Does the total mass of an isolated object include the mass stored in its gravitational field? Since neither the object nor its field could exist without the other, it would seem strange not to include the field energy as part of the object. But how exactly does the accounting go? How is the mass of the system divided between the rest mass and the field mass?
For a Schwarzschild black hole, the mass appears to be shifted completely to the field.
According to Lynden-Bell and Katz, http://adsabs.harvard.edu/full/1985MNRAS.213P..21L, the total energy distributed in the gravitational field of a Schwarzschild black hole is mc^2. In other words, all the mass of the black hole resides outside the event horizon.  
 A: GR does not have a local, tensorial measure of energy density in the gravitational field. This is because of the equivalence principle, which tells us that the Newtonian gravitational field $\textbf{g}$ can't be a tensor. (It vanishes in an inertial frame, and any four-vector that vanishes in one frame vanishes in all frames.)
GR does have various measures of the total mass in an asymptotically flat spacetime, such as the ADM mass. These are not local things, so they don't give us a way to define where the energy resides.

According to Lynden-Bell and Katz, http://adsabs.harvard.edu/full/1985MNRAS.213P..21L, the total energy distributed in the gravitational field of a Schwarzschild black hole is mc^2.

The Schwarzschild spacetime is a vacuum solution, so if you want to attribute its mass-energy to something, you're going to have to attribute it to some property of the empty space..

In other words, all the mass of the black hole resides outside the event horizon.

No, this is wrong. There is no logical connection to the event horizon.
A: First of all, dcgeorge’s contrast of the rest mass vs the field mass is erroneous. The concept of rest mass requires the center-of-momentum frame of reference. Rest mass is indifferent about whether the massive object is a “particle”, a “field”, or both things combined. The problem is that curved spacetime doesn’t admit inertial frames of reference. The only known workaround is to define a reference frame that is inertial asymptotically (on the infinite distance from the object), and this is the way to define masses of strongly gravitating bodies (such as black holes). We can measure lengths, time intervals, and velocities far away of the body, and that’s how we can determine how is it massive.
Also, field energy of gravitation is, in general, very ill-defined concept in General Relativity. One reason is the same as above: it disrupts translational symmetry of the spacetime, and hence hinders the “part of the mass/momentum/energy lies here and another part lies there” discourse. This not only makes a general definition of gravitational energy impossible, but hinders consistent definitions of momentum/energy distribution in a curved spacetime (time is a conjugated to energy, and spatial length – to momentum).
Lynden-Bell and Katz introduce their definition of mass/energy density for some cases. Where the outer space is asymptotically flat, we can unambiguously define the total mass/energy (relatively to this asymptotically flat universe), as well as momentum. But the question “how is this mass/energy distributed inside” does not have a universal solution for several reasons. Should I explicate it? Lynden-Bell and Katz said themselves that “their” mass distribution is different from Penrose’s one.
