Physics:Carminati–McLenaghan invariants

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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 [math]\displaystyle{ C_{abcd} }[/math] and its right (or left) dual [math]\displaystyle{ {{}^\star C}_{ijkl}=(1/2)\epsilon_{klmn}C_{ij}{}^{mn} }[/math], the Ricci tensor [math]\displaystyle{ R_{ab} }[/math], and the trace-free Ricci tensor

[math]\displaystyle{ S_{ab} = R_{ab} - \frac{1}{4} \, R \, g_{ab} }[/math]

In the following, it may be helpful to note that if we regard [math]\displaystyle{ {S^a}_b }[/math] as a matrix, then [math]\displaystyle{ {S^a}_m \, {S^m}_b }[/math] is the square of this matrix, so the trace of the square is [math]\displaystyle{ {S^a}_b \, {S^b}_a }[/math], and so forth.

The real CM scalars are:

  1. [math]\displaystyle{ R = {R^m}_m }[/math] (the trace of the Ricci tensor)
  2. [math]\displaystyle{ R_1 = \frac{1}{4} \, {S^a}_b \, {S^b}_a }[/math]
  3. [math]\displaystyle{ R_2 = -\frac{1}{8} \, {S^a}_b \, {S^b}_c \, {S^c}_a }[/math]
  4. [math]\displaystyle{ R_3 = \frac{1}{16} \, {S^a}_b \, {S^b}_c \, {S^c}_d \, {S^d}_a }[/math]
  5. [math]\displaystyle{ M_3 = \frac{1}{16} \, S^{bc} \, S_{ef} \left( C_{abcd} \, C^{aefd} + {{}^\star C}_{abcd} \, {{}^\star C}^{aefd} \right) }[/math]
  6. [math]\displaystyle{ 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]

The complex CM scalars are:

  1. [math]\displaystyle{ W_1 = \frac{1}{8} \, \left( C_{abcd} + i \, {{}^\star C}_{abcd} \right) \, C^{abcd} }[/math]
  2. [math]\displaystyle{ W_2 = -\frac{1}{16} \, \left( {C_{ab}}^{cd} + i \, {{{}^\star C}_{ab}}^{cd} \right) \, {C_{cd}}^{ef} \, {C_{ef}}^{ab} }[/math]
  3. [math]\displaystyle{ M_1 = \frac{1}{8} \, S^{ab} \, S^{cd} \, \left( C_{acdb} + i \, {{}^\star C}_{acdb} \right) }[/math]
  4. [math]\displaystyle{ M_2 = \frac{1}{16} \, S^{bc} \, S_{ef} \, \left( C_{abcd} \, C^{aefd} - {{}^\star C}_{abcd} \, {{}^\star C}^{aefd} \right) + \frac{1}{8} \, i \, S^{bc} \, S_{ef} \, {{}^\star C}_{abcd} \, C^{aefd} }[/math]
  5. [math]\displaystyle{ 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]

The CM scalars have the following degrees:

  1. [math]\displaystyle{ R }[/math] is linear,
  2. [math]\displaystyle{ R_1, \, W_1 }[/math] are quadratic,
  3. [math]\displaystyle{ R_2, \, W_2, \, M_1 }[/math] are cubic,
  4. [math]\displaystyle{ R_3, \, M_2, \, M_3 }[/math] are quartic,
  5. [math]\displaystyle{ M_4, \, M_5 }[/math] 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

[math]\displaystyle{ R, \, R_1, \, R_2, \, R_3, \, \Re (W_1), \, \Re (M_1), \, \Re (M_2) }[/math]
[math]\displaystyle{ \frac{1}{32} \, S^{cd} \, S^{ef} \, C^{aghb} \, C_{acdb} \, C_{gefh} }[/math]

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

References

External links



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