  According to the Virial Theorem, all gravitational systems in equilibrium sit
on a plane in the 3D parameter space defined by their mass, size and second
moment of the velocity tensor. While these quantities cannot be directly
observed, there are suitable proxies: the luminosity Lk, half-light radius Re
and dispersion sigma_e. These proxies indeed lie on a very tight Fundamental
Plane (FP). How do the black holes in the centers of galaxies relate to the FP?
Their masses are known to exhibit no strong correlation with total galaxy mass,
but they do correlate weakly with bulge mass (when present), and extremely well
with the velocity dispersion through the Mbh = sigma_e^5.4 relation. These
facts together imply that a tight plane must also exist defined by black hole
mass, total galaxy mass and size. Here I show that this is indeed the case
using a heterogeneous set of 230 black holes. The sample includes BHs from zero
to 10 billion solar masses and host galaxies ranging from low surface
brightness dwarfs, through bulge-less disks, to brightest cluster galaxies. The
resulting BH-size-luminosity Mbh=(Lk/Re)^3.8 has the same amount of scatter as
the M-sigma relation and is aligned with the galaxy FP, such that it is just a
re-projection of sigma. The inferred BH-size-mass relation is
Mbh=(M_star/Re)^2.9. These relationships are universal and extend to galaxies
without bulges. This implies that the black hole is primarily correlated with
its global velocity dispersion and not with the properties of the bulge. I show
that the classical bulge--mass relation is a projection of the M-sigma
relation. When the velocity dispersion cannot be measured (at high-z or low
dispersions), the BH-size-mass relation should be used as a proxy for black
hole mass in favor of just galaxy or bulge mass.
