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Articles and Publications

Weyl Tensor Classification in Four-dimensional Manifolds of All Signatures

Author: Carlos Batista 

Abstract: 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 the Weyl tensor in four dimensions it is established that this last approach produces a classification equivalent to the one generated by the bivector method.

Reference: Gen. Rel. Grav. 45 (2013), 785. 

A Generalization of the Goldberg-Sachs Theorem and its Consequences

Author: Carlos Batista 

Abstract: The Goldberg-Sachs theorem is generalized for all four-dimensional manifolds endowed with torsion-free connection compatible with the metric, the treatment includes all signatures as well as complex manifolds. It is shown that when the Weyl tensor is algebraically special severe geometric restrictions are imposed. In particular it is demonstrated that the simple self-dual eigenbivectors of the Weyl tensor generate integrable isotropic planes. Another result obtained here is that if the self-dual part of the Weyl tensor vanishes in a Ricci-flat manifold of (2,2) signature the manifold must be Calabi-Yau or symplectic and admits a solution for the source-free Einstein-Maxwell equations.

Reference: Gen. Rel. Grav. 45 (2013), 1411. 

Spinors and the Weyl Tensor Classification in Six Dimensions

Authors: Carlos Batista and Bruno Carneiro da Cunha

Abstract: 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 Goldberg-Sachs theorem. We work in complexified spaces, showing that the results for any signature can be obtained by taking the desired real slice.

Reference: J. Math. Phys. 54 (2013), 052502.

On the Weyl Tensor Classification in All Dimensions and its Relation with Integrability Properties

Authors: Carlos Batista

Abstract: 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.

Reference: J. Math. Phys. 54 (2013), 042502.

On the Pursuit of Generalizations for the Petrov Classification and the Goldberg-Sachs Theorem

Doctor Thesis Published as a Book

Author: Carlos Batista 

Abstract:The Petrov classification is an important algebraic classification for the Weyl tensor valid in 4-dimensional space-times. In this thesis such classification is generalized to manifolds of arbitrary dimension and signature. This is accomplished by interpreting the Weyl tensor as a linear operator on the bundle of p-forms, for any p, and computing the Jordan canonical form of this operator. Throughout this work the spaces are assumed to be complexified, so that different signatures correspond to different reality conditions, providing a unified treatment. A higher-dimensional generalization of the so-called self-dual manifolds is also investigated. 
The most important result related to the Petrov classification is the Goldberg-Sachs theorem. Here are presented two partial generalizations of such theorem valid in even-dimensional manifolds. One of these generalizations states that certain algebraic constraints on the Weyl operator imply the existence of an integrable maximally isotropic distribution. The other version of the generalized Goldberg-Sachs theorem states that these algebraic constraints imply the existence of a null congruence whose optical scalars obey special restrictions. 
On the pursuit of these results the spinorial formalism in 6 dimensions was developed from the very beginning, using group representation theory. Since the spinors are full of geometric significance and are suitable tools to deal with isotropic structures, it should not come as a surprise that they provide a fruitful framework to investigate the issues treated on this thesis. In particular, the generalizations of the Goldberg-Sachs theorem acquire an elegant form in terms of the pure spinors.

Reference: ISBN: 978-3-659-52065-5, Lambert Academic Publishing, Germany, 2014. 

Pure Subspaces, Generalizing the Concept of Pure Spinors

Author: Carlos Batista

Abstract: The concept of pure spinor is generalized, giving rise to the notion of pure subspaces, spinorial subspaces associated to isotropic vector subspaces of non-maximal dimension. Several algebraic identities concerning the pure subspaces are proved here, as well as some differential results. Furthermore, the freedom in the choice of a spinorial connection is exploited in order to relate twistor equation to the integrability of maximally isotropic distributions.

Reference: Journal of Geometry and Physics 81 (2014), 117-127

Killing-Yano Tensors of Order n-1

Author: Carlos Batista

Abstract: The properties of a Killing-Yano tensor of order n-1 in an n-dimensional manifold are investigated. The integrability conditions are worked out and all metrics admitting a Killing-Yano tensor of order n-1 are found. It is pointed out a connection between such tensors and a generalization of the concept of angular momentum. A theorem on how to generate closed conformal Killing vectors using the symmetries of a manifold is proved and used to find all Killing-Yano tensors of order n-1 of a maximally symmetric space.

