What is the Form of Riemann Tensor in 3D?

sadegh4137
Messages
72
Reaction score
0
hi

Riemann tensor has a definition that independent of coordinate and dimension of manifold where you work with it.

see for example Geometry,Topology and physics By Nakahara Ch.7
In that book you can see a relation for Riemann tensor and that is usual relation according to Christoffel symbol and it's derivative. ( that relation is independent of dimension of manifold)
I don't know why Riemann tensor in 3D has the following form

https://www.physicsforums.com/attachment.php?attachmentid=7457&d=1154966304

In this page
https://www.physicsforums.com/showthread.php?t=128275
you can see a prove for 3D Riemann tensor but i can't understand why it has different form
 
Physics news on Phys.org
Post #7 in that thread has the exact same form of the equation in your attachment. The only difference is samalkhaiat pulled out a negative sign in the Ricci scalar term in the final expression.
 
Last edited:
  • Like
Likes 1 person
Thread 'Can this experiment break Lorentz symmetry?'

Thread 'Can this experiment break Lorentz symmetry?'

1. The Big Idea: According to Einstein’s relativity, all motion is relative. You can’t tell if you’re moving at a constant velocity without looking outside. But what if there is a universal “rest frame” (like the old idea of the “ether”)? This experiment tries to find out by looking for tiny, directional differences in how objects move inside a sealed box. 2. How It Works: The Two-Stage Process Imagine a perfectly isolated spacecraft (our lab) moving through space at some unknown speed V...
View full post »

Thread '1-Way Speed of Light'

Does the speed of light change in a gravitational field depending on whether the direction of travel is parallel to the field, or perpendicular to the field? And is it the same in both directions at each orientation? This question could be answered experimentally to some degree of accuracy. Experiment design: Place two identical clocks A and B on the circumference of a wheel at opposite ends of the diameter of length L. The wheel is positioned upright, i.e., perpendicular to the ground...
View full post »

Thread 'How can a black hole absorb matter?'

According to the General Theory of Relativity, time does not pass on a black hole, which means that processes they don't work either. As the object becomes heavier, the speed of matter falling on it for an observer on Earth will first increase, and then slow down, due to the effect of time dilation. And then it will stop altogether. As a result, we will not get a black hole, since the critical mass will not be reached. Although the object will continue to attract matter, it will not be a...
View full post »

Similar threads

I Calculating Covariant Derivative of Riemann Tensor in Riemann Normal Coordinates
Replies
5
Views
2K
I Number of independent components of the Riemann tensor
Replies
6
Views
4K
I Computing Riemann Tensor: 18 Predicted Non-Trivial Terms
Replies
1
Views
2K
Draft re Ricci vs Riemann tensors
Replies
1
Views
1K
A Re-writing the geodesic deviation eqn in matrix notation (3d only)
Replies
0
Views
1K
I Non-zero components of Riemann curvature tensor with Schwarzschild metric
Replies
14
Views
6K
A Weyl tensor and coordinate acceleration
Replies
12
Views
2K
On the order of indices of the Christoffel symbol of the 1st kind
Replies
19
Views
2K
I Manipulating Christoffel Symbols: Questions & Answers
Replies
2
Views
2K
What is the Geometrical Meaning of the Einstein Tensor?
Replies
8
Views
3K

Hot Threads

Recent Insights

Back
Top