Overview of the Physics[ThreePlusOne] package - Maple Programming Help

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Overview of the Physics[ThreePlusOne] package

Description

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In the general theory of relativity, x1,x2,x3 can be any quantities defining the position of bodies in space and the time coordinate x0 can be defined by an arbitrarily running clock. In order to define simultaneity (synchronizing clocks located at different points in space) as well as determine actual space distances and time intervals in terms of these quantities x0,x1,x2,x3, it is relevant to split the 4D mathematical description of gravity into 3D plus time. This 3 + 1 split description is also an appropriate framework for a Hamiltonian formulation of gravity and for the study of gravitational waves, black holes and neutron stars by running numerical simulations as traditional initial value (Cauchy) problems.

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In this framework, the ThreePlusOne package includes

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Commands to decompose 4D tensorial expressions into 3+1 with respect to its free and contracted indices

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Computational representations for the 3D: metric, covariant derivative, Christoffel symbols, Ricci and Riemann tensors.

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Computational representations for the lapse, shift, unit normal and time vectors and extrinsic curvature related to the ADM equations.

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When ThreePlusOne is loaded:

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An attempt is made to set the letters used to represent space indices to be lowercaselatin_is, that is, the lowercase latin letters between i and s. If those letters are in use, the system searches for one of the other possible letters available to represent space indices and, if no one is available, a message is displayed on the screen requiring your action to free one kind of letters.

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The signature is automatically changed from the Physics package default: - - - +, to: + + + - in order to match the literature for the ADM formalism. The Physics package commands automatically take into account the signature set, that can be any of - - - +, + - - - , + + + -, and - + + + . You can change the value of the signature using Setup(signature = ...) where the right-hand side is any of these four signatures.

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Regarding a 3 + 1 decomposition, the signature - + + + customarily used in the ADM formalism, also in the database of solutions to Einstein's equations in the Maple system, has the computational disadvantage that the space components of a 4D tensor are those in position 2, 3 and 4 instead of 1, 2 and 3, while the time component (the different sign in the signature), is in position 1, and so the value 0 of a spacetime index points to the 1st component of 4D tensors. On the other hand, the signature + + + -, automatically set when ThreePlusOne is loaded, is physically equivalent to - + + + and with it you can refer to the spatial components of 4D tensors more naturally, using the values 1, 2, 3, while the value 0 automatically points to the 4th component. The ordering of space and time components in the 3+1 decomposition performed by the Physics:-Decompose command is according to the position of the timelike component in the signature set.

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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.

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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 ThreePlusOne Package Commands

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The following is a list of available commands.

ADMEquations

Christoffel3

D3_

ExtrinsicCurvature

gamma3_

Lapse

Ricci3

Riemann3

Shift

TimeVector

UnitNormalVector

 

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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.

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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

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The ADMEquations command returns the ADM equations, expressed in terms of the Lapse, Shift, ExtrinsicCurvature and the 3D metric gamma3_.

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The gamma3_ command, a 4D tensor of two indices that projects 4D tensors onto a 3D hypersurface, its 3D space part represents the metric in that 3D hypersurface and so it can be used to lower and raise space indices in 3D tensorial expressions.

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The Christoffel3 command, a 4D object with three indices, defined in terms of the derivatives of the gamma3_ metric, its 3D space part represents the Christoffel symbols in the 3D space hypersurface.

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The D3_ command, a a 4D tensorial operator with one index, the projection of D_ onto the 3D hypersurface, defined in terms of the Christoffel3 symbols, its 3D space part represents the covariant derivative operator in the 3D space.

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The ExtrinsicCurvature command a 4D tensor that is purely spatial, and whose spatial components represent the 3D extrinsic curvature.

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The Lapse command represents the lapse of the ADM formalism, that is the lapse of proper time between two hypersurfaces as measured by normal (Eulerian) observers

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The Ricci3 command, a 4D tensor of two indices, defined in terms of the Christoffel3 symbols and the gamma3_ metric, its 3D space part represents the Ricci tensor in the 3D space.

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The Riemann3 command, a 4D tensor of four indices, defined in terms of the Christoffel3 symbols, its 3D space part represents the Riemann tensor in the 3D space.

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The Shift command, a tensor of 1 index, represents the shift of the ADM formalism, that is the relative velocity between the normal (Eulerian) observers and the lines of constant spatial coordinates.

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The TimeVector command, a tensor of 1 index, represents the time vector of the ADM formalism, that is a vector tangent to the lines of constant spatial coordinates.

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The UnitNormalVector command, a tensor of 1 index, represents the unit normal vector of the ADM formalism, that is a unit vector normal to the spatial hypersurfaces.

See Also

conversions between differentiation notations, PhysicalConstants, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, UsingPackages

References

Textbooks

  

[1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

  

[2] Alcubierre, M., Introduction to 3+1 Numerical Relativity, International Series of Monographs on Physics 140, Oxford University Press, 2008.

  

[3] Baumgarte, T.W., Shapiro, S.L., Numerical Relativity, Solving Einstein's Equations on a Computer, Cambridge University Press, 2010.

  

[4] Gourgoulhon, E., 3+1 Formalism and Bases of Numerical Relativity, Lecture notes, 2007, https://arxiv.org/pdf/gr-qc/0703035v1.pdf.

  

[5] Arnowitt, R., Dese, S., Misner, C.W., The Dynamics of General Relativity, Chapter 7 in Gravitation: an introduction to current research (Wiley, 1962), https://arxiv.org/pdf/gr-qc/0405109v1.pdf