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Piamedes
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Homework Statement
I'm not sure if this belongs in this section or in one of the physics homework sections. If it has been misposted please move it to the proper area.
According to the Theory of Relativity, if an event occurs at a space-time point (x,t) according to an observer, another moving relative to him at speed v (measured in units in which the velocity of light c=1) will ascribe to it the coordinates
[tex] x^{'} = \frac{x-vt}{\sqrt{1-v^2}} [/tex]
[tex] t^{'} = \frac{t-vx}{\sqrt{1-v^2}} [/tex]
Verfiy that s, the space-time interval is same for both:
[tex] s^2 = t^2 - x^2 = t^{'}^2 - x^{'}^2 = s^{'} [/tex]
Show that if we parametrize the transformation terms of the rapidity [tex] \theta [/tex],
[tex] x^{'} = x\cosh{\theta} - t\sinh{\theta} [/tex]
[tex] t^{'} = t\cosh{\theta} - x\sinh{\theta} [/tex]
the space-time interval will be automatically invariant under this transformation thanks to an identity satisfied by hyperbolic functions. Relate [tex] \tanh{\theta} [/tex] to the velocity.
The question has three more parts, but they all just build on this aspect. My major problem here is that I do not understand what the term "rapidity" means. I solved the first part of the question by just substituting the prime values of x and t into the difference of squares equation and showed their equality. However for this second part I don't even know where to start.
I tried plugging in the equations relating rapidity to the remaining variables and came up with nothing. If someone could perhaps explain what the concept of rapidity is I would be most grateful.
I had considered taking a derivative of the rapidity equations, but didn't know if that were possible because I don't know if [tex] \theta [/tex] varies with regard to x or t
Thanks for any help