A Derive 4-Velocity Components: Anyone Know How?

[berlinspeed]

TL;DR Summary

Need help on a simple 4-velocity components derivation.

Anyone know how to derive $u^0=\frac {dt} {d\tau}=\frac {1} {\sqrt {1-\mathbf {v}^2}}$ and $u^j=\frac {dx^j} {d\tau}=\frac {v^j} {\sqrt {1-\mathbf{v}^2}}$?

 

[Orodruin]

What textbooks have you consulted? This should be described in any introductory text on SR.

 

[berlinspeed]

What textbooks have you consulted? This should be described in any introductory text on SR.

I don't own any SR books, which ones would you recommend?

 

[Orodruin]

By textbook I do not necessarily mean a physical textbook, it can just as well be some form of online lecture notes or at least some sort of learning material. You must be getting the information from somewhere. That somewhere should preferably be a didactic material intended to teach SR. If you do not have some sort of learning material and just look things up on, e.g., Wikipedia, that is typically not a useful or effective learning strategy. There is lots of useful information available online, but if you do not find anything else you can try my lecture notes.

 

[berlinspeed]

By textbook I do not necessarily mean a physical textbook, it can just as well be some form of online lecture notes or at least some sort of learning material. You must be getting the information from somewhere. That somewhere should preferably be a didactic material intended to teach SR. If you do not have some sort of learning material and just look things up on, e.g., Wikipedia, that is typically not a useful or effective learning strategy. There is lots of useful information available online, but if you do not find anything else you can try my lecture notes.

Thanks so much! Gonna grind on that for a while now..

 

[PAllen]

I’ll give you an outline of steps you can follow.

Write the definition of timelike interval, or proper time between events in inertial frame.

Divide by delta t.

Take the limit as delta t goes to zero.

Use the rule for derivative of inverse.

That takes care of the first question. For the second, see how to use the chain rule plus this fact.

 

[vanhees71]

Another one is my SRT FAQ

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf