Understanding Kronecker Delta: Why & How?

[TimeRip496]

I know what is it but I can't figure out the rationale behind it. As in why do we need it? Moreover I don't know how to represent it in terms of kronecker delta. How do you do that?

 

[stevendaryl]

I know what is it but I can't figure out the rationale behind it. As in why do we need it? Moreover I don't know how to represent it in terms of kronecker delta. How do you do that?

Levi-Civita is a person who has several different concepts attached to his name, including the Levi-Civita symbol, the Levi-Civita connection, and probably others. So which one are you talking about?

 

[TimeRip496]

Levi-Civita is a person who has several different concepts attached to his name, including the Levi-Civita symbol, the Levi-Civita connection, and probably others. So which one are you talking about?

The Levi Civita symbol

 

[WannabeNewton]

I know what is it but I can't figure out the rationale behind it. As in why do we need it? Moreover I don't know how to represent it in terms of kronecker delta. How do you do that?

$\epsilon_{abcd}$ is a non-vanishing 4-form on a space-time $(M,g_{ab})$ so at the basic level it allows one to define an orientation on the space-time. Indeed a space-time is defined to be orientable if it possesses a non-vanishing 4-form. What makes $\epsilon_{abcd}$ unique (up to a sign) on top of it being an orientation is its definition $\epsilon^{abcd}\epsilon_{abcd} = -4!$ i.e. it is determined by $g_{ab}$ and as such is called a volume element because if $v^a_1,...,v^a_4$ are a set of four arbitrary vectors in $T_pM$ for any $p$, then $\epsilon_{abcd}v^a_1...v^a_4$ is the volume of the parallelepiped determined by these four vectors. So, in summary, we need $\epsilon_{abcd}$ in order to define an orientation on space-time and subsequently do volume integrals on space-time. This of course carries over to other dimensions and to Riemannian manifolds, not just Lorentzian manifolds. C.f. Wald Appendix B.

You cannot express $\epsilon_{abcd}$ by itself in terms of $\delta_{ab}$ but you can for example show that $\epsilon^{abcd}\epsilon_{efgh} = -4! \delta^{[a}{}{}_{e}...\delta^{d]}{}{}_{h}$ and similarly for other dimensions. C.f. Wald Appendix B.

 

[TimeRip496]

$\epsilon_{abcd}$ is a non-vanishing 4-form on a space-time $(M,g_{ab})$ so at the basic level it allows one to define an orientation on the space-time. Indeed a space-time is defined to be orientable if it possesses a non-vanishing 4-form. What makes $\epsilon_{abcd}$ unique (up to a sign) on top of it being an orientation is its definition $\epsilon^{abcd}\epsilon_{abcd} = -4!$ i.e. it is determined by $g_{ab}$ and as such is called a volume element because if $v^a_1,...,v^a_4$ are a set of four arbitrary vectors in $T_pM$ for any $p$, then $\epsilon_{abcd}v^a_1...v^a_4$ is the volume of the parallelepiped determined by these four vectors. So, in summary, we need $\epsilon_{abcd}$ in order to define an orientation on space-time and subsequently do volume integrals on space-time. This of course carries over to other dimensions and to Riemannian manifolds, not just Lorentzian manifolds. C.f. Wald Appendix B.

You cannot express $\epsilon_{abcd}$ by itself in terms of $\delta_{ab}$ but you can for example show that $\epsilon^{abcd}\epsilon_{efgh} = -4! \delta^{[a}{}{}_{e}...\delta^{d]}{}{}_{h}$ and similarly for other dimensions. C.f. Wald Appendix B.

Thanks for the reply. But is there a way you can simplify it? Cause I have difficulty following.

 

[Ibix]

As I noted before, I'm still learning this too, so any criticisms of the following are probably true. :D

You can define spaces with or without certain properties that you might naively think of as intrinsic to the very notion of a space. For example you can define a space without a metric, in which case there is no notion of distance along a path. It's just a set of points. The connection relates vectors in different tangent spaces and ends up defining the geodesics - giving a notion of a "straight line". WBN is saying that it's possible to define a space with such notions as distance and geodesics without necessarily being able to define an elementary volume, basically because that notion relies on orientability, the ordering of the bases. The Levi-Civita symbol defines the notion that one can have an elementary volume by defining the notion of left- and right-handed bases.

Most of the manifolds you'll encounter in GR are equipped with a Levi-Civita symbol. I think someone here said that some exotic space-times like closed timelike curves aren't, which is why I say "most" and not "all", but don't take my word for that.

 

[Fredrik]

I know what is it but I can't figure out the rationale behind it. As in why do we need it?

It simplifies both definitions and calculations. For example, can you find a simpler proof of $\nabla\cdot(\nabla\times f)=0$ than this?
$$\nabla\cdot(\nabla\times f)= \partial_i(\varepsilon_{ijk}\partial_j f) =\varepsilon_{ijk}\partial_i\partial_j f=0.$$

Moreover I don't know how to represent it in terms of kronecker delta. How do you do that?

Not sure what you mean. Are you talking about how to prove identities like $\varepsilon_{ijk}\varepsilon_{ijl}=\delta_{kl}$?