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.
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 the trace-free Ricci tensor In the following, it may be helpful to note that if we regard
The real CM scalars are: The complex CM scalars are: The CM scalars have the following degrees: They can all be expressed directly in terms of the Ricci spinors and Weyl spinors, using Newman–Penrose formalism; see the link below.
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.
Additional invariants may be required for more general spacetimes; determining the exact number (and possible syzygies among the various invariants) is an open problem.