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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 × intensity of the gravitational field

Very 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.^[1]: 32

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 on the surface of the Earth and being in a spaceship in deep space accelerating at 1g and the laws of physics are unable to distinguish these cases.^[1]: 33

History

Galileo compared different materials experimentally to determine that the acceleration due to gravitation is independent of the amount of mass being accelerated.^[2]

Newton, just 50 years after Galileo, developed the idea that gravitational and inertial mass were different concepts and compared the periods of pendulums composed of different materials to verify that these masses are the same. This form of the equivalence principle became known as "weak equivalence".^[2]

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.^[3]: 152 Since the physical laws are the same, Einstein assumed the graviational field and the acceleration were "physically equivalent". Einstein stated this hypothesis as:

we ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.

— Einstein, 1907^[4]

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.^[3]: 153 He connected the equivalence principle to his earlier principle of special relativity:

This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system of reference, just as the usual theory of relativity forbids us to talk of the absolute velocity of a system; and it makes the equal falling of all bodies in a gravitational field seem a matter of course.

— Einstein, 1911^[5]

Immediately after completing his work^[6]: 111 on a theory of gravity (known as general relativity) and in later years Einstein recalled the role of the equivalence principle:

The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly a thought struck me: If a man falls freely, he would not feel his weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity.

— Einstein, 1922^[7]

Since Einstein developed general relativity, there was a need to develop a framework to test the theory against other possible theories of gravity compatible with special relativity. This was developed by Robert Dicke as part of his program to test general relativity. Two new principles were suggested, the so-called Einstein equivalence principle and the strong equivalence principle, each of which assumes the weak equivalence principle as a starting point. These are discussed below.

Definitions

Three main forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong.^[8]: 6 Some studies also create finer divisions or slight alternative.^[9][10]

Weak equivalence principle

The weak equivalence principle, also known as the universality of free fall or the Galilean equivalence principle can be stated in many ways. The strong equivalence principle, a generalization of the weak equivalence principle, includes astronomic bodies with gravitational self-binding energy.^[11] Instead, the weak equivalence principle assumes falling bodies are self-bound by non-gravitational forces only (e.g. a stone). Either way:

• "All uncharged, freely falling test particles follow the same trajectories, once an initial position and velocity have been prescribed".^[8]: 6
• "... 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."^[11]
• "... in a gravitational field the acceleration of a test particle is independent of its properties, including its rest mass."^[12]
• 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).

Uniformity of the gravitational field eliminates measurable tidal forces originating from a radial divergent gravitational field (e.g., the Earth) upon finite sized physical bodies.

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 − (1) the independence of the outcome on relative velocity (local Lorentz invariance) and (2) independence of "where" known as (local positional invariance) − have far reaching consequences. With these constraints alone Einstein was able to predict the gravitational redshift.^[13] 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.^[14]: 9

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.^[14]: 20 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^[15] and Durand^[16]).

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.^[8] 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^[14]: 49 and is incompatible with a fifth force. It is much more restrictive than the Einstein equivalence principle.

Like the Einstein equivalence principle, the strong equivalence principle requires gravity is 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.^[8]

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:^[14]

1. Inertial mass intrinsic to an object, the sum of all of its mass–energy.
2. Passive mass, the response to gravity, the object's weight.
3. Active mass, the mass that determines the objects gravitational effect.

By definition of active and passive gravitational mass, the force on ${\displaystyle M_{1}}$ due to the gravitational field of ${\displaystyle M_{0}}$ 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 ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ are the same distance ${\displaystyle r}$ from ${\displaystyle m_{0}}$ 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 $$Zdroj:https://en.wikipedia.org?pojem=Equivalence_principle
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