How do we calculate the 4 velocity of a particle that is projected radially downwards at velocity u at a radius r(adsbygoogle = window.adsbygoogle || []).push({}); _{a}?

The condition on 4 velocity is that g_{μν}v^{μ}v^{ν}= 1 which implies that at radius r_{a}we have

gSo if we start from x^{a}_{00}(v^{0})^{2}+ g^{a}_{11}(v^{1})^{2}= 1 (eq 1)

^{μ}= (t,r) we get v^{μ}= (1/√g_{00}, 1/√g_{11}) 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 v^{μ}to

vWhere 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.^{μ}= γ(1/√g_{00}, β/√g_{11}) (eq 2)

Do I have to solve the geodeisic equation for v^{μ}?

Thank you!

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# 4 velocity in Schwarzchild metric

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