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The phrase "n-dimensional Levi-Civita image" refers to the fact that the amount of indices to the image n matches the dimensionality from the vector Room in problem, which may be Euclidean or non-Euclidean, such as, R 3 displaystyle mathbb R ^ three

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then left aspect is zero, and correct facet can be zero because two of its rows are equivalent. Similarly for j c = j c + one displaystyle j_ c =j_ c+one

The Levi-Civita tensor is Utilized in the Einstein field equations to precise the curvature of spacetime with regards to the energy and momentum of subject and radiation.

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The next are examples of the final id previously mentioned specialized to Minkowski space (with the detrimental signal arising from the odd amount of negatives inside the signature of the metric tensor in both sign Conference):

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For each sequence of nonempty open up sets You will find a disjoint sequence of nonempty open up sets "underneath" it

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 visit = − two δ ρ σ α β E α β γ δ E ρ σ θ δ = − δ ρ σ θ α β γ . displaystyle start off 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 &=-sixdelta _ 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 ,.finish aligned

The real key defining assets with the symbol is total antisymmetry in the indices. When any two indices are interchanged, equivalent or not, the symbol is negated:

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