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HELP FOR: grcalc
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CALLING SEQUENCE: grcalc ( objects )
PARAMETERS: objects - a GRTensorII object name or sequence of names
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SYNOPSIS:
- Calculates the GRTensorII objects specified by the parameters.
- Objects are specified in the form:
TensorName ( IndexList )
where IndexList is a list of indexes specifying their nature (covariant
or contravariant) and type (coordinate or basis). IndexList entries
can take the form:
dn, up - covariant and contravariant coordinate indices, resp.
bdn, bup - covariant and contravariant basis indices, resp.
Thus the `natural form' of the Riemann tensor calculated from a metric
would be specified as:
R(up,dn,dn,dn)
The components of the same tensor in a basis would be:
R(bup,bdn,bdn,bdn)
A full set of default tensors which are defined automatically
can be found in the help pages ?grt_objects and ?grt_basis.
Tensors can also be defined using the grdef() command.
- The covariant derivatives of a tensor can be calculated automatically
by specifying the indices `cdn' and `cup'. Thus to calculate the
covariant derivative of the covariant Einstein tensor, one could use:
G(dn,dn,cdn)
Similarly, the ordinary partial derivatives can be calculated using the
indices `pup' and `pdn', as in
G(dn,dn,pdn)
- Note that most objects require the calculation of `precursor' objects.
Implicit in a request to calculate an object is a request to calculate
all of these precursors, which GRTensorII will do automatically. The
program will keep track of which tensors have been calculated for a
given metric so that they can always be accessed later and will never
be calculated twice.
Thus, if a request is made to calculate the Riemann tensor, then the
Christoffel symbols will automatically be calculated and stored in the
process.
It is possible, then, to proceed directly from the metric to a calculation
of the Riemann tensor, or other complicated curvature tensors, in one
step. This is discouraged, however, since for complicated spacetimes the
intermediate steps of the calculation (such as the Christoffel symbols)
will often require simplification to reduce their size before any further
calculation can be done. Thus it is recommended that calculations be
carried out in steps with gralter() applied when necessary.
- In addition to the standard tensor-type objects, a number of `operators'
are also defined in the standard GRTensorII library. These act on tensors
which are specified as arguments to the operator in square braces [, ].
For example, the commonly defined d'Alembertian (`Box') operator acting
on a tensor can be calculated as follows:
> grcalc ( Box[R(dn,dn)] ):
> grdisplay ( Box[R(dn,dn)] ):
For a full list of operators defined by the standard library, see
?grt_operators.
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EXAMPLES:
# 1: Covariant Ricci tensor for the Schwarzschild metric.
> qload ( schw ):
> grcalc ( R(dn,dn) ):
Calculated detg for schw. (0.030000 sec.)
Calculated g(up,up) for schw. (0.070000 sec.)
Calculated g(dn,dn,pdn) for schw. (0.070000 sec.)
Calculated Chr(dn,dn,dn) for schw. (0.060000 sec.)
Calculated Chr(dn,dn,up) for schw. (0.100000 sec.)
Calculated R(dn,dn) for schw. (0.100000 sec.)
CPU Time = .430
> grdisplay ( _ ): # note that the `_' indicates `use the objects from the
# last command.'
For the schw spacetime:
Covariant Ricci
R(dn,dn) = All components are zero
# 2: Covariant derivative of the mixed Ricci tensor for the Bondi metric.
> qload ( bondi ):
> grcalc ( R(up,dn,cdn) ):
Created definition for R(up,dn)
bytes used=1766732, alloc=917336, time=1.50
bytes used=1777636, alloc=917336, time=1.61
Created a definition for R(up,dn,cdn)
Calculated detg for bondi. (0.020000 sec.)
Calculated g(up,up) for bondi. (0.050000 sec.)
Calculated g(dn,dn,pdn) for bondi. (0.080000 sec.)
Calculated Chr(dn,dn,dn) for bondi. (0.070000 sec.)
Calculated Chr(dn,dn,up) for bondi. (0.120000 sec.)
Calculated R(dn,dn) for bondi. (0.170000 sec.)
Calculated R(up,dn) for bondi. (0.090000 sec.)
Calculated R(up,dn,cdn) for bondi. (0.510000 sec.)
CPU Time = 1.450
> grdisplay ( _ ):
[ Long output ... ]
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SEE ALSO: grdisplay, grt_objects, grt_basis, grt_operators, grt_invars,
gralter, grcalcalter.
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