The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same trajectories and landing at identical times. The extended form by Albert Einstein requires special relativity to also hold in free fall and requires the weak equivalence to be valid everywhere. This form was a critical input for the development of the theory of general relativity. The strong form requires Einstein's form to work for stellar objects. Highly precise experimental tests of the principle limit possible deviations from equivalence to be very small.
Concept In classical mechanics, Newton's equation of motion in a gravitational field, written out in full, is: inertial mass × acceleration = gravitational mass × gravitational acceleration Careful experiments have shown that the inertial mass on the left side and gravitational mass on the right side are numerically equal and independent of the material composing the masses. The equivalence principle is the hypothesis that this numerical equality of inertial and gravitational mass is a consequence of their fundamental identity. The equivalence principle can be considered an extension of the principle of relativity, the principle that the laws of physics are invariant under uniform motion.
An observer in a windowless room cannot distinguish between being in a uniform gravitational field of 1g and being in a spaceship in deep space accelerating at 1g. Observations of physical phenomena in these cases will be indistinguishable. Einstein, Albert (1920). Relativity; The Special and the General Theory. University Paperbacks, Methuen & Co Ltd London, 1968 edition, sbn 416-67600-6, pp 66-70, translated by R.W.Lawson. See https://www.bartleby.com/lit-hub/relativity-the-special-and-general-theory/xx-the-equality-of-inertial-and-gravitational-mass-as-an-argument-for-the-general-postulate-of-relativity
In Newton's law of gravity, the equivalence principle is viewed as an observation, an accident of nature encoded in empirical law. In Einstein's model of gravity, the equivalence principle is a consequence of physics. The same mass both causes and responds to gravity: mass causes spacetime curvature and then moves through spacetime according to that curvature.
History
By experimenting with the acceleration of different materials, Galileo Galilei determined that gravitation is independent of the amount of mass being accelerated.
Isaac Newton, just 50 years after Galileo, investigated whether gravitational and inertial mass might be different concepts. He compared the periods of pendulums composed of different materials and found them to be identical. From this, he inferred that gravitational and inertial mass are the same thing. The form of this assertion, where the equivalence principle is taken to follow from empirical consistency, later became known as "weak equivalence".
A version of the equivalence principle consistent with special relativity was introduced by Albert Einstein in 1907, when he observed that identical physical laws are observed in two systems, one subject to a constant gravitational field causing acceleration and the other subject to constant acceleration, like a rocket far from any gravitational field. Since the physical laws are the same, Einstein assumed the gravitational field and the acceleration were "physically equivalent". Einstein stated this hypothesis by saying he would:
In 1911 Einstein demonstrated the power of the equivalence principle by using it to predict that clocks run at different rates in a gravitational potential, and light rays bend in a gravitational field. He connected the equivalence principle to his earlier principle of special relativity:
Soon after completing work on his theory of gravity (known as general relativity)Lorentz, Hendrik Antoon, et al. The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity. United Kingdom, Dover Publications, 1923. and then also in later years, Einstein recalled the importance of the equivalence principle to his work:
Einstein's development of general relativity necessitated some means of empirically discriminating the theory from other theories of gravity compatible with special relativity. Accordingly, Robert Dicke developed a test program incorporating two new principles – the , and the – each of which assumes the weak equivalence principle as a starting point.
Definitions
Three main forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong. Some proposals also suggest finer divisions or minor alterations.
Weak equivalence principle The weak equivalence principle, also known as the universality of free fall or the Galilean equivalence principle assumes falling bodies are self-bound by non-gravitational forces only (e.g. a stone). Some ways to define it include:
"All uncharged, freely falling test particles follow the same trajectories, once an initial position and velocity have been prescribed". "... in a uniform gravitational field all objects, regardless of their composition, fall with precisely the same acceleration." "The weak equivalence principle implicitly assumes that the falling objects are bound by non-gravitational forces." "... in a gravitational field the acceleration of a test particle is independent of its properties, including its rest mass." Mass (measured with a balance) and weight (measured with a scale) are locally in identical ratio for all bodies (the opening page to Newton's Philosophiæ Naturalis Principia Mathematica, 1687). "the trajectory of a freely falling 'test' body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition."
