Relationship between gravitational mass and inertial mass and the equivalence principle

Equivalence Principle

The equivalence principle is a key concept of general relativity, linking two fundamental notions: gravitational mass and inertial mass. Although these two types of mass describe different phenomena, they are equivalent.

Albert Einstein (1879-1955) envisioned a hypothetical situation where a person is in a completely closed elevator cabin, unable to see outside. This cabin can either be in free fall in a gravitational field or in an accelerating reference frame in space. He formulated this idea to show that the person could not distinguish whether they feel a gravitational force or an inertial force due to acceleration.

Gravitational Mass and Inertial Mass

Gravitational mass is the mass that determines the force with which an object is attracted in a gravitational field. In contrast, inertial mass measures a body's resistance to any change in its motion. Formally, these two concepts are defined by:

• Gravitational mass: $F_{grav} = m_g \cdot g$, where $m_g$ is the gravitational mass and $g$ is the gravitational acceleration.
• Inertial mass: $F = m_i \cdot a$, where $m_i$ is the inertial mass and $a$ is the acceleration.

According to the equivalence principle, these two masses are equal: $m_g = m_i$. This equivalence has been experimentally verified with high precision.

Consequence of the Equivalence Principle

The equivalence principle states that the effects of a gravitational field are indistinguishable from those of an acceleration in a non-inertial reference frame (amusement park ride, car turning, Earth rotating around its axis, etc.). This leads to a remarkable consequence: all objects, regardless of their mass or composition, fall with the same acceleration in a gravitational field in the absence of air resistance.

Implication in General Relativity

In general relativity, gravitation is no longer considered a classical force, but rather a manifestation of the curvature of space-time. The equivalence principle helps explain why an object in free fall does not feel a gravitational force, as it follows a geodesic in curved space-time.