A new look at the Bondi-Sachs energy-momentum

How does one compute the Bondi mass on an arbitrary cut of null infinity I when it is not presented in a Bondi system? What then is the correct definition of the mass aspect? How does one normalise an asymptotic translation computed on a cut which is not equipped with the unit-sphere metric? These are questions which need to be answered if one wants to calculate the Bondi-Sachs energy-momentum for a space-time which has been determined numerically. Under such conditions there is not much control over the presentation of I so that most of the available formulations of the Bondi energy-momentum simply do not apply. The purpose of this article is to provide the necessary background for a manifestly conformally invariant and gauge independent formulation of the Bondi energy-momentum. To this end we introduce a conformally invariant version of the GHP formalism to rephrase all the well-known formulae. This leads us to natural definitions for the space of asymptotic translations with its Lorentzian metric, for the Bondi news and the mass-aspect. A major role in these developments is played by the “co-curvature”, a naturally appearing quantity closely related to the Gauß curvature on a cut of I.

On the existence of the global conformal gauge in string theory

The global conformal gauge is playing the crucial role in string theory providing the basis for quantization. Its existence for two-dimensional Lorentzian metric is known locally for a long time. We prove that if a Lorentzian metric is given on a plain then the conformal gauge exists globally on the whole $${{\mathbb {R}}}^2$$ R 2 . Moreover, we prove the existence of the conformal gauge globally on the whole worldsheets represented by infinite strips with straight boundaries for open and closed bosonic strings. The global existence of the conformal gauge on the whole plane is also proved for the positive definite Riemannian metric.

On the null-timelike boundary for Maxwell and spin-2 fields in asymptotically flat spaces

We study the Maxwell equation and the spin-2 field equation in Bondi–Sachs coordinates associated with an asymptotically flat Lorentzian metric. We consider the mixed boundary/initial value problem, where the initial data are imposed on a null hypersurface and a boundary value is prescribed on a timelike hypersurface. We establish Sobolev [Formula: see text] space-time estimates for solutions and their asymptotic expansions at the null infinity.

Gravitational Constraints on a Lightlike Boundary

We analyse the boundary structure of general relativity in the coframe formalism in the case of a lightlike boundary, i.e. when the restriction of the induced Lorentzian metric to the boundary is degenerate. We describe the associated reduced phase space in terms of constraints on the symplectic space of boundary fields. We explicitly compute the Poisson brackets of the constraints and identify the first- and second-class ones. In particular, in the 3+1-dimensional case, we show that the reduced phase space has two local degrees of freedom, instead of the usual four in the non-degenerate case.

On Lorentezian almost Para-contact Manifold

In the present paper, we focus on Lorentzian almost para-contact manifold and explain their relationship. In section 1, we have introduced the historical background of a contact manifold. Next, in section 2, we have studied the basic formulae of the Lorentzian metric manifold. Further, in section 3, we introduced a new tensor field h as well as calculated some theorems and lemma. Now in section 4, we have investigated curvature properties and their relationship with the Lorentzian almost para-contact manifold. In the end section, we have discussed the entire work

On the symmetries of Lorentzian four-dimensional generalized symmetric spaces of type C

We consider the four-dimensional generalized symmetric spaces of type C, equipped with a left-invariant Lorentzian metric. We completely describe its affine, homothetic and Killing vector fields. We also obtain a full classification of its Ricci, curvature and matter collineations.

Equiaffine Braneworld

Higher dimensional theories, wherein our four dimensional universe is immersed into a bulk ambient, have received much attention recently, and the directions of investigation had, as far as we can discern, all followed the ordinary Euclidean hypersurface theory’s isometric immersion recipe, with the spacetime metric being induced by an ambient parent. We note, in this paper, that the indefinite signature of the Lorentzian metric perhaps hints at the lesser known equiaffine hypersurface theory as being a possibly more natural, i.e., less customized beyond minimal mathematical formalism, description of our universe’s extrinsic geometry. In this alternative, the ambient is deprived of a metric, and the spacetime metric becomes conformal to the second fundamental form of the ordinary theory, therefore is automatically indefinite for hyperbolic shapes. Herein, we advocate investigations in this direction by identifying some potential physical benefits to enlisting the help of equiaffine differential geometry. In particular, we show that a geometric origin for dark energy can be proposed within this framework.

Weighted Bergman Kernels and Mathematical Physics

We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Bergman kernels of domains Ω ⊂ C n but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is a complex manifold, and turn out to be an efficient computational tool that is useful for the calculation of transition probability amplitudes from a classical state (identified to a coherent state) to another. We review the weighted version (for weights of the form γ = | φ | m on strictly pseudoconvex domains Ω = { φ < 0 } ⊂ C n ) of Fefferman’s asymptotic expansion of the Bergman kernel and discuss its possible extensions (to more general classes of weights) and implications, e.g., such as related to the construction and use of Fefferman’s metric (a Lorentzian metric on ∂ Ω × S 1 ). Several open problems are indicated throughout the survey.

Classification of first order sesquilinear forms

A natural way to obtain a system of partial differential equations on a manifold is to vary a suitably defined sesquilinear form. The sesquilinear forms we study are Hermitian forms acting on sections of the trivial [Formula: see text]-bundle over a smooth [Formula: see text]-dimensional manifold without boundary. More specifically, we are concerned with first order sesquilinear forms, namely, those generating first order systems. Our goal is to classify such forms up to [Formula: see text] gauge equivalence. We achieve this classification in the special case of [Formula: see text] and [Formula: see text] by means of geometric and topological invariants (e.g., Lorentzian metric, spin/spinc structure, electromagnetic covector potential) naturally contained within the sesquilinear form — a purely analytic object. Essential to our approach is the interplay of techniques from analysis, geometry, and topology.

On pseudo H-symmetric Lorentzian manifolds with applications to relativity

In this paper, we introduce a new type of curvature tensor named H-curvature tensor of type (1, 3) which is a linear combination of conformal and projective curvature tensors. First we deduce some basic geometric properties of H-curvature tensor. It is shown that a H-flat Lorentzian manifold is an almost product manifold. Then we study pseudo H-symmetric manifolds (PHS)n (n > 3) which recovers some known structures on Lorentzian manifolds. Also, we provide several interesting results. Among others, we prove that if an Einstein (PHS)n is a pseudosymmetric (PS)n, then the scalar curvature of the manifold vanishes and conversely. Moreover, we deal with pseudo H-symmetric perfect fluid spacetimes and obtain several interesting results. Also, we present some results of the spacetime satisfying divergence free H-curvature tensor. Finally, we construct a non-trivial Lorentzian metric of (PHS)4.