Scalar Curvature - Definition

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

S = g^{ab} (\Gamma^c_{ab,c} - \Gamma^c_{ac,b} + \Gamma^d_{ab}\Gamma^c_{cd} - \Gamma^d_{ac} \Gamma^c_{bd}) = 2g^{ab} (\Gamma^c_{a} + \Gamma^d_{ad})

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

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