# A Four velocity with the Schwarzchild metric

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1. Mar 16, 2017

### 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 Uu in terms of v using the normalization condition for Uu. Do not make any approximations yet.

We are using the metric:

ds2=-(1+2Φ)dt2+(1+2Φ)-1dr2+r22

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

guvUuUv=-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.

2. Mar 16, 2017

### Staff: Mentor

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 $v$.

3. Mar 16, 2017

### Pogags

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