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Tetrads in General Relativity

Appendix A

Levi-Civita Tensor

   In the tetrad formalism, it is natural 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 volume element and its inverse, defined in Volume Element, may be written according to

A.1

\[{\operatorname{I} _{4}} = - \frac{\varepsilon ^{abcd}}{{4!}}{}{{\mathbf{e}}_a} \wedge {{\mathbf{e}}_b} \wedge {{\mathbf{e}}_c} \wedge {{\mathbf{e}}_d}\quad \operatorname{I} ^{4} = \frac{\varepsilon _{abcd}}{{4!}}{}{{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \wedge {{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d}\]

These relations may be inverted to give following useful representations of the unit 4-blades:

A.2

\[{{\mathbf{e}}_a} \wedge {{\mathbf{e}}_b} \wedge {{\mathbf{e}}_c} \wedge {{\mathbf{e}}_d} = {\varepsilon _{abcd}}\operatorname{I} _{4}^{} \quad {{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \wedge {{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d} = - {\varepsilon ^{abcd}}\operatorname{I} ^{4}\]

   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 that 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

A.3

\[{\epsilon_{\mu \nu \kappa \lambda }}: = \sqrt {\left| g \right|} {\varepsilon _{\mu \nu \kappa \lambda }} \qquad {\epsilon^{\mu \nu \kappa \lambda }}: = {\left| g \right|^{ - 1}}{\epsilon_{\mu \nu \kappa \lambda }} = {\left| g \right|^{ - 1/2}}{\varepsilon ^{\mu \nu \kappa \lambda }}\]

These formulae are consistent with the vierbein transformation

A.4

\[{\epsilon_{\mu \nu \kappa \lambda }}  = {e_\mu }^a{e_\nu }^b {e{_\kappa}^c}{e_\lambda }^d{\varepsilon _{abcd}} = \det [{e_\mu }^a]{\varepsilon _{\mu \nu \kappa \lambda }} = \sqrt {\left| g \right|} {\varepsilon _{\mu \nu \kappa \lambda }}\]

and similarly for the second equation in (A.3).

   The above formulae may be extended to spaces of arbitrary dimension $n$ with generalized orthonormal metric ${{\mathbf{e}}_a} \cdot {{\mathbf{e}}_b} = {\eta _{ab}} = {\text{diag}}(1,...,1, - 1,..., - 1)$ of Lorentz signature $(n-s,s)$, where $s$ is the number of negative elements. Indices are raised and lowered with the generalized metric; in particular ${\varepsilon ^{0123...n - 1}} = {\left( { - 1} \right)^s}{\varepsilon _{012...n - 1}}$.

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