In this paper the Weyl tensor is used to define operators that act on the space of forms. These operators are shown to have interesting properties and are used to classify the Weyl tensor, the well known Petrov classification emerging as a special case. Particularly, in the Euclidean signature this classification turns out be really simple. Then it is shown that the integrability condition of maximally isotropic distributions can be described in terms of the invariance of certain subbundles under the action of these operators. Here it is also proved a new generalization of the Goldberg-Sachs theorem, valid in all even dimensions, stating that the existence of an integrable maximally isotropic distribution imposes restrictions on the optical matrix. Also the higher-dimensional versions of the self-dual manifolds are investigated. These topics can shed light on the integrability of Einstein's equation in higher dimensions.
We introduce a new gauge and solution space for three-dimensional gravity. As its name Bondi-Weyl suggests, it leads to non-trivial Weyl charges, and uses Bondi-like coordinates to allow for an arbitrary cosmological constant and therefore spacetimes which are asymptotically locally (A)dS or flat. We explain how integrability requires a choice of integrable slicing and also the introduction of a corner term. After discussing the holographic renormalization of the action and of the symplectic potential, we show that the charges are finite, symplectic and integrable, yet not conserved. We find f…
We consider a Lorentz violating scalar field cosmological model given by the modified Einstein-aether theory defined in Weyl integrable geometry. The existence of exact and analytic solutions is investigated for the case of a spatially flat Friedmann--Lema\^ıtre--Robertson--Walker background space. We show that the theory admits cosmological solutions of special interests. In addition, we prove that the cosmological field equations admit the Lewis invariant as a second conservation law, which indicates the integrability of the field equations.
The integrability conditions for the existence of Killing-Yano tensors or, equivalently, covariantly closed conformal Killing-Yano tensors, in the presence of torsion are worked out. As an application, all metrics and torsions compatible with the existence of a Killing-Yano tensor of order n-1 are obtained. Finally, the issue of defining a maximally symmetric space with respect to connections with torsion is addressed.
The integrability conditions for the existence of a conformal Killing-Yano tensor of arbitrary order are worked out in all dimensions and expressed in terms of the Weyl tensor. As a consequence, the integrability conditions for the existence of a Killing-Yano tensor are also obtained. By means of such conditions, it is shown that in certain Einstein spaces one can use a conformal Killing-Yano tensor of order p to generate a Killing-Yano tensor of order (p-1). Finally, it is proved that in maximally symmetric spaces the covariant derivative of a Killing-Yano tensor is a closed conformal Killing…
All spacetimes for an irrotational collisionless fluid with a purely electric Weyl tensor, with spacetime curvature determined by the exact Einstein field equations, are shown to be integrable. These solutions include the relativistic generalisations of the Zeldovich solutions of Newtonian theory. Thus our result shows the consistency of various studies of "silent" universes (where such consistency was assumed rather than proved).
Assuming the existence of a single rank-2 closed conformal Killing-Yano tensor with a certain symmetry we show that there exist mutually commuting rank-2 Killing tensors and Killing vectors. We also discuss the condition of separation of variables for the geodesic Hamilton-Jacobi equations.
We prove that a Lanczos potential L_abc for the Weyl candidate tensor W_abcd does not generally exist for dimensions higher than four. The technique is simply to assume the existence of such a potential in dimension n, and then check the integrability conditions for the assumed system of differential equations; if the integrability conditions yield another non-trivial differential system for L_abc and W_abcd, then this system's integrability conditions should be checked; and so on. When we find a non-trivial condition involving only W_abcd and its derivatives, then clearly Weyl candidate tenso…
It is well known that the classification of the Weyl tensor in Lorentzian manifolds of dimension four, the so called Petrov classification, was a great tool to the development of general relativity. Using the bivector approach it is shown in this article a classification for the Weyl tensor in all four-dimensional manifolds, including all signatures and the complex case, in an unified and simple way. The important Petrov classification then emerges just as a particular case in this scheme. The boost weight classification is also extended here to all signatures as well to complex manifolds. For…
HyperCR Einstein--Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can be in principle constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions yield new Einstein--Weyl structures.
