# Levi-Civita properties in 4 dimensions

• I
• Phys pilot
In summary, the conversation is about the properties of the Levi-Civita symbol. The first part discusses a formula that doesn't seem to match the expected answer, and the second part involves proving another formula. The conversation ends with a final solution that satisfies the properties.
Phys pilot
first of all english is not my mother tongue sorry. I want to ask if you can help me with some of the properties of the levi-civita symbol.

I am showing that

$$\epsilon_{ijkl}\epsilon_{ijmn}=2!(\delta_{km}\delta_{ln}-\delta_{kn}\delta_{lm})$$

so i have this...

$$\epsilon_{ijkl}\epsilon_{ijmn}=\delta_{ii}\delta_{jj}\delta_{km}\delta_{ln}+\delta_{ij}\delta_{jm}\delta_{kn}\delta_{li}+\delta_{im}\delta_{jn}\delta_{ki}\delta_{lj}+\delta_{in}\delta_{ji}\delta_{kj}\delta_{lm}-\delta_{ii}\delta_{jn}\delta_{km}\delta_{lj}-\delta_{in}\delta_{jm}\delta_{kj}\delta_{li}-\delta_{im}\delta_{jj}\delta_{ki}\delta_{ln}-\delta_{ij}\delta_{ji}\delta_{kn}\delta_{lm}=3^2\delta_{km}\delta_{ln}+\delta_{kn}\delta_{lm}+\delta_{km}\delta_{ln}+\delta_{kn}\delta_{lm}-3\delta_{km}\delta_{ln}-\delta_{km}\delta_{ln}-3\delta_{km}\delta_{ln}-3\delta_{kn}\delta_{lm}=3\delta_{km}\delta_{ln}-3\delta_{kn}\delta_{lm}=3(\delta_{km}\delta_{ln}-\delta_{kn}\delta_{lm})$$

which is not equal to the supposedly correct answer. can the statement be wrong?
if not i can't see my error.

Also i need to prove that:

$$\epsilon_{ijkl}\epsilon_{ijkm}=3!\delta_{lm}$$
i know and i proved that
$$\epsilon_{ijkl}\epsilon_{ijkl}=24=4!$$
so if $l=m$ i have this
$$\epsilon_{ijkl}\epsilon_{ijkl}=3!\delta_{ll}=3!\cdot 3=18$$
which is not equal to 4!=24 so its the statement wrong again? it must be
$$\epsilon_{ijkl}\epsilon_{ijkm}=8\delta_{lm}$$
Actually i proved that:
$$\epsilon_{ijkl}\epsilon_{ijkm}=\delta_{ii}\delta_{jj}\delta_{kk}\delta_{lm}+\delta_{ij}\delta_{jk}\delta_{km}\delta_{li}+\delta_{ik}\delta_{jm}\delta_{ki}\delta_{lj}+\delta_{im}\delta_{ji}\delta_{kj}\delta_{lk}-\delta_{ii}\delta_{jm}\delta_{kk}\delta_{lj}-\delta_{im}\delta_{jk}\delta_{kj}\delta_{li}-\delta_{ik}\delta_{jj}\delta_{ki}\delta_{lm}-\delta_{ij}\delta_{ji}\delta_{km}\delta_{lk}=3^3\delta_{lm}+\delta_{ik}\delta_{km}\delta_{li}+\delta_{ii}\delta_{lj}\delta_{jm}+\delta_{jm}\delta_{kj}\delta_{lk}-3^2\delta_{lm}-\delta_{jj}\delta_{li}\delta_{im}-3\delta_{ii}\delta_{lm}-\delta_{ii}\delta_{lm}=3^3\delta_{lm}+\delta_{lk}\delta_{km}+\delta_{ii}\delta_{lm}+\delta_{lj}\delta_{jm}-3^2\delta_{lm}-3\delta_{lm}-3^2\delta_{lm}-3\delta_{lm}=3^3\delta_{lm}+\delta_{lm}+3\delta_{lm}+\delta_{lm}-3^2\delta_{lm}-3\delta_{lm}-3^2\delta_{lm}-3\delta_{lm}=8\delta_{lm}$$
Even more if i use my solution:

$$\epsilon_{ijkl}\epsilon_{ijmn}=3(\delta_{km}\delta_{ln}-\delta_{kn}\delta_{lm})$$
to prove
$$\epsilon_{ijkl}\epsilon_{ijkm}=3!\delta_{lm}$$
using $m=k$ and $n=m$ i have this
$$3(\delta_{km}\delta_{ln}-\delta_{kn}\delta_{lm})$$
$$3(\delta_{kk}\delta_{lm}-\delta_{km}\delta_{lk})$$
$$3(3\delta_{lm}-\delta_{lm})$$
$$3(2\delta_{lm})$$
$$6(\delta_{lm})=3!(\delta_{lm})$$

## 1. What are Levi-Civita properties in 4 dimensions?

Levi-Civita properties in 4 dimensions refer to a set of mathematical properties that describe the behavior of a mathematical object called a 4-dimensional Levi-Civita tensor. This tensor is used in differential geometry to define the curvature of a 4-dimensional space.

## 2. What is the significance of Levi-Civita properties in 4 dimensions?

The significance of Levi-Civita properties in 4 dimensions lies in their role in the mathematical description of 4-dimensional spaces, which have important applications in physics and cosmology. They also have applications in other fields such as computer graphics and robotics.

## 3. What are some examples of Levi-Civita properties in 4 dimensions?

Examples of Levi-Civita properties in 4 dimensions include the symmetric and antisymmetric nature of the tensor, its relation to the metric tensor, and its transformation properties under coordinate transformations. These properties are essential for understanding the behavior of the tensor in different coordinate systems.

## 4. How are Levi-Civita properties in 4 dimensions related to Einstein's theory of general relativity?

Levi-Civita properties in 4 dimensions are closely related to Einstein's theory of general relativity, which describes the curvature of spacetime in the presence of massive objects. The Levi-Civita tensor is used in the Einstein field equations to express the curvature of spacetime in terms of the energy and momentum of matter and radiation.

## 5. Are there any open questions or areas of research related to Levi-Civita properties in 4 dimensions?

Yes, there are still many open questions and areas of research related to Levi-Civita properties in 4 dimensions. These include further understanding the geometric and physical implications of these properties, as well as their possible generalizations to higher dimensions and other mathematical structures.

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