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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?
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TimeRip496 said: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?
The Levi Civita symbolstevendaryl said: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 said: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?
Thanks for the reply. But is there a way you can simplify it? Cause I have difficulty following.WannabeNewton said:##\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.
It simplifies both definitions and calculations. For example, can you find a simpler proof of ##\nabla\cdot(\nabla\times f)=0## than this?TimeRip496 said:I know what is it but I can't figure out the rationale behind it. As in why do we need it?
Not sure what you mean. Are you talking about how to prove identities like ##\varepsilon_{ijk}\varepsilon_{ijl}=\delta_{kl}##?TimeRip496 said:Moreover I don't know how to represent it in terms of kronecker delta. How do you do that?