# Mass

In physics, mass (from Greek μᾶζα "barley cake, lump [of dough]") is a property of a physical body which determines the body's resistance to being accelerated by a force and the strength of its mutual gravitational attraction with other bodies. The SI unit of mass is the kilogram (kg). As mass is difficult to measure directly, usually balances or scales are used to measure the weight of an object, and the weight is used calculate the object's mass. For everyday objects and energies well-described by Newtonian physics, mass describes the amount of matter in an object. However, at very high speeds or for subatomic particles, General Relativity shows that energy is an additional source of mass. Thus, any body having mass has an equivalent amount of energy, and all forms of energy resist acceleration by a force and have gravitational attraction.

There are several distinct phenomena which can be used to measure mass. Although some theorists have speculated some of these phenomena could be independent of each other,[1] current experiments have found no difference among any of the ways used to measure mass:

• Inertial mass measures an object's resistance to changes in velocity. (the object's acceleration)
• Active gravitational mass measures the gravitational force exerted by an object.
• Passive gravitational mass measures the gravitational force experienced by an object in a known gravitational field.
• Mass-Energy measures the total amount of energy contained within a body, using E=mc²

## Overview

### Mass and applied force

The mass of an object determines its acceleration in the presence of an applied force. This phenomenon is called inertia. According to Newton's second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A body's mass also determines the degree to which it generates or is affected by a gravitational field. If a first body of mass mA is placed at a distance r (center of mass to center of mass) from a second body of mass mB, each body experiences an attractive force Fg = GmAmB/r2, where G = 6.67×10−11 N kg−2m2 is the "universal gravitational constant". This is sometimes referred to as gravitational mass.[note 1] Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are identical; since 1915, this observation has been entailed a priori in the equivalence principle of general relativity.

### Matter

Macroscopically, mass is associated with matter—although matter, unlike mass, is poorly defined in science. On the sub-atomic scale, not only fermions, the particles often associated with matter, but also some bosons, the particles that act as force carriers, have rest mass. Another problem for easy definition is that much of the rest mass of ordinary matter derives from the invariant mass contributed to matter by particles and kinetic energies which have no rest mass themselves (only 1% of the rest mass of matter is accounted for by the rest mass of its fermionic quarks and electrons). From a fundamental physics perspective, mass is the number describing under which the representation of the little group of the Poincaré group a particle transforms. In the Standard Model of particle physics, this symmetry is described as arising as a consequence of a coupling of particles with rest mass to a postulated additional field, known as the Higgs field.

The total mass of the observable universe is estimated at between 1052 kg and 1053 kg, corresponding to the rest mass of between 1079 and 1080 protons.

## Units of mass

The kilogram is one of the seven SI base units; one of three which is defined ad hoc, without reference to another base unit.

The standard International System of Units (SI) unit of mass is the kilogram (kg). The kilogram is 1000 grams (g), which were first defined in 1795 as one cubic centimeter of water at the melting point of ice. Then in 1889, the kilogram was redefined as the mass of the international prototype kilogram, and as such is independent of the meter, or the properties of water. In October 2011, the 24th General Conference on Weights and Measures resolved to "take note of the intention" to redefine the kilogram in terms of the Planck constant, scheduled for 2014.[citation needed]

Other units are accepted for use in SI:

Outside the SI system, a variety of different mass units are used, depending on context, such as the slug (sl), the pound (lb), the Planck mass (mP), and the solar mass.

A mass can sometimes be expressed in terms of length. The mass of a very small particle may be identified with its inverse Compton wavelength (1 cm−13.52×10−41 kg). The mass of a very large star or black hole may be identified with its Schwarzschild radius (1 cm ≈ 6.73×1024 kg).

## Summary of mass concepts and formalisms

In classical mechanics, mass has a central role in determining the behavior of bodies. Newton's second law relates the force F exerted in a body of mass m to the body's acceleration a:

$\boldsymbol{F}=m\boldsymbol{a}$.

