SR-time dilation vs GR-time dilation on rotating Earth

[cianfa72]

TL;DR Summary

About SR vs GR time dilation effect on rotating Earth

Reading this old thread does-special-and-general-relativity-affect-aging-of-apollo-astronauts I came with some question.

To me isn't clear how SR-time dilation by rotation of the Earth is compensated by GR-time dilation effect.

 

[Dale]

Personally, I would not consider the SR effect to be separate from the GR effect. SR is part of GR. Similarly, I would not consider the gravitational potential to be separate from the centrifugal potential.

 

[Sagittarius A-Star]

To me isn't clear how SR-time dilation by rotation of the Earth is compensated by GR-time dilation effect.

I described it in other words here:
https://www.physicsforums.com/threads/gravity-at-earths-poles-vs-equator.1065794/post-7121118

A clock at the equator at sea level is in a higher gravitational potential than a clock at a pole at sea level - with respect to the (non-rotating) ECI frame. In this frame, the clock at the equator is moving (SR-time-dilation). In the (rotating) ECEF frame, the clock at the equator is at rest, but the gravitational potential is modified by the centrifugal force (also SR-time-dilation).

In 1905, when Einstein couldn't consider GR-effects, he wrote:

From this, we conclude that a clock placed at the equator must be slower by a very small amount than a similarly constructed clock which is placed at the pole, all other conditions being identical.

Source (see end of §4):
https://en.wikisource.org/wiki/On_the_Electrodynamics_of_Moving_Bodies_(1920_edition)

 

[Vanadium 50]

I would argue "please reopen this thread just for me" is, well, sounds like a privileged place in the universe, and cosmology tells us thee is no such thing.

More importantly, why don't you calculate it yourself? If you don't know how, reread the thousands of posts PF members have made explaining it to you.

 

[cianfa72]

I described it in other words here:
https://www.physicsforums.com/threads/gravity-at-earths-poles-vs-equator.1065794/post-7121118

A clock at the equator at sea level is in a higher gravitational potential than a clock at a pole at sea level - with respect to the (non-rotating) ECI frame. In this frame, the clock at the equator is moving (SR-time-dilation). In the (rotating) ECEF frame, the clock at the equator is at rest, but the gravitational potential is modified by the centrifugal force (also SR-time-dilation).

In ECI frame the center of the Earth is at rest. One can assume the Earth's center is free-falling. Does an object at fixed $(x,y,z)$ in it in free-fall as well?