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Recently, it was hypothesized that some supermassive black holes (SMBHs) maycouple to the cosmic expansion. The mass of these SMBHs increase as the cubicpower of the cosmic scale factor, leaving the energy density of the SMBHsunchanged when the universe expands. However, following general principles ofgeneral relativity, namely, locality or junction conditions, we show that theinner part of a gravitationally bounded system is unaware of the cosmicexpansion, since the outer solution can be smoothly replaced by anasymptotically flat background. For the same reason, the direct reason that wedo not expand with the expansion of the universe is not that we are boundstates, but rather we are positioned in a greater gravitationally boundedsystem, namely the local group.

Decoupling between gravitationally bounded systems and the cosmic expansion