Riemann & Ricci Curvature Tensors: No Coord. Indices?

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In summary, there is a standard way to denote the Riemann and Ricci curvature tensors without using coordinate indices. This is recommended by Doran and Lasenby in their book "Geometric Algebra for Physicists". The Riemann tensor can be denoted as R(v1^v2), where ^ is the wedge-product, while the Ricci tensor can be denoted as R(v). However, there is some ambiguity with using R as it can also be interpreted as a trilinear or 4-linear map, which can result in different notations. Ultimately, it is up to personal preference and convenience to use R and Ric to denote the tensors, as the raising and lowering of indices is a trivial process.
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Matterwave
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For you math people who like to express objects in a coordinate-free way, how would you denote the Riemann and the Ricci curvature tensors? They are both usually denoted R but with different indices to show which one is which. Is there a standard way to write them without the coordinate indices?
 
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I strongly recommend Doran and Lasenby's "Geometric Algebra for Physicists". They distinguish the two by showing their arguments. The Riemann tensor is a bivector-valued function of a bivector argument, so it can designated R(v1^v2), where ^ is the wedge-product. The Ricci tensor is a vector valued function of a vector argument, so it can be designated R(v).
 
  • #3
Yes, I believe writing R and Ric to denote the respective tensors is standard. R is tricky though because sometimes it is interpreted as a trilinear map R: TM x TM x TM --> TM. In which case we write not R(X,Y,Z) but R(X,Y)Z or RZ(X,Y) and call R the riemann curvature endomorphism because given (X,Y), R(X,Y) is an endomorphism TM-->TM: Z-->R(X,Y)Z. Other times, R is interpreted as a 4-linear map R:TM x TM x TM x T*M-->R. This is related of course to the curvature endomorphism by means of the musical isomorphism (aka raising/lowering the last index).
 
  • #4
Since I'm always working on a manifold with metric, I think I'll just use R and Ric. The raising and lowering of indices is trivial.
 
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Yes, there is a standard way to write the Riemann and Ricci curvature tensors without using coordinate indices. This approach is known as the abstract index notation, where tensors are expressed in terms of abstract indices that do not depend on any specific coordinate system.

In this notation, the Riemann curvature tensor is denoted as R_{abcd} and the Ricci curvature tensor is denoted as R_{ab}. The abstract indices a, b, c, and d represent the abstract components of the tensors, which can be transformed to any coordinate system without changing the underlying mathematical structure.

This notation is widely used in modern differential geometry and general relativity, as it allows for a coordinate-free description of tensors and their properties. It also simplifies calculations and makes it easier to generalize results to different coordinate systems.

In summary, the Riemann and Ricci curvature tensors can be denoted in a coordinate-free way using the abstract index notation, providing a more elegant and versatile approach to expressing these important mathematical objects.
 

FAQ: Riemann & Ricci Curvature Tensors: No Coord. Indices?

1. What are Riemann and Ricci Curvature Tensors?

Riemann and Ricci Curvature Tensors are mathematical objects used in the field of differential geometry to describe the curvature of a space. They are used in Einstein's theory of general relativity to describe the curvature of spacetime.

2. How are Riemann and Ricci Curvature Tensors related?

Riemann and Ricci Curvature Tensors are related by the Ricci tensor, which is obtained by contracting the Riemann tensor. The Ricci tensor contains information about the average curvature of a space, while the Riemann tensor contains information about the local curvature at a specific point.

3. What is the significance of Riemann and Ricci Curvature Tensors in physics?

In physics, Riemann and Ricci Curvature Tensors are used in Einstein's theory of general relativity to describe the curvature of spacetime. They are essential for understanding gravity and how it affects the motion of objects in the universe.

4. How are Riemann and Ricci Curvature Tensors calculated?

Riemann and Ricci Curvature Tensors are calculated using a combination of mathematical equations and components of the metric tensor, which describes the geometry of a space. The calculations can be complex and involve multiple steps, but there are established methods for obtaining these tensors.

5. Can Riemann and Ricci Curvature Tensors be used in other fields of science?

Yes, Riemann and Ricci Curvature Tensors have applications in other fields of science, such as computer science, where they are used in algorithms for machine learning and data analysis. They also have applications in physics beyond general relativity, such as in string theory and quantum gravity.

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