Black dwarf binary systems I know black dwarf stars do not exist yet, and that they are what white dwarfs end up after a long time. The question is really simple: would it be possible to have a binary system in which one of the two is a black dwarf, and the other companion star is dumping mass on the black dwarf, causing the black dwarf to supernova? I guess based on the timescales, the maximal initial mass of the second object could be calculated to see if it is possible that one object is at the black dwarf stage and the other still a giant. Or is the timescale on which the star goes from white dwarf to black dwarf big enough that it will never reach the black dwarf stage while the other one is still an AGB star? 

Second part: If not possible that one is black dwarf and the other AGB star, is it possible if the companion is a brown dwarf, that they are close binary, and that a brown dwarf would dump mass on a black dwarf? This would need the binary system to be a very close binary system at this stage of the system. 

Third part of the question: would the spectrum of the supernova be any different from the normal one with a white dwarf and a companion AGB star? I guess the spectral lines would be different, since the companion would have to be a very low-mass AGB star, since that companion would be about 20billion years or so, when the black dwarf star becomes a black dwarf.
I see two real questions here. First, whether it's possible to have a black dwarf with a companion object. For a given black dwarf, this is unlikely, since the orbits would likely be unstable at the time scale required to produce a black dwarf. Given the size of the universe, however, it's not out of the question. A black dwarf could even capture a companion star after cooling down, although this would be extremely rare.
Assuming it's possible, the next question would be whether the supernova produced would be the same as a normal type Ia supernova. The supernova is generally triggered based on the critical (Chandrasekhar) mass which triggers carbon fusion. Since it's dependent on mass, it would likely be a &quot;normal&quot; type 1A supernova. Keep in mind not all 1A supernovae are identical to begin with, although they can still be &quot;normalized&quot; to function as a fairly reliable standard candle.
One thing to note is, the accretion process would cause the black dwarf to heat up, meaning it would not remain black.