Deviation of geodesics in FLRW spacetime geometries

The geodesic deviation equation (`GDE') provides an elegant tool to investigate the timelike, null and spacelike structure of spacetime geometries. Here we employ the GDE to review these structures within the Friedmann--Lema\^{\i}tre--Robertson--Walker (`FLRW') models, where we assume the sources to be given by a non-interacting mixture of incoherent matter and radiation, and we also take a non-zero cosmological constant into account. For each causal case we present examples of solutions to the GDE and we discuss the interpretation of the related first integrals. The de Sitter spacetime geometry is treated separately.Comment: 17 pages, LaTeX 2.09, 3 *.eps figures, Contribution to the forthcoming Engelbert Sch\"{u}cking Festschrift (Springer Verlag

Sphere packings of discs with two radii

In this paper we will take a look at sphere packings and we will try to find the highest density binary lattice packings in the 2-dimensional space R2. First we start with defining the properties of lattice packings of different convex bodies. Then we transform the problem into an optimization problem, which turns out to be a Non-Convex Quadratic Constrained Quadratic Program that can not be solved normally. Therefore a couple of relaxation techniques are used in order to find an upper bound on the highest density. Eventually only the option with two spheres in a unit tile have been examined for which also a lower bound for the highest density is given. Though the upper bounds and the lower bounds are equal, they are still lower bounds for the actual highest density of binary lattice packings. In this paper we will take a look at sphere packings and we will try to find the highest density binary lattice packings in the 2-dimensional space R2. First we start with defining the properties of lattice packings of different convex bodies. Then we transform the problem into an optimization problem, which turns out to be a Non-Convex Quadratic Constrained Quadratic Program that can not be solved normally. Therefore a couple of relaxation techniques are used in order to find an upper bound on the highest density. Eventually only the option with two spheres in a unit tile have been examined for which also a lower bound for the highest density is given. Though the upper bounds and the lower bounds are equal, they are still lower bounds for the actual highest density of binary lattice packings

Integrability of irrotational silent cosmological models

We revisit the issue of integrability conditions for the irrotational silent cosmological models. We formulate the problem both in 1+3 covariant and 1+3 orthonormal frame notation, and show there exists a series of constraint equations that need to be satisfied. These conditions hold identically for FLRW-linearised silent models, but not in the general exact non-linear case. Thus there is a linearisation instability, and it is highly unlikely that there is a large class of silent models. We conjecture that there are no spatially inhomogeneous solutions with Weyl curvature of Petrov type I, and indicate further issues that await clarification.Comment: Minor corrections and improvements; 1 new reference; to appear Class. Quantum Grav.; 16 pages Ioplpp

Partially locally rotationally symmetric perfect fluid cosmologies

We show that there are no new consistent cosmological perfect fluid solutions when in an open neighbourhood ${\cal U}$ of an event the fluid kinematical variables and the electric and magnetic Weyl curvature are all assumed rotationally symmetric about a common spatial axis, specialising the Weyl curvature tensor to algebraic Petrov type D. The consistent solutions of this kind are either locally rotationally symmetric, or are subcases of the Szekeres dust models. Parts of our results require the assumption of a barotropic equation of state. Additionally we demonstrate that local rotational symmetry of perfect fluid cosmologies follows from rotational symmetry of the Riemann curvature tensor and of its covariant derivatives only up to second order, thus strengthening a previous result.Comment: 20 pages, LaTeX2.09 (10pt), no figures; shortened revised version, new references; accepted for publication in Classical and Quantum Gravit

History and sensitivity comparison of two standard whole-sediment toxicity tests with crustaceans : the amphipod Hyalella azteca and the ostracod Heterocypris incongruens microbiotest

