# Special Relativity With Curved Coordinates

• B
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
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2021 Award
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
Orodruin
Staff Emeritus
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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
Ibix
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