# Ultrarelativistic approximation

1. Jul 6, 2012

### mps

sqrt(1-β2) = sqrt(2*(1-β)).
How do you show this mathematically? I have no idea. Thanks! :)

2. Jul 6, 2012

### pervect

Staff Emeritus
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 $\epsilon^2$.

Last edited: Jul 6, 2012
3. Jul 6, 2012

### mps

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

4. Jul 6, 2012

### 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.

5. Jul 6, 2012

### mps

Thank you! :)