Equivalence principle

In retrospect, the law of equivalence of inertial mass and gravitational mass is independent of the body's nature of the acceleration imparted to it by the gravitational field. For Newton's equation of motion in a gravitational field, written out in full, it is:

(Inertial mass) ${\displaystyle \cdot }$ (Acceleration) ${\displaystyle =}$ (Gravitational mass) ${\displaystyle \cdot }$ (Intensity of the gravitational field).

The numerical equality between inertial mass and gravitational mass and acceleration are independent of the properties of the body.[1][2]

Something like the equivalence principle emerged in the early 17th century, when Galileo expressed experimentally that the acceleration of a test mass due to gravitation is independent of the amount of mass being accelerated.

Johannes Kepler, using Galileo's discoveries, showed knowledge of the equivalence principle by accurately describing what would occur if the Moon were stopped in its orbit and dropped towards Earth. This can be deduced without knowing if or in what manner gravity decreases with distance, but requires assuming the equivalency between gravity and inertia.

If two stones were placed in any part of the world near each other, and beyond the sphere of influence of a third cognate body, these stones, like two magnetic needles, would come together in the intermediate point, each approaching the other by a space proportional to the comparative mass of the other. If the moon and earth were not retained in their orbits by their animal force or some other equivalent, the earth would mount to the moon by a fifty-fourth part of their distance, and the moon fall towards the earth through the other fifty-three parts, and they would there meet, assuming, however, that the substance of both is of the same density.

— Johannes Kepler, "Astronomia Nova", 1609[3]

The 1/54 ratio is Kepler's estimate of the Moon–Earth mass ratio, based on their diameters. The accuracy of his statement can be deduced by using Newton's inertia law F=ma and Galileo's gravitational observation that distance ${\displaystyle D=(1/2)at^{2}}$. Setting these accelerations equal for a mass is the equivalence principle. Noting the time to collision for each mass is the same gives Kepler's statement that D_moon/D_Earth=M_Earth/M_moon, without knowing the time to collision or how or if the acceleration force from gravity is a function of distance.

Newton's gravitational theory simplified and formalized Galileo's and Kepler's ideas by recognizing Kepler's "animal force or some other equivalent" beyond gravity and inertia were not needed, deducing from Kepler's planetary laws how gravity reduces with distance.

The equivalence principle was properly introduced by Albert Einstein in 1907, when he observed that the acceleration of bodies towards the center of the Earth at a rate of 1g (g = 9.81 m/s² being a standard reference of gravitational acceleration at the Earth's surface) is equivalent to the acceleration of an inertially moving body that would be observed on a rocket in free space being accelerated at a rate of 1g. Einstein stated it thus:

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

— Einstein, 1907

That is, being on the surface of the Earth is equivalent to being inside a spaceship (far from any sources of gravity) that is being accelerated by its engines. The direction or vector of acceleration equivalence on the surface of the earth is "up" or directly opposite the center of the planet while the vector of acceleration in a spaceship is directly opposite from the mass ejected by its thrusters. From this principle, Einstein deduced that free-fall is inertial motion. Objects in free-fall do not experience being accelerated downward (e.g. toward the earth or other massive body) but rather weightlessness and no acceleration. In an inertial frame of reference bodies (and photons, or light) obey Newton's first law, moving at constant velocity in straight lines. Analogously, in a curved spacetime the world line of an inertial particle or pulse of light is as straight as possible (in space and time).[4] Such a world line is called a geodesic and from the point of view of the inertial frame is a straight line. This is why an accelerometer in free-fall doesn't register any acceleration; there isn't any between the internal test mass and the accelerometer's body.

As an example: an inertial body moving along a geodesic through space can be trapped into an orbit around a large gravitational mass without ever experiencing acceleration. This is possible because spacetime is radically curved in close vicinity to a large gravitational mass. In such a situation the geodesic lines bend inward around the center of the mass and a free-floating (weightless) inertial body will simply follow those curved geodesics into an elliptical orbit. An accelerometer on-board would never record any acceleration.

