anonanon 152k1212 gold badges233233 silver badges398398 bronze badges $endgroup$ 4 $begingroup$ That is perfect thanks a bunch! I also posted a similar issue below: math.stackexchange.com/inquiries/2135276/… that you might have some Perception on? $endgroup$
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= δ i l ( δ j m δ k n − δ j n δ k m ) − δ i m ( δ j l δ k n − δ j n δ k l ) + δ i n ( δ j l δ k m − δ j m δ k l ) .
or Minkowski Area. The values in the Levi-Civita symbol are independent of any metric tensor and coordinate technique. Also, the specific term "symbol" emphasizes that it's not a tensor as a consequence of how it transforms concerning coordinate units; even so it may be interpreted like a tensor density.
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In almost any arbitrary curvilinear coordinate program and in some cases within the absence of the metric on the manifold, the Levi-Civita symbol as defined earlier mentioned might be considered to be a tensor density discipline in two various ways.
In Einstein notation, the duplication from the i index indicates the sum on i. The earlier is then denoted εijkεimn = δjmδkn − δjnδkm.
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where gab will be the illustration from the metric in that coordinate procedure. We can equally contemplate a contravariant Levi-Civita tensor by increasing the indices Together with the metric as typical,
Do the thing is how this tends to affect the worth of $epsilon^ alphabetagammadelta epsilon_ alpha'beta'gamma'delta' $? $endgroup$