I've never thought of the equivalence principle like this. I knew gravitational fields have their own mass-energy and can cause a runaway effect (black holes), but I never realized this could mean more inertia too. Neat!
I'm curious about what this means as the event horizon is crossed.
Sorry I'm a few days late to this party. There are lots of things I am tempted to comment about other comments, but even though the OP doesn't really have anything to do with black holes directly, I'll try an answer restricting myself to your good, honest question:
> I'm curious about what this means as the event horizon is crossed
tl;dr it means General Relativity is how we answer this; more specifically, we expect that if we do a Galileo-like experiment replacing the Leaning Tower and the ground with a giant scaffold surrounding a black hole, and we carefully drop in a feather, a planet, and a neutron star, neither object's centre of mass wins a race to the horizon (it's a tie, even though the neutron-star is strongly gravitationally self-bound, and the feather is not gravitationally self-bound at all).
In practical terms, and strictly with respect to your question, the results of Archibald et al. [2018] (the work described in the original post) mostly (more below) vindicate the use of a post-Newtonian approximation (PNA) formalism as a short-cut to the results we would get from the full theory of General Relativity. At the horizon of anything but the smallest black hole[1] one expects "no drama" for a freely-falling infaller. This is not terribly surprising, as we could already make a model black hole's mass arbitrarily high, bringing the curvature at the horizon down well below the curvature we experience in laboratories here on Earth.
Clifford Will, who has written many papers on post-Newtonian methods, in (deliberately accessible to non-specialists) Will [2011] [2] writes:
The reason is a remarkable property of general relativity called the Strong Equivalence Principle (SEP). A consequence of this principle is that the internal structure of a body is “effaced,” so that the orbital motion and gravitational radiation emitted by a system of well separated bodies depend only on the total mass of each body and not on its internal structure, apart from standard tidal and spin-coupling effects. In other words, the motion of a normal star or a neutron star or a black hole depends on the body’s total mass and not on the strength of its internal gravitational fields. This behavior was already implicit in the work of Einstein, Infeld, and Hoffman, where only the exterior nearby field of each body was needed, and has been verified theoretically to at least second post-Newtonian order by more modern methods.
(So, for example, SEP means a neutron star's strong internal curvature -- the strongest we can access observationally with current technology -- doesn't cause the neutron star to radiate energy-momentum away as dipole gravitational waves.)
Archibald [2018] provides observational verification for SEP for the inner radio pulsar of the triple system to good sub-leading post-Newtonian order, improving on results from (among others) the Hulse-Taylor binary and (indirectly) LIGO.
Will [2011] uses the "xPN" notation for subleading orders, where 1PN is leading-order, 2PN is next-to-leading order, 3PN is next-to-next-to-leading order, and so forth.
A problem raised in Archibald [2018] is that the most popular formal system (the parameterized post-Newtonian (PPN) formalism) for comparing alternative theories of gravitation which may distinguish inertial mass from gravitational mass is only good to 1PN; any theory that does not match the results of Archibald [2018] at 1PN can be excluded, but anything that differs at subleading order needs a different comparison framework.
Again, this is not tremendously surprising; PPN was explicitly constructed to fully contain weak-field results (mostly within our own solar system), and the region around the central binary in Archibald [2018] is clearly not weak-field. So while this spells trouble for several families of alternatives to General Relativity (GR), there are several others where the results of breaking GR's inertial mass == gravitational mass equality appear only in the strong field limit. An extension of PPN is needed to capture the results of Archibald [2018] into a formal comparison system that may distinguish between such theories and General Relativity, assuming the compared theories match in every other PPN parameter (if they don't we can exclude one on that basis).
Or, in short, General Relativity looks really sound still, and post-Newtonian programmes are not wildly off-track.
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[1] The region near (but outside) the horizon of a small black hole is going to be dramatic for other reasons, ranging from astrophysical ones like the virtual certainty of a hot accretion structure to theoretical ones like the increasing heat of Hawking radiation as one takes the black hole mass to zero. An object like a space capsule with a person inside would be vapourized by the hot matter outside a small black hole well before reaching the horizon. As we make the mass of a black hole larger, the matter just outside the horizon -- at least on average -- is a lot cooler and sparser, so a space capsule could easily freely fall through the horizon.
I'm curious about what this means as the event horizon is crossed.