Carminati–McLenaghan 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 ${\displaystyle C_{abcd}}$ and its right (or left) dual ${\displaystyle {{}^{\star }C}_{ijkl}=(1/2)\epsilon _{klmn}C_{ij}{}^{mn}}$, the Ricci tensor ${\displaystyle R_{ab}}$, and the trace-free Ricci tensor

${\displaystyle S_{ab}=R_{ab}-{\frac {1}{4}}\,R\,g_{ab}}$

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

The real CM scalars are:

1. ${\displaystyle R={R^{m}}_{m}}$ (the trace of the Ricci tensor)
2. ${\displaystyle R_{1}={\frac {1}{4}}\,{S^{a}}_{b}\,{S^{b}}_{a}}$
3. ${\displaystyle R_{2}=-{\frac {1}{8}}\,{S^{a}}_{b}\,{S^{b}}_{c}\,{S^{c}}_{a}}$
4. ${\displaystyle R_{3}={\frac {1}{16}}\,{S^{a}}_{b}\,{S^{b}}_{c}\,{S^{c}}_{d}\,{S^{d}}_{a}}$
5. ${\displaystyle M_{3}={\frac {1}{16}}\,S^{bc}\,S_{ef}\left(C_{abcd}\,C^{aefd}+{{}^{\star }C}_{abcd}\,{{}^{\star }C}^{aefd}\right)}$
6. ${\displaystyle M_{4}=-{\frac {1}{32}}\,S^{ag}\,S^{ef}\,{S^{c}}_{d}\,\left({C_{ac}}^{db}\,C_{befg}+{{{}^{\star }C}_{ac}}^{db}\,{{}^{\star }C}_{befg}\right)}$

The complex CM scalars are:

1. ${\displaystyle W_{1}={\frac {1}{8}}\,\left(C_{abcd}+i\,{{}^{\star }C}_{abcd}\right)\,C^{abcd}}$
2. ${\displaystyle W_{2}=-{\frac {1}{16}}\,\left({C_{ab}}^{cd}+i\,{{{}^{\star }C}_{ab}}^{cd}\right)\,{C_{cd}}^{ef}\,{C_{ef}}^{ab}}$
3. ${\displaystyle M_{1}={\frac {1}{8}}\,S^{ab}\,S^{cd}\,\left(C_{acdb}+i\,{{}^{\star }C}_{acdb}\right)}$
4. ${\displaystyle M_{2}={\frac {1}{16}}\,S^{bc}\,S_{ef}\,\left(C_{abcd}\,C^{aefd}-{{}^{\star }C}_{abcd}\,{{}^{\star }C}^{aefd}\right)+{\frac {1}{8}}\,i\,S^{bc}\,S_{ef}\,{{}^{\star }C}_{abcd}\,C^{aefd}}$
5. ${\displaystyle M_{5}={\frac {1}{32}}\,S^{cd}\,S^{ef}\,\left(C^{aghb}+i\,{{}^{\star }C}^{aghb}\right)\,\left(C_{acdb}\,C_{gefh}+{{}^{\star }C}_{acdb}\,{{}^{\star }C}_{gefh}\right)}$

The CM scalars have the following degrees:

1. ${\displaystyle R}$ is linear,
2. ${\displaystyle R_{1},\,W_{1}}$ are quadratic,
3. ${\displaystyle R_{2},\,W_{2},\,M_{1}}$ are cubic,
4. ${\displaystyle R_{3},\,M_{2},\,M_{3}}$ are quartic,
5. ${\displaystyle M_{4},\,M_{5}}$ 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.

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

${\displaystyle R,\,R_{1},\,R_{2},\,R_{3},\,\Re (W_{1}),\,\Re (M_{1}),\,\Re (M_{2})}$

${\displaystyle {\frac {1}{32}}\,S^{cd}\,S^{ef}\,C^{aghb}\,C_{acdb}\,C_{gefh}}$

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.

• 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

• Carminati J.; McLenaghan, R. G. (1991). "Algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space". J. Math. Phys. 32 (11): 3135–3140. Bibcode:1991JMP....32.3135C. doi:10.1063/1.529470.

• The GRTensor II website Archived 2002-09-14 at the Library of Congress Web Archives includes a manual with definitions and discussions of the CM scalars.
• Implementation in the Maxima computer algebra system