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 ${{}^{\star }C}_{ijkl}=(1/2){ϵ}_{klmn}{C}_{ij}{}^{mn}$, 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:

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. $M_3=\frac{1}{16}{S}^{bc}{S}_{ef}(C_{abcd}{C}^{aefd}+{{}^{\star }C}_{abcd}{{}^{\star }C}^{aefd})$
6. $M_4=-\frac{1}{32}{S}^{ag}{S}^{ef}{S^c}_d({C_{ac}}^{db}{C}_{befg}+{{{}^{\star }C}_{ac}}^{db}{{}^{\star }C}_{befg})$

The complex CM scalars are:

1. $W_1=\frac{1}{8}(C_{abcd}+i{{}^{\star }C}_{abcd})C^{abcd}$
2. $W_2=-\frac{1}{16}({C_{ab}}^{cd}+i{{{}^{\star }C}_{ab}}^{cd}){C_{cd}}^{ef}{C_{ef}}^{ab}$
3. $M_1=\frac{1}{8}{S}^{ab}{S}^{cd}(C_{acdb}+i{{}^{\star }C}_{acdb})$
4. $M_2=\frac{1}{16}{S}^{bc}{S}_{ef}(C_{abcd}{C}^{aefd}-{{}^{\star }C}_{abcd}{{}^{\star }C}^{aefd})+\frac{1}{8}i{S}^{bc}{S}_{ef}{{}^{\star }C}_{abcd}{C}^{aefd}$
5. $M_5=\frac{1}{32}{S}^{cd}{S}^{ef}(C^{aghb}+i{{}^{\star }C}^{aghb})(C_{acdb}{C}_{gefh}+{{}^{\star }C}_{acdb}{{}^{\star }C}_{gefh})$

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

$R,R_1,R_2,R_3,\mathrm{\Re }(W_1),\mathrm{\Re }(M_1),\mathrm{\Re }(M_2)$

$\frac{1}{32}{S}^{cd}{S}^{ef}{C}^{aghb}{C}_{acdb}{C}_{gefh}$

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.

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 (11): 3135–3140. Bibcode:1991JMP....32.3135C. doi:10.1063/1.529470.

External links

• 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