Carminati–McLenaghan Invariants
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In
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
, the Carminati–McLenaghan invariants or CM scalars are a set of 16 scalar
curvature invariant In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature. These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors form ...
s 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 In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal forc ...
C_ and its right (or left) dual _=(1/2)\epsilon_C_^, the
Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
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 In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
) #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
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...
s: #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 In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the thre ...
, using
Newman–Penrose formalism The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Sp ...
; see the link below.


Complete sets of invariants

In the case of
spherically symmetric spacetime In physics, spherically symmetric spacetimes are commonly used to obtain analytic and numerical solutions to Einstein's field equations in the presence of radially moving matter or energy. Because spherically symmetric spacetimes are by definition ...
s 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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...
of invariants for the Riemann tensor. In the case of
vacuum solution In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or n ...
s,
electrovacuum solution In general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass–energy present is the field energy of an electromagnetic field, which must satisfy the ( ...
s and perfect
fluid solution In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid. In astrophysics, fluid solutions are often ...
s, the CM scalars comprise a complete set. Additional invariants may be required for more general spacetimes; determining the exact number (and possible
syzygies Syzygy (from Greek Συζυγία "conjunction, yoked together") may refer to: Science * Syzygy (astronomy), a collinear configuration of three celestial bodies * Syzygy (mathematics), linear relation between generators of a module * Syzygy, ...
among the various invariants) is an open problem.


See also

*
Curvature invariant In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature. These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors form ...
, 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