# No. of field equations and components or Riemann tensor?

• damnedcat
In summary, the number of field equations, M, is always equal to or less than the number of components of the Riemann tensor, N. This implies that the Riemann tensor provides information about the curvature of space, as it has at least as many components as the field equations.
damnedcat
no. of field equations and components or Riemann tensor??

Someone was trying to explain to me about curvature in space. From what I got from what they were saying doesn't make sense to me. I'm not sure I understand what the number of components, N, of R$$\alpha,\beta,\gamma,\delta$$ when compared to
the number of field equations, M, implies about curvature (when i say compare i mean: N>M, N<M, N=M). I thought looking at just the Riemann tensor tells u about curvature. mYBE i Just didn't get what he was saying. Any help?

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Was your friend referring to something like this? http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html See section 8.1.2, first para.

-Ben

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Something like that. He was saying that certain property can only occur for space times of more than 3 spatial dimensions. Since for instance the einstein eqn for vacuum in 1,2 and 3 dimensions have the number of field equations being less than or equal to the number of components of the Riemann tensor. I think he was talking about curvature of the space.

Sorry, but can you clarify by what you mean by alpha/beta/gamma/delta ?

nicksauce said:
Sorry, but can you clarify by what you mean by alpha/beta/gamma/delta ?

I think he means the indicies of the Riemann tensor?

damnedcat said:
Something like that. He was saying that certain property can only occur for space times of more than 3 spatial dimensions. Since for instance the einstein eqn for vacuum in 1,2 and 3 dimensions have the number of field equations being less than or equal to the number of components of the Riemann tensor. I think he was talking about curvature of the space.

I think this must be somewhat garbled. In any number of dimensions, the Riemann tensor, which has 4 indices, has at least as many independent components as the Ricci tensor, which has two indices and is constructed from it. The Einstein tensor has the same number of components as the Ricci tensor. The field equations have the Einstein tensor on one side. So the number of field equations will always be equal to the number of components in the Ricci and Einstein tensors, and less than or equal to the number of components in the Riemann tensor.

## 1. What are the field equations and components of the Riemann tensor?

The Riemann tensor is a mathematical object that describes the curvature of space-time in Einstein's theory of general relativity. It has 256 independent components and is made up of the metric tensor and its first and second derivatives.

## 2. How many field equations make up the Riemann tensor?

The Riemann tensor is described by 20 field equations, known as the Einstein field equations, which relate the curvature of space-time to the distribution of matter and energy.

## 3. What is the significance of the number of components in the Riemann tensor?

The number of components in the Riemann tensor reflects the complexity of the curvature of space-time. It allows us to fully describe the curvature and make predictions about how matter and energy will behave in that space-time.

## 4. Can the Riemann tensor be simplified or reduced in any way?

While the Riemann tensor itself cannot be simplified, it can be contracted with other tensors to create simpler objects, such as the Ricci tensor or the Ricci scalar, which have fewer components and are easier to work with.

## 5. How is the Riemann tensor used in physics?

The Riemann tensor is a fundamental tool in general relativity and is used to calculate the curvature of space-time in the presence of matter and energy. It is also used in other areas of physics, such as in the study of black holes and gravitational waves.

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