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# Category: Spacetime geometries

## (Poster) Static vs Falling: Time slicings of Schwarzschild black holes

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I presented this poster, “Static vs Falling: Time slicings of Schwarzschild black holes” at the GR21 conference in New York City, July 2016. You can download a PDF version. A big thank-you to those who gave feedback, especially to Jiří Podolský who encouraged me to publish it! The section on coordinate vectors has been updated. Also there is a major update to the poster.

## Tetrad for Schwarzschild metric, in terms of e

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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…

## Radial motion in the Schwarzschild metric, in terms of *e*

**Difficulty level:**★ ★ ★

A nice way to parametrise the 4-velocity **u** of a small test body moving radially within Schwarzschild spacetime is by the “energy per unit rest mass” *e*:

For the “±” term, choose the sign based on whether the motion is inwards or outwards. All components are given in Schwarzschild coordinates . The result was derived as follows. In geometric units, the metric is:

By definition , where is the Killing vector corresponding to the independence of the metric from *t*, and has components (Hartle §9.3). For geodesic (freefalling) motion *e* is invariant, however even for accelerated motion *e* is well-defined instantaneously and makes a useful parametrisation.

We want to find say. Rearranging the defining equation for *e* gives . Radial motion means , so the normalised condition yields the remaining component . The resulting formula is valid for all , and for *e*=1 the 4-velocity describes “raindrops” as expected.