Mass

There are several distinct phenomena which can be used to measure mass. Although some theorists have speculated that some of these phenomena could be independent of each other, current experiments have found no difference in results regardless of how it is measured:
 * Inertial mass measures an object's resistance to being accelerated by a force (represented by the relationship F = ma).
 * Active gravitational mass measures the gravitational force exerted by an object.
 * Passive gravitational mass measures the gravitational force exerted on an object in a known gravitational field.

Equivalence principle
By definition of active and passive gravitational mass, the force on $$M_1$$ due to the gravitational field of $$M_0$$ is:


 * $$F_1 = \frac{M_0^\mathrm{act} M_1^\mathrm{pass}}{r^2}$$

Likewise the force on a second object of arbitrary mass2 due to the gravitational field of mass0 is:


 * $$F_2 = \frac{M_0^\mathrm{act} M_2^\mathrm{pass}}{r^2}$$

By definition of inertial mass:


 * $$F = m^\mathrm{inert} a$$

If $$m_1$$ and $$m_2$$ are the same distance $$r$$ from $$m_0$$ then, by the weak equivalence principle, they fall at the same rate (i.e. their accelerations are the same)


 * $$a_1 = \frac{F_1}{m_1^\mathrm{inert}} = a_2 = \frac{F_2}{m_2^\mathrm{inert}}$$

Hence:


 * $$\frac{M_0^\mathrm{act} M_1^\mathrm{pass}}{r^2 m_1^\mathrm{inert}} = \frac{M_0^\mathrm{act} M_2^\mathrm{pass}}{r^2 m_2^\mathrm{inert}}$$

Therefore:


 * $$\frac{M_1^\mathrm{pass}}{m_1^\mathrm{inert}} = \frac{M_2^\mathrm{pass}}{m_2^\mathrm{inert}}$$

In other words, passive gravitational mass must be proportional to inertial mass for all objects.

Furthermore, by Newton's third law of motion:


 * $$F_1 = \frac{M_0^\mathrm{act} M_1^\mathrm{pass}}{r^2}$$

must be equal and opposite to


 * $$F_0 = \frac{M_1^\mathrm{act} M_0^\mathrm{pass}}{r^2}$$

It follows that:


 * $$\frac{M_0^\mathrm{act}}{M_0^\mathrm{pass}} = \frac{M_1^\mathrm{act}}{M_1^\mathrm{pass}}$$

In other words, passive gravitational mass must be proportional to active gravitational mass for all objects.

The dimensionless Eötvös-parameter $$\eta(A,B)$$ is the difference of the ratios of gravitational and inertial masses divided by their average for the two sets of test masses "A" and "B."


 * $$\eta(A,B)=2\frac{ \left(\frac{m_g}{m_i}\right)_A-\left(\frac{m_g}{m_i}\right)_B }{\left(\frac{m_g}{m_i}\right)_A+\left(\frac{m_g}{m_i}\right)_B}$$