(self study) Time Dilation on Earth

[LGPz]

Hello,

Firstly, let me point out that I do not have much knowledge in physics. However, after reading a certain book, I wanted to find out more about time dilation. I spent a few days reading about it. Here is the problem:

Homework Statement

I would like to be able to calculate the time dilation at any point of the Earth relative to another. From what I have seen the following method is the most used and the simplest:

Method 1

- Use the Lorentz factor to calculate time dilation due to velocity.
- Use the Schwarzschild radius to calculate the Gravitational time dilation

My issue with this formula is that if we simple substitute v with the speed of the object, we wouldn't be taking into account the rotation of the Earth. (A place in the poles is moving faster than a place on the equator). Furthermore, we wouldn't account for the difference in speed based on the altitude (higher points move faster).


Can you think of another factor which needs to be taken into account?

How do I account for these factors?

This website provides a formula using the angular velocity of Earth rotation. This might take into consideration the height. Could someone let me know how to calculate the angular velocity?

Methods 2 and 3

Do you think this formula on time dilation due to gravitation and motion together would be my best option? Do you know where can I find example calculations?

Or I can use Kerr metric instead of the Schwarzschild solution. However, I couldn't find examples on how to calculate time dilation using this system. Could someone tell me where can I read more about it?

I understand that you might not have time / be able to provide me with complete answers, however, would you mind giving me useful links where I can continue my readings?

Thank you.

 

[BvU]

Hello LGP, and welcome to PF.
A lot of questions, and nothing to hold on to (you didn't use the template).
What is the distance between someone on the north pole and someone on the equator? Wouldn't you say this distance is a constant in time ? So no relative motion, no time dilation.

Angular velocity is simply angle/time. One revolution per day means an angular velocity of $2\pi\over 24 * 60 * 60$ rad/s.

 

[vanhees71]

Well, that's subtle. Even Einstein got it wrong in his famous paper of 1905, but one cannot blame him for that, because he had not dicovered General Relativity at this time, of course. There's a very nice paper in Scientific American on it:

Alex Harvey and Engelbert Schucking, A small puzzle from 1905, Physics Today (March 2005)
http://dx.doi.org/10.1063/1.1897562

 

[LGPz]

Thank you for your answer, BvU.

While the distance between the two people might not change, this site says that we should take the center of the Earth as an approximation to an inertial frame. Since these two locations move at different speeds relative to the center, there is time dilation. I might be wrong, in which case please correct me.

Your explanation of angular velocity helped me realize what I need to do and I think that I have now solved my problem. Again, thanks for your help.

 

[vanhees71]

This is wrong! The same mistake has been made by Einstein in 1905 in his famous paper, that discovered Special Relativity. He couldn't know that 10 years later he would find another more comprehensive theory, General Theory of Relativity.

Perhaps, you read this marvelous article in Physics Today, I've posted yesterday. There it is pretty well explained that there is no (or negligible) time dilation between the two observers, because the Earth is (approximately) a equipotential surface of the gravitational field (spoken in Newtonian terms, which are well justified because of the weakness of the Earth's gravitational field).

The important point is that a fixed point on Earth is not exactly an inertial frame. This you can even see using Newtonian mechanics and it's demonstrated quite often by Foucault pendulae. Many science museums exhibit one; also some universities have one installed somewhere in their physics departments.

 

[BvU]

I have no problem admitting I was wrong, especially if I end up in the same corner as A.E. (I also haven't really discovered GR yet...).

LGP didn't help me very much by asking for a correction to special relativity (if I may consider it like that) and an expression for angular velocity in one and the same post. Felt like applying a band-aid while doing open heart surgery (on the same patient). Thought I'd start with the band-aid.

Makes me realize how useful the template is. In fact I cordially invite LGPz to show what he's done so far (as mentioned in #4). Might give vanH a chance to help you further (he's the theorist - I'm out of my depth as a mere experimental physicist, so I'll be an interested onlooker from now on).