Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Riemann Tensor and Ricci Scalar

In course

PHYS-427: Relativity and cosmology I

Introduce the students to general relativity and its classical tests. ...

Description

This lecture covers the properties of the Riemann tensor in normal coordinates at a point, the Ricci tensor, Ricci scalar, Weyl tensor, and Einstein tensor. It also discusses symmetries, Killing vectors, and the geodesic equation.

Official source

Ontological neighbourhood

Mathematics

Geometry: Differential geometry

Related concepts (32)

Tensor

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product.

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.

Weyl tensor

In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force.

Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space.

Ricci decomposition

In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. This decomposition is of fundamental importance in Riemannian and pseudo-Riemannian geometry. Let (M,g) be a Riemannian or pseudo-Riemannian n-manifold. Consider its Riemann curvature, as a (0,4)-tensor field.

Related lectures (45)

General Covariance in Gravitational Field

Explores general covariance, Riemann tensor, Bianchi identities, and Einstein tensor in gravitational field.

Flat Space-Time: Conditions and Theorems

Covers the necessary conditions and theorems to have flat space-time and discusses the symmetry properties of the Riemann tensor.

Scalar Gravity: Einstein Equations

Introduces scalar gravity, covering covariant derivatives, Ricci tensor, Einstein Equivalence Principle, and the generalization of Newtonian gravity equations.

General Covariance in Gravity

Explores general covariance in gravity, covering equations, curvature tensor, and static gravity field.

Symmetries and Conservation Laws