Null Rotations Research Articles

Spacetimes with continuous linear isotropies III: null rotations

It is shown that in many of the possible cases local null rotation invariance of the curvature and its first derivatives is sufficient to ensure that there is an isometry group G_r with rge 3 acting on (a neighbourhood of) the spacetime and containing a null rotation isotropy. The exceptions where invariance of the second derivatives is additionally required to ensure this conclusion are Petrov type N Einstein spacetimes, spacetimes containing “pure radiation” (a Ricci tensor of Segre type [(11,2)]), and conformally flat spacetimes with a Ricci tensor of Segre type [1(11,1)] (a “tachyon fluid”).

Holographic integral geometry with time dependence

We write down Crofton formulas — expressions that compute lengths of space- like curves in asymptotically AdS3 geometries as integrals over kinematic space — which apply when the curve and/or the background spacetime is time-dependent. Relative to their static predecessor, the time-dependent Crofton formulas display several new features, whose origin is the local null rotation symmetry of the bulk geometry. In pure AdS3 where null rotations are global symmetries, the Crofton formulas simplify and become integrals over the null planes, which intersect the bulk curve.

Sachs\u2019 free data in real connection variables

We discuss the Hamiltonian dynamics of general relativity with real connection variables on a null foliation, and use the Newman-Penrose formalism to shed light on the geometric meaning of the various constraints. We identify the equivalent of Sachs’ constraint-free initial data as projections of connection components related to null rotations, i.e. the translational part of the ISO(2) group stabilising the internal null direction soldered to the hypersurface. A pair of second-class constraints reduces these connection components to the shear of a null geodesic congruence, thus establishing equivalence with the second-order formalism, which we show in details at the level of symplectic potentials. A special feature of the first-order formulation is that Sachs’ propagating equations for the shear, away from the initial hypersurface, are turned into tertiary constraints; their role is to preserve the relation between connection and shear under retarded time evolution. The conversion of wave-like propagating equations into constraints is possible thanks to an algebraic Bianchi identity; the same one that allows one to describe the radiative data at future null infinity in terms of a shear of a (non-geodesic) asymptotic null vector field in the physical spacetime. Finally, we compute the modification to the spin coefficients and the null congruence in the presence of torsion.

Space-Time Defects and Group Momentum Space

We study massive and massless conical defects in Minkowski and de Sitter spaces in various space-time dimensions. The energy momentum of a defect, considered as an (extended) relativistic object, is completely characterized by the holonomy of the connection associated with its space-time metric. The possible holonomies are given by Lorentz group elements, which are rotations and null rotations for massive and massless defects, respectively. In particular, if we fix the direction of propagation of a massless defect in n+1-dimensional Minkowski space, then its space of holonomies is a maximal Abelian subgroup of the AN(n-1) group, which corresponds to the well known momentum space associated with the n-dimensional κ-Minkowski noncommutative space-time and κ-deformed Poincaré algebra. We also conjecture that massless defects in n-dimensional de Sitter space can be analogously characterized by holonomies belonging to the same subgroup. This shows how group-valued momenta related to four-dimensional deformations of relativistic symmetries can arise in the description of motion of space-time defects.

Invisibility and PT Symmetry: A Simple Geometrical Viewpoint

We give a simplified account of the properties of the transfer matrix for a complex one-dimensional potential, paying special attention to the particular instance of unidirectional invisibility. In appropriate variables, invisible potentials appear as performing null rotations, which lead to the helicity-gauge symmetry of massless particles. In hyperbolic geometry, this can be interpreted, via Möbius transformations, as parallel displacements, a geometric action that has no Euclidean analogy.

Mode localization in composite laminates

We study free vibrations of monolithic and composite thin rectangular plates; the former are made of linear elastic, homogeneous, isotropic materials and the latter of fiber reinforced laminas. The plates are clamped on all four edges and interior points on a transverse normal to the plate midsurface are rigidly tied together and have either null displacements and null rotations (Type-I constraint) or only null transverse displacements (Type-II constraint). Depending upon the location of the point on the midsurface through which the transverse normal passes, modes localize in different regions of the plate. Plates of various aspect ratios (length/width) and stacking sequences of 0°, 45° and 90° leading to symmetric and anti-symmetric configurations about their midsurfaces are considered. The problem is studied using the first order shear deformable (or the Mindlin) plate theory. It is found that both the Type-I and Type-II constraints divide the plate into two vibrating regions with amplitudes of transverse vibration localized in a particular region on either side of the clamped interior points. It is found that the mode localization in laminates is governed by the mode localization characteristics of constituent laminas. For symmetric cross ply laminates the localization of modes is found to decrease with the increase in the number of 0° plies. For anti-symmetric cross ply laminates and those made of all 45° plies the mode localization is found to be independent of the number of plies. For isotropic plates made of a monolithic material the mode localization phenomenon is stronger for Type-I constraint compared to that for Type-II constraints. Also, for these plates the mode localization occurs when lumped masses are placed at these interior points. The significance of the work lies in providing an alternative and an economical way of annulling plate vibrations in selected parts of the plate, and confining the energy of vibration in desired regions of the plate.

