It dawned upon me finally to apply the Group Theory to the Sagnac
Effect. We have the group of rotations on angle FI (being
dimensionless parameter), and we have the group of Lorentz
transformations with the parameter V. If we measure time in meters
(new t=ct, which is very convenient in SR), then the parameter V also
dimensionless. We have a full analogy between FI and V. The time
derivative of FI (the angular velocity) can be measured by the Sagnac
Gyroscope from within the rotating frame. It should be then, that the
time derivative of V (the acceleration) can be measured by the "Sagnac
Accelerometer" (Based on some kind of linear Sagnac Effect) from
within the accelerating frame.

The physical meaning of the "Sagnac Accelerometer" should be
understandable: suppose we have a stick with two mirrors on the ends
and a light signal bouncing between the mirrors infinitely (supported
by a laser). At zero acceleration the frequency of bounces on the
front mirror equals to the one on the rear mirror. Suppose now that
the stick accelerates (important: we are watching this acceleration
from the inertial frame). The frequency of bouncing on the front
mirror will decrease and the frequency of bouncing on the rear mirror
will increase (by the time of inertial frame). The simple calculation
gives dt=2ad^2/c^3 for each act of reflection (a- acceleration, d-
half of the stick length). This compare to dt=4piR^2w/c^2 for the
circular Sagnac Effect (w- angular velocity).

This accelerometer, most likely, will work in a space without gravity
where acceleration is caused by a rocket. What about a free fall in a
gravitation field? We have the contradiction to the Local Equivalence
Principal. Now we have a good opportunity to check it.