Null Vector Field Research Articles

Null scrolls as fluctuating surfaces: a new simple way to construct extrinsic string solutions

We exhibit a surprising phenomenon that happens in the conformal Lorentz Minkowski three space, which has no counterpart in a Riemannian setting. Whenever a curve, no matter its causal character, propagates transversely through a conformal geodesic null vector field, it is generating the worldsheet of an extrinsic Polyakov string solution. Furthermore, the Polyakov extrinsic energy of these solutions only depend on the world- sheet topology and it can be computed not only intrinsically, but also holographically by measuring the hyperbolic angles in the boundary corners. This geometric approach, to provide extrinsic string solutions, can be considered as an alternative to the Pohlmeyer reduced mechanism. Then, we describe how to translate these solutions to the language of Pohlmeyer theory. We will also show that any curve in the conformal boundary of the anti de Sitter 3-space can be viewed as a piece of the generalized Wilson loops associated with an extrinsic string solution obtained by this geometric mechanism.

Examples of Einstein spacetimes with recurrent null vector fields

The Einstein equation on four-dimensional Lorentzian manifolds admitting recurrent null vector fields is discussed. Several examples of a special form are constructed. The holonomy algebras, Petrov types and the Lie algebras of Killing vector fields of the obtained metrics are found.

The geometry of closed conformal vector fields on Riemannian spaces

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T.K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to spaces furnished with a closed, Ricci null conformal vector field; then we prove general Bernstein-type theorems for certain complete, not necessarily cmc, hypersurfaces of Riemannian manifolds furnished with closed conformal vector fields. In particular, we obtain a generalization of theorems J. Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres of Riemannian space forms.

Trapping Horizons in Sultana—Dyer Space-Time

The Sultana—Dyer space-time is suggested as a model describing a black hole embedded in an expanding universe. Recently, its global structure is analyzed and the trapping horizons are shown. In the paper, by directly calculating the expansions of the radial null vector fields normal to the space-like two-spheres foliating the trapping horizons, we find that the trapping horizon outside the event horizon in the Sultana—Dyer space-time is a past trapping horizon. Further, we find that the past trapping horizon is an outer, instantaneously degenerate or inner trapping horizon accordingly when the radial coordinate is less than, equal to or greater than some value.

Singularities of spacelike constant mean curvature surfaces in Lorentz–Minkowski space

AbstractWe study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space L3. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve f0(x), and a real analytic null vector field v(x) parallel to the tangent field of f0, find a conformally parameterized (generalized) CMC H surface in L3 which contains this curve as a singular set and such that the partial derivatives fx and fy are given by df0/dx and v along the curve. Within the class of generalized surfaces considered, the solution is unique and we give a formula for the generalized Weierstrass data for this surface. This gives a framework for studying the singularities of non-maximal CMC surfaces in L3. We use this to find the Björling data – and holomorphic potentials – which characterize cuspidal edge, swallowtail and cuspidal cross cap singularities.

Parallel surfaces in Lorentzian three-manifolds admitting a parallel null vector field

We classify parallel surfaces in Lorentzian three-manifolds admitting a parallel null vector field. Some families of semi-parallel surfaces are also described.

THE KERR–SCHILD ANSATZ REVISED

Kerr–Schild metrics have been introduced as a linear superposition of the flat spacetime metric and a squared null-vector field, say k, multiplied by some scalar function, say H. The basic assumption which led to Kerr solution was that k be both geodesic and shearfree. This condition is relaxed here and Kerr–Schild Ansatz is revised by treating Kerr–Schild metrics as exact linear perturbations of Minkowski spacetime. The scalar function H is taken as the perturbing function, so that Einstein's field equations are solved order-by-order in powers of H. It turns out that the congruence must be geodesic and shearfree as a consequence of third- and second-order equations, leading to an alternative derivation of Kerr solution.