Integration using invariant operators: conformally flat radiation metrics

A new method is presented for obtaining the general conformally flat radiation metric by using the differential operators of Machado Ramos and Vickers (a generalization of those of Geroch, Held and Penrose) which are invariant under null rotations and rescalings. The solution is found by constructing involutive tables of these derivatives applied to the quantities which arise in the Karlhede classification of this class of metrics.

Invariant differential operators and the Karlhede classification of type-N non-vacuum solutions

A spacetime calculus based on a single null direction and therefore invariant under the subgroup of null rotations, is employed to show that a type-N non-vacuum solution of Einstein's equations requires the calculation of at most five covariant derivatives of the curvature for its complete Karlhede classification.

A spacetime calculus based on a single null direction

A spacetime calculus based on a single null direction is developed. The fundamental objects are totally symmetric spinors formed from the Ricci rotation coefficients together with four new differential operators which only depend upon the choice of up to a complex scaling. The formalism combines features of the GHP formalism with those of one which is invariant under null rotations. Einstein's equations and the full Bianchi identities are given in this new formalism.

Invariant differential operators and the Karlhede classification of type N vacuum solutions

A spacetime calculus based on a single null direction, and which is therefore invariant under null rotations, is employed to show that a type N vacuum solution of Einstein's equations requires the calculation of at most five covariant derivatives of the curvature for its complete Karlhede classification.

A spacetime calculus invariant under null rotations

A spacetime calculus which is invariant under null rotations is presented. The fundamental objects are totally symmetric spinors formed from the Ricci rotation coefficients whose components transform covariantly under null rotations, and four new differential operators which are formed from the directional derivatives and the remaining Ricci rotation coefficients which transform ‘badly’ under null rotations. Einstein’s equations and the full Bianchi identities are translated into this new formalism. Because only totally symmetric spinors are used there is no need to explicitly use indices in any of the equations. Several applications of the formalism are mentioned and the geometrical interpretation is briefley discussed.

The Tertiary geodynamical evolution of the Aegean arc: a paleomagnetic reconstruction

Abstract The paleomagnetic results obtained in the last 5–6 years from Tertiary formations in the Aegean domain indicate that the Lower Miocene arc was almost rectilinear with an E-W trend and that its curvature has been acquired tectonically in two major phases. During the Middle Miocene a first phase of deformation is characterized by rotations occurring at the two terminations of the arc, clockwise in the west (Epirus) anticlockwise in the east (southeastern Anatolia). A second phase of rotation occurring in the last 5 Ma about a pole situated in the southern Adriatic Sea has affected only the northwestern part. The data also indicate that the Antalya basin has not been involved in the geodynamical evolution of the arc so that the Bey Daglari represent its easternmost unit. A more complex rotational pattern is observed in the internal zones: the large rotations measured in Evia (48°) and Skyros (26°) probably occurred in the last 5 Ma and result from rotation of fault bounded blocks in a shear zone connecting the North Aegean trough to the external arc. In the Izmir region, in the Karaburun peninsula of western Anatolia and in the island of Lesbos significant counterclockwise, clockwise or null rotations of coherent blocks have taken place during the neotectonic extensional regime. Furthermore, the values of the inclination obtained from both sedimentary and volcanic Oligo-Miocene formations are systematically shallower than expected on the basis of a geocentered dipole field. This suggests that the entire studied area has undergone a northward drift of over 1000 km since the Middle Miocene.

Paleomagnetic evidence for rotation in opposite senses of adjacent blocks in northeastern Aegea and Western Anatolia

Paleomagnetic data from western Anatolia and from the island of Lesbos demonstrate significant counterclockwise, clockwise or null rotations of coherent blocks which occurred during the neotectonic period. These results suggest that the brittle carapace of the lithosphere may not everywhere follow the motion of the ductile lower part but that its movement arises from adjustment to the extensional tectonic regime.

Complex conformal rescalings and complex Lorentz transformations

Complex Lorentz transformations and complex conformal rescalings with independent conformal factorsθ and\(\tilde \theta \) are investigated in terms of elements of the group GL(2,C) ⊗ G\({\tilde L}\)(2,C). It is shown how a general element of this group decomposes into a “standard” conformal rescaling (with\(\tilde \theta \) =θ), a “pure spin transformation,” complex null rotations, and a complex boost-rotation. Of particular interest are the pure spin transformations that leave invariant the metric but transform the permutation spinors. It is these transformations that, when\(\tilde \theta \)≡θ, are responsible for seemingly complicating the transformation law of the derivative operator and of spinors dependent thereon. It has been suggested that to avoid this complication one should allow the rescaled metric to have torsion. It is argued here that simplicity can be achieved even when the torsion-free condition is imposed.

Isometries compatible with gravitational radiation

Isometries compatible with asymptotic flatness and admitting radiation are studied by using Bondi’s formalism. In axially symmetric space-times, the only second allowable symmetry that does not exclude radiation is boost symmetry. The boost-rotation symmetric solutions describe ‘‘uniformly accelerated particles’’ of various kinds. The news function is restricted by a differential equation; however, it need not vanish, as has been claimed in the literature. If two Killing fields corresponding to null rotations at null infinity are present, then it is shown that the vacuum field equations imply a further isometry. The resulting space-time is a plane wave.

The classification of the Ricci tensor in general relativity theory

Tangent space null rotations are used to give a straightforward classification of the Ricci tensor in general relativity theory.