Additionally, mass relates a body's momentum p to its linear velocity v:

$\boldsymbol{p}=m\boldsymbol{v}$,

and the body's kinetic energy K to its velocity:

$K=\dfrac{1}{2}m|\boldsymbol{v}|^2$.

In special relativity, relativistic mass is a formalism which accounts for relativistic effects by having the mass increase with velocity.

$M=\gamma m_0\!$
$E=Mc^2\!$

Since energy is dependent on reference frame (upon the observer) it is convenient to formulate the equations of physics in a way such that mass values are invariant (do not change) between observers, and so the equations are independent of the observer. For a single particle, this quantity is the rest mass; for a system of bound or unbound particles, this quantity is the invariant mass. The invariant mass M of a body is related to its energy E and the magnitude of its momentum p by

$Mc^2=\sqrt{E^2-(|\boldsymbol{p}|c)^2},\!$

where c is the speed of light.

## Definitions of mass

The relation between properties of mass and their associated physical constants. Every massive object is believed to exhibit all five properties. However, due to extremely large or extremely small constants, it is generally impossible to verify more than two or three properties for any object.
• The Schwarzschild radius (rs) represents the ability of mass to cause curvature in space and time.
• The standard gravitational parameter (μ) represents the ability of a massive body to exert Newtonian gravitational forces on other bodies.
• Inertial mass (m) represents the Newtonian response of mass to forces.
• Rest energy (E0) represents the ability of mass to be converted into other forms of energy.
• The Compton wavelength (λ) represents the quantum response of mass to local geometry.

In physical science, one may distinguish conceptually between at least seven different aspects of mass, or seven physical notions that involve the concept of mass:[2]

• The amount of matter in certain types of samples can be exactly determined through electrodeposition[clarification needed] or other precise processes. The mass of an exact sample is determined in part by the number and type of atoms or molecules it contains, and in part by the energy involved in binding it together (which contributes a negative "missing mass," or mass deficit).
• Inertial mass is a measure of an object's resistance to changing its state of motion when a force is applied. It is determined by applying a force to an object and measuring the acceleration that results from that force. An object with small inertial mass will accelerate more than an object with large inertial mass when acted upon by the same force. One says the body of greater mass has greater inertia.
• Active gravitational mass [note 3] is a measure of the strength of an object's gravitational flux (gravitational flux is equal to the surface integral of gravitational field over an enclosing surface). Gravitational field can be measured by allowing a small 'test object' to freely fall and measuring its free-fall acceleration. For example, an object in free-fall near the Moon will experience less gravitational field, and hence accelerate slower than the same object would if it were in free-fall near the Earth. The gravitational field near the Moon is weaker because the Moon has less active gravitational mass.
• Passive gravitational mass is a measure of the strength of an object's interaction with a gravitational field. Passive gravitational mass is determined by dividing an object's weight by its free-fall acceleration. Two objects within the same gravitational field will experience the same acceleration; however, the object with a smaller passive gravitational mass will experience a smaller force (less weight) than the object with a larger passive gravitational mass.
• Energy also has mass according to the principle of mass–energy equivalence. This equivalence is exemplified in a large number of physical processes including pair production, nuclear fusion, and the gravitational bending of light. Pair production and nuclear fusion are processes through which measurable amounts of mass and energy are converted into each other. In the gravitational bending of light, photons of pure energy are shown to exhibit a behavior similar to passive gravitational mass.
• Curvature of spacetime is a relativistic manifestation of the existence of mass. Curvature is extremely weak and difficult to measure. For this reason, curvature was not discovered until after it was predicted by Einstein's theory of general relativity. Extremely precise atomic clocks on the surface of the earth, for example, are found to measure less time (run slower) when compared to similar clocks in space. This difference in elapsed time is a form of curvature called gravitational time dilation. Other forms of curvature have been measured using the Gravity Probe B satellite.

