**Difficulty level:**☆

# Category: Tetrads

## Tetrad for Schwarzschild metric, in terms of e

**Difficulty level:**★ ★ ★

The following is a natural choice of orthonormal tetrad for an observer moving radially in Schwarzschild spacetime with “energy per unit rest mass” *e*:

The components are given in Schwarzschild coordinates. (The ± signs are not independent — they must be either both +1 or both -1. Note that *e* does not distinguish between inward and outward motion. There is additional freedom to define any of these vectors as their negative.)

We normally think of *e* as invariant, where there is a presumption of freely falling / geodesic motion, but even if not we can regard it as an instantaneous value.

is the 4-velocity computed previously. The other vectors can be obtained from substituting and into the tetrad here. is determined from and the equation for *e* above, then *V* follows from inverting . This orthonormal frame is useful for determining the object’s perspective, e.g. tidal forces, visual appearances, etc.

## Tetrad for Schwarzschild metric

**Difficulty level:**★ ★ ★ ★

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…