Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions

We study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated twistor distribution, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with non-zero scalar curvature this twistor distribution has exactly two integral leaves and is ‘maximally non-integrable’ on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-zero scalar curvature and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type D, we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.

Dynamic study of Weyl tensor corrected f(R) gravity

We investigate the cosmological and thermodynamic aspects of Weyl tensor corrected [Formula: see text] gravity. For this purpose, we assume some well-known cosmological bouncing scenarios such as symmetric bounce cosmology, oscillatory cosmology, matter bounce cosmology, little rip cosmology, superbounce cosmology and develop some cosmological parameters. For instance, the equation of state parameter [Formula: see text] describes the quintessence phase for symmetric bounce cosmology, vacuum phase for oscillatory, little rip and matter bounce cosmology while it gives both quintessence and vacuum phases for matter bounce cosmology. It is also observed that the squared speed of sound [Formula: see text] gives positive behavior for all models resulting in that the models assumed are stable. We evaluate generalized second law of thermodynamics which remains valid for all cosmological models except symmetric bounce cosmology. Moreover, we also investigate the thermal equilibrium condition [Formula: see text] and found its validity for all models except symmetric bounce cosmological model.

Comprehensive quasi-Einstein spacetime with application to general relativity

The aim of this paper is to extend the notion of all known quasi-Einstein (QE) manifolds like generalized QE, mixed generalized QE manifold, pseudo generalized QE manifold and many more and name it comprehensive QE manifold [Formula: see text]. We investigate some geometric and physical properties of the comprehensive QE manifolds [Formula: see text] under certain conditions. We study the conformal and conharmonic mappings between [Formula: see text] manifolds. Then we examine the [Formula: see text] with harmonic Weyl tensor. We define the manifold of comprehensive quasi-constant curvature and prove that conformally flat [Formula: see text] is manifold of comprehensive quasi-constant curvature and vice versa. We study the general two viscous fluid spacetime [Formula: see text] and find out some important consequences about [Formula: see text]. We study [Formula: see text] with vanishing space matter tensor. Finally, we prove the existence of such manifolds by constructing nontrivial example.

Asymptotic structure with vanishing cosmological constant

This is the first of two papers [1] devoted to the asymptotic structure of space-time in the presence of a non-negative cosmological constant Λ. This first paper is concerned with the case of Λ = 0. Our approach is fully based on the tidal nature of the gravitational field and therefore on the ‘tidal energies’ built with the Weyl curvature. In particular, we use the (radiant) asymptotic supermomenta computed from the rescaled Weyl tensor at infinity to provide a novel characterisation of radiation escaping from, or entering into, the space-time. Our new criterion is easy to implement and shown to be fully equivalent to the classical one based on the news tensor. One of its virtues is that its formulation can be easily adapted to the case with Λ > 0 covered in the second paper. We derive the general energy-momentum-loss formulae including the matter terms and all factors associated to the choices of arbitrary foliation and of super- translation. We also revisit and present a full reformulation of the traditional peeling behaviour with a neat geometrical construction that leads, in particular, to an asymptotic alignment of the supermomenta in accordance with the radiation criterion.

Polarized image of a Schwarzschild black hole with a thin accretion disk as photon couples to Weyl tensor

We have studied polarized image of a Schwarzschild black hole with an equatorial thin accretion disk as photon couples to Weyl tensor. The birefringence of photon originating from the coupling affect the black hole shadow, the thin disk pattern and its luminosity distribution. We also analyze the observed polarized intensity in the sky plane. The observed polarized intensity in the bright region is stronger than that in the darker region. The stronger effect of the coupling on the observed polarized vector appears only in the bright region close to black hole. These features in the polarized image could help us to understand black hole shadow, the thin accretion disk and the coupling between photon and Weyl tensor.

Free data at spacelike $${\mathscr {I}}$$ and characterization of Kerr-de Sitter in all dimensions

We study the free data in the Fefferman–Graham expansion of asymptotically Einstein $$(n+1)$$ ( n + 1 ) -dimensional metrics with non-zero cosmological constant. We analyze the relation between the electric part of the rescaled Weyl tensor at $${\mathscr {I}}$$ I , D, and the free data at $${\mathscr {I}}$$ I , namely a certain traceless and transverse part of the n-th order coefficient of the expansion $$\mathring{g}_{(n)}$$ g ˚ ( n ) . In the case $$\Lambda <0$$ Λ < 0 and Lorentzian signature, it was known [23] that conformal flatness at $${\mathscr {I}}$$ I is sufficient for D and $$\mathring{g}_{(n)}$$ g ˚ ( n ) to agree up to a universal constant. We recover and extend this result to general signature and any sign of non-zero $$\Lambda $$ Λ . We then explore whether conformal flatness of $${\mathscr {I}}$$ I is also neceesary and link this to the validity of long-standing open conjecture that no non-trivial purely magnetic $$\Lambda $$ Λ -vacuum spacetimes exist. In the case of $${\mathscr {I}}$$ I non-conformally flat we determine a quantity constructed from an auxiliary metric which can be used to retrieve $$\mathring{g}_{(n)}$$ g ˚ ( n ) from the (now singular) electric part of the Weyl tensor. We then concentrate in the $$\Lambda >0$$ Λ > 0 case where the Cauchy problem at $${\mathscr {I}}$$ I of the Einstein vacuum field equations is known to be well-posed when the data at $${\mathscr {I}}$$ I are analytic or when the spacetime has even dimension. We establish a necessary and sufficient condition for analytic data at $${\mathscr {I}}$$ I to generate spacetimes with symmetries in all dimensions. These results are used to find a geometric characterization of the Kerr-de Sitter metrics in all dimensions in terms of its geometric data at null infinity.

Some aspects of Ricci flow on the 4-sphere

In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with L^2 norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the L^p norm for certain p>2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.

An exploration of the black hole entropy in Gauss–Bonnet gravity via the Weyl tensor

Various proposals for gravitational entropy densities have been constructed from the Weyl tensor. In almost all cases, though, these studies have been restricted to general relativity, and little has been done in modified theories of gravity. However, in this paper, we investigate the simplest proposal for an entropy density constructed from the Weyl tensor in five-dimensional Gauss–Bonnet gravity and find that it fails to reproduce the expected entropy of a black hole.