Atkisnon's derivation of \gamma factor.

[MathematicalPhysicist]

So I am rereading David Atkinson's QFT book:
http://books.google.co.il/books/about/Quantum_Field_Theory.html?id=vbAnQAAACAAJ&redir_esc=y

And I am puzzled by what is written in pages 5-6 section 1.3 Special Theory of Relativity.

He writes down the transformation between two inertial frames, one moving at speed v compared to the other.

$$x'=\gamma (x-vt) \ y'=y \ z'=z$$

Now he writes on page 6:

What is gamma?
From the uniformity of space, we see that \gamma may not depend on the coordinates (...).
In the absence of gravity, \gamma is independent of t,x,y,z but it may depend on v.
...
Rotational invariance therefore means that \gamma must be independent of the sign of v, i.e, it's a function only of v^2.

My problem is with the last conclusion, why can't we have \gamma to be any even function of v, like v^4 and so forth?

 

[qbert]

any even function of v is a function of v^2.

v^4 = (v^2)^2

 

[MathematicalPhysicist]

Feel so stupid now... :-D

Thanks.

 

[qbert]

don't feel bad. everybody misses stuff like that at times.

in fact you could come back and ask. really? every even function
of v is a function of v^2? those are two different things in principle,
even functions f(v) = f(-v) and f(v) = h(v^2) for some h.

it's not that hard to prove either.

 

[sardar]

I would respond by saying that Atkinson's derivation of the \gamma factor is based on the principles of special relativity, which state that the laws of physics should be the same in all inertial frames of reference. This leads to the conclusion that \gamma cannot depend on the coordinates, as it would violate the principle of uniformity of space. Additionally, rotational invariance requires \gamma to be independent of the sign of v, which means it can only be a function of v^2.

It is possible that other even functions of v, such as v^4, could satisfy these requirements, but they would not be consistent with the principles of special relativity. Atkinson's derivation is based on these principles and has been extensively tested and verified through experiments, making it a valid and reliable explanation of the \gamma factor.