The Equivalence Principle Explained

Einstein Happiest Thought

Einstein later recalled that the key insight came to him in 1907, which he described as the happiest thought of his life. He imagined a person falling freely from the roof of a building. During the fall, the person would feel weightless, as though gravity had been switched off. Every object falling alongside them would float as if in empty space. Einstein realized that free fall is locally equivalent to the absence of gravity.

This thought experiment reveals something deep about the nature of gravity. If you are sealed inside an elevator with no windows and the cable snaps, you cannot tell from any local experiment whether you are falling toward the Earth or floating in deep space far from any massive object. Your coffee floats in front of you, a dropped ball hangs motionless, and a scale beneath your feet reads zero. The local physics is identical in both situations.

This equivalence between free fall and the absence of gravity is the core of what physicists now call the Einstein equivalence principle. It goes beyond the earlier observation by Galileo and Newton that all objects fall at the same rate. It asserts that all of physics, not just mechanics, behaves the same way in a freely falling frame as it does in the absence of gravity.

The Weak Equivalence Principle

The weak equivalence principle, also called the universality of free fall, states that the trajectory of a freely falling test body depends only on its initial position and velocity, not on its composition or internal structure. A lead ball and a wooden ball dropped from the same height hit the ground at the same time (neglecting air resistance). This principle has been tested to extraordinary precision.

The most precise tests use torsion balances that compare the gravitational acceleration of different materials toward the Sun or Earth. The Eotvos experiment, originally performed by Lorand Eotvos in the early 1900s, confirmed the weak equivalence principle to about one part in a billion. Modern versions, including the MICROSCOPE satellite mission launched in 2016, have confirmed it to about one part in 10^{15{\/sup}, making it one of the best tested principles in all of physics.}

The weak equivalence principle is equivalent to saying that gravitational mass (the property that determines how strongly an object is attracted by gravity) is exactly equal to inertial mass (the property that determines how much an object resists acceleration). Newton assumed this equality without explanation. Einstein elevated it to a fundamental principle and used it as the starting point for a new theory of gravity.

The Einstein Equivalence Principle

The Einstein equivalence principle is stronger than the weak version. It states that in a sufficiently small, freely falling laboratory, the laws of physics reduce to those of special relativity. This means not only that objects fall at the same rate regardless of composition, but that all non-gravitational experiments (electromagnetic, nuclear, chemical, biological) yield the same results in a freely falling frame as they would in the absence of gravity.

The requirement that the laboratory be sufficiently small is crucial. In a large laboratory, tidal effects become detectable. If you are freely falling toward the Earth, two balls separated horizontally will gradually drift toward each other because they are both falling toward the center of the Earth, not on exactly parallel paths. Two balls separated vertically will drift apart because the nearer one experiences slightly stronger gravity. These tidal effects are the signature of genuine spacetime curvature and cannot be eliminated by any choice of reference frame.

The distinction between genuine gravity (spacetime curvature, detected by tidal forces) and apparent gravity (which can be mimicked by acceleration) is fundamental to general relativity. The equivalence principle tells us that apparent gravity can always be transformed away locally by choosing a freely falling frame. Real gravity, the tidal stretching and squeezing described by the Riemann curvature tensor, cannot be transformed away and represents the true physical content of the gravitational field.

From Equivalence to Curved Spacetime

The equivalence principle leads directly to the conclusion that gravity must be described by curved spacetime. The argument proceeds in several steps. First, the principle tells us that freely falling frames are locally equivalent to inertial frames in the absence of gravity. Second, different freely falling frames at different locations are accelerating relative to each other (due to tidal effects). Third, patching together many local inertial frames that are accelerating relative to one another is exactly the mathematical description of a curved manifold.

An analogy helps illustrate this. The surface of the Earth is curved, but any small patch looks flat. You can lay a flat map over a city and it works perfectly. But you cannot cover the entire globe with a single flat map without distortion, because the surface is curved. Similarly, spacetime near a massive object is curved, but any small patch looks like the flat spacetime of special relativity. The equivalence principle guarantees this local flatness, while the global curvature is what we call gravity.

Einstein spent eight years, from 1907 to 1915, developing the mathematics needed to describe this curved spacetime geometry. The result was general relativity, in which the Einstein field equations relate the curvature of spacetime to the distribution of mass and energy. The equivalence principle is the physical foundation upon which the entire mathematical structure rests.

Predictions from the Equivalence Principle

Even before developing the full theory of general relativity, Einstein used the equivalence principle to make several predictions. One was gravitational redshift: light climbing out of a gravitational field loses energy and shifts to longer (redder) wavelengths. This follows directly from the equivalence principle by considering light emitted from the floor of an accelerating rocket toward the ceiling. By the time the light reaches the ceiling, the ceiling is moving faster than the floor was when the light was emitted, so the light is Doppler shifted to a lower frequency. Since acceleration is equivalent to gravity, the same redshift must occur in a gravitational field.

Another prediction was that light should be deflected by gravity. Since a beam of light in an accelerating elevator follows a curved path (the elevator accelerates upward while the light travels horizontally), the equivalence principle implies that light in a gravitational field must also follow a curved path. Einstein initial 1911 calculation of this deflection, based on the equivalence principle alone, gave a value that was later shown to be half the correct general relativistic result, because the full theory includes spatial curvature that the equivalence principle alone does not capture.

Gravitational time dilation also follows from the equivalence principle. Clocks at higher altitude in a gravitational field tick faster than clocks at lower altitude. This has been confirmed experimentally many times, from the Pound-Rebka experiment in 1959 to modern GPS satellites, which must account for gravitational time dilation to maintain their accuracy.