# Can local experiments determine whether spacetime is static or not?

As far as I understand, there is no local experiment that can determine whether one has crossed the event horizon of a black hole or not.

At the same time, spacetime in a black hole is non-static if I understand correctly, and outside the even-horizon it is static.

So the answer to my question is "no", at least in the case of black holes.

Is this a general feature?

What puzzles me is that static spacetime vs. non-static spacetime seems to be quite a far reaching difference for any observer that experiences it, yet it is confusing that no observer should be able to determine that fundamental difference by conducting a local experiment.

• Agree++. "Expanding space stretches photons"....where is the lab experiment ? Commented Feb 19, 2020 at 23:47

What puzzles me is that static spacetime vs. non-static spacetime seems to be quite a far reaching difference for any observer that experiences it, yet it is confusing that no observer should be able to determine that fundamental difference by conducting a local experiment.

One can easily define an inertial frame of reference locally in curved spacetime. Thus special relativity holds locally and no experiment can show any effects of gravity including those you mentioned. For more please see Equivalence principle and local inertial reference frames

• So your "locally" means "arbitrarily small region of space", but if a region is arbitrarily small so we may neglect curvature, then you can no more conduct any physical experiment because our bodies are not point masses. So when we assume we travel in a space ship with, say, 10m×10m×10m and restrict the experiment to be carried out there, then there is no way to determine static vs. non-static? For example you'll see tidal effects in the space ship. Commented Feb 20, 2020 at 14:07
• "So your "locally" means "arbitrarily small region of space"". In principle yes. More precisely as Wikipedia puts it: "The room, therefore, should be small enough that tidal effects can be neglected". This is the very meaning of local regarding Einstein's Equivalence Principle. But if you understand "local" in a wider sense then while observing tidal forces is possible one can't proof experimentally that the spacetime is not static, because this would require the proof that hovering at constant $r$ is not possible.
– timm
Commented Feb 20, 2020 at 16:37

The statement that the Schwarzschild space-time is "static" outside of the black hole should rather be stated as

There are particular families of observers in the exterior of the Schwarzschild black hole to whom the space-time seems static.

On the other hand, the statement that the interior is "not static" should be stated as

There is no family of physical observers in the Schwarzschild interior to whom the space-time seems static.

But notice one thing. The fact that there is some family of observers in the exterior that sees the space-time as static means nothing, really, if you are a generic observer. If you are free-falling into the black hole, I guarantee you your surroundings will look pretty dynamic to you both when you are above and below the horizon.

The point is, of course, that the geometry seems static to observers at the horizon that are infinitely accelerated. So the notion of the space-time being static necessarily breaks down already above the horizon for any experiment with finite resources. You can then see that the operational definition of a horizon will always be somewhat "fuzzy".

• " So the notion of the space-time being static necessarily breaks down already above the horizon for any experiment with finite resources." It breaks down inside. Non-static regarding the interior means that there is an interchange of the r- and t-coordinates as the metric shows. So non-static spacetime means that r decreases inevitably which represents the passage of time. As a result hovering at constant r is not possible, further more does e.g. the r-coordinate of un upwards emitted photon decrease. Nevertheless are these effects not observable locally.
– timm
Commented Feb 20, 2020 at 10:26
• @timm I agree. My point is that the observers that see the space-time as static outside of the black hole are always accelerated. As one approaches the horizon, these observers reach higher and higher accelerations without an upper bound. So considering a real set of referential rockets that see the space-time as static will always fail at a finite distance above the horizon. In other words, the practical experimental measurement of the "staticity" of the exterior of a Schwarzschild black hole is limited to a region that necessarily excludes a small part of the spacetime above the horizon.
– Void
Commented Feb 20, 2020 at 10:44
• in practice this is true. But what matters is that in principle hovering arbitrarily close to the horizon hovering is possible. The theoretical criterion for non-static results from the interchange of the $r$- and $t$-coordinates for $r < 2M$ as mentioned.
– timm
Commented Feb 20, 2020 at 17:00
• @timm I am well aware of that. Do you feel like there is something that still needs to be answered about your original question?
– Void
Commented Feb 20, 2020 at 17:49

Can local experiments determine whether spacetime is static or not?

Yes, if we allow that “local” observations include the ability to calculate local invariants constructed from curvature tensor and its derivatives. An experimenter then can locally determine whether spacetime is (locally) static or not. This could be achieved by calculating appropriate (finite) number of curvature invariants, and finding if there is a timelike direction along which their Lie derivative is zero.

The conflict between the first and second paragraph of the question comes from mixing properties of “general” black hole spacetime and specific static/stationary black hole solutions.

As far as I understand, there is no local experiment that can determine whether one has crossed the event horizon of a black hole or not.

That statement is true for a “general” black hole spacetime about which no apriori information is given to the observer. However, if observer has some “external” information about the spacetime where she is performing experiments, then under certain conditions it would be possible to determine the moment of the horizon crossing.

At the same time, spacetime in a black hole is non-static if I understand correctly, and outside the even-horizon it is static.

“General” black hole spacetimes are neither static nor stationary outside event horizons. Schwarzschild metric is a specific example of static black hole solution, it has a Killing vector field (KVF) that is timelike outside the event horizon while inside the horizon there are no timelike KVF. So, for Schwarzschild black holes this statement is indeed true. But “realistic” black holes cannot be truly static or even stationary, they absorb mass and exist within evolving universe.

Nevertheless, once black hole stabilizes we can say that certain finite region of spacetime around it is approximately stationary (if black hole is rotating) or even approximately static (if nonrotating). Perturbations within this region will decay exponentially with a characteristic timescale of Schwarzschild crossing time $$r_s/c$$, so if there are no continuous external sources of perturbation, black holes would become stationary quite fast (wrt ordinary timescales) and with a high degree of precision.

For static and stationary black holes an observes can locally determine the moment she crosses event horizon by measuring certain curvature invariants. For example, for the Schwarzschild metric, invariant combination $$R^{αβγδ;\epsilon}R_{αβγδ;\epsilon}$$ crosses zero and switches sign as one crosses the horizon. This paper outlines the Gedanken-experiment that could be used to locally measure the invariant and thus detect the horizon. This paper provides invariant polynomial for Kerr metric, and this paper generalizes the technique to an arbitrary stationary black hole spacetime.

If the black hole is not stationary, calculation of such invariants could still be used to locate an approximate position of the event horizon. For a black hole that is stabilized the difference between calculated and actual horizon could be exponentially small (with a factor $$e^{- A \tau c /r_s}$$, where $$\tau$$ is a time after or until measurable perturbation and $$A$$ a constant of order unity).