A spinorial approach to 6-dimensional differential geometry is constructed and used to analyze tensor fields of low rank, with special attention to the Weyl tensor. We perform a study similar to the 4-dimensional case, making full use of the SO(6) symmetry to uncover results not easily seen in the tensorial approach. Using spinors, we propose a classification of the Weyl tensor by reinterpreting it as a map from 3-vectors to 3-vectors. This classification is shown to be intimately related to the integrability of maximally isotropic subspaces, establishing a natural framework to generalize the …
Recently a generalization of the Fefferman-Graham gauge for asymptotically locally AdS spacetimes, called the Weyl-Fefferman-Graham (WFG) gauge, has been proposed. It was shown that the WFG gauge induces a Weyl geometry on the conformal boundary. The Weyl geometry consists of a metric and a Weyl connection. Thus, this is a useful setting for studying dual field theories with background Weyl symmetry. Working in the WFG formalism, we find the generalization of obstruction tensors, which are Weyl-covariant tensors that appear as poles in the Fefferman-Graham expansion of the bulk metric for even…
We obtain scale-invariant scalar and tensor power spectra from a Weyl-invariant scalar-tensor theory in de Sitter spacetime. This implies that the Weyl-invariance guarantees to implement the scale-invariance of power spectrum in de Sitter spacetime. We establish a deep connection between the Weyl-invariance of the action and scale-invariance of power spectrum in de Sitter spacetime.
In this paper, we develop a method based on the analysis of the Kovalewski exponents to study the integrability of anisotropic and homogeneous Universes. The formalism is developed in scalar-tensor gravity, the general relativistic case appearing as a special case of this larger framework. Then, depending on the rationality of the Kovalewski exponents, the different models, both in the vacuum and in presence of a barotropic matter fluid, are classified, and their integrability is discussed.
The algebraic classification of the Weyl tensor in arbitrary dimension n is recovered by means of the principal directions of its "superenergy" tensor. This point of view can be helpful in order to compute the Weyl aligned null directions explicitly, and permits to obtain the algebraic type of the Weyl tensor by computing the principal eigenvalue of rank-2 symmetric future tensors. The algebraic types compatible with states of intrinsic gravitational radiation can then be explored. The underlying ideas are general, so that a classification of arbitrary tensors in general dimension can be achie…
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-Perose approach and of the four eigenvalues of the 4-Weyl tensor. After having introduced the Dirac observables of cano…
We give a general expression for the Weyl tensor two-point function in a general Friedmann-Lemaître-Robertson-Walker spacetime. We work in reduced phase space for the perturbations, i.e., quantize only the dynamical degrees of freedom without adding any gauge-fixing term. The general formula is illustrated by a calculation in slow-roll single-field inflation to first order in the slow-roll parameters $ε$ and $δ$, and the result is shown to have the correct de Sitter limit as $ε, δ\to 0$. Furthermore, it is seen that the Weyl tensor correlation function does not suffer from infrared divergences…
An extension to higher dimensions of the Bel-Debever characterization of the Weyl tensor is considered. This provides algebraic conditions that uniquely determine the multiplicity of a Weyl aligned null direction (WAND), and thus the principal Weyl type, in a frame independent way. The specification of several "subtypes" is also encompassed by the criteria. We further comment on a Cartan-like geometrical interpretation of WANDs in terms of their invariance properties under parallel transport around infinitesimal loops. As a result, restrictions on the algebraic types permitted in spacetimes th…
We explore some new off-shell and on-shell conserved quantities for a scalar field in Minkowski space, using integrability condition. The off-shell conserved tensors are related to the kinematics of the field, while a linear combination of the off-shell and the on-shell conserved tensors ends up with the energy-momentum tensor for the scalar field. In the curved background, using Ricci and Bianchi identities, Brans-Dicke type field equations emerge, without requiring the principle of equivalence. Further, starting from the curvature scalar and using these identities, the field equations for mo…
We discuss the algebraic classification of the Weyl tensor in higher dimensional Lorentzian manifolds. This is done by characterizing algebraically special Weyl tensors by means of the existence of aligned null vectors of various orders of alignment. Further classification is obtained by specifying the alignment type and utilizing the notion of reducibility. For a complete classification it is then necessary to count aligned directions, the dimension of the alignment variety, and the multiplicity of principal directions. The present classification reduces to the classical Petrov classification…
We investigate the spinor classification of the Weyl tensor in five dimensions due to De Smet. We show that a previously overlooked reality condition reduces the number of possible types in the classification. We classify all vacuum solutions belonging to the most special algebraic type. The connection between this spinor and the tensor classification due to Coley, Milson, Pravda and Pravdová is investigated and the relation between most of the types in each of the classifications is given. We show that the black ring is algebraically general in the spinor classification.
Recent results on purely electric (PE) or magnetic (PM) spacetimes in n dimensions are summarized. These include: Weyl types; diagonalizability; conditions under which direct (or warped) products are PE/PM.
