  Motivated by the recent heated debate on whether the masses of local objects,
such as compact stars or black holes (BHs), may be affected by the large-scale,
cosmological dynamics, we analyze the conditions under which, in a general
relativity framework, such a coupling small/large scales is allowed. We shed
light on some controversial arguments, which have been used to rule out the
latter possibility. We find that the cosmological coupling occurs whenever the
energy of the central objects is quantified by the quasi-local Misner-Sharp
mass (MS). Conversely, the decoupling occurs whenever the MS mass is fully
equivalent to the (nonlocal) Arnowitt-Deser-Misner (ADM) mass. Consequently,
for singular BHs embedded in cosmological backgrounds, like the
Schwarzschild-de Sitter or McVittie solutions, we show that there is no
cosmological coupling, confirming previous results in the literature.
Furthermore, we show that nonsingular compact objects couple to the
cosmological background, as quantified by their MS mass. We conclude that
observational evidence of cosmological coupling of astrophysical BHs would be
the smoking gun of their nonsingular nature.
