gordon
decomposition into a convection current and a spin current, and that the spin
current vanishes in the Whitham approximation, which explains the negligible
effect of spin on wave packet solutions, independent of the size of Planck's
constant. We further discuss the classical-quantum correspondence in a curved
spacetime based on both Lagrangian and Hamiltonian formulations of the Whitham
equations. We show that the generalized de Broglie relations in a curved
spacetime are a direct consequence of Whitham's Lagrangian method, and not just
a physical hypothesis as introduced by Einstein and de Broglie, and by many
quantum mechanics textbooks.Comment: PDF, 32 pages in referee format. Added significant material on
canonical forms of Dirac equations. Simplified Theorem 1 for normal Dirac
equations. Added section on Gordon decomposition of the probability current.
Encapsulated main results in the statement of Theorem
Dirac equation from the Hamiltonian and the case with a gravitational field
Starting from an interpretation of the classical-quantum correspondence, we
derive the Dirac equation by factorizing the algebraic relation satisfied by
the classical Hamiltonian, before applying the correspondence. This derivation
applies in the same form to a free particle, to one in an electromagnetic
field, and to one subjected to geodesic motion in a static metric, and leads to
the same, usual form of the Dirac equation--in special coordinates. To use the
equation in the static-gravitational case, we need to rewrite it in more
general coordinates. This can be done only if the usual, spinor transformation
of the wave function is replaced by the 4-vector transformation. We show that
the latter also makes the flat-space-time Dirac equation Lorentz-covariant,
although the Dirac matrices are not invariant. Because the equation itself is
left unchanged in the flat case, the 4-vector transformation does not alter the
main physical consequences of that equation in that case. However, the equation
derived in the static-gravitational case is not equivalent to the standard
(Fock-Weyl) gravitational extension of the Dirac equation.Comment: 27 pages, standard LaTeX. v2: minor style changes, accepted for
publication in Found. Phys. Letter
On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations
The two-dimensional unsteady coupled Burgers' equations with moderate to
severe gradients, are solved numerically using higher-order accurate finite
difference schemes; namely the fourth-order accurate compact ADI scheme, and
the fourth-order accurate Du Fort Frankel scheme. The question of numerical
stability and convergence are presented. Comparisons are made between the
present schemes in terms of accuracy and computational efficiency for solving
problems with severe internal and boundary gradients. The present study shows
that the fourth-order compact ADI scheme is stable and efficient
A spatially-VSL gravity model with 1-PN limit of GRT
A scalar gravity model is developed according the 'geometric conventionalist'
approach introduced by Poincare (Einstein 1921, Poincare 1905, Reichenbach
1957, Gruenbaum1973). In principle this approach allows an alternative
interpretation and formulation of General Relativity Theory (GRT), with
distinct i) physical congruence standard, and ii) gravitation dynamics
according Hamilton-Lagrange mechanics, while iii) retaining empirical
indistinguishability with GRT. In this scalar model the congruence standards
have been expressed as gravitationally modified Lorentz Transformations
(Broekaert 2002). The first type of these transformations relate quantities
observed by gravitationally 'affected' (natural geometry) and 'unaffected'
(coordinate geometry) observers and explicitly reveal a spatially variable
speed of light (VSL). The second type shunts the unaffected perspective and
relates affected observers, recovering i) the invariance of the locally
observed velocity of light, and ii) the local Minkowski metric (Broekaert
2003). In the case of a static gravitation field the model retrieves the
phenomenology implied by the Schwarzschild metric. The case with proper source
kinematics is now described by introduction of a 'sweep velocity' field w: The
model then provides a hamiltonian description for particles and photons in full
accordance with the first Post-Newtonian approximation of GRT (Weinberg 1972,
Will 1993).Comment: v1: 11 pages, GR17 conf. paper, Dublin 2004, v2: WEP issue solved,
section on acceleration transformation added, text improved, more references,
same results, v3: typos removed, footnotes, added and references updated, v4:
appendix added, improved tex
Eu Law as Private International Law? Reconceptualising the Country-of-Origin Principle as Vested-Rights Theory
A method to measure the resonance transitions between the gravitationally bound quantum states of neutrons in the GRANIT spectrometer
We present a method to measure the resonance transitions between the
gravitationally bound quantum states of neutrons in the GRANIT spectrometer.
The purpose of GRANIT is to improve the accuracy of measurement of the quantum
states parameters by several orders of magnitude, taking advantage of long
storage of Ultracold neutrons at specula trajectories. The transitions could be
excited using a periodic spatial variation of a magnetic field gradient. If the
frequency of such a perturbation (in the frame of a moving neutron) coincides
with a resonance frequency defined by the energy difference of two quantum
states, the transition probability will sharply increase. The GRANIT experiment
is motivated by searches for short-range interactions (in particular
spin-dependent interactions), by studying the interaction of a quantum system
with a gravitational field, by searches for extensions of the Standard model,
by the unique possibility to check the equivalence principle for an object in a
quantum state and by studying various quantum optics phenomena
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