Reference: Class. Quantum Grav. 31 (2014) 165019

Integrability Conditions for Killing-Yano Tensors and Conformal Killing-Yano Tensors

Author: Carlos Batista

Abstract: 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-Yano tensor and that every conformal Killing-Yano tensor is uniquely decomposed as the sum of a Killing-Yano tensor and a closed conformal Killing-Yano tensor.

Reference: Phys. Rev. D 91 (2015), 024013 

 

 

Integrability Conditions for Killing-Yano Tensors and Maximally Symmetric Spaces in the Presence of Torsion

Author: Carlos Batista

Abstract: 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.

Reference: Phys. Rev. D 91 (2015), 084036

 

 

Conformally Invariant Spinorial Equations in Six Dimensions

Author: Carlos Batista

Abstract: This work deals with the conformal transformations in six-dimensional spinorial formalism. Several conformally invariant equations are obtained and their geometrical interpretation are worked out. Finally, the integrability conditions for some of these equations are established. Moreover, in the course of the article, some useful identities involving the curvature of the spinorial connection are attained and a digression about harmonic forms and more general massless fields is made.

Reference: Classical and Quantum Garvity 33 (2016), 015002

 

 

Killing Spinors and Related Symmetries in Six Dimensions

Author: Carlos Batista

Abstract: Benefiting from the index spinorial formalism, the Killing spinor equation is integrated in six-dimensional spacetimes. The integrability conditions for the existence of a Killing spinor are worked out and the Killing spinors are classified in two algebraic types, in the first type the scalar curvature of the spacetime must be negative, while in the second type the spacetime must be an Einstein manifold. In addition, the equations that define Killing-Yano (KY) and closed conformal Killing-Yano (CCKY) tensors are expressed in the index notation and, as consequence, all non-vanishing KY and CCKY tensors that can be generated from a Killing spinor are made explicit.

Reference: Phys. Rev. D 93 (2016), 065002.

 

 

A Class of Integrable Metrics

Authors: Andrés Anabalón and Carlos Batista

Abstract: In four dimensions, the most general metric admitting two commuting Killing vectors and a rank-two Killing tensor can be parameterized by ten arbitrary functions of a single variable. We show that picking a special vierbien, reducing the system to eight functions, implies the existence of two geodesic and share-free, null congruences, generated by two principal null directions of the Weyl tensor. Thus, if the spacetime is an Einstein manifold, the Goldberg-Sachs theorem implies it is Petrov type D, and by explicit construction, is in the Carter class. Hence, our analysis provide an straightforward connection between the most general integrable structure and the Carter family of spacetimes.

Reference: Phys. Rev. D  93 (2016), 064079

 

 

Hamiltonian formulation towards minimization of viscous fluid fingering

Authors: Carlos Batista, Eduardo Olímpio R. Dias and José Américo Miranda

Abstract: A variational approach has been recently employed to determine the ideal time-dependent injection rate Q(t) that minimizes fingering formation when a fluid is injected in a Hele-Shaw cell filled with another fluid of much greater viscosity. However, such a calculation is approximate in nature, since it has been performed by assuming a high capillary number regime. In this work, we go one step further, and utilize a Hamiltonian formulation to obtain an analytical exact solution for Q(t), now valid for arbitrary values of the capillary number. Moreover, this Hamiltonian scheme is applied to calculate the corresponding injection rate that minimizes fingering formation in a uniform three-dimensional porous media. An analysis of the improvement offered by these exact injection rate expressions in comparison with previous approximate results is also provided.

Reference: Phys. Rev. E  94 (2016), 013109

 

Generalized Charged Nariai Solutions in Arbitrary Even Dimensions with Multiple Magnetic Charges

Authors: Carlos Batista

Abstract: Higher-dimensional solutions for Einstein-Maxwell equations that generalize the charged Nariai spacetime are obtained. The solutions presented here are made from the direct product of several 2-spaces of constant curvature. These solutions turn out to have many magnetic charges, contrary to the usual higher-dimensional generalization of the Nariai spacetime, which has no magnetic charge at all. These solutions are then used to generate black hole metrics. Finally, it is analyzed how these generalized Nariai solutions are modified in more general theories of gravity.