In all dimensions and arbitrary signature, we demonstrate the existence of a new local potential -- a double (2,3)-form -- for the Weyl curvature tensor, and more generally for all tensors with the symmetry properties of the Weyl curvature tensor. The classical four-dimensional Lanczos potential for a Weyl tensor -- a double (2,1)-form -- is proven to be a particular case of the new potential: its double dual.
We review the theory of alignment in Lorentzian geometry and apply it to the algebraic classification of the Weyl tensor in higher dimensions. This classification reduces to the the well-known Petrov classification of the Weyl tensor in four dimensions. We discuss the algebraic classification of a number of known higher dimensional spacetimes. There are many applications of the Weyl classification scheme, especially in conjunction with the higher dimensional frame formalism that has been developed in order to generalize the four dimensional Newman--Penrose formalism. For example, we discuss hi…
We use a de Sitter breaking graviton propagator to compute the tree order correlator between noncoincident Weyl tensors on a locally de Sitter background. An explicit, and very simple result is obtained, for any spacetime dimension D, in terms of a de Sitter invariant length function and the tensor basis constructed from the metric and derivatives of this length function. Our answer does not agree with the one derived previously by Kouris, but that result must be incorrect because it not transverse and lacks some of the algebraic symmetries of the Weyl tensor. Taking the coincidence limit of o…
The algebraic study of the Bel-Robinson tensor proposed and initiated in a previous work (Gen. Relativ. Gravit. {\bf 41}, see ref [11]) is achieved. The canonical form of the different algebraic types is obtained in terms of Bel-Robinson eigen-tensors. An algorithmic determination of the Weyl tensor from the Bel-Robinson tensor is presented.
We introduce a new Weyl-invariant and generally-covariant vector-tensor theory with higher derivatives. This theory can be induced by extending the mimetic construction to vector fields of conformal weight four. We demonstrate that in gauge-invariant variables this novel theory reduces to the Henneaux-Teitelboim description of the unimodular gravity. Hence, compared with the standard general relativity, our new higher derivative vector-tensor theory has only one new global degree of freedom - the cosmological constant. Finally we discuss potential extensions of this vector-tensor theory.
We consider scalar-tensor theories of gravity defined in Weyl integrable space-time and show that in the ADM formalism Weyl transformations corresponding to change of frames induce canonical transformations between different representations of the phase space. In this context, we discuss the physical equivalence of two distinct Weyl frames at the classical level.
Algebraically special gravitational fields are described using algebraic and differential invariants of the Weyl tensor. A type III invariant is also given and calculated for Robinson-Trautman spaces.
We prove that, in a space-time of dimension n>3 with a velocity field that is shear-free, vorticity-free and acceleration-free, the covariant divergence of the Weyl tensor is zero if the contraction of the Weyl tensor with the velocity is zero. The other way, if the covariant divergence of the Weyl tensor is zero, then the contraction of the Weyl tensor with the velocity has recurrent geodesic derivative. This partly extends a property found in Generalised Robertson-Walker spacetimes, where the velocity is also eigenvector of the Ricci tensor. Despite the simplicity of the statement, the proof…
We investigate the evolution of cosmological perturbations during de Sitter inflation in the Einstein-Chern-Simons-Weyl gravity. Primordial massive gravitational waves are composed of one scalar, two vector and four tensor circularly polarized modes. We show that the vector power spectrum decays quickly like a transversely massive vector in the superhorizon limit $z\to 0$. In this limit, the power spectrum coming from massive tensor modes decays quickly, leading to the conventional tensor power spectrum. Also, we find that in the limit of $m^2 \to 0$ (keeping the Weyl-squared term only), the v…
We determine the leading order fall-off behaviour of the Weyl tensor in higher dimensional Einstein spacetimes (with and without a cosmological constant) as one approaches infinity along a congruence of null geodesics. The null congruence is assumed to "expand" in all directions near infinity (but it is otherwise generic), which includes in particular asymptotically flat spacetimes. In contrast to the well-known four-dimensional peeling property, the fall-off rate of various Weyl components depends substantially on the chosen boundary conditions, and is also influenced by the presence of a cos…
We prove theorems about the Ricci and the Weyl tensors on generalized Robertson-Walker space-times of dimension $n\ge 3$. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmonic if and only if it is annihilated by Chen's vector, and any of the two conditions is necessary and sufficient for the GRW space-time to be a quasi-Einstein (perfect fluid) manifold. Finally, the general structure of the Riemann tensor for Robertson-Walker space-times is given, i…
It is well-known that the conformal structure of a relativistic spacetime is of profound physical and conceptual interest. In this note, we consider the analogous structure for Newtonian theories. We show that the Newtonian Weyl tensor is an invariant of this structure.
