Suppose an observer u moves radially with speed (3-velocity) relative to “stationary” Schwarzschild observers, where we define as inward motion. Then one natural choice of orthonormal tetrad is:
where the components are given in Schwarzschild coordinates. This may be derived as follows.
The Schwarzschild observer has 4-velocity
because the spatial coordinates are fixed, and the t-component follows from normalisation (Hartle §9.2).
Now the Lorentz factor for the relative speed satisfies , and together with normalisation and the assumption that the θ and φ components are zero, this yields given above.
We obtain by orthonormality: and , and again making the assumption the θ and φ components are zero. Note the negative of the r-component is probably an equally natural choice. Then and follow from simply normalising the coordinate vectors.
Strictly speaking this setup only applies for , because stationary timelike observers cannot exist inside a black hole event horizon! Yet remarkably the formulae can work out anyway (MTW …) . An alternate approach is local Lorentz boost described shortly.
Hartle … Also check no “twisting” etc…