Definition
The scalar curvature is usually denoted by S (other notations are Sc, R). It is defined as the trace of the Ricci curvature tensor with respect to the metric:
The trace depends on the metric since the Ricci tensor is a (0,2)-valent tensor; one must first raise an index to obtain a (1,1)-valent tensor in order to take the trace. In terms of local coordinates one can write
where Rij are the components of the Ricci tensor in the coordinate basis:
Given a coordinate system and a metric tensor, scalar curvature can be expressed as follows
where are the Christoffel symbols of the metric.
Unlike the Riemann curvature tensor or the Ricci tensor, which both can be naturally defined for any affine connection, the scalar curvature requires a metric of some kind. The metric can be pseudo-Riemannian instead of Riemannian. Indeed, such a generalization is vital to relativity theory. More generally, the Ricci tensor can be defined in broader class of metric geometries (by means of the direct geometric interpretation, below) that includes Finsler geometry.
Read more about this topic: Scalar Curvature
Famous quotes containing the word definition:
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)