A Riemann Curvature Tensor

  • Thread starter superbat
  • Start date
12
0
Can someone explain mathematically why do we say Riemann Curvature Tensor has all the information about curvature of Space
Thank You
 

andrewkirk

Science Advisor
Homework Helper
Insights Author
Gold Member
3,745
1,370
Statements like that are not mathematically precise, so I wouldn't worry about them too much. After all, the metric tensor also has all the information about curvature, since anything else can be expressed in terms of various partial derivatives of the metric.

I think it's a loose way of saying that in everyday operations of differential geometry, and in particular of general relativity, one can derive any piece of info we want about curvature from the Riemann tensor without having to take additional derivatives (which would take the order of differentiation from two to at least three) or taking fancy tensor products (which the Weyl tensor requires).
 
12
0
Ok Thanks
Since you mention metric, I was also wondering what does contravariant version of metric mean.
Covariant version of metric tells us about distance between 2 points. What does contravariant version of metric physically mean?
 

andrewkirk

Science Advisor
Homework Helper
Insights Author
Gold Member
3,745
1,370
Covariant version has components ##g^{ab}##. It is a linear function that takes two vectors in the tangent space at the relevant manifold point as input and returns a real scalar as output.
Contravariant version has components ##g_{ab}##. It is a linear function that takes two covectors (aka one-forms or dual vectors) in the cotangent space at the relevant manifold point as input and returns a real scalar as output.
 
12
0
Thanks
So do they have any physical significance or should I consider them as just mathematical tools
 

Want to reply to this thread?

"Riemann Curvature Tensor" You must log in or register to reply here.

Related Threads for: Riemann Curvature Tensor

  • Posted
Replies
2
Views
4K
  • Posted
Replies
1
Views
2K
Replies
5
Views
3K
Replies
13
Views
6K
Replies
5
Views
1K
Replies
0
Views
2K
Replies
0
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top