LORENTZIAN SYMMETRIC THREE-SPACES AND THE CLASSIFICATION OF THEIR PARALLEL SURFACES

We describe a global model for Lorentzian symmetric three-spaces admitting a parallel null vector field, and classify completely the surfaces with parallel second fundamental form in all Lorentzian symmetric three-spaces. Interesting differences arise with respect to the Riemannian case studied in [2]. Our results complete the classification of parallel surfaces in all three-dimensional Lorentzian homogeneous spaces.

Gauduchon-Tod structures,Simholonomy and De Sitter supergravity

Solutions of five-dimensional De Sitter supergravity admitting Killing spinors are considered, using spinorial geometry techniques. It is shown that the solutions are defined in terms of a one parameter family of 3-dimensional constrained Einstein-Weyl spaces called Gauduchon-Tod structures. They admit a geodesic, expansion-free, twist-free and shear-free null vector field and therefore are a particular type of Kundt geometry. When the Gauduchon-Tod structure reduces to the 3-sphere, the null vector becomes recurrent, and therefore the holonomy is contained in Sim(3), the maximal proper subgroup of the Lorentz group SO(4,1). For these geometries, all scalar invariants built from the curvature are constant. Explicit examples are discussed.

Second-order symmetric Lorentzian manifolds: I. Characterization and general results

The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor—2–symmetric spacetimes—are characterized and classified. The main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkmann's class in n dimensions. Related issues and applications are considered, and new open questions presented.

Regularity at space-like and null infinity

AbstractWe extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino–Schoen/Chrusciel–Delay. We combine conformal techniques and vector field methods: a naive adaptation of the ‘Morawetz vector field’ to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino–Schoen/Chrusciel–Delay space-times. A good classification of solutions that peel arises from the use of a null vector field that is transverse to null infinity to raise the regularity in the estimates. We obtain a new characterization of solutions admitting a peeling at a given order that is valid for both Schwarzschild and Minkowski space-times. On flat space-time, this allows larger classes of solutions than the characterizations used since Penrose's work. Our results establish the validity of the peeling model at all orders for the scalar wave equation on the Schwarzschild metric and on the corresponding Corvino–Schoen/Chrusciel–Delay space-times.

The general supersymmetric solution of topologically massive supergravity

We find the general fully nonlinear solution of topologically massive supergravity admitting a Killing spinor. It is of plane-wave type, with a null Killing vector field. Conversely, we show that all solutions with a null Killing vector are supersymmetric for one or the other choice of sign for the Chern–Simons coupling constant μ. If μ does not take the critical value, μ = ±1, these solutions are asymptotically regular on a Poincaré patch, but do not admit a smooth global compactification with boundary . In the critical case, the solutions have a logarithmic singularity on the boundary of the Poincaré patch. We derive a Nester–Witten identity, which allows us to identify the associated charges, but we conclude that the presence of the Chern–Simons term prevents us from making a statement about their positivity. The Nester–Witten procedure is applied to the BTZ black hole.

Geometry of all supersymmetric four-dimensional 𝒩 = 1 supergravity backgrounds

We solve the Killing spinor equations of ${\cal N}=1$ supergravity, with four supercharges, coupled to any number of vector and scalar multiplets in all cases. We find that backgrounds with N=1 supersymmetry admit a null, integrable, Killing vector field. There are two classes of N=2 backgrounds. The spacetime in the first class admits a parallel null vector field and so it is a pp-wave. The spacetime of the other class admits three Killing vector fields, and a vector field that commutes with the three Killing directions. These backgrounds are of cohomogeneity one with homogenous sections either $\bR^{2,1}$ or $AdS_3$ and have an interpretation as domain walls. The N=3 backgrounds are locally maximally supersymmetric. There are N=3 backgrounds which arise as discrete identifications of maximally supersymmetric ones. The maximally supersymmetric backgrounds are locally isometric to either $\bR^{3,1}$ or $AdS_4$.