To summarize, every experiment to date has shown the above seven values to be proportional, and in some cases equal, and this proportionality gives rise to the abstract concept of mass. If, in some future experiment, one of the mass-related phenomena is shown to be not proportional to the others, then that specific phenomenon will no longer be considered a part of the abstract concept of mass.

## Weight and amount

### Weight vs. mass

In everyday usage, mass and "weight" are often used interchangeably. For instance, a person's weight may be stated as 75 kg. In normal situations, the weight of an object is proportional to its mass, which usually makes it unproblematic to use the same unit for both concepts. However, the distinction between mass and weight becomes important for measurements with a precision better than a few percent (because of slight differences in the strength of the Earth's gravitational field at different places), and for places far from the surface of the Earth, such as in space or on other planets.In scientific use, however, these concepts are distinct. A measurement of mass (such as kilograms) refers to an intrinsic property of an object, whereas "weight"(measured in newtons) measures an object's resistance to deviating from its natural course of free fall, which can be influenced by the nearby gravitational field. No matter how strong the gravitational field, objects in free fall are weightless, though they still have mass.[3]

The force known as "weight" is proportional to mass and acceleration in all situations where the mass is accelerated away from free fall. For example, when a body is at rest in a gravitational field (rather than in free fall), it must be accelerated by a force from a scale or the surface of a planetary body such as the Earth or the Moon. This force keeps the object from going into free fall. Weight is the opposing force in such circumstances, and is thus determined by the acceleration of free fall. On the surface of the Earth, for example, an object with a mass of 50 kilograms weighs 491 newtons, which means that 491 newtons is being applied to keep the object from going into free fall. By contrast, on the surface of the Moon, the same object still has a mass of 50 kilograms but weighs only 81.5 newtons, because only 81.5 newtons is required to keep this object from going into a free fall on the moon. Restated in mathematical terms, on the surface of the Earth, the weight W of an object is related to its mass m by W = mg, where g = 9.80665 m/s2 is the Earth's gravitational field, (expressed as the acceleration experienced by a free-falling object).

For other situations, such as when objects are subjected to mechanical accelerations from forces other than the resistance of a planetary surface, the weight force is proportional to the mass of an object multiplied by the total acceleration away from free fall, which is called the proper acceleration. Through such mechanisms, objects in elevators, vehicles, centrifuges, and the like, may experience weight forces many times those caused by resistance to the effects of gravity on objects, resulting from planetary surfaces. In such cases, the generalized equation for weight W of an object is related to its mass m by the equation W = –ma, where a is the proper acceleration of the object caused by all influences other than gravity. (Again, if gravity is the only influence, such as occurs when an object falls freely, its weight will be zero).

Weight, by definition, is a measure of the force of gravity on an object.(such as in earth's gravitational pull) Typically, gravitational fields change only slightly over short distances, and the Earth's field is nearly uniform at all locations on the Earth's surface; therefore, an object's weight changes only slightly when it is moved from one location to another, and these small changes went unnoticed through much of history. This may have given early humans the impression that weight is an unchanging, fundamental property of objects in the material world. Also, measurement of weight using a balance scale is unaffected by changes in gravitational field strength, and so this was a non-issue in early times.

### Weight as an amount

Depiction of early balance scales in the Papyrus of Hunefer (dated to the 19th dynasty, ca. 1285 BC). The scene shows Anubis weighing the heart of Hunefer.

