Mass: inertial or gravitational?

The quantity $m$ appearing in the second law $\vv{F} = m\,\vv{a}$ is called inertial mass: it measures a body’s resistance to a change in its own velocity. The greater a body’s inertial mass, the harder it is to accelerate or slow it down with a given force. It is, in this sense, a measure of the body’s “stubbornness” in maintaining its state of motion.

There exists, however, another quantity, apparently entirely different: gravitational mass. This is the one that appears in the law of universal gravitation, in the form $F = G\,m_g M_g / d^2$, and that determines the strength of the gravitational attraction between two bodies. A priori there is no reason why the two masses should coincide: they are defined through conceptually independent experiments. Inertial mass is measured by applying any force and observing the resulting acceleration; gravitational mass is measured by weighing the body, that is, by observing the force with which it is attracted by another body. Nothing in the logical structure of mechanics requires the two numbers to be equal.

Yet, experimentally, the two masses always turn out to be equal, with a truly extraordinary precision: today the equality is verified to within one part in $10^{15}$. This mysterious coincidence has, within Newtonian mechanics, no explanation: it is simply an observed fact. It was precisely this coincidence that led Einstein, in 1915, to formulate general relativity, in which gravity is no longer a force but the curvature of spacetime. In that theory the equality between inertial mass and gravitational mass ceases to be a coincidence and becomes a founding principle — the equivalence principle (Jammer 2000, ch. 8).

Historical context

Max Jammer, in his conceptual history of mass, observes that “the notion of mass, although fundamental to physics, is still shrouded in mystery”. The concept seems elementary, almost obvious, but it actually conceals centuries of philosophical and physical reflection that are not yet entirely concluded (Jammer 2000; Jammer 1957).

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Topics: Dinamica Concepts: Seconda legge di Newton · Forza peso

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