2,316 research outputs found
Classical and semi-classical energy conditions
The standard energy conditions of classical general relativity are (mostly)
linear in the stress-energy tensor, and have clear physical interpretations in
terms of geodesic focussing, but suffer the significant drawback that they are
often violated by semi-classical quantum effects. In contrast, it is possible
to develop non-standard energy conditions that are intrinsically non-linear in
the stress-energy tensor, and which exhibit much better well-controlled
behaviour when semi-classical quantum effects are introduced, at the cost of a
less direct applicability to geodesic focussing. In this article we will first
review the standard energy conditions and their various limitations. (Including
the connection to the Hawking--Ellis type I, II, III, and IV classification of
stress-energy tensors). We shall then turn to the averaged, nonlinear, and
semi-classical energy conditions, and see how much can be done once
semi-classical quantum effects are included.Comment: V1: 25 pages. Draft chapter, on which the related chapter of the book
"Wormholes, Warp Drives and Energy Conditions" (to be published by Springer),
will be based. V2: typos fixed. V3: small typo fixe
Non-unitary representations of the SU(2) algebra in the Dirac equation with a Coulomb potential
A novel realization of the classical SU(2) algebra is introduced for the
Dirac relativistic hydrogen atom defining a set of operators that, besides,
allow the factorization of the problem. An extra phase is needed as a new
variable in order to define the algebra. We take advantage of the operators to
solve the Dirac equation using algebraic methods. To acomplish this, a similar
path to the one used in the angular momentum case is employed; hence, the
radial eigenfuntions calculated comprise non unitary representations of the
algebra. One of the interesting properties of such non unitary representations
is that they are not labeled by integer nor by half-integer numbers as happens
in the usual angular momentum representation.Comment: 20 pages 1 eps figure in a single zipped file, submitted to J. Math.
Phy
An algebraic SU(1,1) solution for the relativistic hydrogen atom
The bound eigenfunctions and spectrum of a Dirac hydrogen atom are found
taking advantage of the Lie algebra in which the radial part of the
problem can be expressed. For defining the algebra we need to add to the
description an additional angular variable playing essentially the role of a
phase. The operators spanning the algebra are used for defining ladder
operators for the radial eigenfunctions of the relativistic hydrogen atom and
for evaluating its energy spectrum. The status of the Johnson-Lippman operator
in this algebra is also investigated.Comment: to appear in Physics Letters A (2005). We corrected a misprint in
page 7, in the paragraph baggining with "With the value of ..." the ground
state should be |\lambda, \lambda>, not |\lambda, \lambda+1
Formation of corner waves in the wake of a partially submerged bluff body
We study theoretically and numerically the downstream flow near the corner of a bluff body partially submerged at a deadrise depth Îh into a uniform stream of velocity U, in the presence of gravity, g. When the Froude number, Fr=U/âgÎh, is large, a three-dimensional steady plunging wave, which is referred to as a corner wave, forms near the corner, developing downstream in a similar way to a two-dimensional plunging wave evolving in time. We have performed an asymptotic analysis of the flow near this corner to describe the wave's initial evolution and to clarify the physical mechanism that leads to its formation. Using the two-dimensions-plus-time approximation, the problem reduces to one similar to dam-break flow with a wet bed in front of the dam. The analysis shows that, at leading order, the problem admits a self-similar formulation when the size of the wave is small compared with the height difference Îh. The essential feature of the self-similar solution is the formation of a mushroom-shaped jet from which two smaller lateral jets stem. However, numerical simulations show that this self-similar solution is questionable from the physical point of view, as the two lateral jets plunge onto the free surface, leading to a self-intersecting flow. The physical mechanism leading to the formation of the mushroom-shaped structure is discussed
New non-unitary representations in a Dirac hydrogen atom
New non-unitary representations of the SU(2) algebra are introduced for the
case of the Dirac equation with a Coulomb potential; an extra phase, needed to
close the algebra, is also introduced. The new representations does not require
integer or half integer labels. The set of operators defined are used to span
the complete space of bound state eigenstates of the problem thus solving it in
an essentially algebraic way
A useful form of the recurrence relation between relativistic atomic matrix elements of radial powers
Recently obtained recurrence formulae for relativistic hydrogenic radial
matrix elements are cast in a simpler and perhaps more useful form. This is
achieved with the help of a new relation between the and the
terms ( is a Dirac matrix and are constants) in the
atomic matrix elements.Comment: 7 pages, no figure
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