We present an expression for the Weyl-Weyl two-point function in de Sitter spacetime, based on a recently calculated covariant graviton two-point function with one gauge parameter. We find that the Weyl-Weyl two-point function falls off with distance like r^{-4}, where r is spacelike coordinate separation between the two points.
The two-point Wightman function of the free photon field defined in a gauge-invariant manner is known to be equivalent to the field-strength two-point function in any spacetime that is topologically trivial. We show that the gauge-invariant graviton two-point function defined in a similar manner is equivalent to the Weyl-tensor two-point function in Minkowski space and in the Poincare patch of de Sitter space. This implies that in the Poincare patch of de Sitter space the gauge-invariant graviton two-point function decays like (distance)^(-4) as a function of coordinate distance for spacelike …
We provide evidence that gravitational radiation in a 4D radiation-dominated universe, with equation-of-state $w=1/3$, consists of two components: helicity-2 gravitons and massless scalar acoustic waves. On physical grounds, we would expect the homogeneous solution of the Weyl tensor components to be a good approximation to its inhomogeneous counterparts, whenever the observer is located well in the far zone of an isolated astrophysical source of cosmological gravitational radiation. We show explicitly that these homogeneous and inhomogeneous solutions both receive contributions from the gauge…
This brief paper investigates the consequences for the metric tensor of space-time when the Weyl tensor (in its conformally invariant form) and the energy-momentum tensor is specified. It is shown that, unless rather special conditions hold, the metric is uniquely determined up to a constant conformal factor.
The usual interpretation of Weyl geometry is modified in two senses. First, both the additive Weyl connection and its variation are treated as (1, 2) tensors under the action of Weyl covariant derivative. Second, a modified covariant derivative operator is introduced which still preserves the tensor structure of the theory. With its help, the Riemann tensor in Weyl geometry can be written in a more compact form. We justify this modification in detail from several aspects and obtain some insights along the way. By introducing some new transformation rules for the variation of tensors under the …
We investigate cosmological inflationary scenarios from a gravitoelectromagnetic theory. Our work is formulated in the light of a recently introduced geometrical Weyl-Invariant scalar-tensor theory of gravity, where the nature of both the electromagnetic potential and the inflaton field is attributed to the space-time geometry. We obtain a Harrison-Zeldovich power spectrum for quantum fluctuations of the inflaton field. In our model the electromagnetic fields have also a nearly scale invariant power spectrum for a power-law inflation. We found that the the seed magnetic fields have a nearly sc…
The peeling behaviour of the Weyl tensor near null infinity is determined for an asymptotically flat higher dimensional spacetime. The result is qualitatively different from the peeling property in 4d. To leading order, the Weyl tensor is type N. The first subleading term is type II. The next term is algebraically general in 6 or more dimensions but in 5 dimensions another type N term appears before the algebraically general term. The Bondi energy flux is written in terms of "Newman-Penrose" Weyl components.
A set of new tensors of purely geometric origin have been investigated, which form a hierarchy. A tensor of a lower rank plays the role of the potential for the tensor of one rank higher. The tensors have interesting mathematical and physical properties. The highest rank-tensor of the hierarchy, possesses all the geometrical properties of the Weyl tensor.
Conformal Weyl and Cotton tensors are dimensionally reduced by a Kaluza-Klein procedure. Explicit formulas are given for reducing from four and three dimensions to three and two dimensions, respectively. When the higher dimensional conformal tensor vanishes because the space is conformallly flat, the lower-dimensional Kaluza-Klein functions satisfy equations that coincide with the Einstein-Weyl equations in three dimensions and kink equations in two dimensions.
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 ($σ_{ab}$) 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 …
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 the Weyl tensor to be algebraically special, and it is a necessary and sufficient condition for the magnetic part to vanish. Some theorems (Derdzinski and Shen, Hall) are extended to the broader hypothesis of Weyl or Riemann compatibility. Weyl compatibility includes conditions t…
A peeling theorem for the Weyl tensor in higher dimensional Lorentzian manifolds is presented. We obtain it by generalizing a proof from the four dimensional case. We derive a generic behavior, discuss interesting subcases and retrieve the four dimensional result.
