Calculating Components

The most common operations in GRTensorIII are the definition, calculation, and simplification of the components of tensors in the spacetime. Definitions are provided for all of the commonly used tensors. A full list can be found on the ?grt_objects help screen.

To indicate a tensor object in GRTensorIII a Maple function expression is used. The name of the function is the tensor name and the arguments specify the number and type of indices. For example the covariant (indices down) metric tensor is g(dn,dn); the contravariant version of the same tensor is g(up,up). A scalar object in GRTensorIII is a name in Maple. For example RicciSq is the Ricci scalar.

To calculate one of the pre-defined tensor objects the command grcalc() is used. The result of the calculation is not automatically displayed (because in some cases the expression may be very large and require simplification), however, it can be displayed with grdisplay(). In cases where the output is expected to be small, a tensor can be calculated and displayed using grcalcd().

Commonly Referenced Objects

Only the index down versions are shown. GRTensorIII can automatically adjust to requests for other index configurations. Adding the terms cdn or cup after the indices will automatically generate the appropriate covariant derivative of the tensor.

Object	Description
g(dn,dn)	metric tensor
Chr(dn,dn,up)	Christoffel symbol of the second kind
R(dn,dn,dn,dn)	Riemann tensor
R(dn,dn)	Ricci tensor
G(dn,dn)	Einstein tensor
C(dn,dn,dn,dn)	Weyl tensor
RicciSq	Ricci scalar
detg	determinant of the metric tensor

(for a more complete list see ?grt_objects. To define new objects see ?grdefine)

Commands

Three commands are used most frequently within a GRTensorIII session:

grcalc() calculates the components of tensors;

grdisplay() displays the components;

gralter() applies simplification routines to tensors;

It is common to want to calculate, then display; or alter, then display. This can be done with the commands: grcalcd() and gralterd().

Thus, to calculate the covariant Ricci tensor for the predefined Kerr metric, we might use the command sequence

grcalc ( R(dn,dn) ):
grdisplay ( R(dn,dn) ):

Image Here

At this point, since no simplifications of the results have been applied, the components of Ricci are not obviously zero as we would expect. The following commands are needed:

gralter ( R(dn,dn), trig ): grdisplay ( R(dn,dn) ):

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The first command applies trigonometric simplification to the components of R(dn,dn). Upon using grdisplay(), we see that the simplification has indeed reduced the components to zero.

Note: For a more complicated metric, it may be better to carry out the computation in stages. First calculate the Christoffel symbols and apply simplifications, then the Ricci tensor, and finally, the Ricci scalar.

A number of alternate simplification routines exist within gralter(), including trigonometric, radical, expansion, and factorization techniques. The command grmap() can also be used to apply any Maple function to the components of a tensor.

The commands listed in this Section are described in more detail in Booklet C: Calculating tensor components.
The calculation of objects in a coordinate basis is described in Booklet E: Bases and tetrads.