Is there a deeper meaning behind how things of different mass fall at the same acceleration? It feels so perfectly balanced...

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    $\begingroup$ Yes, that "deeper meaning" is called Einstein's General relativity (GR) $\endgroup$ – user164843 Apr 19 at 4:26
  • $\begingroup$ Great question. $\endgroup$ – Avantgarde Apr 19 at 4:59

This is a priori an assumption.

In Newtonian physics, they are not assumed to be equal but because empirically they are, we often take them to be the same. For example, Newton's second law says $\vec{F}=m_i\vec{a}$ and Newton's law of gravitation says $\vec{F} = -(GMm_g/r^2)\hat{r}$. The standard (high school) calculation assumes $m_i=m_g$ from empirical results but as stated, the two laws do not say they are the same.

In general relativity, however, it is assumed that gravitational mass and inertial mass is the same: this automatically leads to the result that every object accelerates the same way under gravity, since it is simply a statement that every (point) particle (with no non-gravitational forces acting on them) follows spacetime geodesics. This is closely connected to weak equivalence principle. The fact that general relativity remains the currently best explanation (verified by experiments) for gravitation can be taken as evidence that they should be equal: if deviations were found by ultra-precise experiments, then one will have to forgo general relativity.

Note that unlike Newtonian gravity where the equality is not justified by the framework itself (Newton's laws did not specify they have to), general relativity tells us that we should take them to be equal as an explanation for how gravity works. The way test particles follow geodesic equation in curved spacetime can only make sense if test masses's (inertial/gravitational) mass do not enter the picture. That's why one cannot simply "geometrize" electromagnetism (at least not by itself, as in the case of Kaluza-Klein which geometrizes electromagnetism and gravity).

  • $\begingroup$ Thanks! What does it really mean though? Is it just a matter of us knowing that is the way or how it works without knowledge of any deeper interpretation of it? $\endgroup$ – csp2018 Apr 19 at 5:18
  • $\begingroup$ @csp2018 I think it depends on what you mean. To many, general relativity's assumption that $m_i = m_g$ is a "deep insight": that's how you make gravity geometrical in spacetime. This does not work (naively) for e.g. electromagnetism: the fact that there are neutral particles mean that electric field cannot be made "geometrical" the same way gravity is. For gravity, because $E=mc^2$, even massless particles will "fall" in gravitational field. I think the fact that it can be made to work at all is an insight in itself. $\endgroup$ – Everiana Apr 20 at 7:54
  • $\begingroup$ Newtonian physics doesn't really say that $m_g = m_i$. You could write $m_g = km_i$, for some non-zero $k$, and incorporate $k$ into $G$, so $F=(G/k^2)M_gm_g/r^2$, but it's more convenient to set $k=1$. Of course, using a $k\ne 1$ just shuffles the problem around: why is $k$ the same for all bodies, no matter their size, shape, or what substances they're made from. $\endgroup$ – PM 2Ring Apr 20 at 8:21

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