Intuition regarding Riemann curvature tensor

  • #1
Hill
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"Why" there are terms quadratic in first derivatives of metric?
The Riemann curvature tensor contains second derivatives of metric and squares of the first derivatives. The second derivatives have to be there because they are the ones that cannot be eliminated locally by a choice of coordinates. But other than being a mathematical artifact, is there a meaning for the squares of the first derivatives to be there?
 
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  • #2
The first derivatives tell you directions. Curvature can be different in different directions. And the curvature in one direction can change as you move in a different direction
 
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  • #3
Hill said:
The second derivatives have to be there because they are the ones that cannot be eliminated locally by a choice of coordinates.
If you look at how this actually works, you find that in Riemann normal coordinates, which are the ones that eliminate everything that can be eliminated by a choice of coordinates, the first derivatives are zero at the origin of coordinates, i.e., at the point the coordinates are "centered" on. So at that point, the first derivatives indeed do not appear at all in the Riemann tensor.

But as soon as you move away from the origin, the first derivatives are no longer zero (because the second derivatives weren't zero at the origin), and they will be different depending on which direction you move. So you do need them to capture the complete behavior once you are away from the origin.
 
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  • #4
PeterDonis said:
But as soon as you move away from the origin, the first derivatives are no longer zero (because the second derivatives weren't zero at the origin)
lndeed, a zero second derivatives w.r.t. a coordinate basis at the origin, would mean constant first derivatives in a neighborhood of the origin, but first derivatives by definition vanish at the origin in Riemann normal coordinates centered on it.
 

FAQ: Intuition regarding Riemann curvature tensor

What is the Riemann curvature tensor and why is it important?

The Riemann curvature tensor is a mathematical object that measures the curvature of a Riemannian manifold. It is important because it provides a rigorous way to describe how much and in what manner a space deviates from being flat. This tensor plays a crucial role in the field of differential geometry and is fundamental in the theory of General Relativity, where it describes the gravitational field and the curvature of spacetime.

How can I intuitively understand the components of the Riemann curvature tensor?

Intuitively, the components of the Riemann curvature tensor can be understood by considering how a vector changes as it is parallel transported around a small loop in the manifold. Each component of the tensor measures the difference between the initial and final vectors after this transport. This difference encapsulates the curvature of the space in various directions, providing insight into how the space bends or twists.

What is the geometric meaning of the Riemann curvature tensor?

Geometrically, the Riemann curvature tensor captures the idea that the outcome of parallel transporting a vector around a closed loop depends on the curvature of the space. If the space is flat, the vector returns to its original position unchanged. In a curved space, however, the vector will generally be rotated and/or scaled, indicating the presence of curvature. This geometric interpretation helps in visualizing how curvature affects the shape and structure of the manifold.

How does the Riemann curvature tensor relate to the concept of sectional curvature?

The Riemann curvature tensor is related to sectional curvature, which measures the curvature of two-dimensional sections of the manifold. Specifically, the sectional curvature can be computed from the Riemann curvature tensor by considering two tangent vectors at a point and using the tensor to evaluate the curvature of the plane spanned by these vectors. This relationship allows for a more detailed and localized understanding of curvature in the manifold.

Can you explain the symmetries of the Riemann curvature tensor?

The Riemann curvature tensor has several important symmetries. Firstly, it is antisymmetric in its first two and last two indices: \( R_{abcd} = -R_{bacd} \) and \( R_{abcd} = -R_{abdc} \). Secondly, it is symmetric in the exchange of the first pair of indices with the second pair: \( R_{abcd} = R_{cdab} \). Finally, it satisfies the Bianchi identity: \( R_{abcd} + R_{acdb} + R_{adbc} = 0 \). These symmetries reduce the number of independent components of the tensor and are essential in simplifying many calculations in differential geometry and General Relativity.

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