Principal Invariants
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
- the Kretschmann scalar
- the Chern-Pontryagin scalar
- the Euler scalar
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
(Because, there is no need to define a third principal invariant for the Weyl tensor.)
Read more about this topic: Curvature Invariant (general Relativity)
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