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This is a question about history. To my understanding the equivalence of intertial and gravitational mass for all known particles is a weird empirical coincidence that has been confirmed to high precision countless of times but doesn't really have a 'natural' reason. So it is to be expected that people in the past have speculated about yet unobserved particles for which the two masses are not equal and have tried to use these particles in order to explain yet unexplained, mass-related phenomena (like the observations that lead to the introduction of dark matter).

If people in (say) the 1930s have indeed tried this, then what did these theories look like (e. g. was the ratio gravitational/inertial mass for the hypothetical matter supposed to be greater or smaller than for ordinary matter?) and on which grounds have these theories eventually been rejected?

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    $\begingroup$ "This is a question about history". Perhaps History of Science and Mathematics would be a good place to ask? ;) $\endgroup$ – Danu Nov 17 '15 at 9:09
  • $\begingroup$ Ah, right I didn't know that site existed. $\endgroup$ – Vincent Nov 17 '15 at 9:14
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    $\begingroup$ Possible duplicate of Is there a fundamental reason why gravitational mass is the same as inertial mass? $\endgroup$ – Oscar Bravo Nov 17 '15 at 11:21
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    $\begingroup$ I think it is not really a duplicate in the sense that while for us modern SE readers the relation between the two masses may not be a coincidence any longer it is not clear that it has been this way to people in the past. On the other hand, it seems that General Relativity predates the need for more (at the time) unobserved particles (either as dark matter or in the standard model) so that it might well be true that for the reasons explained in the other question nobody ever seriously considered matter for which the two masses are different and so the other question would settle it. $\endgroup$ – Vincent Nov 17 '15 at 11:42
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    $\begingroup$ @Vincent: Flag your post for moderator attention and ask for migration to History of Science and Mathematics if you think it's appropriate! $\endgroup$ – Martin Nov 17 '15 at 14:13
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To give a qualitative answer, the equivalence of gravitational and inertial mass is not a coincidence. Mass has inertia and this resists its motion through space-time (we're moving into the future at the speed of light!). The resistance leads to a bending of space-time and it is this that we interpret as gravity.

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