(a) In the Newtonian theory of gravity, consider an axisymmetric, spinning body (e g , Earth) with spin angular momentum S j and time independent mass distribution (x), interacting with an externally produced tidal gravitational field jk (e g , that of the Sun and the Moon) Show that the torque aro...

The question seems to involve concepts from Newtonian gravity and general relativity as they apply to an axisymmetric spinning body such as the Earth ...

(a) In the Newtonian theory of gravity, consider an axisymmetric, spinning body (e.g., Earth) with spin angular...

Question:

(a) In the Newtonian theory of gravity, consider an axisymmetric, spinning body (e.g., Earth) with spin angular momentum S_j and time-independent mass distribution ρ(x), interacting with an externally produced tidal gravitational field ε_jk (e.g., that of the Sun and the Moon). Show that the torque around the body’s center of mass, exerted by the tidal field, and the resulting evolution of the body’s spin are

Here

is the body’s mass quadrupole moment, with

the distance from the center of mass.

(b) For the centrifugally flattened Earth interacting with the tidal fields of the Moon and the Sun, estimate in order of magnitude the spin-precession period produced by this torque. [The observed precession period is 26,000 years.]

(c) Show that when rewritten in the language of general relativity, and in frame independent, geometric language, Eq. (25.62) takes the form (25.59) discussed in the text. As part of showing this, explain the meaning of I_βμ in that equation.

Equation 25.59.