By contrast, in Newtonian mechanics, gravity is assumed to be a force. This force draws objects having mass towards the center of any massive body. At the Earth's surface, the force of gravity is counteracted by the mechanical (physical) resistance of the Earth's surface. So in Newtonian physics, a person at rest on the surface of a (non-rotating) massive object is in an inertial frame of reference. These considerations suggest the following corollary to the equivalence principle, which Einstein formulated precisely in 1911:

Whenever an observer detects the local presence of a force that acts on all objects in direct proportion to the inertial mass of each object, that observer is in an accelerated frame of reference.

Einstein also referred to two reference frames, K and K'. K is a uniform gravitational field, whereas K' has no gravitational field but is uniformly accelerated such that objects in the two frames experience identical forces:

We arrive at a very satisfactory interpretation of this law of experience, if we assume that the systems K and K' are physically exactly equivalent, that is, if we assume that we may just as well regard the system K as being in a space free from gravitational fields, if we then regard K as uniformly accelerated. 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

This observation was the start of a process that culminated in general relativity. Einstein suggested that it should be elevated to the status of a general principle, which he called the "principle of equivalence" when constructing his theory of relativity:

As long as we restrict ourselves to purely mechanical processes in the realm where Newton's mechanics holds sway, we are certain of the equivalence of the systems K and K'. But this view of ours will not have any deeper significance unless the systems K and K' are equivalent with respect to all physical processes, that is, unless the laws of nature with respect to K are in entire agreement with those with respect to K'. By assuming this to be so, we arrive at a principle which, if it is really true, has great heuristic importance. For by theoretical consideration of processes which take place relatively to a system of reference with uniform acceleration, we obtain information as to the career of processes in a homogeneous gravitational field.

— Einstein, 1911

Einstein combined (postulated) the equivalence principle with special relativity to predict that clocks run at different rates in a gravitational potential, and light rays bend in a gravitational field, even before he developed the concept of curved spacetime.

So the original equivalence principle, as described by Einstein, concluded that free-fall and inertial motion were physically equivalent. This form of the equivalence principle can be stated as follows. 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. This is not strictly true, because massive bodies give rise to tidal effects (caused by variations in the strength and direction of the gravitational field) which are absent from an accelerating spaceship in deep space. The room, therefore, should be small enough that tidal effects can be neglected.

Although the equivalence principle guided the development of general relativity, it is not a founding principle of relativity but rather a simple consequence of the geometrical nature of the theory. In general relativity, objects in free-fall follow geodesics of spacetime, and what we perceive as the force of gravity is instead a result of our being unable to follow those geodesics of spacetime, because the mechanical resistance of Earth's matter or surface prevents us from doing so.

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. They only differ in whether or not they apply to gravitational experiments.

Another clarification needed is that the equivalence principle assumes a constant acceleration of 1g without considering the mechanics of generating 1g. If we do consider the mechanics of it, then we must assume the aforementioned windowless room has a fixed mass. Accelerating it at 1g means there is a constant force being applied, which = m*g where m is the mass of the windowless room along with its contents (including the observer). Now, if the observer jumps inside the room, an object lying freely on the floor will decrease in weight momentarily because the acceleration is going to decrease momentarily due to the observer pushing back against the floor in order to jump. The object will then gain weight while the observer is in the air and the resulting decreased mass of the windowless room allows greater acceleration; it will lose weight again when the observer lands and pushes once more against the floor; and it will finally return to its initial weight afterwards. To make all these effects equal those we would measure on a planet producing 1g, the windowless room must be assumed to have the same mass as that planet. Additionally, the windowless room must not cause its own gravity, otherwise the scenario changes even further. These are technicalities, clearly, but practical ones if we wish the experiment to demonstrate more or less precisely the equivalence of 1g gravity and 1g acceleration.

Three forms of the equivalence principle are in current use: weak (Galilean), Einsteinian, and strong.