Reference: Gen. Rel. Grav. 48 (2016), 160.

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Integrability of the Dirac Equation on Backgrounds that are the Direct Product of Bidimensional Spaces

Authors: Joás Venâncio  and Carlos Batista

Abstract: The field equation for a spin 1/2 massive charged particle propagating in spacetimes that are the direct product of 2-dimensional spaces is separated. Moreover, we use this result to attain the separability of the Dirac equation in some specific static black hole solutions whose horizons have topology RxS^2x ... x S^2.

Reference: Phys. Rev. D. 95 (2017),  084022.

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A mechanistic model for the evolution of multicellularity

Authors: André Amado, Carlos Batista and Paulo R. A. Campos

Abstract: Through a mechanistic approach we investigate the formation of aggregates of variable sizes, accounting mechanisms of aggregation, dissociation, death and reproduction. In our model, cells can produce two metabolites, but the simultaneous production of both metabolites is costly in terms of fitness. Thus, the formation of larger groups can favor the aggregates to evolve to a configuration where division of labor arises. Assuming that the states of the cells in a group maximizes the per capita reproductive rate, we depict the processes affecting cells as kinetic chemical reactions by means of extensive computer simulations. In the model it is considered that the groups can grow linearly, forming a chain, or compactly keeping a roughly spherical shape. Starting from a population consisting of single-celled organisms, we observe the formation of groups with variable sizes and usually much larger than two-cell aggregates. Natural selection can favor the formation of large groups, which allows the system to achieve new and larger fitness maxima.

Reference: Physica A: Statistical Mechanics and its Applications,

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Class of integrable metrics and gauge fields

Authors: Gabriel Luz and Carlos Batista

Abstract: Starting with the most general four-dimensional spacetime possessing two commuting Killing vectors and a nontrivial Killing tensor, we analytically integrate Einstein-Yang-Mills equations for a completely arbitrary gauge group. It is assumed that the gauge field inherits the symmetries of the background and is aligned with the principal null directions of the spacetime.

Reference: Phys. Rev. D. 96 (2017),  084003 .

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A theoretical approach to the size-complexity rule

Authors: André Amado, Carlos Batista, and Paulo R. A. Campos

Abstract: The so-called size-complexity rule claims the existence of a positive correlation between organism size and number of cell types. In this spirit, here we address the relationship between organism size and number of potential tasks that can be performed. The modeling relies on the assumption that the states of the cells within the aggregates are such that the maximum fitness is realized, but also relies on the existence of tradeoffs among the distinct functions. For group sizes larger than the number of potential tasks, fitness maximization is attained when all cells in group specialize in a given task. Under this scenario, the number of potential tasks equals the number of cell types. We have found that the morphology and the topology of aggregates, as well as the developmental mode, strongly influence the dynamics of body formation. Particularly, it has been observed that more compact structures, such as sphere-like structures, are more likely to follow the claim of the size-complexity rule, whereas more fragile structures such as linear chains, which are more vulnerable to more drastic changes due to division mechanisms, can, in a broad scenario, violate the size-complexity rule.

Reference: Evolution, 72 (2018),  18-29.

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On Spaces with the Maximal Number of Conformal Killing Vectors

Author: Carlos Batista

Abstract: It is natural to expect and simple to prove that every conformally flat space possess the maximal number of conformal Killing vector fields (CKVs). On the other hand, it is interesting to ask whether the converse is true. Is conformal flatness a necessary condition for the existence of the maximal number of CKVs? In this review article it is proven that the answer is yes, a space admits the maximal number of CKVs if, and only if, it is conformally flat.

Reference:  Physical Science International Journal, 20-2  (2018), 1-13.

Quasinormal Modes in Generalized Nariai Spacetimes

Authors: Carlos Batista and Joás Venâncio

Abstract: The perturbations of fields with spin 0, 1/2, and 1 propagating in a higher-dimensional generalization of the charged Nariai spacetime are investigated. The boundary conditions leading to quasinormal modes are analyzed and the quasinormal frequencies are analytically obtained.

Reference:  Physical Rview D, 97 (2018), 105025 .