The concept of amount is very old and predates recorded history, so any description of the early development of this concept is speculative in nature. An early example of this concept is a balance scale, which balances the force of one object's weight against the force of another object's weight. The two sides of a balance scale are close enough that the objects experience similar gravitational fields. Hence, if they have similar masses then their weights will also be similar. This allows the scale, by comparing weights, to also compare masses, and gives it the distinction of being one of the oldest known devices capable of measuring mass. However, one might reasonably assume[weasel words] that humans, at some early era, realized that the weight of a collection of similar objects was directly proportional to the number of objects in the collection:

$W_n \propto n,$

where W is the weight of the collection of similar objects and n is the number of objects in the collection. Proportionality, by definition, implies that two values have a constant ratio:

$\frac{W_n}{n} = \frac{W_m}{m}$, or equivalently $\frac{W_n}{W_m} = \frac{n}{m}.$

Consequently, historical weight standards were often defined in terms of amounts. The Romans, for example, used the carob seed (carat or siliqua) as a measurement standard. If an object's weight was equivalent to 1728 carob seeds, then the object was said to weigh one Roman pound. If, on the other hand, the object's weight was equivalent to 144 carob seeds then the object was said to weigh one Roman ounce (uncia). The Roman pound and ounce were both defined in terms of different sized collections of the same common mass standard, the carob seed. The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was:

$\frac{ounce}{pound} = \frac{W_{144}}{W_{1728}} = \frac{144}{1728} = \frac{1}{12}.$

### Weight as a number of atoms

Various atoms and molecules as depicted in John Dalton's A New System of Chemical Philosophy (1808)

The name atom comes from the Greek ἄτομος/átomos, α-τέμνω, which means uncuttable, something that cannot be divided further. The philosophical concept that matter composed of discrete indivisible units had been around for millennia, but the theory was very abstract, and mostly ignored. However, around the late 18th century, chemists uncovered increasing evidence for the existence of a law of multiple proportions. They noticed that when two or more elements combined to form a compound, the ingredients held to a fixed and definite ratio. For example, the mass ratio of nitrogen to oxygen in nitric oxide is seven eights, and ammonia has a hydrogen to nitrogen mass ratio of three fourteenths.

In order to explain the existence of these ratios, the chemist John Dalton proposed that all matter was made of tiny atoms. His first table of relative atomic weights in 1805 listed six elements: hydrogen, oxygen, nitrogen, carbon, sulfur, and phosphorus; he assigned hydrogen an atomic weight of 1. In 1815, the chemist William Prout concluded that the hydrogen atom was in fact the fundamental mass unit from which all other atomic masses were derived. Prout's hypothesis, however, was somewhat inaccurate. Although elemental masses were near-multiples of the hydrogen mass,(within about 1%) the discrepancies could not be ignored. The light isotope of hydrogen, for example, with a single proton, has an atomic mass of 1.007825 u. The most abundant isotope of iron has 26 protons and 30 neutrons, so one might expect its atomic mass to be 56 times that of the hydrogen atom, but in fact, its atomic mass is only 55.9383 u, which is clearly not an integer multiple of 1.007825 (In this case, iron-56 has a mass 0.89% less than 56 hydrogens).[note 4] Despite its shortcomings, the concepts of Prout's theory, such as atomic mass and amount, continue to play an influential role in chemistry, and the atomic mass unit continues to be the unit of choice for very small mass measurements.

## Gravitational mass

Balance scales allow to directly compare gravitational mass within a gravitational field. Such devices have been in use since at least the Middle Bronze Age (shown is a balance for weighing tobacco dating from the mid-19th century).

The masses of objects are the source of gravitational fields, which govern large-scale structures in the Universe, such as galaxies. Gravity collects clouds of gas and dust into stars and planets, provides pressure for nuclear fusion within stars, and determines the orbits of objects within the Solar System. Since gravitational effects are all around us, it is impossible to pin down the exact date when humans first discovered gravitational mass. However, it is possible to identify some of the significant steps towards our modern understanding of gravitational mass and its relationship to the other mass phenomena. Some terms associated with gravitational mass and its effects are the Gaussian gravitational constant, the standard gravitational parameter and the Schwarzschild radius.

