Differential Geometry, Gauge Theories, and Gravity

We introduce the method of topological quantization for gravitational fields in a systematic manner. First we show that any vacuum solution of Einstein's equations can be represented in a principal fiber bundle with a connection that takes values in the Lie algebra of the Lorentz group. This result is generalized to include the case of gauge matter fields in multiple principal fiber bundles. We present several examples of gravitational configurations that include a gravitomagnetic monopole in linearized gravity, the C-energy of cylindrically symmetric fields, the ReissnerNordström and the Kerr-Newman black holes. As a result of the application of the topological quantization procedure, in all the analyzed examples we obtain conditions implying that the parameters entering the metric in each case satisfy certain discretization relationships.

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“…We will see that it determines a unique connection in P. To this end, we use the following theorem 4 valid for principal fiber bundles.…”

Section: ͑7͒mentioning

confidence: 99%

Topological quantization of gravitational fields

Patiño

Quevedo

2005

Journal of Mathematical Physics

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“…This clearly means that they did not grasp the meaning of the different symbols necessary to be used in an unambiguous presentation the theory of connections, and they are not alone. Statements of the same nature appears also, e.g., in [5,23,24,27,30,41,47,56] and also in [12,13,14,15,16,17].…”

Section: Misunderstandingmentioning

confidence: 81%

An Ambiguous Statement Called the "Tetrad Postulate" and the Correct Field Equations Satisfied by the Tetrad Fields

Rodrigues¹,

Souza²

2005

Int. J. Mod. Phys. D

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The names tetrad, tetrads, cotetrads, have been used with many different meanings in the physical literature, not all of them, equivalent from the mathematical point of view. In this paper we introduce unambiguous definitions for each one of those terms, and show how the old miscellanea made many authors to introduce in their formalism an ambiguous statement called 'tetrad postulate', which has been source of many misunderstandings, as we show explicitly examining examples found in the literature. Since formulating Einstein's field equations intrinsically in terms of cotetrad fields θ a , a = 0, 1, 2, 3 is an worth enterprise, we derive the equation of motion of each θ a using modern mathematical tools (the Clifford bundle formalism and the theory of the square of the Dirac operator). Indeed, we identify (giving all details and theorems) from the square of the Dirac operator some noticeable mathematical objects, namely, the Ricci, Einstein, covariant D'Alembertian and the Hodge Laplacian operators, which permit to show that each θ a satisfies a well defined wave equation. Also, we present for completeness a detailed derivation of the cotetrad wave equations from a variational principal. We compare the cotetrad wave equation satisfied by each θ a with some others appearing in the literature, and which are unfortunately in error. 1 IntroductionIn what follows we identify an ambiguous statement called 'tetrad postulate' (a better name, as we shall see would be 'naive tetrad postulate') that appears often in the Physics literature (see e.g., [5,12,24,47,49,50], to quote only a few examples here). We identify the genesis of the wording 'tetrad postulate' as a result of a deficient identification of some mathematical objects of differential geometry. Note that we used the word ambiguous, not the word wrong. This is because, as we shall show, the equation dubbed 'tetrad postulate' can be rigorously interpreted as meaning that the components of a covariant derivative in the direction of a vector field ∂ µ of a certain tensor field Q (Eq.(34)) are null (see Eq. (80)). This equation is not a postulate. Indeed, it is nothing more than the intrinsic expression of an obvious identity of differential geometry that we dubbed the freshman identity (Eq. (63)). However, if the freshman identity is used naively as if meaning a 'tetrad postulate' misunderstandings may arise, and in what follows we present some of them, by examining some examples that we found in the literature. We comment also on a result called 'Evans Lemma' of differential geometry and claimed in [12] to be as important as the Poincaré lemma. We show that 'Evans Lemma' as presented in [12] is a false statement, the proof offered by that author being invalid because in trying to use the naive tetrad postulate he did incorrect use of some fundamental concepts of differential geometry, as, e.g., 1 his (wrong) Eq.(41E). We explain all that in details in what follows. We observe also that in [12,13,14,15,16,17] it is claimed that 'Evans Lemma' is the basic pil...

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“…The action for the electromagnetic field in a four-dimensional (4-d) spacetime domain Ω 4 with flat metric is given by the integral [1][2][3][4] …”

Section: Fundamentalsmentioning

confidence: 99%

“…1 shows, using the 2-d case as example, the geometrical construction of the barycentric coordinate λ 1 associated to a point u within a 2-simplex (triangle). In this case, the value of λ 1 is simply the ratio between the areas of the triangles defined by [u, u 2 …”

Section: Primal Dual and Decomposition Latticesmentioning

confidence: 99%

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LATTICE MAXWELL'S EQUATIONS (Invited Paper)

Teixeira¹

2014

PIER

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Abstract-We discuss the ab initio rendering four-dimensional (4-d) spacetime of Maxwell's equations on random (irregular) lattices. This rendering is based on casting Maxwell's equations in the framework of the exterior calculus of differential forms, and a translation thereof to a simplicial complex whereby fields and causative sources are represented as differential p-forms and paired with the oriented pdimensional geometrical objects that comprise the set of spacetime lattice cells (simplices). We pay particular attention to the case of simplicial spacetime lattices because these can serve as building blocks of lattices made of more generic cells (polygons). The generalized Stokes' theorem is used to construct discrete calculus operations on the lattice based upon combinatorial relations depending solely on the connectivity and relative orientation among simplices. This rendering provides a natural factorization of (lattice) 4-d spacetime Maxwell's equations into a metric-free part and a metric-dependent part. The latter is encoded by discrete Hodge star operators which are built using Whitney forms, i.e., canonical interpolants for discrete differential forms. The derivation of Whitney forms is illustrated here using a geometrical construction based on the concept of barycentric coordinates to represent a point on a simplex, and the generalization thereof to represent higher-dimensional objects (lines, areas, volumes, and hypervolumes) in 4-d. We stress the role of the primal lattice, the barycentric dual lattice, and the barycentric decomposition lattice in achieving a complete description of the lattice theory. Lattice Maxwell's equations based on the exterior calculus of differential forms and on the use of Whitney forms as field interpolants inherits the symplectic structure and discrete analogues of conservation laws present in the continuum theory, such as energy and charge conservation. It also provides precise localization rules for the degrees of freedom associated with the different fields and sources on the lattice, and design principles for constructing consistent numerical solution methods that are free from spurious modes, spectral pollution, and (unconditional) numerical instabilities. We also briefly consider the relationship between lattice 4-d Maxwell's equations and some incarnations of discretization schemes for Maxwell's equations in (3 + 1)-d, such as finite-differences and finite-elements.

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Differential Geometry, Gauge Theories, and Gravity

Cited by 202 publications

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Topological quantization of gravitational fields

Topological quantization of gravitational fields

An Ambiguous Statement Called the "Tetrad Postulate" and the Correct Field Equations Satisfied by the Tetrad Fields

LATTICE MAXWELL'S EQUATIONS (Invited Paper)

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