We have investigated the strong gravitational lensing for the photons coupled to Weyl tensor in a Schwarzschild black hole spacetime. We find that in the four-dimensional black hole spacetime the equation of motion of the photons depends not only on the coupling between photon and Weyl tensor, but also on the polarization direction of the photons. It is quite different from that in the case of the usual photon without coupling to Weyl tensor in which the equation of motion is independent of the polarization of the photon. Moreover, we find that the coupling and the polarization direction modif…
We apply singularity analysis to investigate the integrability properties of the gravitational field equations in Weyl Integrable Spacetime for a spatially flat Friedmann--Lemaître--Robertson--Walker background spacetime induced with an ideal gas. We find that the field equations possess the Painlevé property in the presence of the cosmological constant and the analytic solution is given by a left Laurent expansion.
The problem of obtaining an explicit representation for the fourth invariant of geodesic motion (generalized Carter constant) of an arbitrary stationary axisymmetric vacuum spacetime generated from an Ernst Potential is considered. The coupling between the non-local curvature content of the spacetime as encoded in the Weyl tensor, and the existence of a Killing tensor is explored and a constructive, algebraic test for a fourth order Killing tensor suggested. The approach used exploits the variables defined for the Bäckland transformations to clarify the relationship between Weyl curvature, con…
Penrose suggested that issues with the origin of the second law of thermodynamics and the remarkable homogeneous and isotropic nature of the universe on large scales can be resolved with a concept of an entropy for gravitational fields captured by the Weyl curvature tensor. This has led to various proposals for an entropy density for gravity constructed from the Weyl tensor. Most work investigating such entropy densities have been restricted to general relativity and little work has been done to them in modified theories of gravity. To remedy this, we investigate the simplest proposal for an e…
Symmetries of geometrical and physical quantities in general relativity provide important information about the curvature structure of the spacetimes. Symmetries of the curvature and the Weyl tensors, known as curvature and Weyl collineations respectively, are two of such important symmetries. Some results on these symmetries for Bianchi type V spacetimes are discussed.
It is shown that (except for two well defined cases), the necessary and sufficient condition for any spherically symmetric distribution of fluid to leave the state of equilibrium (or quasi-equilibrium), is that the Weyl tensor changes with respect to its value in the state of equilibrium (or quasi-equilibrium).
In this note we wish to complement some recent work in the cosmological literature concerning the Weyl conformal curvature tensor and its parts. In particular, we shall give a clear-cut definition of the Newtonian limits of electric and magnetic parts of the Weyl tensor. We also discuss that only a subset of the relativistic equations is needed to obtain a closed system of equations in the Newtonian limit.
Stationary and axially symmetric space-times play an important role in astrophysics, particularly in the theory of neutron stars and black holes. The static vacuum sub-class of these space-times is known as Weyl's class, and contains the Schwarzschild space-time as its most prominent example. This paper is going to study the space of Killing tensor fields of valence 3 for space-times of Weyl's class. Killing tensor fields play a crucial role in physics since they are in correspondence to invariants of the geodesic motion (i.e. constants of the motion). It will be proven that in static and axia…
We study the geometrical properties of null congruences generated by an aligned null direction of the Weyl tensor (WAND) in spacetimes of the Weyl and Ricci type N (possibly with a non-vanishing cosmological constant) in an arbitrary dimension. We prove that a type N Ricci tensor and a type III or N Weyl tensor have to be aligned. In such spacetimes, the multiple WAND has to be geodetic. For spacetimes with type N aligned Weyl and Ricci tensors, the canonical form of the optical matrix in the twisting and non-twisting case is derived and the dependence of the Weyl and the Ricci tensors and Ric…
We derive the evolution equations for the electric and magnetic parts of the Weyl tensor for cold dust from both general relativity and Newtonian gravity. In a locally inertial frame at rest in the fluid frame, the Newtonian equations agree with those of general relativity. We give explicit expressions for the electric and magnetic parts of the Weyl tensor in the Newtonian limit. In general, the magnetic part does not vanish, implying that the Lagrangian evolution of the fluid is not purely local.
In an interesting recent paper on the growth of inhomogeneity through the effect of gravity [1], Bertschinger and Hamilton derive equations for the electric and magnetic parts of the Weyl tensor for cold dust for both General Relativity and Newtonian theory. Their conclusion is that both in General Relativity and in Newtonian theory, in general the magnetic part of the Weyl tensor does not vanish, implying that the Lagrangian evolution of the fluid is not local. We show here that the `Newtonian' theory discussed by them is in fact not Newtonian theory {\it per se}, but rather a plausible relat…
