# Question 17 in section 2.9 in Schutz's A First Course in GR

1. Dec 23, 2015

### MathematicalPhysicist

The question is as follows:

Prove that any timelike vector for which $U^0>0$ and $\vec{U}\cdot \vec{U} = -1$ is the 4-velocity of some world line.

I need to show that there exists a 4-vector $\vec{x}$, s.t $\frac{d\vec{x}}{d\tau} = \vec{U}$, where $\vec{x}$ is a world line of some particle.

So far what I have done is: since $\vec{U}\cdot \frac{d\vec{U}}{d\tau} = 0$, so we have:

$$dU^0/d\tau = U^1/U^0 dU^1/d\tau + U^2/U^0dU^2/d\tau + U^3/U^0 dU^3/d\tau$$

I used the fact that $U^0>0$, so I can divide by it above.

We also have: $dx^\mu/d\tau = U^\mu$. How to continue? Any ideas?

This is not for homework, I am self studying GR and QFT.

2. Dec 23, 2015

### Staff: Mentor

Even if it's not technically homework for you, it's a problem that is often assigned as homework, so I have moved this thread to the homework forum.