Static and stationary

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A spacetime is static or stationary if it does not change over time, loosely speaking. “Static” is a stronger condition than “stationary”, and means a reversal of time does not change the spacetime. As Carroll (§5.2) explains,

You should think of stationary as meaning “doing exactly the same thing at every time,” while static means “not doing anything at all.”

Technically, a spacetime is stationary if it possesses a timelike Killing vector field at infinity (or in a given region / subset, see later). A spacetime is static if in addition the timelike Killing vector field is orthogonal to a family of hypersurfaces.

Examples:

  • Minkowski space is static, and hence also stationary
  • Schwarzschild spacetime is static (but in the black hole case, this only holds for outside the event horizon: r>2M. The interior is not even stationary)
  • A Kerr black hole is stationary (only outside the ergosphere: r>M+\sqrt{M^2-a^2\cos^2\theta}) but not static because it is rotating
  • Friedmann-Lemaître-Robertson-Walker (FLRW) universe models are in general neither static nor stationary, roughly speaking because the scale factor is time-dependent. However special cases such as de Sitter space and Einstein’s original “static” universe are static

Observers

We can normalise the Killing vector field at each point to obtain “stationary” or “static” observers, which are considered to be at rest. If the timelike Killing vector field is \fvec\xi, then \fvec\xi/\sqrt{-\fvec\xi\cdot\fvec\xi} has norm -1, and is a normalised timelike 4-velocity.

Coordinates

For a stationary spacetime, we can choose a “time” coordinate x^0\equiv t, such that the Killing vector field is \partial_t and the metric components are independent of t. For a static spacetime, we can additionally impose no time-space cross-terms dt\,dx^i where i=1,2,3. We could then call coordinates with these properties stationary and static coordinates respectively when they make manifest these underlying properties of the spacetime. (Examples of this terminology: Francis & Kosowsky 2004, Kraus & Wilczek 1994).

Examples:

  • Schwarzschild spacetime (for r>2M): Schwarzschild coordinates are static. Gullstrand-Painlevé coordinates are only stationary, because of the cross-term
  • de Sitter space: de Sitter’s original “static” coordinates are static within any given Hubble sphere. FLRW coordinates are not even stationary, because the scale factor R(t)=e^{Ht} appears in the metric
  • Minkowski space: the usual inertial/Cartesian coordinates are static. Even Rindler coordinates, which correspond to accelerating objects, are static (for x>t, considering only one quadrant of the Rindler wedge)! This is apparently because Minkowski space is highly symmetric.

Further details

  • Static ⇒ stationary
  • Spherically symmetric + stationary ⇒ static

The timelike Killing vector field gives a natural splitting of spacetime into space and time.

Naturally we expect the choice of Killing vector field to be unique. The usual definition (c.f. Carroll) of timelike “at infinity” ensures this, in the Schwarzschild and Kerr spacetimes at least. However this definition needs to be broadened, because it excludes de Sitter space for instance. I extend the definition to “timelike in a given region / subset”. An observer “at infinity” is considered preferred or objective, but we can generalise this to a “fiducial” observer at some fiducial location. We expect them to have many of the same properties, such as being static/stationary, freely falling (geodesic), and possibly local inertial coordinates at that location. They are also considered to be free from gravitational effects. For example, in de Sitter space there is no preferred choice in general, however in my forthcoming “galaxy cable” paper, there is a natural choice given a choice of origin. A set of observers are given, and there is a unique field parallel to them.

References

Carroll (§5.2, p.203 onwards) gives a helpful overview

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