This may be nothing more then a bad drain seal. Let's check and see. Close the stopper to the tub and fill the tub with a few inches of water. Let it set for awhile and check with the apartment downstairs for moisture. If yes then you have a ruptured drain seal and nit can be repaired without opening up the ceiling below. If no moisture moisture appears then let the water drain out and keep checking for moisture. If you see any then you have a leak in the trap area and the ceiling will have to be opened up.
Another question is I live in NYC co-op apartment. Is it my responsibility or the co-ops responsibility in this case
Since this is not considered to be a "common area" the expense will ne on you. Good luck and let me know what you find and we'll go for a repair. Tom