general relativity - Is the event horizon locally detectable?

There are two conflicting ideas.

According to the traditional view, determining the location of the event horizon of a black hole requires the knowledge of the whole future behaviour of the black hole solution. Namely, given a partial Cauchy surface, one cannot find where the event horizon is without solving the Cauchy problem for the whole future development of the surface. (Cf. Hawking - Large Scale Structure of Space-Time, p. 328). Therefore, locally one cannot tell whether one is passing through the horizon.

However, according to Karlhede and arXiv:1404.1845, there is a certain scalar, now called the Karlhede's Invariant, which changes sign as one crosses the event horizon. A local measurement of this scalar can tell whether one has encountered an event horizon as of yet or not.

So, who is right? If both are right, how do we reconcile the apparent conflict?

Thanks.

Answer

Toth is a competent person who does good work, but IMO this is one of those cases where sometimes a good scientist writes a bad paper.

If you give me global information about my spacetime, plus the ability to measure local information about my own environment, I can find out all kinds of things about my location, and I don't need to resort to the Karlhede invariant. For example, if you tell me that I'm in the Schwarzschild spacetime with mass $m$, I can measure the Carminati-McLenaghan invariant $W_1$, and because I know that $W_1=6m^2r^{-6}$ for the Schwarzschild spacetime, I can immediately determine my $r$, even if I'm locked inside a closet. If $r=2m$, I know I'm at the horizon.

So the information processing looks like this:

Tell me I'm in the Schwarzschild spacetime with mass $m$, and let me do local measurements on the gravitational field --> I can find out if I'm at the horizon (by measuring $W_1$).

Tell me I'm in the Schwarzschild spacetime, and let me do local measurements on the gravitational field --> I can find out if I'm at the horizon (by measuring the Karlhede invariant).

So the only difference between using the Karlhede invariant and using some other invariant is that I need less global information -- but I still need global information (that I'm in the Schwarzschild spacetime). And keep in mind that the Schwarzschild spacetime does not actually exist. It's not the metric of an astrophysical black hole.

Curvature scalars give you only very limited information about what's going on in a spacetime. For example, all curvature scalars vanish for a gravitational plane wave, so although LIGO can tell you there's a wave passing through your location, you'll never get that information from curvature scalars.

Curvature scalars are also difficult to measure and do not affect laboratory physics except with extremely sensitive hypothetical measurements that we can't actually do. (We currently do not have any technology capable of doing a practical measurement of any curvature scalar in any gravitational environment that we have access to.) So it's absurd IMO to attribute violent physical effects like a firewall to the behavior of a particular curvature scalar.