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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.

Motivated by Aoki et al.’s recent research on conserved charges and entropy current, we reinvestigated the conservation of relativistic Ertel’s current, which has received little attention outside the field of geophysical fluid dynamics. Ertel’s charge is an important indicator of the correlation between vortex vectors and entropy gradient fields in Earth’s meridional heat transport. We first show that in the generalized Hamiltonian structure of baroclinic fluids, the duality between the total energy and the Casimir as a function of Ertel’s charge plays an important role in the nonrelativistic case. Then, by extending the result to relativistic cases, we show that this finding has far-reaching implications not only for space–time issues in cosmology but also for the foundation of quantum field theory. An especially important finding is that, as an unreported dual form of the Einstein field equation, we identify a special equation satisfied not only by the vortex tensor field generated by the conserved charge but also by the Weyl tensor in interpreting the physical nature of the metric tensor $g^{\mu \nu }$, which appears in the cosmological term $\mathrm{\Lambda }{g}^{\mu \nu }$.

$1+3$ covariant approach to cosmological perturbation theory often employs the electric part ($E_{ab}$), the magnetic part ($H_{ab}$) of the Weyl tensor or the shear tensor (${\sigma }_{ab}$) in a phenomenological description of gravitational waves. The Cotton–York tensor is rarely mentioned in connection with gravitational waves in this approach. This tensor acts as a source for the magnetic part of the Weyl tensor which should not be neglected in studies of gravitational waves in the $1+3$ formalism. The tensor is only mentioned in connection with studies of “silent model” but even there the connection with gravitational waves is not exhaustively explored. In this study, we demonstrate that the Cotton–York tensor encodes contributions from both electric and magnetic parts of the Weyl tensor and in directly from the shear tensor. In our opinion, this makes the Cotton–York tensor arguably the natural choice for linear gravitational waves in the $1+3$ covariant formalism. The tensor is cumbersome to work with but that should negate its usefulness. It is conceivable that the tensor would equally be useful in the metric approach, although we have not demonstrated this in this study. We contend that the use of only one of the Weyl tensor or the shear tensor, although phenomenologically correct, leads to loss of information. Such information is vital particularly when examining the contribution of gravitational waves to the anisotropy of an almost-Friedmann–Lamitre–Robertson–Walker (FLRW) universe. The recourse to this loss is the use Cotton–York tensor.

Local conformal symmetry introduces the conformal curvature (Weyl tensor) that gets split into its (gravito-) electric and magnetic (tensor) parts. Newtonian tidal forces are expected from the gravitoelectric field, whereas general-relativistic frame-dragging effects emerge from the gravitomagnetic field. The symmetric, traceless gravitoelectric and gravitomagnetic tensor fields can be visualized by their eigenvectors and eigenvalues. In this paper, we depict the gravitoelectric and gravitomagnetic fields around a slowly rotating black hole. This suggests that the phenomenon of ultra-fast outflows observed at the centers of active galaxies may give evidence for the gravitomagnetic fields of spinning supermassive black holes. We also question whether the current issues in our contemporary observations might be resolved by the inclusion of gravitomagnetism on large scales in a perturbed FLRW model.

We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. First, we show that the alignment condition is equivalent to the principal null direction equation. In 4 dimensions this recovers the usual Petrov types are recovered. For higher dimensions we prove that, in general, a Weyl tensor does not possess any aligned directions. We then go on to describe a number of additional algebraic types for the various alignment configurations. For the case of second-order symmetric (Ricci) tensors, we perform the classification by considering the geometric properties of the corresponding alignment variety.

We find the Hamiltonian expression in the York basis of canonical ADM tetrad gravity of the 4-Weyl tensor of the asymptotically Minkowskian space-time. Like for the 4-Riemann tensor we find a radar tensor (whose components are 4-scalars due to the use of radar 4-coordinates), which coincides with the 4-Weyl tensor on-shell on the solutions of Einstein's equations. Then, by using the Hamiltonian null tetrads, we find the Hamiltonian expression of the Weyl scalars of the Newman–Penrose approach and of the four eigenvalues of the 4-Weyl tensor. After having introduced the Dirac observables (DOs) of canonical gravity, whose determination requires the solution of the super-Hamiltonian and super-momentum constraints, we discuss the connection of the DOs with the notion of 4-scalar Bergmann observables (BOs). Due to the use of radar 4-coordinates these two types of observables coincide in our formulation of canonical ADM tetrad gravity. However, contrary to Bergmann proposal, the Weyl eigenvalues are shown not to be BOs, so that their relevance is only in their use (first suggested by Bergmann and Komar) for giving a physical identification as point-events of the mathematical points of the space-time 4-manifold. Finally we give the expression of the Weyl scalars in the Hamiltonian post-Minkowskian linearization of canonical ADM tetrad gravity in the family of (non-harmonic) 3-orthogonal Schwinger time gauges.

