Overview of the Physics:-Tetrads Subpackage
Description
Tetrads is a package that implements computational representations and related operations for classifying metrics (Petrov, Plebanski-Petrov and Segre, classifications) and computing with tensors in a local system of references (the tetrad system). Many calculations in General Relativity can be dramatically simplified by defining an orthonormal tetrad and by expressing the metric and all other tensors in terms of this orthonormal tetrad.
Mathematical notation: When Physics or Physics:-Vectors are loaded in the Standard Graphical User Interface, with Typesetting level set to Extended (default), anticommutative and noncommutative variables are displayed in different colors, non-projected vectors and unit vectors are respectively displayed with an arrow and a hat on top, the vectorial differential operators (Nabla, Laplacian, etc.) with an upside down triangle, and most of the other Physics commands are displayed as in textbooks.
Examples illustrating the use of the package's commands are found in Physics examples (this page opens only in the Standard Graphical User Interface).
List of Tetrads Package Commands
The following is a list of available commands.
e_
eta_
gamma_
IsTetrad
l_
lambda_
m_
mb_
n_
NullTetrad
OrthonormalTetrad
PetrovType
SegreType
TransformTetrad
WeylScalars
Inert forms of these commands, representing the operations, having the same display and mathematical properties under differentiation, simplification etc., but holding the computations, consist of the same commands' names prefixed by the % character. The inert computations constructed with these commands can be activated when desired using the value command.
Further information relevant to the use of these commands is found under Conventions used in the Physics package (spacetime tensors, not commutative variables and functions, quantum states and Dirac notation).
Brief description of each command
The e_ command represents the tetrad (also called vierbein), it is a tensor with two indices, one of spacetime type the other of tetrad type.
The eta_ command represents the metric in the tetrad (local) system of references, it is a tensor with two indices, both of tetrad type.
The l_, n_, m_ and mb_ commands represent the four null vectors of the Newman-Penrose formalism, all of them tensors with one index of spacetime type.
The gamma_ command represents and computes the Ricci rotation coefficients in the tetrad (local) system of references, it is a tensor with three indices of tetrad type.
The lambda_ command represents a linear combination of Ricci rotation coefficients in the tetrad (local) system of references, it is a tensor with three indices of tetrad type.
The IsTetrad command returns true or false according to whether a given matrix is a null or orthonormal tetrad
The NullTetrad command computes different forms of a null tetrad for the spacetime metric set
The OrthonormalTetrad command computes different forms of an orthonormal tetrad for the spacetime metric set
The PetrovType command indicates the Petrov type of the spacetime metric set, optionally computing the principal roots
The SegreType command indicates the Plebanski-Petrov and the Segre types for the Ricci tensor
The TransformTetrad command performs, on a tetrad, any of a set of predefined transformations you indicate, or a general transformation, or automatically applies the required transformations so that the tetrad is in canonical form.
The WeylScalars command computes the Weyl scalars for any given 4 x 4 matrix representing a tetrad; that is one for which IsTetrad returns true.
You are very welcome to contribute ideas for improvements. There is a great deal of scope for changing and improving things; let us know your suggestions by writing to physics@maplesoft.com.
See Also
conventions used in the Physics package, conversions between differentiation notations, PhysicalConstants, Physics, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, UsingPackages
References
Textbooks
Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.
Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. Herlt, E. Exact Solutions of Einstein's Field Equations, Cambridge Monographs on Mathematical Physics, second edition. Cambridge University Press, 2003.
Hawking, S.W., Israel, W., Chandrasekhar, S. General Relativity: an Einstein Centenary Survey, Chapter 7, Cambridge University Press, 2010.
Compatibility
The Physics:-Tetrads package was introduced in Maple 2015.
For more information on Maple 2015 changes, see Updates in Maple 2015.
The Physics:-Tetrads package was updated in Maple 2017.
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