I am trying to solve the following problem but have gotten stuck.(adsbygoogle = window.adsbygoogle || []).push({});

Consider a massive particle moving in the radial direction above the Earth, not necessarily on a geodesic, with instantaneous velocity

v = dr/dt

Both θ and φ can be taken as constant. Calculate the components of the four-velocity U^{u}in terms of v using the normalization condition for U^{u}. Do not make any approximations yet.

We are using the metric:

ds^{2}=-(1+2Φ)dt^{2}+(1+2Φ)^{-1}dr^{2}+r^{2}dΩ^{2}

I have determined that U^{0}is going to be (1+2Φ)^{(-1/2)}by using the equation

g_{uv}U^{u}U^{v}=-1

However when I try to solve for U^{1}I get

g_{11}*(U^{1})^{2}=-1

Which is clearly not correct as this would yield

U^{1}=(-(1+2Φ))^{(-1/2)}.

What am I doing wrong?

Apologies for formatting.

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# A Four velocity with the Schwarzchild metric

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