Accelerated obserevers in special (not general) theory of relativity

[Fredrik]

Yes, I know that $$t_0$$ is the moment in the system S. But, what is wrong. I have think that it is correct formula for the proper time of bird i.e. time of the bird's clock from its pocket (the clock flying together with the bird). If this formula is incorrect what is correct.

I didn't say that it's wrong. It's not. I just explained what t₀ is because you didn't.

Yes, I know it, and I am expected this question. It is related to this remark:

Yes, JesseM explained it very well.

I am interesting for following: If I use the same ideas for computing frame-invariant quantities in another examples, can I obtain incorrect results?

I'm not sure exactly what you mean by "the same ideas", but it clearly doesn't matter what frame you use to compute something coordinate independent like "proper time".

 

[Nedeljko]

I'm not sure exactly what you mean by "the same ideas", but it clearly doesn't matter what frame you use to compute something coordinate independent like "proper time".

Let we assume that the observer's motion law is given in inertial system S. The ideas are:

1. Computing the time of inertial system as a function depending on the observer's proper time.
2. Computing values in inertial system S' co-moving with the observer in any fixed moment.
3. Considering these values as values measured by the observer in this fixed moment of the observer's time.
4. Expressing these values as functions depending on the observer's time.

That are brief description of the ideas. Consult the attached PDF for the example of computation. Can I use these ideas safely?