Spacetime vector analysis

Sobczyk, Garret

doi:10.1016/0375-9601(81)90586-7

Cited by 38 publications

(24 citation statements)

References 3 publications

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“…One example for such a complexification is based on the R 3,0 paravector algebra, which has been considered by Sobczyk and Baylis [30,31] for the representation of relativistic vectors. Baylis has shown that the theory of electrodynamics can be fully expressed in terms of this algebra.…”

Section: Introductionmentioning

confidence: 99%

Representations of Clifford Algebras with Hyperbolic Numbers

Ulrych¹

2007

AACA

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Section: Introductionmentioning

confidence: 99%

Representations of Clifford Algebras with Hyperbolic Numbers

Ulrych¹

2007

AACA

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“…This is called the even subalgebra Cℓ + 1,3 , and it is well-known that it is isomorphic to the three-dimensional geometric algebra Cℓ 3 . We will find it useful in dealing with the Lorentz group [2], and in constructing a (3 + 1)-dimensional "complex" space [22,23]. The latter discussion will allow us to construct null tetrads, which in turn leads us to the Petrov classification of the Weyl tensor, which we will examine in Section VI.…”

Section: A Paravectors and "Complex" 4-spacementioning

confidence: 99%

“…In the case of vacuum solutions, the Weyl tensor is the gravitational field, so being able to discuss the form of the solution without first solving the equations can help us understand the underlying physical configuration. The classification of the Weyl tensor due to Petrov, which has been translated into the language of geometric algebra by Sobczyk [4,23,26] and Hestenes and Sobczyk [3], provides one such way of describing the fields, independent of a coordinate system. We examine an algebraic technique using the eigenvalue problem in Section VI A, then we apply the null tetrad formalism in Section VI B to provide a geometric interpretation of certain of these classes.…”

Section: Classification Of the Weyl Tensormentioning

confidence: 99%

Geometric algebra techniques for general relativity

Francis¹,

Kosowsky²

2004

Annals of Physics

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Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor.

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“…As real algebras, STA operates in a linear space of sixteen dimensions, whereas the linear space of APS has eight dimensions. APS can also be viewed as the 4-dimensional algebra of complex paravectors [15]. It is also isomorphic to complex quaternions, which have along history of applications in relativity, [16,17] The formulation of relativity in STA can be characterized as absolute, in that physical paths, fields, and other properties of objects are expressed independent of any observer.…”

Section: Introductionmentioning

confidence: 99%

“…The formulation in APS can be either absolute [15] or relative [12,13,14]. In both APS versions, the relationship between APS and STA + is expressed as an algebra isomorphism together with an operation called hermitian conjugation.…”

Section: Introductionmentioning

confidence: 99%

Relativity in Clifford's Geometric Algebras of Space and Spacetime

Baylis

Sobczyk

2004

International Journal of Theoretical Physics

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Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships between formulations of special relativity in the spacetime algebra (STA) Cℓ 1,3 of the underlying Minkowski vector space, and in the algebra of physical space (APS) Cℓ 3 . STA lends itself to an absolute formulation of relativity, in which paths, fields, and other physical properties have observer-independent representations. Descriptions in APS are related by a one-to-one mapping of elements from APS to the even subalgebra STA + of STA. With this mapping, reversion in APS corresponds to hermitian conjugation in STA. The elements of STA + are all that is needed to calculate physically measurable quantities (called measurables) because only they entail the observer dependence inherent in any physical measurement. As a consequence, every relativistic physical process that can be modeled in STA also has a representation in APS, and vice versa. In the presence of two or more inertial observers, two versions of APS present themselves. In the absolute version, both the mapping to STA + and hermitian conjugation are observer dependent, and the proper basis vectors of any observer are persistent vectors that sweep out timelike planes in spacetime. To compare measurements by different inertial observers in APS, we express them in the proper algebraic basis of a single observer. This leads to the relative version of APS, which can be related to STA by assigning every inertial observer in STA to a single absolute frame in STA. The equivalence of inertial observers makes this permissible. The mapping and hermitian conjugation are then the same for all observers. Relative APS gives a covariant representation of relativistic physics with spacetime multivectors represented by multiparavectors in APS. We relate the two versions of APS as consistent models within the same algebra.

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Spacetime vector analysis

Cited by 38 publications

References 3 publications

Representations of Clifford Algebras with Hyperbolic Numbers

Representations of Clifford Algebras with Hyperbolic Numbers

Geometric algebra techniques for general relativity

Relativity in Clifford's Geometric Algebras of Space and Spacetime

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