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Tetrads in General Relativity

VII. Fermi-Walker Transport

FW Bivector

   To count as an observer frame, an accelerated tetrad must conform to the condition of ‘non-rotation’ as stated in Subsection [Observer Tetrad]. The spatial basis at each point is still orthonormal, with ${{\text{e}}_i} \cdot {\mathbf{u}} = 0$, but must be transformed, from point to point, in a precise sense by a sequence of infinitesimal Lorentz boosts, without additional rotations.

   This requirement is uniquely met by specifying the rotation vector in the equation of motion (7.5) for the tetrad $\{ {\operatorname{e} _a}\}$, as the so-called acceleration bivector, in this particular case called the Fermi-Walker (FW) bivector:

7.7

\[  \frac{{D{\operatorname{e} _a}}}{{d\lambda }} = {{\mathbf{\Omega }}_{\text{F}}} \cdot {\operatorname{e} _a}{\quad} {{\mathbf{\Omega }}_{\text{F}}}: = ({\mathbf{\dot u}} \wedge {\mathbf{u}})  \]

The base vectors are then carried along from one instant to the next by a pure boost in the ${\mathbf{u}}$-frame; they are said to be Fermi-Walker transported. If the proper 4-acceleration ${\mathbf{\dot u}}(\lambda )$, defined in (7.4), would be zero, the observer would be freely falling and would be parallel transporting his tetrad as in (7.1). 

FW bivector

  1. respects both the orthogonality ${\mathbf{u}} \cdot {\mathbf{\dot u}} = 0$ and the normalization ${\mathbf{u}} \cdot {\mathbf{u}} = {{\text{e}}_0} \cdot {{\text{e}}_0} = 1$;
  2. generates the appropriate Lorentz boosts in the timelike plane spanned by the 4-velocity and the 4-acceleration;
  3. excludes a rotation in any other plane, in particular, any spacelike plane.

   Physically, the FW bivector ${{\mathbf{\Omega }}_{\text{F}}}$ represents the acceleration projected into the instantaneous rest frame of the observer defined by ${\mathbf{u}}$. In this frame the projected bivector is strictly timelike and the generator of a pure boost.