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By "charge" I mean some kind of unique conserved property, similar to electric charge, color charge, baryon number, etc.

The no-hair theorem states that black holes can only have three macroscopic properties: mass, angular momentum, and electric charge. This implies that if dark matter had some unique "charge", we could throw all that dark matter into the black hole and the "charge" would disappear. Therefore, "charge" cannot be conserved, and this conclusion is independent of everything and anything that might describe "charge". For that matter, the no-hair theorem means that there can be no other "charge" for any kind of physical object, not just dark matter.

For example, suppose I postulate a fifth force between dark matter particles that obeys the inverse square law:

$F_{new} = k \frac{c_1c_2}{r^2}$

where $c_1$ and $c_2$ are the "charges" of the two objects. Then I must have that $c_1$ and $c_2$ cannot be conserved (in contrast to Newton's force law and Coulomb's law, where they are), or the theory is dead before it even begins.

Is this correct? If so, it sounds like a very powerful result, affecting not just the known but also the unknown.

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    $\begingroup$ You can have additional charges, that are conserved even in the presence of black holes. Just add more forces, and you'll have analogues of electric charge. Also, even for charges whose conservation is violated by black holes, that isn't really an important effect in most models. $\endgroup$ – knzhou Jan 8 '20 at 0:40
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That theorem is only valid for the known properties that stuff can have. If there were additional kinds of charges which we don't know of they might also be preserved, for example see the hypothetical NUT charge. If there were magnetic monopole charges they would also be preserved, like in the full form of the Kerr Newman metric.

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  • $\begingroup$ NUT charge is not a very good example, as it is not a "charge" in the same sense as say electric charge. Moreover, while one can write "black hole" solutions with NUT charge, these are not proper vacuum solutions of the Einstein equations. (They carry a distributional energy-momentum supported on their axis of symmetry.) $\endgroup$ – mmeent Jan 8 '20 at 8:07

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