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 left (or right) dual $$}^star C}_{acdb}, 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
#$R\; =\; \{R^m\}\_m$ (the trace of the Ricci tensor)
#$R\_1\; =\; frac\{1\}\{4\}\; ,\; \{S^a\}\_b\; ,\; \{S^b\}\_a$
#$R\_2\; =\; -frac\{1\}\{8\}\; ,\; \{S^a\}\_b\; ,\; \{S^b\}\_c\; ,\; \{S^c\}\_a$
#$R\_3\; =\; frac\{1\}\{16\}\; ,\; \{S^a\}\_b\; ,\; \{S^b\}\_c\; ,\; \{S^c\}\_d\; ,\; \{S^d\}\_a$
#$M\_3\; =\; frac\{1\}\{16\}\; ,\; S^\{bc\}\; ,\; S\_\{ef\}\; left(\; C\_\{abcd\}\; ,\; C^\{aefd\}\; +$}^star C}_{abcd} , }^star C}^{aefd} ight)
#$M\_4\; =\; -frac\{1\}\{32\}\; ,\; S^\{ag\}\; ,\; S^\{ef\}\; ,\; \{S^c\}\_d\; ,\; left(\; \{C\_\{ac$^{db} , C_{befg} + {}^star C}_{ac^{db} , }^star C}_{befg} ight)The complex CM scalars are
#$W\_1\; =\; frac\{1\}\{8\}\; ,\; left(\; C\_\{abcd\}\; +\; i\; ,$}^star C}_{abcd} ight) , C^{abcd}
#$W\_2\; =\; -frac\{1\}\{16\}\; ,\; left(\; \{C\_\{ab$^{cd} + i , {}^star C}_{ab^{cd} ight) , {C_{cd^{ef} , {C_{ef^{ab}
#$M\_1\; =\; frac\{1\}\{8\}\; ,\; S^\{ab\}\; ,\; S^\{cd\}\; ,\; left(\; C\_\{acdb\}\; +\; i\; ,$}^star C}_{acdb} ight)
#$M\_2\; =\; frac\{1\}\{16\}\; ,\; S^\{bc\}\; ,\; S\_\{ef\}\; ,\; left(\; C\_\{abcd\}\; ,\; C^\{aefd\}\; -$}^star C}_{acdb} , }^star C}^{aefd} ight) + frac{1}{8} , i , S^{bc} , S_{ef} , }^star C}_{abcd} , C^{aefd}
#$M\_5\; =\; frac\{1\}\{32\}\; ,\; S^\{cd\}\; ,\; S^\{ef\}\; ,\; left(\; C^\{aghb\}\; +\; i\; ,$}^star C}^{aghb} ight) , left( C_{acdb} , C_{gefh} + }^star C}_{acdb} , }^star C}_{gefh} ight)

The CM scalars have the following degrees:
#$R$ is linear,
#$R\_1,\; ,\; W\_1$ are quadratic,
#$R\_2,\; ,\; W\_2,\; ,\; M\_1$ are cubic,
#$R\_3,\; ,\; M\_2,\; ,\; M\_3$ are quartic,
#$M\_4,\; ,\; M\_5$ 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 :$R,\; ,\; R\_1,\; ,\; R\_2,\; ,\; R\_3,\; ,\; Re\; (W\_1),\; ,\; Re\; (M\_1),\; ,\; Re\; (M\_2)$:$frac\{1\}\{32\}\; ,\; S^\{cd\}\; ,\; S^\{ef\}\; ,\; C^\{aghb\}\; ,\; C\_\{acdb\}\; ,\; C\_\{gefh\}$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.

ee 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

*cite journal | author=Carminati, J.; and McLenaghan, R. G. | title=Algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space | journal=J. Math. Phys. | year=1991 | volume=32 | pages=3135–3140 | doi=10.1063/1.529470

External links

*The [http://grtensor.phy.queensu.ca/ GRTensor II website] includes a manual with definitions and discussions of the CM scalars.