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 $C_$ and its right (or left) dual $_=(1/2)\epsilon_C_^$, the Ricci tensor $R_$, and the ''trace-free Ricci tensor''
:$S_ = R_ - \frac \, R \, g_$
In the following, it may be helpful to note that if we regard $_b$ as a matrix, then $_m \, _b$ is the ''square'' of this matrix, so the ''trace'' of the square is $_b \, _a$, and so forth.

The real CM scalars are:
#$R = _m$ (the trace of the Ricci tensor)
#$R_1 = \frac \, _b \, _a$
#$R_2 = -\frac \, _b \, _c \, _a$
#$R_3 = \frac \, _b \, _c \, _d \, _a$
#$M_3 = \frac \, S^ \, S_ \left( C_ \, C^ + _ \, ^ \right)$
#$M_4 = -\frac \, S^ \, S^ \, _d \, \left( ^ \, C_ + ^ \, _ \right)$
The complex CM scalars are:
#$W_1 = \frac \, \left( C_ + i \, _ \right) \, C^$
#$W_2 = -\frac \, \left( ^ + i \, ^ \right) \, ^ \, ^$
#$M_1 = \frac \, S^ \, S^ \, \left( C_ + i \, _ \right)$
#$M_2 = \frac \, S^ \, S_ \, \left( C_ \, C^ - _ \, ^ \right) + \frac \, i \, S^ \, S_ \, _ \, C^$
#$M_5 = \frac \, S^ \, S^ \, \left( C^ + i \, ^ \right) \, \left( C_ \, C_ + _ \, _ \right)$

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
:$R, \, R_1, \, R_2, \, R_3, \, \Re (W_1), \, \Re (M_1), \, \Re (M_2)$
:$\frac \, S^ \, S^ \, C^ \, C_ \, C_$
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

*

External links

*Th
GRTensor II website
includes a manual with definitions and discussions of the CM scalars.
Implementation in the Maxima computer algebra system
{{DEFAULTSORT:Carminati-McLenaghan invariants
Tensors in general relativity