# Special Relativity With Curved Coordinates

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• sqljunkey

#### 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

## Answers and Replies

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.

vanhees71
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.

vanhees71 and PeroK
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.

vanhees71