Frame Fields in General Relativity - Nonspinning and Inertial Frames

Nonspinning and Inertial Frames

Some frames are nicer than others. Particularly in vacuum or electrovacuum solutions, the physical experience of inertial observers (who feel no forces) may be of particular interest. The mathematical characterization of an inertial frame is very simple: the integral curves of the timelike unit vector field must define a geodesic congruence, or in other words, its acceleration vector must vanish:

It is also often desirable to ensure that the spatial triad carried by each observer does not rotate. In this case, the triad can be viewed as being gyrostabilized. The criterion for a nonspinning inertial (NSI) frame is again very simple:

This says that as we move along the worldline of each observer, their spatial triad is parallel-transported. Nonspinning inertial frames hold a special place in general relativity, because they are as close as we can get in a curved Lorentzian manifold to the Lorentz frames used in special relativity (these are special nonspinning inertial frames in the Minkowski vacuum).

More generally, if the acceleration of our observers is nonzero, we can replace the covariant derivatives

with the (spatially projected) Fermi-Walker derivatives to define a nonspinning frame.

Given a Lorentzian manifold, we can find infinitely many frame fields, even if we require additional properties such as inertial motion. However, a given frame field might very well be defined on only part of the manifold.