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Pawel Nurowski

Publications and source records attributed to Pawel Nurowski.

At least 37 records · Page 2Linked to original sources
arXiv2016

How the green light was given for gravitational wave search

The recent detection of gravitational waves by the LIGO/VIRGO team is an incredibly impressive achievement of experimental physics. It is also a tremendous success of the theory of General Relativity. It confirms the existence of black holes; shows that binary black holes exist; that they may collide and that during the merging process gravitational waves are produced. These are all predictions of General Relativity theory in its fully nonlinear regime. The existence of gravitational waves was predicted by Albert Einstein in 1916 within the framework of linearized Einstein theory. Contrary to common belief, even the very \emph{definition} of a gravitational wave in the fully nonlinear Einstein theory was provided only after Einstein's death. Actually, Einstein had arguments against the existence of nonlinear gravitational waves (they were erroneous but he did not accept this), which virtually stopped development of the subject until the mid 1950s. This is what we refer to as the \emph{Red Light} for gravitational waves research. In the following years, the theme was picked up again and studied vigorously by various experts, mainly Herman Bondi, Felix Pirani, Ivor Robinson and Andrzej Trautman, where the theoretical obstacles concerning gravitational wave existence were successfully overcome, thus giving the `Green Light' for experimentalists to start designing detectors, culminating in the recent LIGO/VIRGO discovery. In this note we tell the story of this theoretical breakthrough. Particular attention is given to the fundamental 1958 papers of Trautman, which seem to be lesser known outside the circle of General Relativity experts. A more detailed technical description of these 2 papers is given in the Appendix.

physics.hist-ph↗
arXiv2016

Ring Type Structures in the Planck map of the CMB

We present the results of the quest for ring-type structures on the maps observed by the Planck satellite. The results show that the vicinity of one radius (γ = 0.14 radians) of the rings is distinguished. Twisting the circles into deformed ellipses gives a pronounced drop of significance with the increase of twisting.

astro-ph.CO↗
arXiv2016

Sp(3,R) Monge geometries in dimension 8

We study a geometry associated with rank 3 distributions in dimension 8, whose symbol algebra is constant and has a simple Lie algebra sp(3,R) as Tanaka prolongation. We restrict our considerations to only those distributions that are defined in terms of a systems of ODEs of the form $\dot{z}_{ij}=\frac{\partial^2 f(\dot{x}_1,\dot{x}_2)}{\partial \dot{x}_i\partial \dot{x}_j}$, $i\leq j=1,2$. For them we built the full system of local differential invariants, by solving an equivalence problem a'la Cartan, in the spirit of his 1910's five variable paper. The considered geometry is a parabolic geometry, and we show that its main invariant - the harmonic curvature - is a certain quintic. In the case when this quintic is maximally degenerate but nonzero, we use Cartan's reduction procedure and reduce the EDS governing the invariants to 11, 10 and 9 dimensions. As a byproduct all homogeneous models having maximally degenerate harmonic curvature quintic are found. They have symmetry algebras of dimension 11 (a unique structue), 10 (a 1-parameter family of nonequivalent structures) or 9 (precisely two nonequivalent structures).

math.DG↗
arXiv2016

New relations between $G_2$-geometries in dimensions 5 and 7

There are two well-known parabolic split $G_2$-geometries in dimension five, $(2,3,5)$-distributions and $G_2$-contact structures. Here we link these two geometries with yet another $G_2$-related contact structure, which lives on a seven-manifold. We present a natural geometric construction of a Lie contact structure on a seven-dimensional bundle over a five-manifold endowed with a $(2,3,5)$-distribution. For a class of distributions the induced Lie contact structure is constructed explicitly and we determine its symmetries. We further study the relation between the canonical normal Cartan connections associated with the two structures. In particular, we show that the Cartan holonomy of the induced Lie contact structure reduces to $G_2$. Moreover, the curved orbit decomposition associated with a $\mathrm{G}_2$-reduced Lie contact structure on a seven-manifold is discussed. It is shown that in a neighbourhood of each point on the open curved orbit the structure descends to a $(2,3,5)$-distribution on a local leaf space, provided an additional curvature condition is satisfied. The closed orbit carries an induced $G_2$-contact structure.