Active, passive, and inertial masses

By definition of active and passive gravitational mass, the force on ${\displaystyle M_{1}}$ due to the gravitational field of ${\displaystyle M_{0}}$ is:

$${\displaystyle F_{1}={\frac {M_{0}^{\mathrm {act} }M_{1}^{\mathrm {pass} }}{r^{2}}}}$$

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

$${\displaystyle F_{2}={\frac {M_{0}^{\mathrm {act} }M_{2}^{\mathrm {pass} }}{r^{2}}}}$$

By definition of inertial mass:

$${\displaystyle F=m^{\mathrm {inert} }a}$$

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)

$${\displaystyle a_{1}={\frac {F_{1}}{m_{1}^{\mathrm {inert} }}}=a_{2}={\frac {F_{2}}{m_{2}^{\mathrm {inert} }}}}$$

Hence:

$${\displaystyle {\frac {M_{0}^{\mathrm {act} }M_{1}^{\mathrm {pass} }}{r^{2}m_{1}^{\mathrm {inert} }}}={\frac {M_{0}^{\mathrm {act} }M_{2}^{\mathrm {pass} }}{r^{2}m_{2}^{\mathrm {inert} }}}}$$

Therefore:

$${\displaystyle {\frac {M_{1}^{\mathrm {pass} }}{m_{1}^{\mathrm {inert} }}}={\frac {M_{2}^{\mathrm {pass} }}{m_{2}^{\mathrm {inert} }}}}$$

In other words, passive gravitational mass must be proportional to inertial mass for all objects.

Furthermore, by Newton's third law of motion:

$${\displaystyle F_{1}={\frac {M_{0}^{\mathrm {act} }M_{1}^{\mathrm {pass} }}{r^{2}}}}$$

must be equal and opposite to

$${\displaystyle F_{0}={\frac {M_{1}^{\mathrm {act} }M_{0}^{\mathrm {pass} }}{r^{2}}}}$$

It follows that:

$${\displaystyle {\frac {M_{0}^{\mathrm {act} }}{M_{0}^{\mathrm {pass} }}}={\frac {M_{1}^{\mathrm {act} }}{M_{1}^{\mathrm {pass} }}}}$$

In other words, passive gravitational mass must be proportional to active gravitational mass for all objects.

The dimensionless Eötvös-parameter ${\displaystyle \eta (A,B)}$ 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".

$${\displaystyle \eta (A,B)=2{\frac {\left({\frac {m_{g}}{m_{i}}}\right)_{A}-\left({\frac {m_{g}}{m_{i}}}\right)_{B}}{\left({\frac {m_{g}}{m_{i}}}\right)_{A}+\left({\frac {m_{g}}{m_{i}}}\right)_{B}}}}$$

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, ideally in a vacuum environment, e.g., inside the Fallturm Bremen drop tower.

Researcher	Year	Method	Result
John Philoponus	6th century	Said that by observation, two balls of very different weights will fall at nearly the same speed	no detectable difference
Simon Stevin[8]	~1586	Dropped lead balls of different masses off the Delft churchtower	no detectable difference
Galileo Galilei	~1610	Rolling balls of varying weight down inclined planes to slow the speed so that it was measurable	no detectable difference
Isaac Newton	~1680	Measure the period of pendulums of different mass but identical length	difference is less than 1 part in 103
Friedrich Wilhelm Bessel	1832	Measure the period of pendulums of different mass but identical length	no measurable difference
Loránd Eötvös	1908	Measure the torsion on a wire, suspending a balance beam, between two nearly identical masses under the acceleration of gravity and the rotation of the Earth	difference is 10±2 part in 109 (H2O/Cu)[9]
Roll, Krotkov and Dicke	1964	Torsion balance experiment, dropping aluminum and gold test masses	${\displaystyle |\eta (\mathrm {Al} ,\mathrm {Au} )|=(1.3\pm 1.0)\times 10^{-11}}$[10]
David Scott	1971	Dropped a falcon feather and a hammer at the same time on the Moon	no detectable difference (not a rigorous experiment, but very dramatic being the first lunar one[11])
Braginsky and Panov	1971	Torsion balance, aluminum and platinum test masses, measuring acceleration towards the Sun	difference is less than 1 part in 1012
Eöt-Wash group	1987–	Torsion balance, measuring acceleration of different masses towards the Earth, Sun and Galactic Center, using several different kinds of masses	${\displaystyle \eta ({\text{Earth}},{\text{Be-Ti}})=(0.3\pm 1.8)\times 10^{-13}}$[12]

