Einstein Equivalence Principle

Equivalent forms

weak eep

m_i \ddot{\vec{x}} = m_g \vec{g},\quad m_i = m_g

redshift form

\frac{\Delta \nu}{\nu} = \frac{g h}{c^2}

The deepest free lunch in physics: equating two a-priori unrelated masses turns gravity from a force into the curvature of spacetime.

Symbol	Name	SI unit	Dimension	Range
$m_i$	inertial mass	kg	M	—
$m_g$	gravitational mass	kg	M	—
$\vec{g}$	local gravitational field	9.8 m/s^2	L T^{-2}	—
$\vec{a}$	frame accelerationoutput	m/s^2	L T^{-2}	—

Discovery

Albert Einstein (after Galileo, Eotvos) · 1907

Einstein called it 'the happiest thought of my life' (1907): a person in free fall feels no weight. Galileo had already argued all bodies fall alike; Eotvos torsion-balance experiments (1885-1909) verified m_i = m_g to 1 part in 10^9. Einstein elevated this coincidence to a principle, the seed of general relativity.

Research status: stable

Real-world applications

Free-fall 'zero-g' aircraft and drop towers reproduce weightlessness by removing the floor's normal force, not gravity.
Gravitational redshift of clocks $(\delta -\nu /\nu = gh/c^2)$ follows directly and underlies GPS timing.
Inertial navigation accelerometers cannot separate gravity from acceleration — the principle in engineering form.
MICROSCOPE satellite (2017/2022) tested EEP to 1 part in 10^15.

Common misconceptions

'Astronauts float because there's no gravity.' — Gravity at the $ISS is \sim 90$ % of surface value; they float because they are in free fall.
'Equivalence is exact everywhere.' — Only locally: tidal forces (curvature) reveal a real gravitational field over extended regions.
'It just means heavy things fall fast.' — It means they fall at the SAME rate regardless of mass.

Experimental verification

Eotvos torsion balance (1 in 10^9), Lunar Laser Ranging, and MICROSCOPE (1 in 10^15) all confirm

m_i = m_g

; gravitational redshift confirmed by Pound-Rebka (1959).

Derivation

Newton's law in a gravitational field: $m_i a = m_g g$ .
Observation (Galileo, Eotvos): $m_i = m_g$ to extreme precision for every material.
Therefore $a = g$ independent of mass or composition — all bodies share one trajectory.
A uniformly accelerating frame (rocket, a) is then locally indistinguishable from rest in a gravitational field $(g =$ -a).
Promoting this to all (non-gravitational) physics gives the Einstein Equivalence Principle, and demanding it hold over extended regions forces spacetime to be curved.

Limiting cases

Small local region⟶ gravity == accelerationFree-fall frame is indistinguishable from inertial; special relativity holds locally.

Extended region⟶ tidal forces appearSpacetime curvature cannot be transformed away — the true signature of gravity.

What if…

What if m_i did not equal m_g?

Different materials would fall at different rates; a torsion balance would twist, and general relativity's geometric picture would collapse.

What if the elevator were very tall?

Tidal effects appear — two coins released apart drift together (curvature). Equivalence is only local.

1

The dropped-coin elevator

Given · An elevator accelerates upward at

a = 9.8\,\mathrm{m/s}^2

in deep space (no gravity). You release a coin.

Find · The coin's motion relative to the elevator floor

Steps

In the inertial frame the coin coasts at constant velocity once released.
The floor accelerates up at 9.8 m/s^2 toward the coin.
Relative to the elevator the coin 'falls' at 9.8 m/s^2.

Result · Identical to dropping a coin on Earth — you could not tell which situation you are in from inside.

Gravitational Redshift→Gravitational Time Dilation→Schwarzschild Radius→General Relativity: Light Bending & Curved Spacetime→