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HELP FOR: GRTensorII Operators
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- GRTensorII `operators' are objects which require the specification of
  an argument in order to be calculated (via grcalc()).

- The argument is specified in square braces, [, ]. For some operators,
  such as the vector operators listed below, the argument must be of a
  particular form, such as a single index tensor.

- Operators currently fall into two main groups, the differential operators
  and vector operators, listed seperately below.


I. Differential operators:

- These operators take tensors or scalars with any number of indices as
  arguments. They can be used in grcalc() with the object to be acted
  upon in square braces:

                  > grcalc ( Box[ R(dn,dn) ] ):
                  > grdisplay ( Box[ R(dn,dn) ] ):

  They can also be used in the same manner in grdef() in defining new
  tensors:
                  > grdef ( `bR{a b} := Box[R{a b}]` ):

  In the definitions below, `object' refers to a GRTensorII object with any
  number of indices in any configuration (eg. R(up,dn,dn,dn), R(bdn,bdn),
  NPmu), `vector' refers to a single-index object, and `scalar' refers to
  an un-indexed object (eg. Ricciscalar, NPmu).

  The standard GRTensorII differential operators are:
===============================================================================
GRTensorII name         Description
-------------------------------------------------------------------------------
  LieD[vector,object] - Lie derivative of `object' along `vector'

  Box[object]         - D'Alembertian, (\nabla^a \nabla_a)[ object ]

  Dl[object]          - derivatives along the null basis vectors, (*)
  Dn[object]            eg. Dl[object] := (l^a \partial_a) [ object ]
  Dm[object]
  Dmbar[object]

  Dsq[scalar]         - ordinary derivative of a scalar contracted with itself,
                        Dsq[scalar] := \partial^a(scalar)\partial_a (scalar)

  CDsq[scalar]        - covariant derivative of a scalar contracted with self,
                        CDsq[scalar] := \nabla^a(scalar) \nabla_a(scalar)
-------------------------------------------------------------------------------
(*) In standard NP notation, these are commonly referred to as
    {D, Delta\, delta, delta^*} respectively.
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II. Vector operators

- A number of operators have been defined which calculate tensors which
  are dependent on a vector field defined on the spacetime.

- To use the vector operators, you must first define a single-index object
  using grdef(), for example,

                  > grdef ( `v{^a} := [v,r,0,0]` ):

  The operators defined below would then take the name `v' of the object
  as an argument in square braces, for example,

                  > grcalc ( shear[v](dn,dn) )

The vector operators are:
===============================================================================
GRTensorII name         Description
-------------------------------------------------------------------------------
  h[v](dn,dn)         - projection of the metric onto the subspace orthogonal
                        to v
  acc[v](up)          - acceleration vector
  expsc[v]            - expansion scalar
  Opexpsc[v]          - optical expansion scalar (*)
  vor[v](dn,dn)       - vorticity tensor
  vor[v](up)          - vorticity vector
  vor[v]              - vorticity scalar
  Opvor[v]            - optical vorticity scalar (*)
  shear[v](dn,dn)     - shear tensor
  shear[v]            - shear scalar
  Opshear[v]          - optical shear scalar (*)
  RayEqn[v]           - Raychaudhuri's equation
  E[v](dn,dn)         - `electric part' of the Weyl tensor
  H[v](dn,dn)         - `magnetic part' of the Weyl tensor
-------------------------------------------------------------------------------
(* - these definitions are valid only for null vector fields.)
-------------------------------------------------------------------------------

- if the vector field is non-null, it can be normalized using the
  grnormalize() command. See the ?grnormalize for more details.

-------------------------------------------------------------------------------
EXAMPLES:

> grcalc ( Box[R(dn,dn)] ):

> grdef ( `u{^a} := [E^(-nu(t,r)/2), 0, 0, 0]` ):

> grcalc ( vor[u](dn,dn), shear[u](dn,dn) ):
> grdisplay ( _ ):

> grcalc ( LieD[u,R(dn,dn)] ):

> grcalc ( Dl[ NPmu ] ):

> grdef ( `ldR{a b} := LieD[u,R{a b}]` ):

> grdef ( `B1 := -Dmbar[Psi0] + Dl[Psi1] + (4*NPalpha - NPpi)*Psi0
   -2*(2*NPrho + NPepsilon)*Psi1 + 3*NPkappa*Psi2` ):

-------------------------------------------------------------------------------
SEE ALSO: grt_objects, grt_basis, grcalc, grdef, grnormalize.
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