Ultrarelativistic approximation

[mps]

I read that when v≈c,
sqrt(1-β²) = sqrt(2*(1-β)).
How do you show this mathematically? I have no idea. Thanks! :)

 

[pervect]

I read that when v≈c,
sqrt(1-β²) = sqrt(2*(1-β)).
How do you show this mathematically? I have no idea. Thanks! :)

Let $\epsilon = 1 - \beta$ , then substitute $\beta = 1 - \epsilon$. You get

sqrt($1 - (1 -2 \beta \epsilon + \epsilon^2)$) = sqrt($1 - (1 - 2 (1 - \epsilon) \epsilon + \epsilon^2)$

Take the limit as $\epsilon$ goes to zero. You might need a bit of calculus to do that. bit basically you keep all the terms proportioanl to $\epsilon$ and trhow out all high order temrs proportional to $\epsilon^2$.

 

[mps]

sqrt($1 - (1 -2 \beta \epsilon + \epsilon^2)$)

I think you might have an extraneous β here, but thanks a lot! I get it now :)

 

[Parlyne]

It's even easier to see than that. Just remember that $1-\beta^2 = (1+\beta)*(1-\beta)$. Then, since $\beta$ is very close to 1, the first term is only very slightly smaller than 2.

 

[mps]

Thank you! :)