Question

In the text book Gravity written by Hartle (p.302), the precession angle per orbit of a gyroscope, initially lied in the equatorial plane and in the radial directon) moving in a circular orbit in the equatorial plane around a central spherical non-rotating mass) as measured by a stationary observer, is calculated (eq. (14.18)). In the next paragraph it is claimed that the precession angle of the gyro measured by the comoving observer is the same. Honestly, I do not understand the reasoning. So I think I'd better calculate it directly for the comoving observer and see its equality with (14.18). Would you please help me in the calculation? Or any explanation to understand the Hartle's reasoning?

A) Calculate the precession angle for the comoving observer by considering the gyroscope's motion relative to the observer's frame and compare it directly with equation (14.18).
B) Discuss the principles of general relativity and how they affect the measurement of precession angles from different observer frames.
C) Ignore the comoving observer's perspective as it does not influence the calculation of precession angles for a gyroscope orbiting around a non-rotating mass.
D) Consult additional sources to understand Hartle's reasoning as it might involve complex relativistic concepts beyond direct calculation.

Answer 1

The precession angle of a gyroscope in a gravitational field is consistent across both stationary and comoving observers due to the principles of general relativity, which ensures that measurements of physical phenomena like gyroscope precession remain equivalent between different reference frames.

Step-by-step explanation:

The concept of gyroscope precession in a gravitational field, as described by Hartle's textbook, involves complex principles of general relativity. In the textbook Gravity, by Hartle, the precession angle per orbit calculated for a gyroscope moving in a circular orbit around a central spherical non-rotating mass can be measured both by a stationary observer and a comoving observer. Calculating the precession angle for the comoving observer involves considerations of the gyroscope's relative motion, as seen from the observer's own frame of reference.

A key principle in general relativity is the equivalence of observational measurements between different frames of reference. Thus, if Hartle states the precession angle measured by a comoving observer is the same as that measured by a stationary observer, it's because the physical effects (the curvatures of spacetime around the mass) affecting the gyroscope are the same regardless of the observer's motion. The calculations can therefore be expected to yield the same result, consistent with Equation (14.18) from Hartle's book.

The general relativity framework ensures that the precession angles measured by any observer, whether stationary or comoving, will be consistent as long as they account for the proper relativistic effects. Therefore, it is unnecessary to recalculate the precession angle for the comoving observer if the stationary observer's calculation has already accounted for the relativistic effects.