Lorentz transformations btw rotating frames of reference

[geordief]

If  a space craft is approaching the Earth with a speed of v and the Earth is rotating around its axes are there simple  transformations btw a frame of reference in the craft and a  frame with its origin on the surface of the (completely spherical earth)?

 

Or are we just talking about an approximation  and does the Earth drag its frame of reference ,even if unnoticeably so?

[Genady]

13 minutes ago, geordief said:

If  a space craft is approaching the Earth with a speed of v and the Earth is rotating around its axes are there simple  transformations btw a frame of reference in the craft and a  frame with its origin on the surface of the (completely spherical earth)?

 

Or are we just talking about an approximation

Simple - no, but not an approximation either. It is an exact matrix multiplication.

 

15 minutes ago, geordief said:

does the Earth drag its frame of reference

Frame dragging comes with gravity, not with coordinate transformation.

[joigus]

I'd say that given that both \( \omega_{\textrm{Earth}}R_{\textrm{Earth}}\ll c \) --slow-rotating Earth-- and \( \frac{GM_{\textrm{Earth}}}{R_{\textrm{Earth}}}\ll c^{2} \) --not very intense gravitational field--, you're quite safe using Lorentz transformations that factor out into a boost --jump to a constant-velocity frame-- and a slow rotation.

For finer effects you would want to consider GR --Lens-Thirring effect, and such.

Does that answer you question?

[geordief]

18 minutes ago, joigus said:

I'd say that given that both ωEarthREarth≪c --slow-rotating Earth-- and GMEarthREarth≪c2 --not very intense gravitational field--, you're quite safe using Lorentz transformations that factor out into a boost --jump to a constant-velocity frame-- and a slow rotation.

For finer effects you would want to consider GR --Lens-Thirring effect, and such.

Does that answer you question?

Oh yes ,I was just  thinking about the subject and would know very little about it.

It is just to give myself another viewpoint on relativity overall.

You gave me a couple of terms that might make sense if I come across them another time.

 

(At least this thread ,short as it is is controversy free

[joigus]

2 minutes ago, geordief said:

(At least this thread ,short as it is is controversy free

Fingers crossed.

 

[md65536]

14 hours ago, geordief said:

If  a space craft is approaching the Earth with a speed of v and the Earth is rotating around its axes are there simple  transformations btw a frame of reference in the craft and a  frame with its origin on the surface of the (completely spherical earth)?

The transformations are between the coordinates in the respective reference frames, and I think you can/must choose the coordinates for Earth here. For a spherically symmetric rotating Earth with nothing else in the universe, the Kerr metric should work (an exact solution), and Kerr–Schild coordinates is apparently a good choice. I think this would be coordinates where the Earth is rotating, not a rotating frame where Earth is completely stationary, but I suppose an additional transformation between those could be used. But I think you could choose different metrics, including ones without exact solutions, and different coordinate systems that I think could include or not some of the negligible relativistic effects.

 

For the transformation, would you basically derive the Lorentz transformation using Kerr–Schild coordinates?

Edited February 26, 2023 by md65536