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HELP FOR: GRTensorII Objects
===============================================================================

- The standard objects in the table below are loaded by default with
  GRTensorII or automatically when their calculation is requested.

- Index permutations of these objects are generated automatically. Thus
  the index configuration R(up,dn,dn,up) can be calculated using grcalc()
  even though this index configuration is not listed below.

- Similarly covariant derivatives can calculated using grcalc() by
  adding indices `cdn' or `cup' after the standard index list. Thus
  the covariant derivative of R(dn,dn) could be calculated using the
  command
                       grcalc ( R(dn,dn,cdn) ):

  Similarly, the ordinary partial derivative can be calculated using
  the indices `pup' and `pdn', as in

                       grcalc ( R(dn,dn,pdn) ):

- For a table of standard objects defined on a set of basis vectors, see
  the help page for ?grt_basis.

- For a list of coordinate invariant scalars see ?grt_invars

- New tensors can be defined using grdef().

===============================================================================
Curvature tensors:
===============================================================================
GRTensorII names(s)	Description
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x(up)			Coordinates

g(dn,dn), metric	metric

g(up,up), invmetric	metric (contravariant components)

ds			line-element form of the metric

kdelta			Kronecker delta (see ?kdelta)

Info			metric text description (see ?Info)

detg			metric determinant

dimension		spacetime dimension

sig			spacetime signature

Chr(dn,dn,dn), Chr1	Christoffel symbol (first kind)
Chr(dn,dn,up), Chr2	Christoffel symbol (second kind)

R(dn,dn,dn,dn),		Riemann tensor
Riemann

R(dn,dn), Ricci		Ricci tensor

S(dn,dn)		trace-free Ricci tensor

C(dn,dn,dn,dn)		Weyl tensor (*)
			                                                  ef
Cstar(dn,dn,dn,dn)	Dual of the Weyl tensor, C*   := (1/detg) eps    C
			                          abcd               abef   cd
Ricciscalar		Ricci scalar

G(dn,dn), Einstein	Einstein tensor (no cosmological constant)

			                                                 ab
RicciSq			Ricci tensor contracted with itself, RicciSq := R  R
			                                                    ab
			           abcd
RiemSq			RiemSq := R    R
			                abcd
			           abcd
WeylSq			WeylSq := C    C
			                abcd
			  a     a  c
S2(up,dn)		S2  := S  S
			   b     c  b

			  a     a  c  d
S3(up,dn)		S3  := S  S  S
			   b     c  d  b

			  a     a  c  d  e
S4(up,dn)		S4  := S  S  S  S
			   b     c  d  e  b

			  ab      ab   ef
C2(up,up,dn,dn)		C2    := C    C
			    cd      ef   cd

			             cd
CS(dn,dn)		CS  := C    S
			  ab    acdb
					 cd
CSstar(dn,dn)		CSstar  := C*   S
			      ab    abcd

LevCS(dn,dn,dn)		Levi-Civita symbol (3 dimensional)
LevCS(dn,dn,dn,dn)	Levi-Civita symbol (4 dimensional)

LevC(dn,dn,dn)		Levi-Civita tensor (3 dimensional)
LevC(dn,dn,dn,dn)	Levi-Civita tensor (4 dimensional)

------------------------------------------------------------------------------
(*) See also ?grt_operators for the `electric' and `magnetic' parts of the
    Weyl tensor.
===============================================================================

EXAMPLES:

> qload ( schw ):
> grcalc ( R(dn,dn) ):
> grdisplay ( _ ):
 
                            For the schw spacetime:
 
                                Covariant Ricci
 
                       R(dn,dn) = All components are zero

> qload ( npschw ):
> grcalc ( R(bdn,bdn) ):
> grdisplay ( _ ):
 
                            For the schw spacetime:
 
                                Covariant Ricci
 
                      R(bdn,bdn) = All components are zero


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SEE ALSO:  grcalc, grt_invars, grt_basis, grt_operators, killing, kdelta,
  Info, grdef, grt_commands, grdisplay, grlib.
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