Relativistic momentum (Lorentz boost) low velocity limit

[Neutrinos02]

Hello,

If I have a momenta p_μ=(E,p_x,p_y,p_z) and transform it via lorentz boost in x-direction with velocity v I'll get for the new 0th component E′=γE+γvp_x why is this in the limit of low velocities the same as transforming the energy by a galilei transformation with velocity v? For γvp_x i get something like vp_x+O(v³) and with a galilei transformation I'll have terms (with p_x=mu) like 1/2m(u−v)²=1/2mu²−vp_x+1/2mv². So in the relativistic case I got the wrong sing for vp_x+O(v³)and lost the 1/2mv²? Did I make a mistake?

Thanks for help
Neutrinos

 

[PAllen]

For the sign error, if if you have (u-v) in the Galilean case, you need E′=γE-γvpx for the corresponding relativistic case. The rest is just algebra, but I am not disposed to work it out right now.