Scalar curvature

The scalar curvature R is one of the invariants of a Riemannian manifold obtained by the convolution of the Ricci tensor with the metric tensor :

R\,=g^{\mu \nu }\,R_{\mu \nu }

{\ displaystyle R \, = g ^ {\ mu \ nu} \, R _ {\ mu \ nu}}

R \, = g ^ {\ mu \ nu} \, R _ {\ mu \ nu}

Thus, scalar curvature is a trace of the Ricci tensor.

Content

1 Equations of the gravitational field
2 Properties
3 See also
4 notes

Gravitational Field Equations

In the general theory of relativity , the action functional for the gravitational field is expressed through the integral over the four-dimensional volume of the scalar curvature:

$S_{G}=\varkappa \int \limits _{M}R{\sqrt {-g}}d\Omega$ ${\ displaystyle S_ {G} = \ varkappa \ int \ limits _ {M} R {\ sqrt {-g}} d \ Omega}$

Therefore, the equations of the gravitational field can be obtained by taking the Euler - Lagrange derivative of the scalar density of curvature ${\sqrt {-g}}\,R$ ${\ displaystyle {\ sqrt {-g}} \, R}$ ^[1] .

Properties

For two-dimensional Riemannian manifolds, the scalar curvature coincides with the doubled Gaussian curvature of the manifold.
- The integral over the Gaussian curvature is equal to the Euler characteristic of the surface times $2\pi$ ${\ displaystyle 2 \ pi}$ - This statement is the essence of the Gauss - Bonnet theorem .

Notes

↑ Science Network >> Relativity Theory for Astronomers

[1] Science Network >> Relativity Theory for Astronomers

Scalar curvature

Content

Gravitational Field Equations

Properties

See also

Notes

More articles: