We know the existence of dark matter because we can test its gravitational mass (e.g. in gravitational lens) but, since we cannot see this matter, how we can be sure that it has an inertial mass, and it is the same as the gravitational mass? In other words, does the principle of equivalence apply also to dark matter?
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If the inertial mass is not equal to the gravitational mass, it would be equivalent to the gravitational constant, G, being different, or perhaps non-constant, for dark matter. If that were so, the manner in which dark matter would orbit the milky way would be different, leading to a different distribution of dark matter
We know from the rotation curve of the milky way roughly how dark matter is distributed in the milky way. Our models are consistent with the inertial mass of dark matter being equal to the gravitational mass.
This is, of course, far from "proof". As we don't even know what dark matter is, it is impossible to be certain of any of its properties. And the equivalence of inertial and graviational mass is not "proved" even for normal matter (but no experiment has ever detected a difference) But it would be exceedingly surprising if gravity affected dark matter differently, and there would need to be strong evidence for that. As it stands, there is none.
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2$\begingroup$ Than you . My idea is that if we have a cluster of galaxies with a gravitational lens (as Abell 1689) we can use two methods to estimate the mass of the dark matter: The mass from the lens effect, that is the gravitational mass ( or not?) and the virial theorem that use the inertial mass to calculate the kinetic energy, so we can compare the two results and see if they are compatible. It is not correct? $\endgroup$ Commented Mar 27, 2016 at 8:59
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$\begingroup$ Yes, what I'm saying is that observations of galaxies are consistent with dark matter having exactly the same gravitational as inertial mass. If it doesn't you have broken General Relativity, and lots of other things too. Breaking the link between gravity and inertial mass is not a minor adjustment, you would need to rebuild a lot of theory, and there is no evidence of any difference. Our models of dark matter fit the observations . $\endgroup$– James KCommented Mar 27, 2016 at 9:07