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! :)