High School Trying to calculate proper time of worldlines using rotating frames

[PeterDonis]

I simply took the Minkowski metric in polar coordinates and transformed it into a rotating frame

That is Born coordinates.

While the resulting metric has a cross-term similar to what appears in Born coordinates, I did not impose rigid rotation or the Born coordinate conditions.

I don't know what you mean by this. A number $\omega$ appears in your metric; that appears to be a fixed number (whose physical interpretation is the angular velocity of rotation of the coordinates relative to an inertial frame). That is "rigid rotation".

 

[Karin Helene Elise]

Oh, I see. Thanks!

I realize now that by using a fixed $\omega$ for all points and transforming the Minkowski metric via $\theta' = \theta - \omega t$, I unwittingly ended up using Born coordinates, and so hadn’t explicitly framed it that way or emphasized the rigid-rotation interpretation.

 

[Karin Helene Elise]

Every time I learn something new about relativity, it’s pretty confusing at first. Once I get it, though, it often doesn’t seem all that hard. But if I don’t work with it regularly, coming back to it can be confusing again. I find that very typical for relativity.

 

[Karin Helene Elise]

I'm not sure the OP is actually using the same Born coordinates for both observers. The calculation in post #7 looks like it's calculating p2's proper time in Born coordinates in which p2 is at rest.

If the OP's intent was to calculate p2's proper time in Born coordinates in which p1 is at rest, then I agree there will be a limit beyond which p2's worldline can't be described in p1's Born coordinates.

Well, my intent was to calculate p2’s proper time (in Born coordinates) in which p1 is at rest, but with $\omega r \ll c$, so the velocity limit isn’t an issue.

 

[PeterDonis]

my intent was to calculate p2’s proper time (in Born coordinates) in which p1 is at rest, but with $\omega r \ll c$, so the velocity limit isn’t an issue.

The velocity limit won't be, but you won't be able to integrate over p2's worldline past a certain point, because, as has been pointed out, only a limited portion of p2's worldline can be covered by Born coordinates in which p1 is at rest.

However, you can take the approach that @JimWhoKnew suggested in post #27, and calculate the ratio of the rates at which the proper times of p1 and p2 "tick" relative to coordinate time in Born coordinates in which p1 is at rest. Since that coordinate time is the same as coordinate time in an inertial frame, that calculation will end up giving you the information you need.

 

[Karin Helene Elise]

Allright. I don't really understand the why, but I'll figure it out, reading about it.

Good to know where I got it incorrect.

My thanks to everyone involved.