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Tetrads in General Relativity

IV. Spin Connection

Connection Bivector

   In GA, the generators of tetrad Lorentz transformations are bivectors. This statement may be explicated by introducing the spin connection bivector

4.4

\[  {{\mathbf{\omega}}_\mu}: = \frac{1}{2} {\Sigma _{\mu ab}}{{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b}  \]

which has the spin (Lorentz) connection coefficients as its components. In terms of this bivector, the connection relation (4.1) may be written in the form

4.5

\[  {D_\mu }{{\mathbf{e}}_a} = \frac{1}{2}[{{\mathbf{\omega }}_\mu },{{\mathbf{e}}_a}] = {{\mathbf{\omega }}_\mu } \cdot {{\mathbf{e}}_a}  \]

It is a general theorem in GA that the commutator of a bivector with a vector is their scalar product. For the proof one may use the identities (3.6)(3.7).

   Equation (4.5) describes a set of orthonormal basis vectors that are rotated from point to point in spacetime. Because the connection coefficients ${\Sigma _{\mu ab}}$ map vectors to bivectors, there can only be 4×6 = 24 degrees of freedom, corresponding to 3 rotations with generators ${{\mathbf{e}}^i} \wedge {{\mathbf{e}}^j}$, and 3 boosts with generators ${{\mathbf{e}}^0} \wedge {{\mathbf{e}}^j}$, in each of the 4 possible directions of displacement ${{\mathbf{g}}_\mu }$. Thus, the connection bivector has a clear geometric meaning as the generator of the Lorentz transformations displacing a set of orthonormal basis vectors in a particular direction. Its value corresponds to the angular velocity of rotation.

   The replacement ${\partial _\mu } \to {\partial _\mu } + {{\mathbf{\omega }}_\mu }$ in curved spacetime suggest a similarity between the bivector connection and a gauge field in gauge theories. The gauge transformations in the case of general relativity would then correspond to the symmetry group of local orthogonal rotations (Lorentz transformations) that express the equivalence of observers associated with the tetrad frames; see section VI.