Carminati–McLenaghan invariants
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
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:
- (the trace of the Ricci tensor)
The complex CM scalars are:
The CM scalars have the following degrees:
- is linear,
- are quadratic,
- are cubic,
- are quartic,
- 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
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
- curvature invariant, for more about curvature invariants in (semi)-Riemannian geometry in general
- curvature invariant (general relativity), for other curvature invariants which are useful in general relativity
References
- Carminati J.; McLenaghan, R. G. (1991). "Algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space". J. Math. Phys. 32: 3135–3140. Bibcode:1991JMP....32.3135C. doi:10.1063/1.529470.
External links
- The GRTensor II website includes a manual with definitions and discussions of the CM scalars.
- Implementation in the Maxima computer algebra system