Black Dwarf Stars are hypothetical stars that don't yet exist but are predicted to in a far off future. When a star dies and turns into a White Dwarf Star, it will emit light for an unimaginable length of time but once it has run out of all energy, it will no longer produce light. When it has stopped producing light, it becomes black and no longer emits light.
A White Dwarf will not die in the same way a star dies, it just runs out of energy to produce and emit light. White Dwarf is like the embers of a fire which is slowly going out. When the embers gone cool and not longer appear on fire, that's the equivalence of a white to black dwarf.
Stars are huge spherical objects of gas that through chemical interaction with one another produce heat and energy. A Black Dwarf Star is believed to be solid unlike the gaseous early stages of its life. The dead star will be composed of carbon, even a diamond once it has turned into its final stage. Harvard
Even though it is solid, you'd probably wouldn't want to touch or walk on the surface. It'll still probably be radio-active.
A black dwarf is the ultimate end of many stars including our own Sun. Because of the length of time it takes to go from a white dwarf to a black dwarf, the Universe is too young for any to exist as yet. After an unfathomable amount of time, the black dwarf may well decline and disappear into nothing. At the very end of the universe will be nothing, just like before the start of the universe.
A black hole and a Black Dwarf Star are two completely different things. Black holes do not contain a Black Dwarf star at the heart of it. Whilst they are both stellar remnants, they are both formed differently and will disappear differently.
According to Medium, the coolest white dwarves have only lost 0.2% of their heat, so if the white dwarf was 20,000K, now it'd still be only 19,960K, still a very long way to go before it'll have cooled down sufficiently,
A Black Dwarf Star is by their very name, black and would be hard to detect because we wouldn't be able to see them. They may give off a little heat or radiation which might give themselves away.
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