References

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Misner C. W., Thorne K. S. and Wheeler J. A. Gravitation (San Francisco, W. H. Freeman & Co., 1973).

3

Hawking S. W. and Ellis G. F. R. The Large-Scale Structure of Space-Time (Cambridge University Press, 1973).

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APPENDIX

A. Summary of operators and commands in RICCIR

<expr> := <expr> assignment statement

{<expr> => <expr> } rule list

@ operator of partial differentiation

D operator of covariant differentiation

LD operator of Lie differentiation

ANTISYMMETRIZE antisymmetrizes in given indices

CHRISTOFFEL SYMBOL declares the Christoffel symbol

CLEAR clears tensor and symbol identifiers clears assignment statements, LET rules

CLEAR CHRISTOFFEL SYMBOL removes the Christoffel symbol

CLEAR INDEX clears identifiers as abstract indices

CLEAR KRNONECKER DELTA removes the identity map

CLEAR LEVI!-CIVITA SYMBOL removes the Levi-Civita symbol

CLEAR LEVI!-CIVITA TENSOR removes the Levi-Civita tensor

CLEAR METRIC TENSOR removes the metric tensor

CLEAR RIEMANN TENSOR removes the Riemann tensor (Ricci tensor, curvature scalar)

CLEAR SCALAR clears the scalar value of tensors and symbols

CLEAR SIGNATURE Clears the signature of the metric tensor

CLEAR SPACEDIM clears the dimension of space-time

CLEAR SYMMETRY removes symmetry properties

CLEAR WEYL TENSOR removes the Weyl tensor

CONTRAV boolean operator: true, if its argument is a contravariant index

DUMMY boolean operator: true, if its argument is a dummy index

FOR ALL <vars> [SUCH THAT <boolean expr> ] clears LET rules, assign-

$CLEAR$ ment statements

FOR ALL <vars> [SUCH THAT <boolean expr> ] LET rule

FOR ALL <vars> [SUCH THAT <boolean expr> ] assignment statement

INDEX declares identifiers as abstract indices

KRONECKER DELTA declares the identity map

LEVI!-CIVITA SYMBOL declares the Levi-Civita symbol

LEVI!-CIVITA TENSOR declares the Levi-Civita tensor

METRIC TENSOR declares the metric tensor

RELATION switches relations on and off keyword in WHERE statements for applying relations locally

RIEMANN TENSOR declares the Riemann tensor (Ricci tensor, curvature scalar

SIGNATURE declares the signature of the metric tensor

SPACEDIM declares the dimension of space-time

STATUS shows introduced identifiers and activated relations, switches

SYMBOL declares identifiers for symbols

SYMMETRIZE symmetrizes in given indices

SYMMETRY defines symmetry properties shows symmetry properties of the objects

TENSOR declares identifiers for tensors

WEYL TENSOR declares the Weyl tensor

B. Summary of switches in RICCIR

LEVI!-CIVITA product of Levi-Civita symbols (or tensors) will be converted to expressions consisting of metric tensors

METRIC contracted metric tensor will be used to raise or to lower the indices