Spinorial equivalent of tensors. V μ is a vector, S μν is a trace-less symmetric tensor, B μν is a 2-vector, T μνρ is a 3-vector, and C μνρσ is a tensor with the symmetries of a Weyl tensor.
Some algebraically special types for the Weyl tensor. The labels come from: Repeated(R), Simple(S), and Non-Simple(NS). Here , and . The types , , and can obviously be defined, as well as many other special types.
Definition of the CMPP types. The second row says which components of the Weyl tensor should vanish according to the boost weight, b. For example, when the type is I all components of boost weight two must vanish in some null frame.
The boost weight of the various components of
The spinorial representation of a bivector basis. The last four lines the inverse relation of the 3 short lines at the center.
This table displays the relation between Weyl tensor's components in a null frame and its spinorial equivalent. The missing components of the Weyl tensor can be obtained by making the changes 1 ↔ 4, 2 ↔ 5, and 3 ↔ 6 on the vectorial indices while swapping the upper and the lower indices of Ψ. The first two rows contain the components of the Weyl tensor with boost weight b = 2, the next ten rows present the components with b = 1, the other rows have the components with zero boost weight.
Article metrics loading...
Full text loading...