The equivalence principle

Special relativity describes observers travelling in a straight line at constant velocity. But the world is full of accelerations: the lift going down, the car braking and, most decisively, free fall in the Earth’s gravitational field. The bridge from special to general relativity begins with a seemingly innocent question about the nature of mass.

When we write $\vec F = m\vec a$, mass is an inertial property: it measures a body’s resistance to being accelerated. When we write $\vec F_g = m\vec g$, the same letter instead denotes a gravitational property: it measures how strongly the body responds to a gravitational field. A priori these are two logically distinct quantities: inertial mass $m_\text{in}$ enters $F = m_\text{in}\,a$, gravitational mass $m_g$ enters $F_g = m_g\,g$. Nothing, in principle, forces the two numbers to coincide.

Yet they do coincide. Newton had already noticed this: experimentally $m_\text{in} = m_g$ within the uncertainties of his time. At the start of the twentieth century Eötvös made the check quantitative with a torsion balance, obtaining $m_\text{in}/m_g = 1$ to a precision of about $10^{-9}$. Today space-based experiments (the MICROSCOPE mission) have confirmed the equality to about $10^{-15}$. Einstein took the bold step: he elevated this experimental fact to the status of a principle.

Equivalence principle (Einstein)

Locally, no experiment — mechanical, optical or electromagnetic — can distinguish motion in a uniform gravitational field from motion in a uniformly accelerated reference frame (Stannard 2008).

The word “locally” is crucial, not a technical detail. Over a sufficiently large region a real gravitational field is not uniform: tidal forces (the field pointing towards the Earth’s centre converges, it is not parallel) again make the two situations distinguishable. The equivalence holds exactly only in the limit of an infinitesimally small region.

From Galileo to Einstein

Galileo had observed, as far back as the (never truly completed) experiments at Pisa, that “all bodies fall with the same acceleration”. Newton explained this by postulating $m_\text{in} = m_g$: a formal but unmotivated equality, two a priori distinct quantities coinciding by pure coincidence. In 1907 Einstein took it up again and asserted that it was not a coincidence at all, but a profound fact: there is no local difference between gravity and acceleration. This is the key to the passage from Newtonian gravity — an instantaneous force acting at a distance — to general relativity, the local geometry of spacetime. A three-and-a-half-century story, to give physical meaning to an equality that had been sitting in plain sight all along.

The consequence is radical: general relativity, formulated by Einstein in 1915, states that gravity is not a force but a geometric property of spacetime, curved by the presence of mass-energy. A rigorous treatment requires the tools of differential geometry, but the key ideas can already be grasped with elementary physics.

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Topics: Relatività ristretta Concepts: Legge di gravitazione universale

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