# Kronecker Delta & Levi Civita manipulation

Homework Statement:
I know what I need to do but I am stuck on one of the steps
Relevant Equations:
Levi-civita and Kronecker delta identities
εikl εjmngkmMkn =

εikl εjknMkn = (in book it changed sign to -εikl εjknMkn - Why? )

By identity
εikl εnjkMln = inδkj - δijδkn)Mkn = ?

I then get ..

Mji - δij Mnn ( is this correct ?)

There 's more to the question but if can get this part right, I should be able to complete the rest.

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## Answers and Replies

nrqed
Homework Helper
Gold Member
Homework Statement:: I know what I need to do but I am stuck on one of the steps
Relevant Equations:: Levi-civita and Kronecker delta identities

εikl εjmngkmMkn =

εikl εjknMkn = (in book it changed sign to -εikl εjknMkn - Why? )
Oops I had not read carefully. What you wrote as the first expression can't be right! Note that you have three indices "k", which is incorrect. Check what the correct answer is.

Last edited:
MichaelAlexDavM
This is a bit confusing because normally using the metric should raise or lower the indices, but here all your indices are downstairs, so I am not sure what the convention is, here.

Yes, those steps are correct.

Thanks very much, I have just finished that part of the question and it worked out. I was not confident of what I was doing but I am now.

The fact that there were only lower indices confused me too, I vaguely remember my lecturer saying because its Minkowski space, contravariant and covariant are the same, might have heard that wrong though.

nrqed
Homework Helper
Gold Member
Thanks very much, I have just finished that part of the question and it worked out. I was not confident of what I was doing but I am now.

The fact that there were only lower indices confused me too, I vaguely remember my lecturer saying because its Minkowski space, contravariant and covariant are the same, might have heard that wrong though.
AH ok. But the very first expression you wrote was incorrect, right? It had three times the index "k".

Dr Transport
Gold Member
OP,

Go back and look at your expressions carefully, from the beginning, you've made errors in copying from the problem.

strangerep
I vaguely remember my lecturer saying because its Minkowski space, contravariant and covariant are the same, might have heard that wrong though.
I'd guess you misheard. They're not the same in general.

What convention is your lecturer using for the Minkowski metric? Is it diag##(-,+,+,+)## or diag##(+,-,-,-)## ?

I'd guess you misheard. They're not the same in general.

What convention is your lecturer using for the Minkowski metric? Is it diag##(-,+,+,+)## or diag##(+,-,-,-)## ?
Now I see that the contravariant and covariant metric tensor are not the same. x0 = x0 and xa = - xa with the (+,-,-,-) metric we used. The same lecturer also teaches classical mechanics and it was during this class, when I think he said there is no difference between the contravariant and covariant indices. He may have mentioned flat space and I took this to mean (without thinking about it enough) Minkowski space.
These are basic mistakes I am making and I am quite sure I have no natural talent for this subject but I still love it.
Thanks for your knowledge, now I understand a little more.

Gaussian97
Homework Helper
Minkowski space is flat, so he probably didn't say flat space, I would bet that he said Euclidean space.

MichaelAlexDavM