Explaination on hyperbolic functions and Relativity?

[Endy0816]

Trying to get more information.

 

I've already looked into it somewhat. I noticed: after eyeballing the equation in that other thread, I am but still trying to wrap my head around all the ins and outs. Rapidity is of particular interest, not sure I have a clear understanding as of yet.

[ajb]

You can use Cosh and Sinh and the rapidity to represent the Lorentz transformations neatly in 1+1 dimensions. It also allows very simple representations of the addition of velocities law.

 

I think you may know this already, so do you have specific questions?

[Endy0816]

 

I think you may know this already

 

Oh, ye of great faith.

 

You can use Cosh and Sinh and the rapidity to represent the Lorentz transformations neatly

 

I was able to grasp this much. Mainly just became annoyed after typing/writing out the whole tedious equation multiple times. While trying to simplify, recognized the rough similarity to "1=sin²(x) + cos²(x)" then went to the Wiki with that and saw that the hyperbolic functions were the actual solution I was after.

 

Tutorials were somewhat useful, but didn't help much with applying newfound knowledge back to an improved understanding of Relativity.

 

Specific questions would be:

 

What is rapidity(besides being an angle) and is it tied to acceleration in some fashion?

Does anything else on the graph have ties back to other physical aspects?

Edited August 23, 2015 by Endy0816

[ajb]

What is rapidity(besides being an angle) and is it tied to acceleration in some fashion?

The definition of rapidity is

 

[math]\varphi = \arctan(v/c)[/math],

 

where v is the velocity and c the speed of light. Notice now that as -c < v < c the rapidity now takes any value on the real line.

 

The important thing with rapidity is that (in one dimension) we have a simple way to add velocity between frames; you just add the rapidities. The notion is often used in relativistic scattering.

 

 

You can show, and it is really just using a change of variables that the proper acceleration is given by

 

[math]\alpha = \frac{d \varphi}{d \tau}[/math],

 

where [math]\tau[/math] is the proper time of the particle in question. So rapidity gives you the proper acceleration quite neatly.

[Endy0816]

Are there any units associated with any of those? I wouldn't think there would be but not certain. Basing this off of unit analysis and the fact that some span X, can be measured as being different lengths based on the frame.

[ajb]

Yes, I have set c=1. You should multiply the right hand side of what I wrote by c to put all the units back in.