math.DG↗
arXiv2015

The dancing metric, $\mathrm G_2$-symmetry and projective rolling

The "dancing metric" is a pseudo-riemannian metric $\pmb{g}$ of signature $(2,2)$ on the space $M^4$ of non-incident point-line pairs in the real projective plane $\mathbb{RP}^2$. The null-curves of $(M^4,\pmb{g})$ are given by the "dancing condition": the point is moving towards a point on the line, about which the line is turning. We establish a dictionary between classical projective geometry (incidence, cross ratio, projective duality, projective invariants of plane curves...) and pseudo-riemannian 4-dimensional conformal geometry (null-curves and geodesics, parallel transport, self-dual null 2-planes, the Weyl curvature,...). There is also an unexpected bonus: by applying a twistor construction to $(M^4,\pmb{g})$, a $\mathrm G_2$-symmetry emerges, hidden deep in classical projective geometry. To uncover this symmetry, one needs to refine the "dancing condition" by a higher-order condition, expressed in terms of the osculating conic along a plane curve. The outcome is a correspondence between curves in the projective plane and its dual, a projective geometry analog of the more familiar "rolling without slipping and twisting" for a pair of riemannian surfaces.

math.DG↗
arXiv2015

Conformal transformations and the beginning of the Universe

We show that two consecutive Universes with positive cosmological constants filled with perfect fluids and conformal to the positive-spatial-curvature FLRW metric by an analytic conformal transformation have the following features implied by the Einstein equations: a) the fluids can only belong to 4 classes (radiation, dust and two other classes with negative pressures corresponding to a gas of strings and a gas of domain walls), b) the field equations on both sides of the future/eternity hypersurface exhibit certain duality with the Einstein Universe at the background, and c) both eons (one at the end, and the other at the beginning) are almost critical, so that the future eon is dominated by radiation and resembles the beginning of our Universe.

gr-qc↗
arXiv2015

A Goldberg-Sachs theorem in dimension three

We prove a Goldberg-Sachs theorem in dimension three. To be precise, given a three-dimensional Lorentzian manifold satisfying the topological massive gravity equations, we provide necessary and sufficient conditions on the tracefree Ricci tensor for the existence of a null line distribution whose orthogonal complement is integrable and totally geodetic. This includes, in particular, Kundt spacetimes that are solutions of the topological massive gravity equations.

gr-qc↗
arXiv2015

Explicit ambient metrics and holonomy

We present three large classes of examples of conformal structures for which the equations for the Fefferman-Graham ambient metric to be Ricci-flat are linear PDEs, which we solve explicitly. These explicit solutions enable us to discuss the holonomy of the corresponding ambient metrics. Our examples include conformal pp-waves and, more importantly, conformal structures that are defined by generic rank 2 and 3 distributions in respective dimensions 5 and 6. The corresponding explicit Fefferman-Graham ambient metrics provide a large class of metrics with holonomy equal to the exceptional non-compact Lie group $\mathbf{G}_2$ as well as ambient metrics with holonomy contained in $\mathbf{Spin}(4,3)$.

math.DG↗
arXiv2014

Generalised Ricci Solitons

We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equations in differential geometry. Among them there are: the Ricci soliton equations, the vacuum near-horizon geometry equations in general relativity, special cases of Einstein-Weyl equations and their projective counterparts, equations for homotheties and Killing's equation. We also prolong the generalised Ricci soliton equations and, by computing differential constraints, we find a number of necessary conditions for a (pseudo)-Riemannian manifold $(M, g)$ to locally admit non-trivial solutions to the generalised Ricci soliton equations in dimensions 2 and 3. The paper provides also a collection of explicit examples of generalised Ricci solitons in dimensions 2 and 3 (in some cases).

math.DG↗
arXiv2014

Non-Rigid Parabolic Geometries of Monge Type

In this paper we study a novel class of parabolic geometries which we call parabolic geometries of Monge type. These parabolic geometries are defined by special gradings of simple Lie algebras, namely, gradings with the property that their -1 component contains a nonzero co-dimension 1 abelian subspace whose bracket with its complement is non-degenerate. We completely classify the simple Lie algebras with such gradings in terms of elementary properties of the defining set of simple roots. We then characterize those parabolic geometries of Monge type which are non-rigid in the sense that, apart from the flat models, they have nonzero harmonic curvatures in positive weights. Standard models of all non-rigid parabolic geometries of Monge type are then described by under-determined ODE systems. The full symmetry algebras for these under-determined ODE systems are explicitly calculated; surprisingly, these symmetries are all just prolonged point symmetries.