A Class of Integrable Metrics II

Authors: Carlos Batista and Gabriel Luz Almeida

Abstract: Starting with a subclass of the four-dimensional spaces possessing two commuting Killing vectors and a non-trivial Killing tensor, we fully integrate Einstein's vacuum equation with a cosmological constant. Although most of the solutions happen to be already known, we have found a solution that, as far as we could search for, has not been attained before. We also characterize the geometric properties of this new solution, it is a Kundt spacetime of Petrov type II possessing a null Killing vector field and an isometry algebra that is three-dimensional and abelian. In particular, such solution becomes a pp-wave spacetime when the cosmological constant is set to zero.

Reference:  Physical Review D 98 (2018), 044030 .

A Class of Higher-Dimensional Solutions of Einstein's Vacuum Equation

Authors: Carlos Batista and Gabriel Luz Almeida

Abstract: A new class of higher-dimensional exact solutions  of Einstein's vacuum equation is presented. These metrics are written in terms of the exponential of a symmetric matrix and when this matrix is diagonal the solution reduces to higher-dimensional generalizations of Kasner spacetime with a cosmological constant. On the other hand, the metrics attained when such matrix is non-diagonal have more intricate algebraic structures. Such solutions have not been presented in the literature yet.

Reference: The European Physical Journal C volume 79, Article number: 11 (2019) 

Spin 2 Quasinormal Modes in Generalized Nariai Spacetimes

Authors: Carlos Batista and Joás Venâncio

Abstract: In this work we analytically obtain the quasinormal spectrum for the gravitational perturbation on a higher-dimensional generalization of the Nariai spacetime that is comprised of the direct product of the two-dimensional de Sitter space with several two-spheres. A key step in order to attain this result is to use a suitable basis for the angular functions depending on the rank of the tensorial degree of freedom that one needs to describe. Here we define such a basis, which is a generalization of the tensor spherical harmonics that is suited for spaces that are the product of several spaces of constant curvature.

Reference: Phys. Rev. D 101, 084037 (2020)

On the Equivalence of Spacetimes, the Cartan-Karlhede Algorithm

Authors: Carlos Batista and  Thiago M. Mergulhão

Abstract: It is well known that in general relativity theory two spacetimes whose metrics are related by a coordinate transformation are physically equivalent. However, given two line elements, it is virtually impossible to implement the most general coordinate transformation in order to check the equivalence of the spacetimes. In this paper we present the so-called Cartan-Karlhede algorithm, which provides a finite sequence of steps to decide whether or not two metrics are equivalent. The point of this note is to illustrate the method through several simple examples, so that the reader can learn the fundamentals and details of the algorithm in practice.

Reference: Rev. Bras. Ensino Fís. vol.42  (2020) 

On Conserved Quantities for the Free Motion of Particles with Spin

Authors: Carlos Batista and Esdras Barbosa dos Santos

Abstract: In the early 80's, R. Rüdiger published a pair of articles in which it was found the most general conserved charges associated to the motion of particles with spin moving in curved spacetime. In particular, it was shown that besides the well-known conserved quantity associated to Killing vectors, it is also possible to have another conserved quantity that is linear in the spin of the particle if the spacetime admits a Killing-Yano tensor. However, in these papers it was proved that in order for this new scalar to be conserved two obscure conditions involving the Killing-Yano tensor and the curvature must be obeyed. In the present paper we try to shed light over these conditions and end up proving that this conserved quantity is useless for most physically relevant spacetimes. Notably, for particles moving in vacuum (Einstein spacetimes) this conserved scalar constructed with the Killing-Yano tensor will not help on the integration of the equations of motion. Moreover, we prove that, as a consequence of these obscure conditions, the Killing-Yano tensor must be covariantly constant.

Reference: Phys. Rev. D 101, 104049 (2020)

Six-dimensional Index Spinorial Formalism using Quaternions

Authors: Carlos Batista and Joás Venâncio

Abstract: In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with SL(2;H), which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects built from the tensor products of the fundamental representations of SL(2;H) do not carry a representation of the group, due to the non-commutativity of the quaternions. The Lie algebra of the spin group is obtained and its connection with the Lie algebra of SO(5,1) is presented, providing a physical interpretation for the elements of SL(2;H). Finally, we present a bridge between this quaternionic spinorial formalism for six-dimensional spacetimes and the four-component spinorial formalism over the complex field that comes from the fact that the spin group in six-dimensional Euclidean spaces is given by SU(4).

Reference: ...

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