Carminati–McLenaghan invariants

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In general relativity, the Carminati–McLenaghan invariants or CM scalars are a set of 16 scalar curvature invariants for the Riemann tensor. This set is usually supplemented with at least two additional invariants.

Mathematical definition[edit]

The CM invariants consist of 6 real scalars plus 5 complex scalars, making a total of 16 invariants. They are defined in terms of the Weyl tensor and its right (or left) dual , the Ricci tensor , and the trace-free Ricci tensor

In the following, it may be helpful to note that if we regard as a matrix, then is the square of this matrix, so the trace of the square is , and so forth.

The real CM scalars are:

  1. (the trace of the Ricci tensor)

The complex CM scalars are:

The CM scalars have the following degrees:

  1. is linear,
  2. are quadratic,
  3. are cubic,
  4. are quartic,
  5. are quintic.

They can all be expressed directly in terms of the Ricci spinors and Weyl spinors, using Newman–Penrose formalism; see the link below.

Complete sets of invariants[edit]

In the case of spherically symmetric spacetimes or planar symmetric spacetimes, it is known that

comprise a complete set of invariants for the Riemann tensor. In the case of vacuum solutions, electrovacuum solutions and perfect fluid solutions, the CM scalars comprise a complete set. Additional invariants may be required for more general spacetimes; determining the exact number (and possible syzygies among the various invariants) is an open problem.

See also[edit]

References[edit]

External links[edit]