Riemann curvature coefficients using Cartan structure equation

In summary, the Riemann coefficient for a metric can be calculated using the second Cartan's structure equation and the tetrad formalism. Once the coefficients for the curvature tensor are obtained, they can be plugged into the Riemannian junk to obtain the Riemannian tensor. However, the notation used may lead to confusion as the basis one forms and the Riemannian junk need to be identified before calculations can be made.
  • #1
snypehype46
12
1
Homework Statement
Computing the Riemann curvature tensor using Cartan's structure equations
Relevant Equations
$$\frac{1}{2} \Omega_{ab} (\theta^a \wedge \theta^b) = -\frac{1}{4} R_{ijkl} (dx^i \wedge dx^j)(dx^k \wedge dx^l)$$
To calculate the Riemann coefficient for a metric ##g##, one can employ the second Cartan's structure equation:

$$\frac{1}{2} \Omega_{ab} (\theta^a \wedge \theta^b) = -\frac{1}{4} R_{ijkl} (dx^i \wedge dx^j)(dx^k \wedge dx^l)$$

and using the tetrad formalism to compute the coefficients of the curvature tensor.

Now I'm trying to properly understand this method, I was doing this exercise for which I obtained:

$$\frac{1}{2} \Omega_{ab} (\theta^a \wedge \theta^b) = -\frac{1}{4} A (dx \wedge du)^2 -\frac{1}{2}B(dx \wedge du) (dy \wedge du) - \frac{1}{4} (dy \wedge du)^2$$

However, from here I'm not quite how I would read the coefficient for the Riemann tensor. From here it seems for example that we have:

$$-\frac{1}{4} A (dx \wedge du)^2 = -\frac{1}{4} R_{xuxu}(dx \wedge du)^2$$
$$-\frac{1}{2}B(dx \wedge du) (dy \wedge du) = -\frac{1}{4}R_{xuyu}(dx \wedge du)(dy \wedge du)$$

The answer is supposed to be ##R_{xuxu} = \frac{1}{2}A## and ##R_{xuyu}= \frac{1}{2} B##, however I don't quite understand how this would be obtained from the equation above. I assume it has something to do with the symmetries of the Riemann tensor but not quite sure.
 
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  • #3
Also, to add onto this, your notation (if it's from a book, id recommend reviewing the above wiki before going back) is probably leading you to so many issues. You have to PICK which a, b you are using in order to calculate it. But the other side of the equation is ijkl? So let's say you let a = 1, and b = 3. What's the ijkl in your notation? If you wrote it yourself, stop. You can't just randomly throw different names for indices and hope it sticks. If you have ##\Omega^a_b##, then you better have an a in an upper index somewhere, and and b in a lower index somewhere. This will make life so much easier for you.

Here are the steps:
1) Identify basis one forms
2) Use first structure equation to calculate connection coefficients
3) Use second structure equation to get your curvature (your ##\Omega^a_b##)

Once you have these calculated, THEN you set it equal to your Riemannian junk, and whalla, you have something you calculated = Riemannian junk times one forms. You can identify what you ##R^a_{bcd}## is easily, but you first have to calculate what your ##\Omega^a_b## is, which comes from taking exterior derivatives of you connection coefficients, which involves taking exterior derivatives of your basis one forms. And in another thread you talked about why do some terms "not matter". It isn't that they "don't matter", it could be that when you calculate those connection coefficients, it's zero.

EDIT: I responded to your other thread, they weren't zero in this case!
 
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Related to Riemann curvature coefficients using Cartan structure equation

1. What is the significance of Riemann curvature coefficients?

The Riemann curvature coefficients, also known as the Riemann curvature tensor, are important in the study of differential geometry and general relativity. They describe the curvature of a manifold, which is a mathematical space that can be curved. These coefficients help us understand how objects move and interact in curved spaces, and are essential in the development of Einstein's theory of general relativity.

2. How are Riemann curvature coefficients calculated?

The Riemann curvature coefficients can be calculated using the Cartan structure equation, which relates the curvature of a manifold to the connection coefficients and their derivatives. This equation takes into account the local geometry of the manifold and allows us to calculate the curvature at any point on the manifold.

3. What is the Cartan structure equation?

The Cartan structure equation is a mathematical equation that relates the curvature of a manifold to the connection coefficients and their derivatives. It is an important tool in the study of differential geometry and is used to calculate the Riemann curvature coefficients of a manifold.

4. How are Riemann curvature coefficients used in general relativity?

In general relativity, Riemann curvature coefficients are used to describe the curvature of spacetime. They are essential in Einstein's field equations, which relate the curvature of spacetime to the distribution of matter and energy. These coefficients help us understand the gravitational effects of massive objects and the dynamics of the universe as a whole.

5. What is the physical interpretation of Riemann curvature coefficients?

The physical interpretation of Riemann curvature coefficients is that they represent the tidal forces experienced by objects in a curved space. These forces are caused by the curvature of the space and can be observed in phenomena such as gravitational lensing and the bending of light near massive objects. The Riemann curvature coefficients allow us to quantify and understand these effects in a mathematical way.

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