Simultaneity in General Relativity and Problem with coordinate systems concepts.

[MManuel Abad]

Hi there, Physics lovers. I'm studying "The Classical Theory of Fields" from the "Course of Theoretical Physics" book series by Lev D. Landau, and I'm stuck with simultaneity in General Relativity.

In page 251 of the Fourth "revised" english edition, by Butterworth Heinemann, There begins the section 84 on "Distances and time intervals". At the beginning we obtein the proper time in terms of the curvilinear coordinates x⁰, x¹, x² and x³. I don't get it: Are this coordinates the coordinates with respect to what system? Any system?

Then in the next page Landau explains fig. 18. x⁰ is the time when the light signal arrives at point A. As measured by that arbitrary system? or by B?

And at last, at page 254 Landau talks about simultaneity, and considers as simultaneous with the moment x⁰ at point A that reading of the clock at B which is half-way between the moments of departure and return of the signal at that point. Now I'm really confused:

Again, with respect to what system? That very same arbitrary system? Why would x⁰ be simultaneous with x⁰+$$\Delta$$x⁰, if they're both times measured at that arbitrary system and they're different? And why $$\Delta$$x⁰ is equal to that "mean" of those differentials of x⁰??

PLEASE, I NEED YOUR HELP!

 

[bcrowell]

At the beginning we obtein the proper time in terms of the curvilinear coordinates x⁰, x¹, x² and x³. I don't get it: Are this coordinates the coordinates with respect to what system? Any system?

General Relativity allows essentially any coordinates whatsoever.

 

[MManuel Abad]

Yeahh, thank you so much... but Landau may be considering in the other precise situations which I also explain, a reference frame centered on B or something like that.

 

[Mentz114]

In page 251 of the Fourth "revised" english edition, by Butterworth Heinemann, There begins the section 84 on "Distances and time intervals". At the beginning we obtein the proper time in terms of the curvilinear coordinates x0, x1, x2 and x3. I don't get it: Are this coordinates the coordinates with respect to what system? Any system?

(my bold)
I don't have the text in front of me, but maybe this will help.

The 'system' here seems to be a curved spacetime where infinitesimal distances are defined - so there must be a metric. We have to assume that the space is endowed with coordinates, and there is a choice when we write the metric.

So, to calculate the proper intervals along worldlines, use the coordinates in which the metric is expressed, over the range in which the coordinates are valid.

 

[MManuel Abad]

Thanks a lot! It really helped! :D