DifferentialGeometry/Tensor/RicciSpinor - Maple Help
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Tensor[RicciSpinor] - compute the spinor form of the trace-free Ricci tensor

Calling Sequences

     RicciSpinor(, R)

Parameters

      - a solder form

   R   - (optional) the Ricci tensor for the metric determined by the solder form

 

Description

Examples

See Also

Description

• 

Let  be a metric tensor. The trace-free Ricci tensor for  is defined by  , where  is the Ricci tensor and  the Ricci scalar of .

• 

The command RicciSpinor(s) first computes the metric tensor  defined by the solder form s. The trace-free Ricci tensor  for  is then computed and converted, using the solder form  to a rank 4 covariant spinor with index type . (See convert/DGspinor.) Finally, a scalar factor of  is introduced according to standard conventions. See Stewart, page 85.

• 

If the Ricci tensor  for the metric  has been previously computed, then the Ricci spinor will be computed more quickly using the second calling sequence RicciSpinor(, R).

• 

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form RicciSpinor(..) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-RicciSpinor.

Examples

 

Example 1.

First create a vector bundle  with base coordinates and fiber coordinates .

M > 

(2.1)

 

Define a metric  on the base. For this example we use the Godel metric. (See (12.26) in Exact Solutions to Einstein's Field Equations.) Note that we have adjusted the metric to conform to the signature conventions  used by the spinor formalism in the DifferentialGeometry package. See SpacetimeConventions.

M > 

(2.2)

 

Use DGGramSchmidt to calculate an orthonormal frame  for the metric .

M > 

(2.3)

 

Use SolderForm to compute the solder form  from the frame .

M > 

(2.4)

 

Calculate the Ricci spinor from the solder form .

M > 

(2.5)

 

Example 2.

In this example we first calculate the Ricci tensor of the metric  and then use the second calling sequence for RicciSpinor.

M > 

(2.6)
M > 

(2.7)

 

Example 3.

We can check the result of Example 1 by direct computation, starting from the solder form . First use the command SpinorInnerProduct to calculate the metric  from . (Note that  coincides with the original metric .)

M > 

(2.8)

 

Second, calculate the curvature tensor , the Ricci tensor , and the Ricci scalar .

M > 

(2.9)
M > 

(2.10)
M > 

(2.11)

 

Calculate the trace-free Ricci tensor .

M > 

(2.12)

 

Convert  to a spinor .

M > 

(2.13)

 

Rearrange the indices of  and scale by  to arrive at the Ricci spinor  (or ).

M > 

(2.14)

See Also

DifferentialGeometry, Tensor, Convert, CurvatureTensor, Physics[Riemann],  RicciTensor, Physics[Ricci],  SolderForm, SpinorInnerProduct, WeylSpinor, Physics[Weyl]


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