Einstein equivalence principle
What is now called the "Einstein equivalence principle" states that the weak equivalence principle holds, and that:
Here local means that experimental setup must be small compared to variations in the gravitational field, called tidal forces. The test experiment must be small enough so that its gravitational potential does not alter the result.
The two additional constraints added to the weak principle to get the Einstein form the independence of the outcome on relative velocity (local Lorentz invariance) and independence of "where" (known as local positional invariance) have far reaching consequences.
With these constraints alone Einstein was able to predict the gravitational redshift. Theories of gravity that obey the Einstein equivalence principle must be "metric theories", meaning that trajectories of freely falling bodies are geodesics of symmetric metric.
Around 1960 Leonard I. Schiff conjectured that any complete and consistent theory of gravity that embodies the weak equivalence principle implies the Einstein equivalence principle; the conjecture can't be proven but has several plausibility arguments in its favor. Nonetheless, the two principles are tested with very different kinds of experiments.
The Einstein equivalence principle has been criticized as imprecise, because there is no universally accepted way to distinguish gravitational from non-gravitational experiments (see for instance Hadley and Durand).
Strong equivalence principle The strong equivalence principle applies the same constraints as the Einstein equivalence principle, but allows the freely falling bodies to be massive gravitating objects as well as test particles. The strong equivalence principle includes astronomic bodies with gravitational self-binding energy. Thus this is a version of the equivalence principle that applies to objects that exert a gravitational force on themselves, such as stars, planets, black holes or Cavendish experiments. It requires that the gravitational constant be the same everywhere in the universe. It is much more restrictive than the Einstein equivalence principle.
Like the Einstein equivalence principle, the strong equivalence principle requires gravity to be geometrical by nature, but in addition it forbids any extra fields, so the metric alone determines all of the effects of gravity. If an observer measures a patch of space to be flat, then the strong equivalence principle suggests that it is absolutely equivalent to any other patch of flat space elsewhere in the universe. Einstein's theory of general relativity (including the cosmological constant) is thought to be the only theory of gravity that satisfies the strong equivalence principle. A number of alternative theories, such as Brans–Dicke theory and the Einstein-aether theory add additional fields.
Active, passive, and inertial masses Some of the tests of the equivalence principle use names for the different ways mass appears in physical formulae. In nonrelativistic physics three kinds of mass can be distinguished: Inertial mass intrinsic to an object, the sum of all of its mass–energy. Passive mass, the response to gravity, the object's weight. Active mass, the mass that determines the object's gravitational effect. By definition of active and passive gravitational mass, the force on due to the gravitational field of is:
Likewise the force on a second object of arbitrary mass2 due to the gravitational field of mass0 is:
By definition of inertial mass:if and are the same distance from then, by the weak equivalence principle, they fall at the same rate (i.e. their accelerations are the same).
Hence:
Therefore:
In other words, passive gravitational mass must be proportional to inertial mass for objects, independent of their material composition if the weak equivalence principle is obeyed.
The dimensionless Eötvös-parameter or Eötvös ratio is the difference of the ratios of gravitational and inertial masses divided by their average for the two sets of test masses "A" and "B".
Values of this parameter are used to compare tests of the equivalence principle.
A similar parameter can be used to compare passive and active mass. By Newton's third law of motion:
must be equal and opposite to
It follows that:
In words, passive gravitational mass must be proportional to active gravitational mass for all objects. The difference,
is used to quantify differences between passive and active mass.
Experimental tests Tests of the weak equivalence principle Tests of the weak equivalence principle are those that verify the equivalence of gravitational mass and inertial mass. An obvious test is dropping different objects and verifying that they land at the same time. Historically this was the first approach – though probably not by Galileo's Leaning Tower of Pisa experiment but instead earlier by Simon Stevin, who dropped lead balls of different masses off the Delft churchtower and listened for the sound of them hitting a wooden plank.
Newton measured the period of pendulums made with different materials as an alternative test giving the first precision measurements. Loránd Eötvös's approach in 1908 used a very sensitive torsion balance to give precision approaching 1 in a billion. Modern experiments have improved this by another factor of a million.
A popular exposition of this measurement was done on the Moon by David Scott in 1971. He dropped a falcon feather and a hammer at the same time, showing on video that they landed at the same time.