Grassmann Algebra

TheInteriorProduct.nb 26
6.7 The Induced Metric Tensor
� Calculating induced metric tensors
In the last chapter we saw how we could use the GrassmannAlgebra function Metric�[m] to
calculate the matrix of metric tensor elements induced on � by the metric tensor declared on � .
m 1
For example, we can declare a general metric in a 3-space:
�3; DeclareMetric���
���1,1, �1,2, �1,3�, ��1,2, �2,2, �2,3�, ��1,3, �2,3, �3,3��
and then ask for the metric induced on � 2 .
Metric��2�
2
����1,2 � �1,1 �2,2, ��1,2 �1,3 � �1,1 �2,3, ��1,3 �2,2 � �1,2 �2,3�,
2
���1,2 �1,3 � �1,1 �2,3, ��1,3 � �1,1 �3,3, ��1,3 �2,3 � �1,2 �3,3�,
2
���1,3 �2,2 � �1,2 �2,3, ��1,3 �2,3 � �1,2 �3,3, ��2,3 � �2,2 �3,3��
� Using scalar products to construct induced metric tensors
An alternative way of viewing and calculating the metric tensor on � is to construct the matrix
m
of inner products of the basis elements of � with themselves, reduce these inner products to
m
scalar products, and then substitute for each scalar product the corresponding metric tensor
element from �. In what follows we will show an example of how to reproduce the result in the
1
previous section.
First, we calculate the basis of � 2 .
�3; B� Basis��2�
�e1 � e2, e1 � e3, e2 � e3�
Second, we use the Mathematica function Outer to construct the matrix of all combinations of
interior products of these basis elements.
M1 � Outer�InteriorProduct, B, B�; MatrixForm�M1�
� e1 � e2 � e1 � e2 e1 � e2 � e1 � e3 e1 � e2 � e2 � e3
�������
��������
e1 � e3 � e1 � e2 e1 � e3 � e1 � e3 e1 � e3 � e2 � e3
� e2 � e3 � e1 � e2 e2 � e3 � e1 � e3 e2 � e3 � e2 � e3 �
Third, we use the GrassmannAlgebra function DevelopScalarProductMatrix to
convert each of these inner products into matrices of scalar products.
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