Weyl tensor

Riemannian geometry | Tensors in general relativity

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. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. This tensor has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free: metric contraction on any pair of indices yields zero. It is obtained from the Riemann tensor by subtracting a tensor that is a linear expression in the Ricci tensor. In general relativity, the Weyl curvature is the only part of the curvature that exists in free space—a solution of the vacuum Einstein equation—and it governs the propagation of gravitational waves through regions of space devoid of matter. More generally, the Weyl curvature is the only component of curvature for Ricci-flat manifolds and always governs the characteristics of the field equations of an Einstein manifold. In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions ≥ 4, the Weyl curvature is generally nonzero. If the Weyl tensor vanishes in dimension ≥ 4, then the metric is locally conformally flat: there exists a local coordinate system in which the metric tensor is proportional to a constant tensor. This fact was a key component of Nordström's theory of gravitation, which was a precursor of general relativity. (Wikipedia).

What is General Relativity? Lesson 49: Constructing the Weyl tensor I

What is General Relativity? Lesson 49: Constructing the Weyl tensor I We calculate the conformally invariant part of the Riemann tensor. This is a complex calculation that is not generally done in textbooks. The motivation for the Weyl tensor will come in a later lesson, but I think the d

From playlist What is General Relativity?

What is a Tensor? Lesson 11: The metric tensor

What is a Tensor 11: The Metric Tensor

From playlist What is a Tensor?

Calculus 3: Tensors (1 of 28) What is a Tensor?

Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is a tensor. A tensor is a mathematical representation of a scalar (tensor of rank 0), a vector (tensor of rank 1), a dyad (tensor of rank 2), a triad (tensor or rank 3). Next video in t

From playlist CALCULUS 3 CH 10 TENSORS

What is a Tensor 3: Dual Spaces

What is a Tensor? Dual Spaces

From playlist What is a Tensor?

What is General Relativity? Lesson 48: Ricci tensor and conformal transformations

What is General Relativity? Lesson 48: Ricci tensor and conformal transformations We introduce the Ricci tensor, curvature scalar, and begin the difficult derivation of the Weyl tensor. Please consider supporting this channel via Patreon: https://www.patreon.com/XYLYXYLYX and discussin

From playlist What is General Relativity?

What Is A Tensor Lesson #1: Elementary vector spaces

We define a vector space and lay the foundation of a solid understanding of tensors.

From playlist What is a Tensor?

What is General Relativity? Lesson 68: The Einstein Tensor

What is General Relativity? Lesson 68: The Einstein Tensor The Einstein tensor defined! Using the Ricci tensor and the curvature scalar we can calculate the curvature scalar of a slice of a manifold using the Einstein tensor. Please consider supporting this channel via Patreon: https:/

From playlist What is General Relativity?

Weyl Anomalies and Cosmology by Atish Dabholkar

11 January 2017 to 13 January 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru String theory has come a long way, from its origin in 1970's as a possible model of strong interactions, to the present day where it sheds light not only on the original problem of strong interactions, but

From playlist String Theory: Past and Present

What is a Tensor 5: Tensor Products

What is a Tensor 5: Tensor Products Errata: At 22:00 I write down "T_00 e^0 @ e^1" and the correct expression is "T_00 e^0 @ e^0"

From playlist What is a Tensor?

Tensor Calculus Lecture 7d: The Voss-Weyl Formula

This course will eventually continue on Patreon at http://bit.ly/PavelPatreon Textbook: http://bit.ly/ITCYTNew Errata: http://bit.ly/ITAErrata McConnell's classic: http://bit.ly/MCTensors Table of Contents of http://bit.ly/ITCYTNew Rules of the Game Coordinate Systems and the Role of Te

From playlist Introduction to Tensor Calculus

David Zywina, Computing Sato-Tate and monodromy groups.

VaNTAGe seminar on May 5, 2020. License: CC-BY-NC-SA Closed captions provided by Jun Bo Lau.

From playlist The Sato-Tate conjecture for abelian varieties

What is a Tensor? Lesson 29: Transformations of tensors and p-forms (part review)

What is a Tensor? Lesson 29: Tensor and N-form Transformations This long lesson begins with a review of tensor product spaces and the relationship between coordinate transformations on spacetime and basis transformations of tensor fields. Then we do a full example to introduce the idea th

From playlist What is a Tensor?

Why Tensor Calculus?

http://bit.ly/PavelPatreon Textbook: http://bit.ly/ITCYTNew Errata: http://bit.ly/ITAErrata McConnell's classic: http://bit.ly/MCTensors Table of Contents of http://bit.ly/ITCYTNew Rules of the Game Coordinate Systems and the Role of Tensor Calculus Change of Coordinates The Tensor Desc

From playlist Introduction to Tensor Calculus

Tensor Calculus Lecture 7c: The Levi-Civita Tensors

From playlist Introduction to Tensor Calculus

Tensor Calculus 5a: The Tensor Property

From playlist Introduction to Tensor Calculus

Tensor Calculus 4a: The Tensor Notation

From playlist Introduction to Tensor Calculus

Tensor Calculus Lecture 10a: The Covariant Surface Derivative in Its Full Generality

From playlist Introduction to Tensor Calculus

An Alert for My Tensors Textbook

From playlist Introduction to Tensor Calculus

What is General Relativity? Lesson 50: Constructing the Weyl Tensor II

What is General Relativity? Lesson 50: Constructing the Weyl Tensor II We calculate the traceless part of the Riemann tensor. We will discover that the result is the same as the calculation of the conformally invariant part discovered in Lesson 49! This begins a diversion into understand

From playlist What is General Relativity?

Weyl tensor

Related pages