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Tetrads in General Relativity

VII. Fermi-Walker Transport

Observer Tetrad

   Let an (ideal) observer, undergoing arbitrary, but smooth, motion along a timelike curve ${\mathbf{x}}(\lambda )$, carry an orthonormal tetrad of basis vector fields $\{ {\operatorname{e} _a}({\mathbf{x}}(\lambda ));a = 0,1,2,3\} $; the parameter $\lambda $ is the proper time of the observer.
The time axis is chosen as the time axis of the comoving frame in which the observer is momentarily at rest. That is, the zeroth base vector is at each instant identified with the 4-velocity: ${\operatorname{e} _0} = {\mathbf{u}}$, ${\mathbf{u}} := d{\mathbf{x}}/d\lambda$. The space-like elements $\{ {\operatorname{e} _i};i = 1,2,3\} $, subject to the constraint ${\operatorname{e} _i} \cdot {\mathbf{u}} = 0 $, provide a Cartesian basis ${\operatorname{e} _i} \cdot {\operatorname{e} _j} = {\delta _{ij}}$ in the observer rest-frame.

   Two conditions on the moving observer frame are imposed:

  1. basis vectors of the tetrad must remain orthonormal;
  2. spatial base vectors should be kept ‘nonrotating’, i.e., they should remain ‘as constant as possible’ subject to the constraint ${\operatorname{e} _i} \cdot {\mathbf{u}} = 0$, without additional rotation.

The tetrad $\{ {\operatorname{e} _a}({\mathbf{x}})\} $ so constructed at each spacetime point may be interpreted as constituting a local observer frame, to be used as a tool to extract physical, measurable quantities from geometric, coordinate-free objects in general relativity.

   In the case that the observer follows a geodesic, the criterion for a non-spinning inertial frame is simply ${\mathbf{u}} \cdot D{\operatorname{e} _i} = 0$, i.e. the whole triad is parallel-transported along the worldline of the observer:

7.1

\[  \frac{{D{\operatorname{e} _a}}}{{d\lambda }} = {\mathbf{u}} \cdot D{\operatorname{e} _a} = 0  \]

Non-spinning local Lorentz frames hold a special place in GRT, because they are as close as one can get in a curved Lorentzian manifold to the inertial frames of special relativity.