[raattgift]

> But a black hole is a purely gravitational object

Are you calling compact objects from stellar collapse that have an apparent horizon something other than a "black hole"?

> The mass of a black hole is something that's really only defined from a distance away

The stress-energy in the region of spacetime where one finds a black hole totally determines the Einstein tensor in that region. Solve the Einstein Field Equations for T_munu in \Sigma \subset \M. Extract the metric tensor. Solve the geodesic equations for that region, and you have the orbit for your Earthlike planet. You can make it simpler and consider the surface stress-energy on a shell within the horizon, or even outside in the case of your orbiting Earthlike planet problem, for which we don't care about the region inside the shell.

You are skipping the first steps, treating the metric tensor as a known, or worse, something that you can extract from a single geodesic.

But let's think about that anyway. Do you really know the metric you source? Sure, your stress-energy is localized so you can deploy a large shell to partition the "inside" and "outside" stress-energy. "Outside" it's zero, and nobody will really care about your small deviations from exact spherical symmetry, so your metric is therefore Schwarzschild, right? No, we can't make that inference based on the far field outside the shell; we need an Israel junction condition. [Synge 1960, Relativity: The General Theory (Amsterdam: North-Holland), ch. IV § 6 goes into this in detail, contrasting "realist" vs "agonist" and "creator" positions; yours is the "creator" position in that you are happy with an exact solution of the EFEs, and so you might enjoy reading what Synge wrote as he walks towards the mathematics of junctions :-) .] In a model can ignore that because we get it "for free" by laying down an exact generator of the vacuum Schwarzschild solution and not worrying whether the stress-energy is physical.

[improbable22]

My first point about mass was that some are tempted to imagine the mass of the BH as residing in the matter of the neutron star or whatever that formed it. This is misleading, it is better to think of that matter as no longer existing, and just deal with the fact that pure Schwarzschild looks identical to a lump of matter, from a distance.

How you measure the mass, well my orbit example is admittedly crude, ADM mass is I think the right asymptotic concept.

By purely gravitational object I mean this: everything we're discussing here about merging and waves and accretion disks all concerns only the exterior. This is all vacuum Einstein, pure gravity. (Whether we can say anything sensible about the interior is another whole different rabbit-hole.)