After we generate the Levi-Civita symbol to be a determinant like we did earlier mentioned, the absolutely antisymmetric property that it possess gets to be obvious: swapping any two indices corresponds to interchanging their corresponding rows while in the matrix because of which the determinant, which happens to be the Levi-Civita by itself, modifications indicator.
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or Minkowski Place. The values on the Levi-Civita image are unbiased of any metric tensor and coordinate procedure. Also, the precise phrase "symbol" emphasizes that it's not a tensor as a result of how it transforms concerning coordinate devices; even so it can be interpreted as a tensor density.
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The Levi-Civita tensor is Employed in the Einstein field equations to specific the curvature of spacetime with regard to the Power and momentum of issue and radiation.
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but observe that If your metric signature contains an odd quantity of adverse eigenvalues q, then the indication from the factors of the tensor vary in the normal 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 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 &=-twodelta _ rho sigma ^ alpha beta E^ alpha beta gamma delta E_ rho sigma theta delta &=-delta _ rho sigma theta ^ alpha beta gamma ,.end aligned
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