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Carminati–McLenaghan invariants

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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.

Contents

Mathematical definition

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 C a b c d and its right (or left) dual ⋆ C i j k l = ( 1 / 2 ) ϵ k l m n C i j m n , the Ricci tensor R a b , and the trace-free Ricci tensor

S a b = R a b − 1 4 R g a b

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

The real CM scalars are:

  1. R = R m m (the trace of the Ricci tensor)
  2. R 1 = 1 4 S a b S b a
  3. R 2 = − 1 8 S a b S b c S c a
  4. R 3 = 1 16 S a b S b c S c d S d a
  5. M 3 = 1 16 S b c S e f ( C a b c d C a e f d + ⋆ C a b c d ⋆ C a e f d )
  6. M 4 = − 1 32 S a g S e f S c d ( C a c d b C b e f g + ⋆ C a c d b ⋆ C b e f g )

The complex CM scalars are:

  1. W 1 = 1 8 ( C a b c d + i ⋆ C a b c d ) C a b c d
  2. W 2 = − 1 16 ( C a b c d + i ⋆ C a b c d ) C c d e f C e f a b
  3. M 1 = 1 8 S a b S c d ( C a c d b + i ⋆ C a c d b )
  4. M 2 = 1 16 S b c S e f ( C a b c d C a e f d − ⋆ C a c d b ⋆ C a e f d ) + 1 8 i S b c S e f ⋆ C a b c d C a e f d
  5. M 5 = 1 32 S c d S e f ( C a g h b + i ⋆ C a g h b ) ( C a c d b C g e f h + ⋆ C a c d b ⋆ C g e f h )

The CM scalars have the following degrees:

  1. R is linear,
  2. R 1 , W 1 are quadratic,
  3. R 2 , W 2 , M 1 are cubic,
  4. R 3 , M 2 , M 3 are quartic,
  5. M 4 , M 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

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

R , R 1 , R 2 , R 3 , ℜ ( W 1 ) , ℜ ( M 1 ) , ℜ ( M 2 ) 1 32 S c d S e f C a g h b C a c d b C g e f h

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.

References

Carminati–McLenaghan invariants Wikipedia
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