Relation To Other Invariants
Another possible invariant (which has been employed for example in writing the gravitational term of the Lagrangian for some higher-order gravity theories) is
where is the Weyl tensor, the conformal curvature tensor which is also the completely traceless part of the Riemann tensor. In dimensions this is related to the Kretschmann invariant by
where is the Ricci curvature tensor and is the Ricci scalar curvature (obtained by taking successive traces of the Riemann tensor).
The Kretschmann scalar and the Chern-Pontryagin scalar
where is the left dual of the Riemann tensor, are mathematically analogous (to some extent, physically analogous) to the familiar invariants of the electromagnetic field tensor
Read more about this topic: Kretschmann Scalar
Famous quotes containing the words relation to and/or relation:
“Much poetry seems to be aware of its situation in time and of its relation to the metronome, the clock, and the calendar. ... The season or month is there to be felt; the day is there to be seized. Poems beginning “When” are much more numerous than those beginning “Where” of “If.” As the meter is running, the recurrent message tapped out by the passing of measured time is mortality.”
—William Harmon (b. 1938)
“There is the falsely mystical view of art that assumes a kind of supernatural inspiration, a possession by universal forces unrelated to questions of power and privilege or the artist’s relation to bread and blood. In this view, the channel of art can only become clogged and misdirected by the artist’s concern with merely temporary and local disturbances. The song is higher than the struggle.”
—Adrienne Rich (b. 1929)