See:[13]

Year	Investigator	Sensitivity	Method
500?	Philoponus[14]	"small"	Drop tower
1585	Stevin[15]	5×10−2	Drop tower
1590?	Galileo[16]	2×10−2	Pendulum, drop tower
1686	Newton[17]	10−3	Pendulum
1832	Bessel[18]	2×10−5	Pendulum
1908 (1922)	Eötvös[19]	2×10−9	Torsion balance
1910	Southerns[20]	5×10−6	Pendulum
1918	Zeeman[21]	3×10−8	Torsion balance
1923	Potter[22]	3×10−6	Pendulum
1935	Renner[23]	2×10−9	Torsion balance
1964	Dicke, Roll, Krotkov[10]	3x10−11	Torsion balance
1972	Braginsky, Panov[24]	10−12	Torsion balance
1976	Shapiro, et al.[25]	10−12	Lunar laser ranging
1981	Keiser, Faller[26]	4×10−11	Fluid support
1987	Niebauer, et al.[27]	10−10	Drop tower
1989	Stubbs, et al.[28]	10−11	Torsion balance
1990	Adelberger, Eric G.; et al.[29]	10−12	Torsion balance
1999	Baessler, et al.[30][31]	5×10−14	Torsion balance
2017	MICROSCOPE[32][33]	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. Future satellite experiments[34] – STEP (Satellite Test of the Equivalence Principle), and Galileo Galilei – will test the weak equivalence principle in space, to much higher accuracy.

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.[35]

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.[36] Currently envisioned tests of the weak equivalence principle are approaching a degree of sensitivity such that non-discovery of a violation would be just as profound a result as discovery of a violation. Non-discovery of equivalence principle violation in this range would suggest that gravity is so fundamentally different from other forces as to require a major reevaluation of current attempts to unify gravity with the other forces of nature. A positive detection, on the other hand, would provide a major guidepost towards unification.[36]

One challenge to the equivalence principle is the Brans–Dicke theory. Self-creation cosmology is a modification of the Brans–Dicke theory.

In August 2010, researchers from the University of New South Wales, Swinburne University of Technology, and Cambridge University published a paper titled "Evidence for spatial variation of the fine-structure constant", whose tentative conclusion is that, "qualitatively, [the] results suggest a violation of the Einstein Equivalence Principle, and could infer a very large or infinite universe, within which our 'local' Hubble volume represents a tiny fraction."[46]

Dutch physicist and string theorist Erik Verlinde has generated a self-contained, logical derivation of the equivalence principle based on the starting assumption of a holographic universe. Given this situation, gravity would not be a true fundamental force as is currently thought but instead an "emergent property" related to entropy. Verlinde's entropic gravity theory apparently leads naturally to the correct observed strength of dark energy; previous failures to explain its incredibly small magnitude have been called by such people as cosmologist Michael Turner (who is credited as having coined the term "dark energy") as "the greatest embarrassment in the history of theoretical physics".[47] These ideas are far from settled and still very controversial.

• University of Washington[48]
• Lunar Laser Ranging[49][50]
• Galileo-Galilei satellite experiment[51]
• Satellite Test of the Equivalence Principle (STEP)[52]
• MICROSCOPE[53]
• Satellite Energy Exchange (SEE)[54]
• "...Physicists in Germany have used an atomic interferometer to perform the most accurate ever test of the equivalence principle at the level of atoms..."[55]

Wikiquote has quotations related to Equivalence principle.

• 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