[improbable22]

Certainly normal matter (like the earth) has mass and warps spacetime. But a black hole is a purely gravitational object, there need not be any normal matter involved.

The mass of a black hole is something that's really only defined from a distance away. If a planet of 1 earth-mass orbits it at the same speed & circumference as we do the sun, then we say the black hole has 1 solar mass.

[raverbashing]

> But a black hole is a purely gravitational object, there need not be any normal matter involved.

There's no "purely gravitational object" it only "looks like that" from behind the event horizon

We know the escape velocity > c and that's pretty much it, for GR it's a singularity (which usually means the theory is incomplete in that circumstance) and we don't know how QM work when squeezed harder than a Neutron star

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

If you mean "we can sense the gravitational field of something having mass X at a distance D larger than the event horizon" I agree.

But I'd rather say "we don't know what happens there" instead of "singularity" (which is what the current theories say it's inside and from the point of view of Relativity they're not wrong)

[evanb]

> There's no "purely gravitational object" it only "looks like that" from behind the event horizon

It depends on what framework you're working in. If you're working in general relativity, there's a singularity and absolutely nothing else---no matter at all. If you're talking about the "real world" you can ask: Is GR reliable for these situations, especially in light of quantum-gravitational puzzles? I think you agree that we don't know the answer. But we can say what the GR answers are.

[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.)