The equivalence principle is a family of statements concerning gravitation, inertia, and the behavior of physical experiments in freely falling laboratories. Its simplest form asserts that test bodies with different compositions follow the same free-fall trajectories when released under identical conditions. Its broader formulations connect gravity with the local laws of special relativity and underpin the geometric description of gravity in general relativity. The weak, Einstein, and strong equivalence principles have distinct scopes and require different experimental tests. (link.springer.com)
Mass and universal free fall
In classical mechanics, mass plays two conceptually different roles. Inertial mass, (m_i), measures resistance to acceleration under an applied force, as expressed by Newton’s second law, (F=m_i a). Passive gravitational mass, (m_g), measures how strongly a body responds to an external gravitational field. Combining these relations gives
[ a=\frac{m_g}{m_i}g, ]
where (g) represents the external field strength. If (m_g/m_i) is identical for all bodies, their gravitational acceleration is independent of composition. With conventional normalization, this universal ratio is set to one. Passive gravitational mass is distinct from active gravitational mass, which determines how a body produces a gravitational field. (einstein-online.info)
The weak equivalence principle (WEP), also called the universality of free fall, expresses this composition-independent motion. It concerns ideal test bodies whose own gravity is negligible and excludes nongravitational disturbances, such as air resistance or electromagnetic forces. Equality of falling accelerations is therefore not a claim that every real object falls identically under arbitrary laboratory conditions. (link.springer.com)
Einstein’s formulation and the falling laboratory
In 1907, Albert Einstein used the equivalence of gravitation and acceleration as a starting point for developing a relativistic theory of gravity. His reasoning shifted attention from falling objects to the physical laws observed inside a falling laboratory. (link.springer.com)
An elevator thought experiment illustrates the idea. In an enclosed cabin accelerating through otherwise gravity-free space, released objects appear to fall toward the floor. Conversely, objects inside a freely falling cabin near Earth float relative to the cabin. Their apparent weightlessness does not mean Earth’s gravitational influence has disappeared: the cabin and its contents are falling together. Within a sufficiently small region and observation interval, such a laboratory behaves like an inertial reference frame in special relativity. (einstein-online.info)
The Einstein equivalence principle (EEP) combines WEP with two further requirements:
- Local Lorentz invariance: the outcomes of local nongravitational experiments are independent of the velocity of the freely falling laboratory.
- Local position invariance: those outcomes are independent of where and when the experiments are performed.
Together, these requirements extend the principle beyond mechanical free fall to local nongravitational physics. They do not include experiments measuring the gravitational interaction between bodies inside the laboratory. (link.springer.com)
Locality, curvature, and light
The qualification “local” is essential. Real gravitational fields vary across space. Two freely falling bodies can approach or separate because they experience different gravitational influences. These tidal effects remain observable inside a falling laboratory and prevent the complete removal of gravity over an extended region. Reducing the laboratory’s dimensions and the duration of measurements makes their effects correspondingly smaller. (einstein-online.info)
In the geometric description, gravity is associated with the curvature of spacetime. Freely falling test bodies follow geodesics, the spacetime counterparts of straight paths. A falling observer can eliminate the common apparent gravitational acceleration locally, but not the relative acceleration of separated trajectories that reveals curvature. Thus, the equivalence principle does not identify a curved spacetime globally with an accelerating laboratory in flat spacetime. (einstein-online.info)
The principle also applies to light. A light pulse traveling across an accelerating cabin appears to follow a bent path because the cabin moves during the pulse’s transit. Equivalent reasoning predicts gravitational deflection, although calculating the full deflection around a massive body requires the spacetime geometry supplied by a gravitational theory. (einstein-online.info)
Combined with special relativity, equivalence arguments also predict gravitational redshift: light exchanged between stationary observers at different gravitational potentials can have different measured frequencies. The principle supplies constraints on gravitational theories rather than uniquely determining the Einstein field equations. (arxiv.org)
Strong equivalence principle
The strong equivalence principle (SEP) extends equivalence to self-gravitating bodies and local gravitational experiments. In particular, gravitational binding energy should contribute to gravitational response in the same way as other forms of energy. This distinguishes SEP from tests involving small bodies whose self-gravity is negligible. (nature.com)
Lunar laser ranging tests this aspect by examining whether Earth and the Moon fall toward the Sun at the same rate. Their different fractional gravitational self-energies provide sensitivity to a possible violation, which would produce a characteristic perturbation of the lunar orbit. Composition-dependent effects must also be considered when interpreting such measurements. (nature.com)
Experimental tests
WEP tests compare the accelerations of bodies made from different materials. A common measure is the Eötvös parameter,
[ \eta=\frac{2(a_A-a_B)}{a_A+a_B}, ]
where (a_A) and (a_B) are accelerations toward the same source. Exact universality predicts (\eta=0). (journals.aps.org)
The MICROSCOPE mission compared titanium-alloy and platinum-alloy test masses aboard a drag-compensated satellite. Its final results, published on September 14, 2022, gave
[ \eta(\mathrm{Ti},\mathrm{Pt}) =[-1.5\pm2.3\text{ (statistical)} \pm1.5\text{ (systematic)}]\times10^{-15}. ]
No WEP violation was detected within those uncertainties. This result constrains differential acceleration for the tested materials; it is not a proof of exact equivalence in every physical regime. Tests of velocity independence, clock behavior, and gravitational self-energy address other components of the broader principle. (journals.aps.org)