Quasi-local mass in the covariant Newtonian space-time

In general relativity, quasi-local energy-momentum expressions have been constructed from various formulae. However, Newtonian theory of gravity gives a well known and an unique quasi-local mass expression (surface integration). Since geometrical formulation of Newtonian gravity has been established in the covariant Newtonian space-time, it provides a covariant approximation from relativistic to Newtonian theories. By using this approximation, we calculate Komar integral, Brown-York quasi-local energy and Dougan-Mason quasi-local mass in the covariant Newtonian space-time. It turns out that Komar integral naturally gives the Newtonian quasi-local mass expression, however, further conditions (spherical symmetry) need to be made for Brown-York and Dougan-Mason expressions.Comment: Submit to Class. Quantum Gra

Two dimensional Sen connections and quasi-local energy-momentum

The recently constructed two dimensional Sen connection is applied in the problem of quasi-local energy-momentum in general relativity. First it is shown that, because of one of the two 2 dimensional Sen--Witten identities, Penrose's quasi-local charge integral can be expressed as a Nester--Witten integral.Then, to find the appropriate spinor propagation laws to the Nester--Witten integral, all the possible first order linear differential operators that can be constructed only from the irreducible chiral parts of the Sen operator alone are determined and examined. It is only the holomorphy or anti-holomorphy operator that can define acceptable propagation laws. The 2 dimensional Sen connection thus naturally defines a quasi-local energy-momentum, which is precisely that of Dougan and Mason. Then provided the dominant energy condition holds and the 2-sphere S is convex we show that the next statements are equivalent: i. the quasi-local mass (energy-momentum) associated with S is zero; ii.the Cauchy development $D(\Sigma)$ is a pp-wave geometry with pure radiation ($D(\Sigma)$ is flat), where $\Sigma$ is a spacelike hypersurface whose boundary is S; iii. there exist a Sen--constant spinor field (two spinor fields) on S. Thus the pp-wave Cauchy developments can be characterized by the geometry of a two rather than a three dimensional submanifold.Comment: 20 pages, Plain Tex, I

Discharge-mechanical method of rock breakage

The electric discharge and mechanical technology of hard rock breakage was developed on the ground of mechanical and electrical pulse methods and it was tested for purposes of deep drilling. It was demonstrated that, due to breakage of the rock surface by electric discharges, the rock excavation volume (breakage performance) is significantly improved as compared to conventional mechanical methods

On the construction of a geometric invariant measuring the deviation from Kerr data

This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed, to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation ---the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant ---however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.Comment: 36 pages. Updated references. Technical details correcte

Two dimensional Sen connections in general relativity

The two dimensional version of the Sen connection for spinors and tensors on spacelike 2-surfaces is constructed. A complex metric $\gamma_{AB}$ on the spin spaces is found which characterizes both the algebraic and extrinsic geometrical properties of the 2-surface $\$$. The curvature of the two dimensional Sen operator $\Delta_e$ is the pull back to $\$$ of the anti-self-dual part of the spacetime curvature while its `torsion' is a boost gauge invariant expression of the extrinsic curvatures of $\$$. The difference of the 2 dimensional Sen and the induced spin connections is the anti-self-dual part of the `torsion'. The irreducible parts of $\Delta_e$ are shown to be the familiar 2-surface twistor and the Weyl--Sen--Witten operators. Two Sen--Witten type identities are derived, the first is an identity between the 2 dimensional twistor and the Weyl--Sen--Witten operators and the integrand of Penrose's charge integral, while the second contains the `torsion' as well. For spinor fields satisfying the 2-surface twistor equation the first reduces to Tod's formula for the kinematical twistor.Comment: 14 pages, Plain Tex, no report numbe

Gravitational Energy in Spherical Symmetry

Various properties of the Misner-Sharp spherically symmetric gravitational energy E are established or reviewed. In the Newtonian limit of a perfect fluid, E yields the Newtonian mass to leading order and the Newtonian kinetic and potential energy to the next order. For test particles, the corresponding Hajicek energy is conserved and has the behaviour appropriate to energy in the Newtonian and special-relativistic limits. In the small-sphere limit, the leading term in E is the product of volume and the energy density of the matter. In vacuo, E reduces to the Schwarzschild energy. At null and spatial infinity, E reduces to the Bondi-Sachs and Arnowitt-Deser-Misner energies respectively. The conserved Kodama current has charge E. A sphere is trapped if E>r/2, marginal if E=r/2 and untrapped if E<r/2, where r is the areal radius. A central singularity is spatial and trapped if E>0, and temporal and untrapped if E<0. On an untrapped sphere, E is non-decreasing in any outgoing spatial or null direction, assuming the dominant energy condition. It follows that E>=0 on an untrapped spatial hypersurface with regular centre, and E>=r_0/2 on an untrapped spatial hypersurface bounded at the inward end by a marginal sphere of radius r_0. All these inequalities extend to the asymptotic energies, recovering the Bondi-Sachs energy loss and the positivity of the asymptotic energies, as well as proving the conjectured Penrose inequality for black or white holes. Implications for the cosmic censorship hypothesis and for general definitions of gravitational energy are discussed.Comment: 23 pages. Belatedly replaced with substantially extended published versio

Editing of the urease gene by CRISPR-Cas in the diatom Thalassiosira pseudonana

Background: CRISPR-Cas is a recent and powerful addition to the molecular toolbox which allows programmable genome editing. It has been used to modify genes in a wide variety of organisms, but only two alga to date. Here we present a methodology to edit the genome of Thalassiosira pseudonana, a model centric diatom with both ecological significance and high biotechnological potential, using CRISPR-Cas. Results: A single construct was assembled using Golden Gate cloning. Two sgRNAs were used to introduce a precise 37 nt deletion early in the coding region of the urease gene. A high percentage of bi-allelic mutations (≤61.5%) were observed in clones with the CRISPR-Cas construct. Growth of bi-allelic mutants in urea led to a significant reduction in growth rate and cell size compared to growth in nitrate. Conclusions: CRISPR-Cas can precisely and efficiently edit the genome of T. pseudonana. The use of Golden Gate cloning to assemble CRISPR-Cas constructs gives additional flexibility to the CRISPR-Cas method and facilitates modifications to target alternative genes or species

Curved spaces admiting solutions to twistor equations

This thesis comprises three sections. In the first, real space-times admitting a solution to the two-index twistor (Killing spinor) equation are constructed and the separability properties of solutions to zero rest-mass field equations within these space-times are derived. In addition conditions for differential relations between components of opposite helicity are derived. In the second section class of half algebraically-special spaces is investigated, a class which may be characterised as those admitting a certain partial two-index twistor (Killing spinor). A formalism based on their structure as a two-dimensional family of flat null two-planes is developed and used to provide an explicit integration of the curvature equations, to discuss Hertz-type potentials for zero rest-mass fields and to examine a class of Einstein-Maxwell spaces and their perturbations. In the third section we examine the relation between the space of solutions to the twistor equation restricted to a general space--like two-surface of spherical topology (two-surface twistor space) and its dual. This is then used to discuss the nature of the local embedding of the two-surface in a space conformally related to complex Minkowski space