Frame Fields in General Relativity - Example: Hagihara Observers in The Schwarzschild Vacuum

Example: Hagihara Observers in The Schwarzschild Vacuum

In the same way that we found the Lemaître observers, we can boost our static frame in the direction by an undetermined parameter (depending on the radial coordinate), compute the acceleration vector, and require that this vanish in the equatorial plane . The new Hagihara frame describes the physical experience of observers in stable circular orbits around our massive object. It was apparently first discussed by the distinguished (and mathematically gifted) astronomer Yusuke Hagihara.

In the static polar spherical chart, the Hagihara frame is

which in the equatorial plane becomes

The tidal tensor where turns out to be given (in the equatorial plane) by

Thus, compared to a static observer hovering at a given coordinate radius, a Hagihara observer in a stable circular orbit with the same coordinate radius will measure radial tidal forces which are slightly larger in magnitude, and transverse tidal forces which are no longer isotropic (but slightly larger orthogonal to the direction of motion).

Note that the Hagihara frame is only defined on the region . Indeed, stable circular orbits only exist on, so the frame should not be used inside this locus.

Computing Fermi derivatives shows that the frame field just given is in fact spinning with respect to a gyrostabilized frame. The principal reason why is easy to spot: in this frame, each Hagihara observer keeps his spatial vectors radially aligned, so rotate about as the observer orbits around the central massive object. However, after correcting for this observation, a small precession of the spin axis of a gyroscope carried by a Hagihara observer still remains; this is the de Sitter precession effect (also called the geodetic precession effect).