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| In [[general relativity]], the '''Carminati–McLenaghan invariants''' or '''CM scalars''' are a set of 16 scalar [[curvature invariant]]s for the [[Riemann tensor]]. This set is usually supplemented with at least two additional invariants.
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| ==Mathematical definition==
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| 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]] <math>C_{abcd}</math> and its left (or right) dual <math>{{}^\star C}_{acdb}</math>, the [[Ricci tensor]] <math>R_{ab}</math>, and the ''trace-free Ricci tensor''
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| :<math> S_{ab} = R_{ab} - \frac{1}{4} \, R \, g_{ab}</math>
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| In the following, it may be helpful to note that if we regard <math>{S^a}_b</math> as a matrix, then <math>{S^a}_m \, {S^m}_b</math> is the ''square'' of this matrix, so the ''trace'' of the square is <math>{S^a}_b \, {S^b}_a</math>, and so forth.
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| The real CM scalars are
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| #<math>R = {R^m}_m</math> (the trace of the [[Ricci tensor]])
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| #<math>R_1 = \frac{1}{4} \, {S^a}_b \, {S^b}_a</math>
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| #<math>R_2 = -\frac{1}{8} \, {S^a}_b \, {S^b}_c \, {S^c}_a</math>
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| #<math>R_3 = \frac{1}{16} \, {S^a}_b \, {S^b}_c \, {S^c}_d \, {S^d}_a</math>
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| #<math>M_3 = \frac{1}{16} \, S^{bc} \, S_{ef} \left( C_{abcd} \, C^{aefd} + {{}^\star C}_{abcd} \, {{}^\star C}^{aefd} \right)</math>
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| #<math>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)</math>
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| The complex CM scalars are
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| #<math>W_1 = \frac{1}{8} \, \left( C_{abcd} + i \, {{}^\star C}_{abcd} \right) \, C^{abcd}</math>
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| #<math>W_2 = -\frac{1}{16} \, \left( {C_{ab}}^{cd} + i \, {{{}^\star C}_{ab}}^{cd} \right) \, {C_{cd}}^{ef} \, {C_{ef}}^{ab}</math>
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| #<math>M_1 = \frac{1}{8} \, S^{ab} \, S^{cd} \, \left( C_{acdb} + i \, {{}^\star C}_{acdb} \right)</math>
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| #<math>M_2 = \frac{1}{16} \, S^{bc} \, S_{ef} \, \left( C_{abcd} \, C^{aefd} - {{}^\star C}_{acdb} \, {{}^\star C}^{aefd} \right) + \frac{1}{8} \, i \, S^{bc} \, S_{ef} \, {{}^\star C}_{abcd} \, C^{aefd}</math>
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| #<math>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)</math>
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| The CM scalars have the following [[Degree of a polynomial|degree]]s:
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| #<math>R</math> is linear,
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| #<math>R_1, \, W_1</math> are quadratic,
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| #<math>R_2, \, W_2, \, M_1</math> are cubic,
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| #<math>R_3, \, M_2, \, M_3</math> are quartic,
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| #<math>M_4, \, M_5</math> are quintic.
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| They can all be expressed directly in terms of the [[Ricci spinors]] and [[Weyl spinors]], using [[Newman–Penrose formalism]]; see the link below.
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| ==Complete sets of invariants==
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| In the case of [[spherically symmetric spacetime]]s or planar symmetric spacetimes, it is known that
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| :<math>R, \, R_1, \, R_2, \, R_3, \, \Re (W_1), \, \Re (M_1), \, \Re (M_2)</math> | |
| :<math>\frac{1}{32} \, S^{cd} \, S^{ef} \, C^{aghb} \, C_{acdb} \, C_{gefh}</math>
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| comprise a [[complete set]] of invariants for the Riemann tensor. In the case of [[vacuum solution (general relativity)|vacuum solution]]s, [[electrovacuum solution]]s and perfect [[fluid solution]]s, the CM scalars comprise a complete set. Additional invariants may be required for more general spacetimes; determining the exact number (and possible [[Syzygy (mathematics)|syzygies]] among the various invariants) is an open problem.
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| ==See also==
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| *[[curvature invariant]], for more about curvature invariants in (semi)-Riemannian geometry in general
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| *[[curvature invariant (general relativity)]], for other curvature invariants which are useful in general relativity
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| ==References==
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| *{{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|bibcode = 1991JMP....32.3135C }}
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| ==External links==
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| *The [http://grtensor.phy.queensu.ca/ GRTensor II website] includes a manual with definitions and discussions of the CM scalars.
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| {{DEFAULTSORT:Carminati-McLenaghan invariants}}
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| [[Category:Tensors in general relativity]]
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