Special Relativity With Curved Coordinates

[sqljunkey]

Someone told me that I don't need the whole mechanics of GR to be able to calculate the proper time in an accelerated frame of reference. I can just use SR but with curved coordinates and then integrate for time. But he didn't give me a reference where I could find the formula to do this. How do you use Lorentz transformation with curved coordinates? Is this true? Anyone has a reference for it?

Thanks

 

[Dale]

Someone told me that I don't need the whole mechanics of GR to be able to calculate the proper time in an accelerated frame of reference. I can just use SR but with curved coordinates and then integrate for time. But he didn't give me a reference where I could find the formula to do this.

You can use equation 2 here:

https://en.m.wikipedia.org/wiki/Proper_time

You need to have $ds^2$ which is known as the arc length, as well P, both in terms of the chosen coordinates.

 

[Orodruin]

How do you use Lorentz transformation with curved coordinates?

You don't. The Lorentz transformation is, by definition, a transformation between inertial frames. However, you do not even need curvilinear coordinates to compute proper times of accelerated observers. You can just apply
$$
\tau = \int_{t_1}^{t_2} \sqrt{1 - v(t)^2/c^2}\, dt.
$$
However, you can of course define a coordinate system where your accelerated observer is at rest, but you do not need to.

 

[Ibix]

In case Orodruin's answer isn't clear, he is noting that if your velocity in some inertial coordinate system is $v$ at coordinate time $t$ then in the elementary time from $t$ to $t+dt$ your clock advances $d\tau=dt/\gamma$, then integrating. Or you can start from the expression for the interval and get the same result.