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XI. Field Equations
At this point, it is useful to introduce the completely
anti-symmetric Levi-Civita symbol ${\varepsilon
_{abcd}}$, invariant under ${\text{SO}}(1,3)$, with values $ \pm 1$.
Its normalization is chosen to be ${\varepsilon _{0123}} = 1 = -
{\varepsilon ^{0123}}$, i.e. the Levi-Civita symbol equals the sign
of a permutation when the indices are all unequal. De minus sign
reflects the metric signature of the Lorentzian manifold.
By means of the Levi-Civita symbol, the unit pseudoscalar volume and
its inverse, defined in Pseudoscalar
Volume, may be written according to
\[{\operatorname{I} _{4}} = -
\frac{\varepsilon ^{abcd}}{{4!}}{}{{\mathbf{e}}_a} \wedge
{{\mathbf{e}}_b} \wedge {{\mathbf{e}}_c} \wedge
{{\mathbf{e}}_d}\quad \operatorname{I} _{4}^{ - 1} =
\frac{\varepsilon _{abcd}}{{4!}}{}{{\mathbf{e}}^a} \wedge
{{\mathbf{e}}^b} \wedge {{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d}\]
These relations may be inverted to give following useful
representations of the unit 4-blades:
\[{{\mathbf{e}}_a} \wedge
{{\mathbf{e}}_b} \wedge {{\mathbf{e}}_c} \wedge {{\mathbf{e}}_d} =
{\varepsilon _{abcd}}\operatorname{I} _{4}^{} \quad {{\mathbf{e}}^a}
\wedge {{\mathbf{e}}^b} \wedge {{\mathbf{e}}^c} \wedge
{{\mathbf{e}}^d} = - {\varepsilon ^{abcd}}\operatorname{I} _{4}^{ -
1}\]
The term 'symbol' emphasizes that the Levi-Civita symbol does not
behave as a tensor: under a general coordinate change, the
components of the permutation tensor are multiplied by the Jacobian
of the transformation matrix. Objects that transform this way are
known as tensor densities. [Wikipedia:
Levi-Civita symbol]
In curved spacetime, one may construct a covariant Levi-Civita
tensor (also known as the Riemannian volume form) by defining
\[{\epsilon_{\mu \nu \kappa
\lambda }}: = \sqrt {\left| g \right|} {\varepsilon _{\mu \nu \kappa
\lambda }} \qquad {\epsilon^{\mu \nu \kappa \lambda }}: = {\left| g
\right|^{ - 1}}{\epsilon_{\mu \nu \kappa \lambda }} = {\left| g
\right|^{ - 1/2}}{\varepsilon ^{\mu \nu \kappa \lambda }}\]
These formulae are consistent with the vierbein transformation
\[{\epsilon_{\mu \nu \kappa
\lambda }} = {e_\mu }^a{e_\nu }^b {e{_\kappa}^c}{e_\lambda
}^d{\varepsilon _{abcd}} = \det [{e_\mu }^a]{\varepsilon _{\mu \nu
\kappa \lambda }} = \sqrt {\left| g \right|} {\varepsilon _{\mu \nu
\kappa \lambda }}\]
and similarly for the second equation in (11.14).
\[