### Planetary motion and precursors to universal mass

In 1600 AD, Johannes Kepler sought employment with Tycho Brahe, who had some of the most precise astronomical data available. Using Brahe's precise observations of the planet Mars, Kepler spent the next five years developing his own method for characterizing planetary motion. In 1609, Johannes Kepler published his three laws of planetary motion, explaining how the planets follow elliptical orbits under the influence of the Sun. In Kepler's final planetary model, he described planetary orbits as following elliptical paths with the Sun at a focal point of the ellipse. Kepler discovered that the square of the orbital period of each planet is directly proportional to the cube of the semi-major axis of its orbit, or equivalently, that the ratio of these two values is constant for all planets in the Solar System.[note 5]

On 25 August 1609, Galileo Galilei demonstrated his first telescope to a group of Venetian merchants, and in early January of 1610, Galileo observed four dim objects near Jupiter, which he mistook for stars. However, after a few days of observation, Galileo realized that these "stars" were in fact orbiting Jupiter. These four objects (later named the Galilean moons in honor of their discoverer) were the first celestial bodies observed to orbit something other than the Earth or Sun. Galileo continued to observe these moons over the next eighteen months, and by the middle of 1611 he had obtained remarkably accurate estimates for their periods.

### Galilean gravitational field

Galileo Galilei 1636
Distance traveled by a freely falling ball is proportional to the square of the elapsed time

Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects falling under the influence of Earth's gravity, and he was actively attempting to characterize these motions. Galileo was not the first to investigate Earth's gravitational field, nor was he the first to accurately describe its fundamental characteristics. However, Galileo's reliance on scientific experimentation to establish physical principles would have a profound effect on future generations of scientists. Galileo used a number of scientific experiments to characterize free fall motion. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo,[4] but the results obtained from these experiments were both realistic and compelling. A biography by Galileo's pupil Vincenzo Viviani stated that Galileo had dropped balls of the same material, but different masses, from the Leaning Tower of Pisa to demonstrate that their time of descent was independent of their mass.[note 6] In support of this conclusion, Galileo had advanced the following theoretical argument: He asked if two bodies of different masses and different rates of fall are tied by a string, does the combined system fall faster because it is now more massive, or does the lighter body in its slower fall hold back the heavier body? The only convincing resolution to this question is that all bodies must fall at the same rate.[5]

A later experiment was described in Galileo's Two New Sciences published in 1638. One of Galileo's fictional characters, Salviati, describes an experiment using a bronze ball and a wooden ramp. The wooden ramp was "12 cubits long, half a cubit wide and three finger-breadths thick" with a straight, smooth, polished groove. The groove was lined with "parchment, also smooth and polished as possible". And into this groove was placed "a hard, smooth and very round bronze ball". The ramp was inclined at various angles to slow the acceleration enough so that the elapsed time could be measured. The ball was allowed to roll a known distance down the ramp, and the time taken for the ball to move the known distance was measured. The time was measured using a water clock described as follows:

"a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results."[6]

Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time:

$g = \frac{\text{Distance}}{\text{Time}^2} \propto \text{Gravitational field}$

Galileo Galilei died in Arcetri, Italy (near Florence), on 8 January 1642. Galileo had shown that objects in free fall under the influence of the Earth’s gravitational field have a constant acceleration, and Galileo’s contemporary, Johannes Kepler, had shown that the planets follow elliptical paths under the influence of the Sun’s gravitational mass. However, the relationship between Galileo’s gravitational field and Kepler’s gravitational mass wasn’t comprehended during Galileo’s lifetime.

