A Four velocity with the Schwarzchild metric

[Pogags]

I am trying to solve the following problem but have gotten stuck.
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:

ds²=-(1+2Φ)dt²+(1+2Φ)^-1dr²+r²dΩ²

I have determined that U⁰ is going to be (1+2Φ)^(-1/2) by using the equation

g_uvU^uU^v=-1

However when I try to solve for U¹ I get

g₁₁*(U¹)²=-1

Which is clearly not correct as this would yield

U¹=(-(1+2Φ))^(-1/2).

What am I doing wrong?

Apologies for formatting.

 

[PeterDonis]

I have determined that U⁰ is going to be (1+2Φ)(-1/2) by using the equation

g_uvU^uU^v=-1

Do you realize that this equation contains both $U^0$ and $U^1$? You can't get an equation for just $U^0$ from it. The notation $g_{uv} U^u U^v$ means you have to sum over all possible values for the indexes $u$ and $v$.

 

[Pogags]

This clears up the problem thank you. That slipped my mind