Curvature invariant (general relativity)

As one might expect from the Ricci decomposition of the Riemann tensor into the Weyl tensor plus a sum of fourth-rank tensors constructed from the second rank Ricci tensor and from the Ricci scalar, these two sets of invariants are related (in d=4):

${\displaystyle K_{1}=I_{1}+2\,R_{ab}\,R^{ab}-{\tfrac {1}{3}}\,R^{2}}$

${\displaystyle K_{3}=-I_{1}+2\,R_{ab}\,R^{ab}-{\tfrac {2}{3}}\,R^{2}}$

In four dimensions, the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field ${\displaystyle {\vec {X}}}$, not necessarily geodesic or hypersurface orthogonal, consists of three pieces

1. the electrogravitic tensor ${\displaystyle E[{\vec {X}}]_{ab}=R_{ambn}\,X^{m}\,X^{n}}$
2. the magnetogravitic tensor ${\displaystyle B[{\vec {X}}]_{ab}={{}^{\star }R}_{ambn}\,X^{m}\,X^{n}}$
3. the topogravitic tensor ${\displaystyle L[{\vec {X}}]_{ab}={{}^{\star }R^{\star }}_{ambn}\,X^{m}\,X^{n}}$

Because these are all transverse (i.e. projected to the spatial hyperplane elements orthogonal to our timelike unit vector field), they can be represented as linear operators on three-dimensional vectors, or as three by three real matrices. They are respectively symmetric, traceless, and symmetric (6,8,6 linearly independent components, for a total of 20). If we write these operators as E, B, L respectively, the principal invariants of the Riemann tensor are obtained as follows:

• ${\displaystyle K_{1}/4}$ is the trace of E² + L² - 2 B B^T,
• ${\displaystyle -K_{2}/8}$ is the trace of B ( E - L ),
• ${\displaystyle K_{3}/8}$ is the trace of E L - B².

In terms of the Weyl scalars in the Newman–Penrose formalism, the principal invariants of the Weyl tensor may be obtained by taking the real and imaginary parts of the expression

${\displaystyle I_{1}-i\,I_{2}=16\,\left(3\Psi _{2}^{2}+\Psi _{0}\,\Psi _{4}-4\,\Psi _{1}\Psi _{3}\right)}$

(But note the minus sign!)

The principal quadratic invariant of the Ricci tensor, ${\displaystyle R_{ab}\,R^{ab}}$, may be obtained as a more complicated expression involving the Ricci scalars (see the paper by Cherubini et al. cited below).