math.DG↗
arXiv2012

Twistor space for rolling bodies

On a natural circle bundle T(M) over a 4-dimensional manifold M equipped with a split signature metric g, whose fibers are real totally null selfdual 2-planes, we consider a tautological rank 2 distribution D obtained by lifting each totally null plane horizontally to its point in the fiber. Over the open set where g is not antiselfdual, the distribution D is (2,3,5) in T(M). We show that if M is a Cartesian product of two Riemann surfaces (S1,g1) and (S2,g2), and if g=g1--g2, then the circle bundle T(S1 x S2) is just the configuration space for the physical system of two solid bodies B1 and B2, bounded by the surfaces S1 and S2 and rolling on each other. The condition for the two bodies to roll on each other `without slipping or twisting' identifies the restricted velocity space for such a system with the tautological distribution D on T(S1 x S2). We call T(S1 x S2) the twistor space, and D the twistor distribution for the rolling bodies. Among others we address the following question: "For which pairs of bodies does the restricted velocity distribution (which we identify with the twistor distribution D) have the simple Lie group G2 as its group of symmetries?" Apart from the well known situation when the boundaries S1 and S2 of the two bodies have constant curvatures whose ratio is 1:9, we unexpectedly find three different types of surfaces, which when bounding a body that rolls `without slipping or twisting' on a plane, have D with the symmetry group G2. Although we have found the differential equations for the curvatures of S1 and S2 that gives D with G2 symmetry, we are unable to solve them in full generality so far.

math.DG↗
arXiv2012

Calculus and invariants on almost complex manifolds, including projective and conformal geometry

We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In particular we are able to construct an invariant and efficient calculus for conformal almost Hermitian geometries, and also for almost complex structures that are equipped with a projective structure. In the latter case we find a projectively invariant tensor the vanishing of which is necessary and sufficient for the existence of an almost complex connection compatible with the path structure. In both the conformal and projective setting we give torsion characterisations of the canonical connections and introduce certain interesting higher order invariants.

math.DG↗
arXiv2011

Analog of selfduality in dimension nine

We introduce a type of Riemannian geometry in nine dimensions, which can be viewed as the counterpart of selfduality in four dimensions. This geometry is related to a 9-dimensional irreducible representation of ${\bf SO}(3) \times {\bf SO} (3)$ and it turns out to be defined by a differential 4-form. Structures admitting a metric connection with totally antisymmetric torsion and preserving the 4-form are studied in detail, producing locally homogeneous examples which can be viewed as analogs of self-dual 4-manifolds in dimension nine.

math.DG↗
arXiv2011

Conformal pure radiation with parallel rays

We define pure radiation metrics with parallel rays to be n-dimensional pseudo-Riemannian metrics that admit a parallel null line bundle K and whose Ricci tensor vanishes on vectors that are orthogonal to K. We give necessary conditions in terms of the Weyl, Cotton and Bach tensors for a pseudo-Riemannian metric to be conformal to a pure radiation metric with parallel rays. Then we derive conditions in terms of tractor calculus that are equivalent to the existence of a pure radiation metric with parallel rays in a conformal class. We also give an analogous result for n-dimensional pseudo-Riemannian pp-waves.

math.DG↗
arXiv2010

Is dark energy meaningless?

We show that there are isometrically nonequivalent Robertson-Walker metrics which have the same set of geodesics. While one of these metrics satisfies the Einstein equations of pure dust without a cosmological constant, all the other describe pure dust with additional energy momentum tensor of cosmological constant type. Since each of these metrics have the same geodesics it is not clear how to distinguish experimentally between the Universes whose energy momentum tensor includes or not the cosmological constant type term.

math.DG↗
arXiv2010

Projective vs metric structures

We present a number of conditions which are necessary for an n-dimensional projective structure (M,[nabla]) to include the Levi-Civita connection nabla of some metric on M. We provide an algorithm, which effectively checks if a Levi-Civita connection is in the projective class and, in the positive, which finds this connection and the metric. The article also provides a basic information on invariants of projective structures, including the treatment via Cartan's normal projective connection. In particular we show that there is a number of Fefferman-like conformal structures, defined on a subbundle of the Cartan bundle of the projective structure, which encode the projectively invariant information about (M,[nabla]).

math.DG↗