IIB backgrounds with five-form flux

We investigate all N = 2 supersymmetric IIB supergravity backgrounds with non-vanishing five-form flux. The Killing spinors have stability subgroups Spin ( 7 ) ⋉ R 8 , SU ( 4 ) ⋉ R 8 and G 2 . In the SU ( 4 ) ⋉ R 8 case, two different types of geometry arise depending on whether the Killing spinors are generic or pure. In both cases, the backgrounds admit a null Killing vector field which leaves invariant the SU ( 4 ) ⋉ R 8 structure, and an almost complex structure in the directions transverse to the lightcone. In the generic case, the twist of the vector field is trivial but the almost complex structure is non-integrable, while in the pure case the twist is non-trivial but the almost complex structure is integrable and associated with a relatively balanced Hermitian structure. The G 2 backgrounds admit a time-like Killing vector field and two spacelike closed one-forms, and the seven directions transverse to these admit a co-symplectic G 2 structure. The Spin ( 7 ) ⋉ R 8 backgrounds are pp-waves propagating in an eight-dimensional manifold with holonomy Spin ( 7 ) . In addition we show that all the supersymmetric solutions of simple five-dimensional supergravity with a time-like Killing vector field, which include the AdS 5 black holes, lift to SU ( 4 ) ⋉ R 8 pure Killing spinor IIB backgrounds. We also show that the LLM solution is associated with a co-symplectic co-homogeneity one G 2 manifold which has principal orbit S 3 × S 3 .

Propagation Equations of Electric and Magnetic Parts of Weyl Tensor in Einstein-Cartan Theory

The electric part and magnetic part of the Weyl tensor with respect to the Kerr-Newman metric in Einstein-Cartan theory of gravitation are obtained, and their propagation equations are studied. The necessary and sufficient conditions for the electric part and the magnetic part to be propagated along the null vector field li are obtained. For particular types of free-gravitational field we tried to explore the meaning of these propagation equations with the help of NewmanPenrose, Jogia and Griffiths null tetrad formalism.

Geometry of type II common sectorN= 2 backgrounds

We describe the geometry of all type II common sector backgrounds with two supersymmetries. In particular, we determine the spacetime geometry of those supersymmetric backgrounds for which each copy of the Killing spinor equations admits a Killing spinor. The stability subgroups of both Killing spinors are $Spin(7)\ltimes \bR^8$, $SU(4)\ltimes \bR^8$ and $G_2$ for IIB backgrounds, and $Spin(7)$, SU(4) and $G_2\ltimes \bR^8$ for IIA backgrounds. We show that the spacetime of backgrounds with spinors that have stability subgroup $K\ltimes \bR^8$ is a pp-wave propagating in an eight-dimensional manifold with a $K$-structure. The spacetime of backgrounds with $K$-invariant Killing spinors is a fibre bundle with fibre spanned by the orbits of two commuting null Killing vector fields and base space an eight-dimensional manifold which admits a $K$-structure. Type II T-duality interchanges the backgrounds with $K$- and $K\ltimes\bR^8$-invariant Killing spinors. We show that the geometries of the base space of the fibre bundle and the corresponding space in which the pp-wave propagates are the same. The conformal symmetry of the world-sheet action of type II strings propagating in these N=2 backgrounds can always be fixed in the light-cone gauge.