We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications. In particular, it is shown that the existence of a Weyl compatible vector implies that the Weyl tensor is algebraically special, and it is a necessary and sufficient condition for the magnetic part to vanish. Some theorems (Derdziński and Shen [11], Hall [15]) are extended to the broader hypothesis of Weyl or Riemann compatibility. Weyl compatibility includes conditions that were investigated in the literature of general relativity (as in McIntosh et al. [16, 17]). A simple example of Weyl compatible tensor is the Ricci tensor of an hypersurface in a manifold with constant curvature.

Conformally recurrent pseudo-Riemannian manifolds of dimension $n\ge 5$ are investigated. The Weyl tensor is represented as a Kulkarni–Nomizu product. If the square of the Weyl tensor is non-zero, a covariantly constant symmetric tensor is constructed, that is quadratic in the Weyl tensor. Then, by Grycak’s theorem, the explicit expression of the traceless part of the Ricci tensor is obtained, up to a scalar function. The Ricci tensor has at most two distinct eigenvalues, and the recurrence vector is an eigenvector. Lorentzian conformally recurrent manifolds are then considered. If the square of the Weyl tensor is non-zero, the manifold is decomposable. A null recurrence vector makes the Weyl tensor of algebraic type IId or higher in the Bel–Debever–Ortaggio classification, while a time-like recurrence vector makes the Weyl tensor purely electric.

We prove that an almost pseudo ${𝒵}$ symmetric spacetime with Codazzi type of ${𝒵}$ tensor is perfect fluid whose velocity vector field $B^i$ is parallel. The energy density $\sigma$ of such perfect fluid spacetime is constant and the state equation is obtained. Also, this spacetime is shown to be a static spacetime. Next, it is shown that such spacetime is a GRW spacetime with Einstein fiber. This kind of spacetime is investigated in f (R) theory of gravity; in this case, we find the forms of the isotropic pressure p and the energy density $\sigma$ which are constant. Further, some energy conditions are studied in this spacetime. Finally, a concrete example of almost pseudo ${𝒵}$ symmetric spacetimes is considered.

In this paper, we study the PNDP-manifold, which has shown to be effective in applications, particularly in the field of general relativity, by introducing a new topological approach to the emergent electric or magnetic parts associated to the Weyl tensor. The working hypothesis in this study is that the electric or magnetic parts of the Weyl tensor are the dimensions of the PNDP-manifold and the interaction that emerges as “virtual” point-like manifolds and as a manifold with positive/ negative dimensions results in the generation of the purely electric or purely magnetic parts of the Weyl tensor. The PNDP metric for the electric or magnetic parts of the Weyl tensor is also derived. The generalized Lorentz transformations on the PNDP manifold are introduced. These can give rise to novel forms of gauge theories if we consider local Lorentz transformations. Also we briefly show that principal fiber bundles with PNDP manifolds as base manifolds can be constructed.

Due to cosmic structures in the Universe, the inhomogeneities influence the evolution of the background Universe. To characterize this influence, we extend the Kullback-Leibler relative entropy in information theory to cosmology. We study the relation between the relative entropy and the averaging problem in the perturbed Universe, and explore the temporal evolution in linear cosmological perturbation theory and beyond. The possible relationship of the relative entropy and the Weyl tensor is also discussed briefly.

Lie symmetries of various geometrical and physical quantities in general relativity play an important role in understanding the curvature structure of manifolds. The Riemann curvature and Weyl tensors are two fourth-rank tensors in the theory. Interrelations between the symmetries of these two tensors (known as collineations) are studied. Some illustrative examples are also provided.

We discuss the peeling bahaviour of the Weyl tensor near null infinity for an asymptotivally flat higher dimensional spacetime and describe how these results can be used to write the Bondi energy flux in terms of “Newman-Penrose” Weyl components.