The review first details the development of the test procedures with Hyalella azteca which historically emerged as one of the recommended test species for whole-sediment assays and its gradual standardization and endorsement by national and international organizations. The sensitivity and precision of the H. azteca test for application on chemicals and on real world sediments is discussed. The review subsequently addresses the development of the whole sediment microbiotest with the ostracod crustacean Heterocypris incongruens with larvae of this test species hatched from dormant eggs (cysts), rendering this assay stock culture/maintenance free. The application of the 6-day ostracod microbiotest on sediments in Canada and in Belgium is discussed, as well as its endorsement by the ISO subsequent to an extensive international inter-laboratory ring test. The sensitivity of the amphipod and ostracod tests is compared by data from studies in which both assays were applied in parallel. A comparison of more than 1000 ostracod/amphipod data pairs of a 12-year river sediment monitoring study in Flanders/Belgium confirmed that both whole-sediment assays have a similar sensitivity and that the 6-day ostracod microbiotest is a valuable and cost-effective alternative to the 10-14 day amphipod test for evaluation of the toxic hazard of polluted sediments

Quasi-Newtonian dust cosmologies

Exact dynamical equations for a generic dust matter source field in a cosmological context are formulated with respect to a non-comoving Newtonian-like timelike reference congruence and investigated for internal consistency. On the basis of a lapse function $N$ (the relativistic acceleration scalar potential) which evolves along the reference congruence according to $\dot{N} = \alpha \Theta N$ ($\alpha = {const}$), we find that consistency of the quasi-Newtonian dynamical equations is not attained at the first derivative level. We then proceed to show that a self-consistent set can be obtained by linearising the dynamical equations about a (non-comoving) FLRW background. In this case, on properly accounting for the first-order momentum density relating to the non-relativistic peculiar motion of the matter, additional source terms arise in the evolution and constraint equations describing small-amplitude energy density fluctuations that do not appear in similar gravitational instability scenarios in the standard literature.Comment: 25 pages, LaTeX 2.09 (10pt), to appear in Classical and Quantum Gravity, Vol. 15 (1998

General relativistic analysis of peculiar velocities

We give a careful general relativistic and (1+3)-covariant analysis of cosmological peculiar velocities induced by matter density perturbations in the presence of a cosmological constant. In our quasi-Newtonian approach, constraint equations arise to maintain zero shear of the non-comoving fundamental worldlines which define a Newtonian-like frame, and these lead to the (1+3)-covariant dynamical equations, including a generalized Poisson-type equation. We investigate the relation between peculiar velocity and peculiar acceleration, finding the conditions under which they are aligned. In this case we find (1+3)-covariant relativistic generalizations of well-known Newtonian results.Comment: 8 pages, LaTeX2e (iopart); minor changes, matches version accepted for publication by Classical and Quantum Gravit

Local freedom in the gravitational field

In a cosmological context, the electric and magnetic parts of the Weyl tensor, E_{ab} and H_{ab}, represent the locally free curvature - i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of the derivatives of E_{ab} and H_{ab}, we show that the parts of the derivative of the curvature which are locally free (i.e. not pointwise determined by the matter via the Bianchi identities) are exactly the symmetrised trace-free spatial derivatives of E_{ab} and H_{ab} together with their spatial curls. These parts of the derivatives are shown to be crucial for the existence of gravitational waves.Comment: New results on gravitational waves included; new references added; revised version (IOP style) to appear Class. Quantum Gra

1+1+2 Electromagnetic perturbations on non-vacuum LRS class II space-times: Decoupling scalar and 2-vector harmonic amplitudes

We use the covariant and gauge-invariant 1+1+2 formalism of Clarkson and Barrett \cite{Clarkson2003} to analyze electromagnetic (EM) perturbations on non-vacuum {\it locally rotationally symmetric} (LRS) class II space-times. Ultimately, we show how to derive six real decoupled equations governing the total of six EM scalar and 2-vector harmonic amplitudes. Four of these are new, and result from expanding the complex EM 2-vector which we defined in \cite{Burston2007} in terms of EM 2-vector harmonic amplitudes. We are then able to show that there are four precise combinations of the amplitudes that decouple, two of these are polar perturbations whereas the remaining two are axial. The remaining two decoupled equations are the generalized Regge-Wheeler equations which were developed previously in \cite{Betschart2004}, and these govern the two EM scalar harmonic amplitudes. However, our analysis generalizes this by including a full description and classification of energy-momentum sources, such as charges and currents.Comment: 9 page