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I'm utilizing the Einstein notation, so multiple indices are summed) or they offer two three-indices symbols with unique indicator
These properties are essential for comprehending the habits from the tensor in several coordinate systems.
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This tensor shouldn't be puzzled Using the tensor density field stated over. The presentation During this section carefully follows Carroll 2004.
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In almost any arbitrary curvilinear coordinate procedure and also while in the absence of a metric to the manifold, the Levi-Civita image as defined previously mentioned could possibly be regarded as being a tensor density area in two different ways.
but discover that if the metric signature consists of an odd quantity of negative eigenvalues q, then the signal with the factors of the tensor differ through the regular Levi-Civita symbol:
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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 check here 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 = − 6 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 σ θ α β γ . displaystyle begin 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 ,.conclusion aligned
The crucial element defining home from the symbol is whole antisymmetry in the indices. When any two indices are interchanged, equivalent or not, the symbol is negated:
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