Commentary 2.5 Metric, Rocket and 4-velocity (four-velocity)

The observer, abandoned by the rocket.

Before equation (2.51) near the end of section 2.5, Carroll writes: "Take a very simple example, featuring an observer with four-velocity U and a rocket flying past with 4-velocity V. What does the observer measure as the ordinary three-velocity, v, of the rocket? In special relativity the answer is straightforward. Work in inertial coordinates (globally not locally) such that the observer is in the rest frame and the rocket is moving along the x-axis. Then the four-velocity of the observer is

U^μ  =  (1 , 0 , 0 , 0) 

and the four-velocity of the rocket is

V^μ =(γ , γv , 0 , 0)

where v is the three-velocity and

γ = 1 / (√(1 - v² ) ) 

so that

v = √(1 - γ^-2)

"

It is easy enough to understand the first and last equations but the two middle ones were not so easy.

His (2.51) itself starts

γ =  -η_μν U^μV^ν

"since η₀₀ = -1"

that seems to come out of nowhere and had me thinking.

It all ends well with the gloriously tensorial equation

v = √(1 - (U_ν V^ν)² )

which allows me to proceed to section 2.6 on An Expanding Universe!

Details are in Commentary 2.5 Metric Rocket 4-velocity.pdf. 

I reviewed this June 2021 and realised that this is really an example of  Riemann normal coordinates which is super useful! Details in the pdf, I have not modified this very old, rather quaint post.