**Machian definition**

The concept of inertial mass was revolutionized by Mach’s work. He was able to eliminate metaphysical elements that persisted in classical mechanics, reformulating the definition of mass in an operationally precise way and without logical contradictions. From this redefinition, general relativity started, even if Einstein himself was not able to include Mach principle within general relativity.

The Machian definition is based on the principle of action-reaction, leaving the principle of proportionality to define the force later. Consider an isolated system consisting of two interacting (point-like) bodies. Whatever is the force acting between the two bodies, it is observed experimentally that the accelerations undergone by the two bodies are always proportional and in constant ratio between them:

\[\vec{a}_2=-\mu_{12}\vec{a}_1\]

What is particularly relevant is that the ratio \(\mu_{12}\) between the two instantaneous accelerations is not only constant over time, but does not depend on the initial state of the system: it is therefore associated with an intrinsic physical property of the two bodies under consideration. By changing one of the two bodies, the constant of proportionality also varies. Let’s suppose then to use three bodies, and carry out separately three experiments with the three possible pairs (it is always assumed the absence of external forces). In this way we can measure the constants \(\mu_{12},\mu_{23},\mu_{31}\). Note that by definition:

\[\mu_{ab}=\dfrac{1}{\mu_{ba}}\]

Comparing the values of the observed constants, one will invariably find that they satisfy the relation \(\mu_{12}\cdot \mu_{23}\cdot \mu_{31}=1\). Thus the product \(\mu_{12}\cdot \mu_{31}\) does not depend on the nature of body 1, since it is equal to the inverse of \(\mu_{23}\), namely \(\mu_{32}\), which is independent of it due to the independence of \(\mu_{23}\). From this it follows that any coefficient \(\mu_{ij}\) must be able to be expressed as a product of two constants, each dependent on only one of the two bodies.

\[\mu_{ab}=\nu_{b}\cdot m_{a}=\dfrac{1}{\nu_a m_b}=\dfrac{1}{\mu_{ba}} \quad \Rightarrow\begin{cases} \nu_a=\dfrac{1}{m_a} \\ \nu_b=\dfrac{1}{m_b} \end{cases}\]

\[\mu_{ij}=\dfrac{m_i}{m_j}\quad\Rightarrow\quad m_i\vec{a}_i=-m_j\vec{a}_j\]

at any instant of time, for any pair of bodies. The quantity \(m\) that results so defined (unless a constant factor, which corresponds to the choice of the unit of measurement) is called inertial mass of the body: it is therefore possible to measure the mass of a body by measuring the accelerations due to interactions between this and another body of known mass, without needing to know what are the forces acting between the two points (provided that the system formed by the two bodies can be considered isolated, ie not subject to external forces). The link between the masses is given by:

\[m_2=\dfrac{a_1}{a_2}m_1\]