Curvature invariant (general relativity)

The principal invariants of the Riemann and Weyl tensors are certain quadratic polynomial invariants (i.e., sums of squares of components).

The principal invariants of the Riemann tensor of a four-dimensional Lorentzian manifold are

1. the Kretschmann scalar ${\displaystyle K_{1}=R_{abcd}\,R^{abcd}}$
2. the Chern–Pontryagin scalar ${\displaystyle K_{2}={{\star }R}_{abcd}\,R^{abcd}}$
3. the Euler scalar ${\displaystyle K_{3}={{\star }R^{\star }}_{abcd}\,R^{abcd}}$

These are quadratic polynomial invariants (sums of squares of components). (Some authors define the Chern–Pontryagin scalar using the right dual instead of the left dual.)

The first of these was introduced by Erich Kretschmann. The second two names are somewhat anachronistic, but since the integrals of the last two are related to the instanton number and Euler characteristic respectively, they have some justification.

The principal invariants of the Weyl tensor are

1. ${\displaystyle I_{1}=C_{abcd}\,C^{abcd}}$
2. ${\displaystyle I_{2}={{\star }C}_{abcd}\,C^{abcd}}$

(Because ${\displaystyle {{\star }C^{\star }}_{abcd}=-C_{abcd}}$, there is not a third independent third principal invariant for the Weyl tensor.)

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 ${\displaystyle {\vec {X}}}$ produces three components

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 ${\displaystyle L[{\vec {X}}]_{ab}={{\star }R^{\star }}_{ambn}\,X^{m}\,X^{n}}$

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

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

(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).