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anonanon 152k1212 gold badges233233 silver badges398398 bronze badges $endgroup$ 4 $begingroup$ That is best many thanks a bunch! I also posted the same problem in this article: math.stackexchange.com/questions/2135276/… that you might have some insight on? $endgroup$

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ε i j k ε l m n =

ε i j ε i n = δ j n displaystyle varepsilon _ ij varepsilon ^ in = delta _ j ^ n

These Qualities are essential for comprehending the behavior of your tensor in numerous coordinate systems.

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The Levi-Civita tensor is Employed in the Einstein subject equations to specific the curvature of spacetime in terms of the Power and momentum of matter and radiation.

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The subsequent are samples of the general identity above specialised to Minkowski Area (Along with the unfavorable indicator arising with the odd variety of negatives while in the signature on the metric tensor in both signal convention):

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On the pseudo-Riemannian manifold, just one may well outline a coordinate-invariant covariant tensor subject whose coordinate illustration agrees with the Levi-Civita image anywhere the coordinate program is this sort of that The premise of your tangent Room is orthonormal with respect for the metric and matches a selected orientation.

The great site true secret defining home from the image is whole antisymmetry inside the indices. When any two indices are interchanged, equivalent or not, the image is negated:

Do the thing is how this will have an effect on the worth of $epsilon^ alphabetagammadelta epsilon_ alpha'beta'gamma'delta' $? $endgroup$

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