The spinorial geometry of supersymmetric heterotic string backgrounds

We determine the geometry of supersymmetric heterotic string backgrounds for which all parallel spinors with respect to the connection $\hat\nabla$ with torsion $H$, the NS$\otimes$NS three-form field strength, are Killing. We find that there are two classes of such backgrounds, the null and the timelike. The Killing spinors of the null backgrounds have stability subgroups $K\ltimes\bR^8$ in $Spin(9,1)$, for $K=Spin(7)$, SU(4), $Sp(2)$, $SU(2)\times SU(2)$ and $\{1\}$, and the Killing spinors of the timelike backgrounds have stability subgroups $G_2$, SU(3), SU(2) and $\{1\}$. The former admit a single null $\hat\nabla$-parallel vector field while the latter admit a timelike and two, three, five and nine spacelike $\hat\nabla$-parallel vector fields, respectively. The spacetime of the null backgrounds is a Lorentzian two-parameter family of Riemannian manifolds $B$ with skew-symmetric torsion. If the rotation of the null vector field vanishes, the holonomy of the connection with torsion of $B$ is contained in $K$. The spacetime of time-like backgrounds is a principal bundle $P$ with fibre a Lorentzian Lie group and base space a suitable Riemannian manifold with skew-symmetric torsion. The principal bundle is equipped with a connection $\lambda$ which determines the non-horizontal part of the spacetime metric and of $H$. The curvature of $\lambda$ takes values in an appropriate Lie algebra constructed from that of $K$. In addition $dH$ has only horizontal components and contains the Pontrjagin class of $P$. We have computed in all cases the Killing spinor bilinears, expressed the fluxes in terms of the geometry and determine the field equations that are implied by the Killing spinor equations.

The G2 spinorial geometry of supersymmetric IIB backgrounds

We solve the Killing spinor equations of supersymmetric IIB backgrounds which admit one supersymmetry and the Killing spinor has stability subgroup G2 in Spin(9, 1) × U(1). We find that such backgrounds admit a timelike Killing vector field and the geometric structure of the spacetime reduces from Spin(9, 1) × U(1) to G2. We determine the type of G2 structure that the spacetime admits by computing the covariant derivatives of the spacetime forms associated with the Killing spinor bilinears. We also solve the Killing spinor equations of backgrounds with two supersymmetries and -invariant spinors, and four supersymmetries with - and with G2-invariant spinors. We show that the Killing spinor equations factorize in two sets, one involving the geometry and the 5-form flux, and the other the 3-form flux and the scalars. In the and cases, the spacetime admits a parallel null vector field and so the spacetime metric can be locally described in terms of Penrose coordinates adapted to the associated rotation free, null, geodesic congruence. The transverse space of the congruence is a Spin(7) and a SU(4) holonomy manifold, respectively. In the G2 case, all the fluxes vanish and the spacetime is the product of a three-dimensional Minkowski space with a holonomy G2 manifold.

Three-dimensional Lorentz manifolds admitting a parallel null vector field

Curvature properties of three-dimensional Lorentz manifolds admitting a parallel degenerate line field are examined. A complete characterization of those manifolds being locally symmetric or locally conformally flat is obtained. The results of this study show nice families of examples of such properties within the Lorentzian setting.

New Hamiltonian formalism and quasilocal conservation equations of general relativity

I describe the Einstein's gravitation of $3+1$ dimensional spacetimes using the (2,2) formalism without assuming isometries. In this formalism, quasilocal energy, linear momentum, and angular momentum are identified from the four Einstein's equations of the divergence-type, and are expressed geometrically in terms of the area of a two-surface and a pair of null vector fields on that surface. The associated quasilocal balance equations are spelled out, and the corresponding fluxes are found to assume the canonical form of energy-momentum-flux as in standard field theories. The remaining non-divergence-type Einstein's equations turn out to be the Hamilton's equations of motion, which are derivable from the nonvanishing Hamiltonian by the variational principle. The Hamilton's equations are the evolution equations along the out-going null geodesic whose affine parameter serves as the time function. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasilocal quantities reduce to the Bondi energy, linear momentum, and angular momentum, and the corresponding fluxes become the Bondi fluxes. The quasilocal angular momentum turns out to be zero for any two-surface in the flat Minkowski spacetime. I also present a candidate for quasilocal rotational energy which agrees with the Carter's constant in the asymptotic region of the Kerr spacetime. Finally, a simple relation between energy-flux and angular momentum-flux of a generic gravitational radiation is discussed, whose existence reflects the fact that energy-flux always accompanies angular momentum-flux unless the flux is an $s$-wave.