# What is the inertial mass of a black hole?

Or the inertial mass of any spherically symmetric object, can I calculated by measuring very accurately the spacetime distortion this object produces in its surroundings?

With 'inertial mass' I mean the resistance to be accelerated. For example, if a spherically symmetric object (not necessarily a black hole) has a Schwarzchild radius $2GM/c^2$ (determined from measuring the spacetime distortions outside of the object), has an electric charge $q$, and is at rest; would it be the case that the instantaneous acceleration at time zero when we turn on an electric field $E$ is given by $qE/M$ with the same $M$ as before?

• The honest answer is "we don't know" because this has, obviously, never been tested directly and it's not clear how one could hope to test it for anything larger than the size of a large planet. I am not aware that orbital data from neutron star binaries and potential normal star or neutron star-black hole binaries has indicated any deviation from the natural assumption that black holes behave like all other inertial masses. It's an interesting question, though. E.g. V404 Cygni orbital data may hold the answer, if there are obvious deviations, but it can't test for forces other than gravity. – CuriousOne Jan 5 '16 at 17:44
• From my point of view, this is like a modern version of the awkward equivalence posed by Newton's theory of gravitation between the inertial and gravitational mass. Gravitational mass in this new case that $M$ in the Schwarzchild radius. – Enredanrestos Jan 5 '16 at 17:48
• What's awkward about the equivalence principle? Does it have to be tested with the highest possible precision? Absolutely. Does nature allow us to test everything? Nope. That's life and we simply have to live with these limitations. General relativity is not a holy cow, by the way. If orbital data of extreme objects will indicate deviations, then we will modify or replace the theory "just like that". – CuriousOne Jan 5 '16 at 17:51
• In Newtonian mechanics, the gravitational self-interaction with the body's own gravitational field reduces the mass of the body. Since gravitational potential energy is not consistent with GR, how is the gravitational self interaction dalt with in computing mass? – Peter R Jan 5 '16 at 21:18