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A The Volume Tensor

Here we list a number of general identities for the completely antisymmetric volume tensor in n dimensions. The most useful are those involving the tensor product (including, as needed, contractions over indices), of the volume tensor with itself [114]:

μ1...μn s [μ1 μn ] ε εν1...νn = (− 1) n!δ ν1 ⋅⋅⋅δ νn, (360 ) εμ1...μjμj+1...μnε = (− 1)s (n − j)!j! δ[μj+1 ⋅⋅⋅δ μn], (361 ) μμ1....μ..jμνj+1...νn s νj+1 νn ε1 nεμ1...μn = (− 1) n!, (362 )
where s is the number of minus signs in the metric (e.g. s = 1 for spacetime). We have used the variation of the volume tensor with respect to the metric in the actions principle presented in Sections 8, 9, and 10. We will here derive this variation using the identities above as applied to four-dimensional spacetime (s = 1 and n = 4).

Start by writing Equation (360View Equation) as

s gμ1λ1gμ2λ2gμ3λ3gμ4λ4ελ1λ2λ3λ4εν1ν2ν3ν4 = (− 1) n! δ[μ1ν1 ⋅⋅⋅δμn]νn, (363 )
vary it with respect to the metric, and then contract the result with εμ1μ2μ3μ4 to find
1 μ μ μ μ στ δεν1ν2ν3ν4 = --εν1ν2ν3ν4 (ε 1 2 3 4δεμ1μ2μ3μ4 + 4!g δgστ), (364 ) 4!
where we have used
( ) 0 = δ (δμν) = δ gμλg λν ⇒ δgμν = − gμλgνρδgλρ. (365 )
If we now contract with εν1ν2ν3ν4 we find
4! εμ1μ2μ3μ4δ εμ1μ2μ3μ4 = − --gσρδgσρ (366 ) 2
and thus
1 σρ δεν1ν2ν3ν4 = -εν1ν2ν3ν4g δg σρ. (367 ) 2

The last thing we need is the variation of the determinant of the metric, since it enters directly in the integrals of the actions. Treating the metric as a 4 × 4 matrix, and “normalizing” the ε by dividing by its one independent component, the determinant is given by

1 g = ---0123-2εμ1μ2μ3μ4εν1ν2ν3ν4gμ1ν1gμ2ν2gμ3ν3gμ4ν4. (368 ) 4!(ε )
The right-hand-side is proportional to the left-hand-side of Equation (362View Equation) and thus
√ --- 0123 --1-- ε0123 = − g, ε = √ − g-. (369 )
It is not difficult to show
√--- 1√ --- μν δ − g = -- − gg δgμν. (370 ) 2


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