Carminati–McLenaghan invariants explained

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

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 tensorS _{ab} = R _{ab} - (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} = (1)/(4) {S ^{a}} _{b} {S ^{b}} _{a} R _{2} =- (1)/(8) {S ^{a}} _{b} {S ^{b}} _{c} {S ^{c}} _{a} R _{3} = (1)/(16) {S ^{a}} _{b} {S ^{b}} _{c} {S ^{c}} _{d} {S ^{d}} _{a} M _{3} = (1)/(16) S ^{bc} S _{ef} \left( C _{abcd} C ^{aefd} + {{} ^{\star} C} _{abcd} {{} ^{\star} C} ^{aefd} \right) M _{4} =- (1)/(32) S ^{ag} S ^{ef} {S ^{c}} _{d} \left( {C _{ac} }^ \, C_ + ^ \, _ \right)The complex CM scalars are:W _{1} = (1)/(8) \left( C _{abcd} + i {{} ^{\star} C} _{abcd} \right) C ^{abcd} W _{2} =- (1)/(16) \left( {C _{ab} }^ + i \, ^ \right) \, ^ \, ^M _{1} = (1)/(8) S ^{ab} S ^{cd} \left( C _{acdb} + i {{} ^{\star} C} _{acdb} \right) M _{2} = (1)/(16) S ^{bc} S _{ef} \left( C _{abcd} C ^{aefd} - {{} ^{\star} C} _{abcd} {{} ^{\star} C} ^{aefd} \right) + (1)/(8) i S ^{bc} S _{ef} {{} ^{\star} C} _{abcd} C ^{aefd} M _{5} = (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:

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)} (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. . Raymond McLenaghan. Algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space . J. Math. Phys. . 1991 . 32 . 11 . 3135–3140 . 10.1063/1.529470. 1991JMP....32.3135C .

External links

• The GRTensor II website includes a manual with definitions and discussions of the CM scalars.
• Implementation in the Maxima computer algebra system