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Tetrads in General Relativity

VII. Fermi-Walker Transport

FW Transport Equation

   From the FW transport equation (7.7) it is possible, at least in principle, to determine the unit space vectors $\{ {\operatorname{e} _i}(\lambda )\}$ at each point of the curve by solving the differential equations

7.8

\[  \frac{{D{\operatorname{e} _i}}}{{d\lambda }} = ({\mathbf{\dot u}} \wedge {\mathbf{u}}) \cdot {\operatorname{e} _i} = - \left( {{\operatorname{e} _i} \cdot \frac{{D{\mathbf{u}}}}{{d\lambda }}} \right) {\mathbf{u}}  \]

for some orthonormal initial vectors ${\operatorname{e} _i}(0)$. Equation (7.8) determines the transport of the orthonormal basis $\left\{ {{\mathbf{u}},{{\text{e}}_i}} \right\}$ along an arbitrary timelike curve ${\mathbf{x}}(\lambda )$, such that ${{\text{e}}_0} = {\mathbf{u}}$ is always tangent to this curve and the spatial base vectors non-precessing. The initial spatial frame can be arbitrarily chosen on some space-like hypersurface, e.g. the initial data surface. The unique solution obtained this way is called the Fermi–Walker transported Lorentz frame along the given curve ${\mathbf{x}}(\lambda )$.

   In calculations it is useful to know the connection coefficients. In the FW transport equation, they are only needed on the observer’s worldline. Some of the connection coefficients are determined by the FW bivector which has the components ${({\mathbf{\dot u}} \wedge {\mathbf{u}})_{ab}} = 2{\dot u_{[a}}{u_{b]}}$ . Comparing with the rotation bivector (7.6), one finds the Ricci rotation coefficients

7.9

\[ {{\mathbf{\Omega }}_{ab}} = {\omega _0}_{ab} = 2{\dot u_{[a}}{u_{b]}}  \]

   FW-transport may be physically realized in the observer frame by a system of three inertial gyroscopes attached to the spatial directions. The three spacelike tetrad fields should keep their orientation aligned to the system of gyroscopes.