### Newtonian gravitational mass

Isaac Newton 1689
Earth's Moon Mass of Earth
Semi-major axis Sidereal orbital period
0.002 569 AU 0.074 802 sidereal year $1.2\pi^2\cdot10^{-5}\frac{\text{AU}^3}{\text{y}^2}=3.986\cdot10^{14}\frac{\text{m}^3}{\text{s}^2}$
9.806 65 m/s2 6 375 km

Robert Hooke had published his concept of gravitational forces in 1674, stating that, all Cœlestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers [and] they do also attract all the other Cœlestial Bodies that are within the sphere of their activity. He further states that gravitational attraction increases by how much the nearer the body wrought upon is to their own center.[7] In a correspondence of 1679–1680 between Robert Hooke and Isaac Newton, Hooke conjectures that gravitational forces might decrease according to the double of the distance between the two bodies.[8] Hooke urged Newton, who was a pioneer in the development of calculus, to work through the mathematical details of Keplerian orbits to determine if Hooke's hypothesis was correct. Newton's own investigations verified that Hooke was correct, but due to personal differences between the two men, Newton chose not to reveal this to Hooke. Isaac Newton kept quiet about his discoveries until 1684, at which time he told a friend, Edmond Halley, that he had solved the problem of gravitational orbits, but had misplaced the solution in his office.[9] After being encouraged by Halley, Newton decided to develop his ideas about gravity and publish all of his findings. In November 1684, Isaac Newton sent a document to Edmund Halley, now lost but presumed to have been titled De motu corporum in gyrum (Latin for "On the motion of bodies in an orbit").[10] Halley presented Newton's findings to the Royal Society of London, with a promise that a fuller presentation would follow. Newton later recorded his ideas in a three book set, entitled Philosophiæ Naturalis Principia Mathematica (Latin: "Mathematical Principles of Natural Philosophy"). The first was received by the Royal Society on 28 April 1685–6, the second on 2 March 1686–7, and the third on 6 April 1686–7. The Royal Society published Newton’s entire collection at their own expense in May 1686–7.[11]:31

Isaac Newton had bridged the gap between Kepler’s gravitational mass and Galileo’s gravitational acceleration, and proved the following relationship:

$\boldsymbol{g}=-\mu\frac{\boldsymbol{\hat{R}}}{|\boldsymbol{R}|^2}$

where g is the apparent acceleration of a body as it passes through a region of space where gravitational fields exist, μ is the gravitational mass (standard gravitational parameter) of the body causing gravitational fields, and R is the radial coordinate (the distance between the centers of the two bodies).

By finding the exact relationship between a body's gravitational mass and its gravitational field, Newton provided a second method for measuring gravitational mass. The mass of the Earth can be determined using Kepler's method (from the orbit of Earth's Moon), or it can be determined by measuring the gravitational acceleration on the Earth's surface, and multiplying that by the square of the Earth's radius. The mass of the Earth is approximately three millionths of the mass of the Sun. To date, no other accurate method for measuring gravitational mass has been discovered.[12]

#### Newton's cannonball

A cannon on top of a very high mountain shoots a cannonball in a horizontally. If the speed is low, it quickly falls back on Earth (A,B). At intermediate speeds, it will revolving around Earth along an elliptical orbit.(C,D) At high speeds, the cannonball will leave Earth altogether.(E)

Newton's cannonball was a thought experiment used to bridge the gap between Galileo's gravitational acceleration and Kepler's elliptical orbits. It appeared in Newton's 1728 book A Treatise of the System of the World. According to Galileo's concept of gravitation, a dropped stone falls with constant acceleration down towards the Earth. However, Newton explains that when a stone is thrown horizontally (meaning sideways or perpendicular to Earth's gravity) it follows a curved path. "For a stone projected is by the pressure of its own weight forced out of the rectilinear path, which by the projection alone it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground. And the greater the velocity is with which it is projected, the farther it goes before it falls to the Earth."[11]:513 Newton further reasons that if an object were "projected in an horizontal direction from the top of a high mountain" with sufficient velocity, "it would reach at last quite beyond the circumference of the Earth, and return to the mountain from which it was projected."[citation needed]

#### Universal gravitational mass and amount

Newton's cannonball illustrated the relationship between the Earth's gravitational mass and its gravitational field; however, a number of other ambiguities still remained. Robert Hooke had asserted in 1674 that: "all Celestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers", but Hooke had neither explained why this gravitating attraction was unique to celestial bodies, nor had he explained why the attraction was directed towards the center of a celestial body.

