Equivalence principle

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thumb|upright=1.5|A falling object behaves exactly the same on a planet or in an equivalent accelerating frame of reference.

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.

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".

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) 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

thumb|During the [[Apollo 15 mission in 1971, astronaut David Scott showed that Galileo was right: acceleration is the same for all bodies subject to gravity on the Moon, even for a hammer and a feather.]]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".
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."

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. 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.

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.

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 <math>M_1</math> due to the gravitational field of <math>M_0</math> is:

<math display="block">F_1 = \frac{M_0^\mathrm{act} M_1^\mathrm{pass{r^2}</math>

Likewise the force on a second object of arbitrary mass2 due to the gravitational field of mass0 is:

<math display="block">F_2 = \frac{M_0^\mathrm{act} M_2^\mathrm{pass{r^2}</math>

By definition of inertial mass:<math display="block">F = m^\mathrm{inert} a</math>if <math>m_1</math> and <math>m_2</math> are the same distance <math>r</math> from <math>m_0</math> then, by the weak equivalence principle, they fall at the same rate (i.e. their accelerations are the same).

<math display="block">a_1 = \frac{F_1}{m_1^\mathrm{inert = a_2 = \frac{F_2}{m_2^\mathrm{inert</math>

Hence:

<math display="block">\frac{M_0^\mathrm{act} M_1^\mathrm{pass{r^2 m_1^\mathrm{inert = \frac{M_0^\mathrm{act} M_2^\mathrm{pass{r^2 m_2^\mathrm{inert</math>

Therefore:

<math display="block">\frac{M_1^\mathrm{pass{m_1^\mathrm{inert = \frac{M_2^\mathrm{pass{m_2^\mathrm{inert</math>

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 <math>\eta(A,B)</math> 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".

<math display="block">\eta(A,B)=2\frac{ \left(\frac{m_{\textrm pass{m_{\textrm inert\right)_A-\left(\frac{m_{\textrm pass{m_{\textrm inert\right)_B }{\left(\frac{m_{\textrm pass{m_{\textrm inert\right)_A+\left(\frac{m_{\textrm pass{m_{\textrm inert\right)_B}.</math>

Values of this parameter are used to compare tests of the equivalence principle.

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. that they landed at the same time.

{| class="wikitable"

|+ Chronology of weak equivalence principles tests

!Year

!Investigator

!Sensitivity

!Method

||500?

||John Philoponus

||"small"

||Drop tower

||1585

||Simon Stevin

||Torsion balance

||1972

||Braginsky, Panov

||Torsion balance

||2008

||Schlamminger, et al.

||Torsion balance

||2017

||MICROSCOPE

||10−15

||Earth orbit

Experiments are still being performed at the University of Washington which have placed limits on the differential acceleration of objects towards the Earth, the Sun and towards dark matter in the Galactic Center.

Proposed tests

Proposals for satellite experiments to test the weak equivalence principle in space to much higher accuracy include Satellite Test of the Equivalence Principle and GG (for Galileo Galilei).

With the first successful production of antimatter, in particular anti-hydrogen, a new approach to test the weak equivalence principle has been proposed. Experiments to compare the gravitational behavior of matter and antimatter are currently being developed.

Proposals that may lead to a quantum theory of gravity such as string theory and loop quantum gravity predict violations of the weak equivalence principle because they contain many light scalar fields with long Compton wavelengths, which should generate fifth forces and variation of the fundamental constants. Heuristic arguments suggest that the magnitude of these equivalence principle violations could be in the 10−13 to 10−18 range.

Tests of the Einstein equivalence principle

In addition to the tests of the weak equivalence principle, the Einstein equivalence principle requires testing the local Lorentz invariance and local positional invariance conditions.

Testing local Lorentz invariance amounts to testing special relativity, a theory with a vast number of existing tests. For example, Webb et al. reported detection of variation (at the 10−5 level) of the fine-structure constant from measurements of distant quasars. Other researchers dispute these findings.

The present best limits on the variation of the fundamental constants have mainly been set by studying the naturally occurring Oklo natural nuclear fission reactor, where nuclear reactions similar to ones we observe today have been shown to have occurred underground approximately two billion years ago. These reactions are extremely sensitive to the values of the fundamental constants.

{| class="wikitable"

|+ Tests of changes in fundamental constants Up to the limit of one part in 1013 there is no Nordtvedt effect.

A tight bound on the effect of nearby gravitational fields on the strong equivalence principle comes from modeling the orbits of binary stars and comparing the results to pulsar timing data. If there is any departure from the strong equivalence principle, it is no more than two parts per million.

Most alternative theories of gravity predict a change in the gravity constant over time. Studies of Big Bang nucleosynthesis, analysis of pulsars, and the lunar laser ranging data have shown that G cannot have varied by more than 10% since the creation of the universe. The best data comes from studies of the ephemeris of Mars, based on three successive NASA missions, Mars Global Surveyor, Mars Odyssey, and Mars Reconnaissance Orbiter.

This does not point to any defect in the equivalence principle itself, only to oversimplified statements of the principle. A charged particle together with its associated electromagnetic field cannot in general be treated as confined to an arbitrarily small neighborhood, since the field extends beyond the local region in which the equivalence principle applies. As the charged and neutral balls are released side by side, the balls will initially accelerate identically. But after a short period of time, the charged ball will experience a back reaction from its own electromagnetic fields. Quantitative treatment of electromagnetic self-force in curved spacetime is technically difficult.

References

External links

Gravity and the principle of equivalence – The Feynman Lectures on Physics
Introducing The Einstein Principle of Equivalence from Syracuse University
The Equivalence Principle at MathPages
The Einstein Equivalence Principle at Living Reviews on General Relativity
"...Physicists in Germany have used an atomic interferometer to perform the most accurate ever test of the equivalence principle at the level of atoms..."

Equivalence principle

Concept

History

Definitions

<span class="anchor" id="Galilean"></span><span class="anchor" id="Weak"></span> Weak equivalence principle

Strong equivalence principle

Experimental tests

Tests of the weak equivalence principle

Proposed tests

Tests of the Einstein equivalence principle

See also

References

Further reading

External links