# Weyl Tensor Notation in Dimension 4

• jfy4
In summary, the conversation involves the use of anti-symmetric bracket notation and the Weyl Tensor in dimension 4. The correct formula for Wabcd is given and confirmed by contracting with gac.
jfy4
Hi,

I'm getting used to the anti-symmetric bracket notation used with indices and I can't seem to find the Weyl Tensor written fully out. So I want to make sure I get it. Here is my attempt in dimension 4.

$$W_{abcd}=R_{abcd}-\frac{1}{2}\left[g_{ac}R_{db}-g_{ad}R_{cb}-g_{bc}R_{da}+g_{bd}R_{ca}\right]+\frac{1}{6}R\left(g_{ac}g_{db}-g_{ad}g_{cb}\right)$$

It is written on wiki as

$$W_{abcd}=R_{abcd}-\frac{2}{n-2}\left(g_{a[c}R_{d]b}-g_{b[c}R_{d]a}\right)+\frac{2}{(n-1)(n-2)}R\, g_{a[c}g_{d]b}$$

Did I do it correct for n=4?

Thanks,

Looks good to me.

Looks right. Try contracting, gacWabcd, and see if you get 0.

cristo said:
Looks good to me.

Bill_K said:
Looks right. Try contracting, gacWabcd, and see if you get 0.

Thanks, I got 0 when I contracted.

## What is the Weyl tensor?

The Weyl tensor is a mathematical quantity used in the theory of general relativity. It describes the curvature of spacetime in four dimensions.

## Why is the Weyl tensor important in dimension 4?

In four dimensions, the Weyl tensor is the only remaining component of the Riemann curvature tensor. This means that it is responsible for the gravitational effects in general relativity.

## What does the notation for the Weyl tensor in dimension 4 look like?

The Weyl tensor in dimension 4 is denoted by the symbol Cabcd. It is a rank-4 tensor with 20 independent components.

## How is the Weyl tensor related to other tensors in general relativity?

The Weyl tensor is related to other tensors in general relativity through the Einstein field equations. It is also closely connected to the Ricci tensor and the Ricci scalar, which describe the local curvature of spacetime.

## What are some applications of the Weyl tensor in dimension 4?

The Weyl tensor is used in various applications, such as studying the formation and evolution of black holes, understanding gravitational waves, and investigating the large-scale structure of the universe.

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