Curvature (mathematics) | Differential geometry | Riemannian manifolds | Tensors in general relativity | Riemannian geometry
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.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation. (Wikipedia).
Curvature of a Riemannian Manifold | Riemannian Geometry
In this lecture, we define the exponential mapping, the Riemannian curvature tensor, Ricci curvature tensor, and scalar curvature. The focus is on an intuitive explanation of the curvature tensors. The curvature tensor of a Riemannian metric is a very large stumbling block for many student
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Connections part 5: Riemannian Curvature Tensor and Faraday Tensor
This video was hacked together. Apologies.
From playlist Connections, Curvature and Covariant Derivatives
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?
What is General Relativity? Lesson 52: Scalar Curvature Part I
What is General Relativity? Lesson 52: Scalar Curvature Part I This is the first of a few lectures about the Scalar Curvature and its interpretation. The goal is to get us to a point where we can have an interpretation of the Einstein Tensor and therefore an interpretation of the Einstein
From playlist What is General Relativity?
Lecture 15: Isometries, Rigidity, and Curvature
CS 468: Differential Geometry for Computer Science
From playlist Stanford: Differential Geometry for Computer Science (CosmoLearning Computer Science)
Relativity 7b4 - Riemann and Ricci tensors
The Riemann curvature tensor tells you everything there is to know about the curvature of spacetime. The Ricci tensor is derived from the Riemann tensor and describes changes in volume. It is the key object in Einstein's "field equations of general relativity."
From playlist Relativity - appendix videos
What is General Relativity? Lesson 40: Round Trip Riemann Tensor
What is General Relativity? Lesson 40: Round Trip Riemann Tensor The curvature tensor is presented as a construction in differential geometry. Please consider supporting this channel via Patreon: https://www.patreon.com/XYLYXYLYX and discussing the material on the forums: https://www.
From playlist What is General Relativity?
Tensor Calculus Ep. 15 | Riemann Curvature Tensor
Todays episode explores the concept of curvature, and we finally arrive at the Riemann Curvature Tensor. Eigenchris's video: https://www.youtube.com/watch?v=-Il2FrmJtcQ&t=1364s&ab_channel=eigenchris This series is based off of the book "Tensor Calculus for Physics" by Dwight Neuenschwand
From playlist New To Tensors? Start Here
Tensor Calculus Lecture 8e: The Riemann Christoffel Tensor & Gauss's Remarkable Theorem
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
Lec 8b - Phys 237: Gravitational Waves with Kip Thorne
Watch the rest of the lectures on http://www.cosmolearning.com/courses/overview-of-gravitational-wave-science-400/ Redistributed with permission. This video is taken from a 2002 Caltech on-line course on "Gravitational Waves", organized and designed by Kip S. Thorne, Mihai Bondarescu and
From playlist Caltech: Gravitational Waves with Kip Thorne - CosmoLearning.com Physics
The Einstein Field Equations | Tensor Calc Finale
Today we use all the tools we've got in our back pocket to "derive" the Einstein Field Equations of general relativity. It's more of a motivation than a derivation, I suppose. This series is based off the book "Tensor Calculus for Physics" by Dwight Neuenschwander: https://amzn.to/3rEema3
From playlist New To Tensors? Start Here
The Bianchi Identities | Tensor Calculus Ep. 17
Today we derive the differential and contracted Bianchi Identities. Video relating metric to gravity (newtonion limit): https://www.youtube.com/watch?v=kIi-qcSBa4w&t=69s&ab_channel=AndrewDotson Deriving the Riemann Curvature Tensor: https://www.youtube.com/watch?v=L8ISLzDYl68&t=1348s&ab_
From playlist New To Tensors? Start Here
Thomas Baumgarte (2) - Numerical relativity: Mathematical formulation
PROGRAM: NUMERICAL RELATIVITY DATES: Monday 10 Jun, 2013 - Friday 05 Jul, 2013 VENUE: ICTS-TIFR, IISc Campus, Bangalore DETAL Numerical relativity deals with solving Einstein's field equations using supercomputers. Numerical relativity is an essential tool for the accurate modeling of a wi
From playlist Numerical Relativity
What is General Relativity? Lesson 54 - Scalar Curvature Part 3: Riemann Normal Coordinates
What is General Relativity? Lesson 54 -Scalar Curvature Part 3 Riemann Normal Coordinates This is the second of a few lectures about the Scalar Curvature and its interpretation. The goal is to get us to a point where we can have an interpretation of the Einstein Tensor and therefore an i
From playlist What is General Relativity?
11. More on spacetime curvature.
MIT 8.962 General Relativity, Spring 2020 Instructor: Scott Hughes View the complete course: https://ocw.mit.edu/8-962S20 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP629n_3fX7HmKKgin_rqGzbx Variants on the Riemann curvature tensor: the Ricci tensor and Ricci scalar,
From playlist MIT 8.962 General Relativity, Spring 2020
What is General Relativity? Lesson 46: Symmetries of the Riemann Tensor
What is General Relativity? Lesson 46: Symmetries of the Riemann Tensor Here we review material that shows up so frequently in general relativity mathematics that was simply must push through it and become comfortable with it. Please consider supporting this channel via Patreon: https:/
From playlist What is General Relativity?
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?
Tensor Calculus Lecture 14a: Non-hypersurfaces
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
What is General Relativity? Lesson 56 - Scalar curvature Part 5: More Riemann Normal Coordinates
What is General Relativity? Lesson 56 - Scalar curvature Part 5: More Riemann Normal Coordinates In this lecture we re-work the Riemann Normal Coordinate formalism from the point of view of a coordinate transformation and leaning on the Einstein Equivalence Principle. We also point out wh
From playlist What is General Relativity?