Consider Gibbons and Hawkings paper wherein a Riemannian metric $\overset{\mathcal{R}}{g}_{\mu\nu}$ and everywhere well defined normalized line field $l_{\mu}$ on spacetime $M$ may be used to construct a Lorentzian metric $\overset{\mathcal{L}}{g}_{\mu\nu}$ on M.
$$\overset{\mathcal{L}}{g}_{\mu\nu}=\overset{\mathcal{R}}{g}_{\mu\nu}-2l_{\mu}l_{\nu}$$
Suppose we choose everwhere orthonormal tetrads (or frame fields, a la Cartans' method du repere mobile ) for both our Riemannian and Lorentzian metrics.
$$\begin{array}{ccc} \overset{\mathcal{L}}{g}_{\mu\nu}=e_{\mu}^{a}\eta_{ab}e_{\nu}^{b} & , & \overset{\mathcal{R}}{g}_{\mu\nu}=E_{\mu}^{a}\mathbb{I}_{ab}E_{\nu}^{b}\end{array}$$
Where $\eta_{ab}$ and $\mathbb{I}_{ab}$ are the Minkowskian and Euclidean metric components respectively. The problem I'm working on is: I would like to find an expression relating a local Lorentzian coframe to it's Euclidean counterpart $e^{a}$ and $E^{a}$ respectively.
We could also choose to represent our line field in terms of the flat space basis:
$$l_{\mu}=E_{\mu}^{a}l_{a}$$
Then we have for expression [eq:1]:
$$e_{\mu}^{a}\eta_{ab}e_{\nu}^{b}=E_{\mu}^{a}\left(\mathbb{I}_{ab}-2l_{a}l_{b}\right)E_{\nu}^{b}$$
But in flat space basis, locally we can also use Gibbons and Hawkings formula and write:
$$\eta_{ab}=\left(\mathbb{I}_{ab}-2l_{a}l_{b}\right)$$
Implying that we can always choose $e_{\mu}^{a}=E_{\mu}^{a}$ locally. Then the problem seems to boil down to the Clifford algebra underlying the flat space metric components? Clearly in an adapted frame, three of the four components are unaffected by the change. For the fourth, I keep getting spinor like quantities..
Since multiplying a clifford matrix by $i$ changes the diagonal metric component sign, this seems to involve complexifying the $\gamma^0$ component or in a general coordinate system, complexifying the Whole Clifford algebra itself which I suppose makes sense since then the Lorentzian and Euclidean Clifford algebras are isomorphic. I'm guessing we're then working with complexifications of the clifford algebra (and hence any structure groups of frame bundles we're looking at) Any help is appreciated.
Another viewpoint is that we might possibly consider the line field as a complex $su(2)$ field so that in making metric components we get a negative sign??? We might then view the field as deforming the Euclidean frame to a lorentzian one?
