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 $$C_{abcd}$$ and its right (or left) dual $${{}^\star C}_{ijkl}=(1/2)\epsilon_{klmn}C_{ij}{}^{mn}$$, the Ricci tensor $$R_{ab}$$, and the trace-free Ricci tensor
 * $$ S_{ab} = R_{ab} - \frac{1}{4} \, R \, g_{ab}$$

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: The complex CM scalars are:
 * 1) $$R = {R^m}_m$$ (the trace of the Ricci tensor)
 * 2) $$R_1 = \frac{1}{4} \, {S^a}_b \, {S^b}_a$$
 * 3) $$R_2 = -\frac{1}{8} \, {S^a}_b \, {S^b}_c \, {S^c}_a$$
 * 4) $$R_3 = \frac{1}{16} \, {S^a}_b \, {S^b}_c \, {S^c}_d \, {S^d}_a$$
 * 5) $$M_3 = \frac{1}{16} \, S^{bc} \, S_{ef} \left( C_{abcd} \, C^{aefd} + {{}^\star C}_{abcd} \, {{}^\star C}^{aefd} \right)$$
 * 6) $$M_4 = -\frac{1}{32} \, S^{ag} \, S^{ef} \, {S^c}_d \, \left( {C_{ac}}^{db} \, C_{befg} + {{{}^\star C}_{ac}}^{db} \, {{}^\star C}_{befg} \right)$$
 * 1) $$W_1 = \frac{1}{8} \, \left( C_{abcd} + i \, {{}^\star C}_{abcd} \right) \, C^{abcd}$$
 * 2) $$W_2 = -\frac{1}{16} \, \left( {C_{ab}}^{cd} + i \, {{{}^\star C}_{ab}}^{cd} \right) \, {C_{cd}}^{ef} \, {C_{ef}}^{ab}$$
 * 3) $$M_1 = \frac{1}{8} \, S^{ab} \, S^{cd} \, \left( C_{acdb} + i \, {{}^\star C}_{acdb} \right)$$
 * 4) $$M_2 = \frac{1}{16} \, S^{bc} \, S_{ef} \, \left( C_{abcd} \, C^{aefd} - {{}^\star C}_{abcd} \, {{}^\star C}^{aefd} \right) + \frac{1}{8} \, i \, S^{bc} \, S_{ef} \, {{}^\star C}_{abcd} \, C^{aefd}$$
 * 5) $$M_5 = \frac{1}{32} \, S^{cd} \, S^{ef} \, \left( C^{aghb} + i \, {{}^\star C}^{aghb} \right) \, \left( C_{acdb} \, C_{gefh} + {{}^\star C}_{acdb} \, {{}^\star C}_{gefh} \right)$$

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.
 * 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.

Complete sets of invariants
In the case of spherically symmetric spacetimes or planar symmetric spacetimes, it is known that
 * $$R, \, R_1, \, R_2, \, R_3, \, \Re (W_1), \, \Re (M_1), \, \Re (M_2)$$
 * $$\frac{1}{32} \, S^{cd} \, S^{ef} \, C^{aghb} \, C_{acdb} \, C_{gefh}$$

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.