When we generate the Levi-Civita image like a determinant like we did higher than, the thoroughly antisymmetric house that it possess will become apparent: swapping any two indices corresponds to interchanging their corresponding rows in the matrix due to which the determinant, which is the Levi-Civita by itself, improvements indication.
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RexcirusRexcirus 43544 silver badges1515 bronze badges $endgroup$ two $begingroup$ Your final expression will not make A lot sense: The LHS has totally free indices $i,j,k$ Whilst your RHS has free indices $i,j,l$. $endgroup$
I'm utilizing the Einstein notation, so several indices are summed) or they provide two 3-indices symbols with distinct signal
These properties are important for being familiar with the behavior from the tensor in numerous coordinate units.
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This tensor shouldn't be bewildered Along with the tensor density area outlined above. The presentation During this section carefully follows Carroll 2004.
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In any arbitrary curvilinear coordinate method and also inside the absence of a metric within the manifold, the Levi-Civita symbol as outlined previously mentioned could possibly be thought to be a tensor density subject in two various ways.
but see that In the event the metric signature incorporates an odd number of damaging eigenvalues q, then the sign in the elements of the tensor differ in the typical Levi-Civita image:
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E α β γ δ E ρ σ μ ν = − g α ζ g β η g γ θ g δ ι δ ρ σ μ ν ζ η θ ι E α β γ δ E ρ σ μ ν = − g α ζ g β η g γ θ g δ ι δ ζ η θ ι ρ σ μ ν E α β γ δ E α β γ δ = − 24 E α β γ δ E ρ β γ δ = − six δ ρ α E α β γ δ E ρ σ γ δ = − 2 δ ρ σ α β E α β γ δ E ρ σ θ δ = − δ ρ more info σ θ α β γ . displaystyle commence aligned E_ alpha beta gamma delta E_ rho sigma mu nu &=-g_ alpha zeta g_ beta eta g_ gamma theta g_ delta iota delta _ rho sigma mu nu ^ zeta eta theta iota E^ alpha beta gamma delta E^ rho sigma mu nu &=-g^ alpha zeta g^ beta eta g^ gamma theta g^ delta iota delta _ zeta eta theta iota ^ rho sigma mu nu E^ alpha beta gamma delta E_ alpha beta gamma delta &=-24E^ alpha beta gamma delta E_ rho beta gamma delta &=-6delta _ rho ^ alpha E^ alpha beta gamma delta E_ rho sigma gamma delta &=-2delta _ rho sigma ^ alpha beta E^ alpha beta gamma delta E_ rho sigma theta delta &=-delta _ rho sigma theta ^ alpha beta gamma ,.conclude aligned
E α β γ δ = g α ζ g β η g γ θ g δ ι E ζ η θ ι . displaystyle E^ alpha beta gamma delta =g^ alpha zeta g^ beta eta g^ gamma theta g^ delta iota E_ zeta eta theta iota ,.
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