  We derive a new metric form of the complete family of black hole spacetimes
(without a cosmological constant) presented by Plebanski and Demianski in 1976.
It further improves the convenient representation of this large family of exact
black holes found in 2005 by Griffiths and Podolsky. The main advantage of the
new metric is that the key functions are considerably simplified, fully
explicit, and factorized. All four horizons are thus clearly identified, and
degenerate cases with extreme horizons can easily be discussed. Moreover, the
new metric depends only on six parameters with direct geometrical and physical
meaning, namely m, a, l, alpha, e, g which characterize mass, Kerr-like
rotation, NUT parameter, acceleration, electric and magnetic charges of the
black hole, respectively. This general metric reduces directly to the familiar
forms of either (possibly accelerating) Kerr-Newman, charged Taub-NUT solution,
or (possibly rotating and charged) C-metric by simply setting the corresponding
parameters to zero, without the need of any further transformations. In
addition, it shows that the Plebanski-Demianski family does not involve
accelerating black holes with just the NUT parameter, which were discovered by
Chng, Mann and Stelea in 2006. It also enables us to investigate various
physical properties, such as the character of singularities, horizons,
ergoregions, global conformal structure including the Penrose diagrams, cosmic
strings causing the acceleration of the black holes, their rotation,
pathological regions with closed timelike curves, or explicit thermodynamic
properties. It thus seems that our new metric is a useful representation of
this important family of black hole spacetimes of algebraic type D in the
asymptotically flat settings.
