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RICCI CALCULUS PACKAGE IN REDUCE

JÓZSEF KADLECSIK

Central Research Institute for Physics

H-1525 Budapest 114, P.O.B. 49, Hungary

Electronic mail: kadlec@sunserv.kfki.hu

10pt

Ricci calculus is performed in the framework of the computer algebra system REDUCE by the package being presented. The program called RICCIR is specialized in the requirements of general relativity.

PROGRAM SUMMARY

Title of program: RICCIR

Computer: Any system capable of running REDUCE

Programming language used: computer algebra system REDUCE 3.5

High speed storage required: 340 kbyte as loadable module

Peripherals used: hard disk, terminal

No. of lines in distributed program, including test data, etc.: 11060

Keywords: Ricci calculus, general relativity, computer algebra

Nature of physical problem: Calculations in general relativity using Ricci calculus according to the restrictions of the program.

Method of solution: Implementation of Ricci calculus in computer algebra

Typical running time: This depends on both the problem and machine (see test and log files).

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Introduction

Ricci calculus[1] is a well-known method in general relativity[2, 3] for applying indexed expressions in terms of tensors and symbols. However, due to the enormous number of the indices and the large expressions, the method can be an extremely cumbersome and mechanical procedure involving application of symmetry properties, substitution and derivation of indexed quantities, collection of terms by choosing appropriate dummy indices, etc. These parts of a calculation could be executed by a computer algebraic program[4, 5, 6].

Since the indexed expressions can be viewed in various ways, different kinds of expressions should be handled by such a program:

• The true tensor expressions with basis-independent meaning are the most important ones. They are valid in all coordinate systems and one is not forced to choose a special basis, e.g.

  $T$_i;j + T_j;i = 0.

  The abstract index notation here ($Ti$₁&cdots;i_k\>\> \>\> j₁&cdots;j_l)[7] is attributed to the meaning that the indices denote the number and the type of the variables of the given object (tensor of type $(k,l)$) on which that acts. The objects $Ti$₁&cdots;i_k\>\> \>\> j₁&cdots;j_l are not viewed as basis components, they only mirror the components of tensors as though a basis was introduced.

• Component equations can be obtained by choosing special basis (e.g. coordinate basis) and now $Ti$₁&cdots;i_k\>\> \>\> j₁&cdots;j_l means the components of a tensor of type $(k,l)$ according to the given basis:

  $Ri$_{\>\> jkl} = Γⁱ_{\>\> jl,k} - Γⁱ_{\>\> jk,l} + Γⁱ_{\>\> km}Γ^m_{\>\> jl} - Γⁱ_{\>\> lm}Γ^m_{\>\> jk}.

  Concrete elements have not been introduced yet and the symbolic objects $Ti$₁&cdots;i_k\>\> \>\> j₁&cdots;j_l should be manipulated. (Although the equations may be like the true tensorial expression to all appearances, the difference of principle exists.)

• In the case of component expressions there may be found equations in terms of certain elements or subtensors of the indexed objects. The best example is a projection from a manifold onto a submanifold, when the quantities from the manifold can be expressed in terms of the quantities of the submanifold:

  $g$₀₀ = f, g_0i = f ω_i, g_ij = - {1f} h_ij + f ω_i ω_j.

  Here there are two groups of the indices: the concrete idex $0$ and the abstract indices $i,\; j$ which cannot be equal to $0$.
• Finally one would like to compute the expressions in concrete components and to obtain the value of the elements of the indexed objects by prescribing for example the metric:

  $ds2=-V(dt\; -\; w\; d\phi )2+\; \{1V\}(\rho 2d\phi 2+\; e2\gamma (d\rho 2+dz2)),R$_ij = {( ? \ }

The computer algebraic system RICCIR written in REDUCE[8] can handle expressions in the first three cases; the last one has not been implemented yet. The indexed expressions are manipulated according to the rules of the tensor algebra. The correct handling of the dummy indices (which means to convert the tensor multiplications in different forms but with the same value to a standard form as well) is the strength of the system. The usual algebraic operations are supplemented by contraction, and by partial, covariant and Lie derivatives. Symmetry properties with respect to the indices of tensors and symbols can be defined and are automatically applied during the calculations. Large variety of the relations from the general relativity is built-in and is used under the full controll of the user. There is a restriction, however: the built-in relations are valid only in the case of Riemann manifolds without torsion.

The developing of the system is still on-going and any comments and suggestions would be greatly appreciated.

• Declarations and notations
  • Tensors
  • Symbols
  • Indices
  • Covariant and contravariant indices
  • Symmetries
  • Special objects
  • Dimension and signature
• Operations
  • Addition, subtraction
  • Contraction
  • Multiplication, division, raising to a power
  • Partial derivation
  • Covariant derivation
  • Lie derivation
  • Symmetrization and antisymmetrization
  • Functions, user defined operators
• Assignments, LET commands and rule lists
  • The exceptions or what is going on in the background
• Built-in relations
  • METRIC switch
  • LEVI!-CIVITA switch
  • RELATION command
• References
• About this document ...