Unraveling the Mysteries of the Riemann Curvature Tensor

In summary, the conversation is about the curvature tensor and its symmetry properties as described by Dirac in his book on General Relativity. The problem at hand is understanding how to derive the number of independent components of the curvature tensor based on its symmetry constraints. The equations provided include the definition of the curvature tensor and Dirac's symmetry constraints, which include equations (1), (2), (3), and (4). In attempting to solve the problem, the poster discusses their general approach and asks for clarification on certain aspects, such as whether to treat equation (4) as three conditions and how to interpret the cyclic sum in equation (2). Another poster responds with their own interpretation of the symmetry constraints and offers some advice.
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Homework Statement



(My first post on this forum)

Background: I am teaching myself General Relativity using Dirac's (very thin) 'General Theory of Relativity' (Princeton, 1996). Chapter 11 introduces the (Riemann) curvature tensor (page 20 in my edition).

Problem: Dirac lists several symmetry properties of the curvature tensor from its definition in terms of the Christoffel Symbols and then claims without proof that from these symmetry properties, the curvature tensor has 20 independent components (reduced from 256).

a) Any hints on how he got that?

b) How do you work out the number of independent tensor components in general, given whatever symmetry constraints on the indices?

Homework Equations



Definition: [tex]R^{\beta}_{\nu\rho\sigma} = \Gamma^{\beta}_{\nu\sigma,\rho} - \Gamma^{\beta}_{\nu\rho,\sigma} + \Gamma^{\alpha}_{\nu\sigma}\Gamma^{\beta}_{\alpha\rho} - \Gamma^{\alpha}_{\nu\rho}\Gamma^{\beta}_{\alpha\sigma}[/tex]

Dirac's symmetry constraints:

(1) [tex]R^{\beta}_{\nu\rho\sigma} = -R^{\beta}_{\nu\sigma\rho}[/tex]

(2) [tex]R^{\beta}_{\nu\rho\sigma} + R^{\beta}_{\rho\sigma\nu} + R^{\beta}_{\sigma\nu\rho} = 0[/tex]

(3) [tex]R_{\mu\nu\rho\sigma} = -R_{\nu\mu\rho\sigma}[/tex]

(4) [tex]R_{\mu\nu\rho\sigma} = R_{\rho\sigma\mu\nu} = R_{\sigma\rho\nu\mu}[/tex]

The Attempt at a Solution



I tried to work out the general case first. A tensor of 2 indices in [tex]N[/tex] dimensions has [tex]N^{2}[/tex] components. If it is a symmetric tensor then it has [tex]\frac{1}{2}N(N+1)[/tex] independent components and hence symmetry constrains [tex]\frac{1}{2}N(N-1)[/tex] components. Similarly, if the tensor is antisymmetric then that constrains [tex]\frac{1}{2}N(N+1)[/tex] components.

[tex]N = 4[/tex] here. Assuming what I said above is correct equations (1) and (3) will constrain 20 components in total.

Now I'm stuck because:

a) Should I treat equation (4) as three conditions?

b) If I do treat equation (4) as three conditions, am I correct in saying that [tex]R_{\rho\sigma\mu\nu} = R_{\sigma\rho\nu\mu}[/tex] constrains 12 components because there are two pairs of symmetric indices?

c) How do I interpret things like [tex]R_{\mu\nu\rho\sigma} = R_{\rho\sigma\mu\nu}[/tex] in the above context?

d) How should I interpret the cyclic sum of equation (2) in terms of what I said above?

Thanks!
 
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  • #2


1st: I don’t think all the symmetry constrains equations are independent. As: lower the upper indices in (1) using the metric tensor. Now apply 2nd equality of (4) to RHS and one gets (3). Hence I think you should only use (1), (2) & (4) or the set (2), (3) & (4) to derive the # of independent tensor elements.
2nd: I think (2) should be interpreted like a supermarket deal, ie “buy 2 get one free”. ie pick any two of the tensors, and then the third tensor is independent (or “free”) if you know (or have “bought”) the other two.

Hope this helps,
Mischa
 

1. What is the Riemann Curvature Tensor?

The Riemann Curvature Tensor is a mathematical object in differential geometry that describes the curvature of a space. It is used to study the geometry of curved surfaces and higher dimensional spaces.

2. Why is understanding the Riemann Curvature Tensor important?

Understanding the Riemann Curvature Tensor is essential for understanding the geometry of curved spaces, such as the surface of a sphere or the fabric of spacetime in general relativity. It also has applications in physics, engineering, and computer graphics.

3. How is the Riemann Curvature Tensor calculated?

The Riemann Curvature Tensor is calculated using a specific formula that involves taking derivatives of the metric tensor, which describes the distance between points in a space. This calculation can be quite complex and requires advanced mathematical techniques.

4. What are the implications of the Riemann Curvature Tensor for our understanding of the universe?

The Riemann Curvature Tensor is an important tool in the study of general relativity, which describes the behavior of gravity on a large scale. It helps us understand the curvature of spacetime and how it is affected by massive objects, such as planets and stars.

5. Are there any real-world applications of the Riemann Curvature Tensor?

Yes, the Riemann Curvature Tensor has many practical applications in fields such as physics, astronomy, and engineering. It is used in the design of space missions, the development of GPS technology, and the analysis of data from gravitational wave detectors.

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