Proving 4-vector Analog Formula for Lorentz Boost

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cosmic_tears
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Homework Statement


Hello everyone, thanks for reading.
This might be a little more math than physics (don't run away though!), but it's an exercise on my General relativity text :)
I need to prove that there exists an analog formula, like a*b = abcos(theta) for 3-vectors, only for 4-vectors, in which:
a*b = abcosh(theta), where a and b are 4-vectors, a and b are defined: a = (-a*a)^-0.5, b = (-b*b)^-0.5, and theta is a parameter that describes lorenz boost between the frame where an observer whose world line points along a is at rest and the frame where an observer whose world line points along b is at rest.
I have no idea how to work with this theta :-\

Thanks!

p.s. a and b are time-like 4 vectors.


Homework Equations



a*b = -a0*b0 + a1*b1 + a2*b2 = a3*b3



The Attempt at a Solution



I just tried to look for examples in the book and work with the definition... But I got nowhere :-\
 
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Hi cosmic_tears! :smile:

(btw, its ^1/2, not ^-1/2)

Let's rephrase the question:

For every two timelike 4-vectors a and b, you know how to make the dot-product a.b.

Since they are timelike, there will be two observers with velocities for which a and b, respectively, are at rest.

Let their relative speed be v.

Find a.b as an expression in a b and v, show how the v part can be written as cosh of something, and explain why that's an advantage. :smile:
 
Thank you very very much!
It's done!