How do we calculate the 4 velocity of a particle that is projected radially downwards at velocity u at a radius ra? The condition on 4 velocity is that gμνvμvν = 1 which implies that at radius ra we have ga00(v0)2 + ga11(v1)2 = 1 (eq 1) So if we start from xμ = (t,r) we get vμ = (1/√g00 , 1/√g11) but this is only for a particle that fell from rest at infinity correct? If we want to give it a velocity u which suggests we Lorentz boost vμ to vμ = γ(1/√g00 , β/√g11) (eq 2) Where β is the velocity we boost it at. This does solve eq 1 but I don't think it's correct to Lorentz boost because we are not traveling between two frames in SR. Do I have to solve the geodeisic equation for vμ? Thank you!