An apple experiences gravitational fields directed towards every part of the Earth; however, the sum total of these many fields produces a single gravitational field directed towards the Earth's center

To answer these questions, Newton introduced the entirely new concept that gravitational mass is "universal": meaning that every object has gravitational mass, and therefore, every object generates a gravitational field. Newton further assumed that the strength of each object's gravitational field would decrease according to the square of the distance to that object. With these assumptions in mind, Newton calculated what the overall gravitational field would be if a large collection of small objects were formed into a giant spherical body. Newton found that a giant spherical body (like the Earth or Sun, with roughly uniform density at each given radius), would have a gravitational field which was proportional to the total mass of the body,[11]:397 and inversely proportional to the square of the distance to the body's center.[11]:221

Newton's concept of universal gravitational mass is illustrated in the image to the left. Every piece of the Earth has gravitational mass and every piece creates a gravitational field directed towards that piece. However, the overall effect of these many fields is equivalent to a single powerful field directed towards the center of the Earth. The apple behaves as if a single powerful gravitational field were accelerating it towards the Earth's center.

Newton's concept of universal gravitational mass puts gravitational mass on an equal footing with the traditional concepts of weight and amount. For example, the ancient Romans had used the carob seed as a weight standard. The Romans could place an object with an unknown weight on one side of a balance scale and place carob seeds on the other side of the scale, increasing the number of seeds until the scale was balanced. If an object's weight was equivalent to 1728 carob seeds, then the object was said to weigh one Roman pound.

According to Newton's theory of universal gravitation, each carob seed produces gravitational fields. Therefore, if one were to gather an immense number of carob seeds and form them into an enormous sphere, then the gravitational field of the sphere would be proportional to the number of carob seeds in the sphere. Hence, it should be theoretically possible to determine the exact number of carob seeds that would be required to produce a gravitational field similar to that of the Earth or Sun. And since the Roman weight units were all defined in terms of carob seeds, then knowing the Earth's, or Sun's "carob seed mass" would allow one to calculate the mass in Roman pounds, or Roman ounces, or any other Roman unit.

Vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated into position next to the small balls by a pulley from outside. Figure 1 of Cavendish's paper.

This possibility extends beyond Roman units and the carob seed. The British avoirdupois pound, for example, was originally defined to be equal to 7,000 barley grains. Therefore, if one could determine the Earth's "barley grain mass" (the number of barley grains required to produce a gravitational field similar to that of the Earth), then this would allow one to calculate the Earth's mass in avoirdupois pounds. Also, the original kilogram was defined to be equal in mass to a liter of pure water. Thus, the mass of the Earth in kilograms could theoretically be determined by ascertaining how many liters of pure water would be required to produce gravitational fields similar to those of the Earth. In fact, it is a simple matter of abstraction to realize that any traditional mass unit can theoretically be used to measure gravitational mass.

Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice. According to Newton's theory all objects produce gravitational fields and it is theoretically possible to collect an immense number of small objects and form them into an enormous gravitating sphere. However, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure. And if one were to collect an immense number of objects, the resulting sphere would probably be too large to construct on the surface of the Earth, and too expensive to construct in space. Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units, the Cavendish experiment, did not occur until 1797, over a hundred years later. Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water. As of 2009, the Earth's mass in kilograms is only known to around five digits of accuracy, whereas its gravitational mass is known to over nine significant figures.

## Inertial and gravitational mass

Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. In classical mechanics, Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but the classical theory offers no compelling reason why the gravitational mass has to equal the inertial mass. That it does is merely an empirical fact.

Albert Einstein developed his general theory of relativity starting from the assumption that this correspondence between inertial and (passive) gravitational mass is not accidental: that no experiment will ever detect a difference between them (the weak version of the equivalence principle). However, in the resulting theory, gravitation is not a force and thus not subject to Newton's third law, so "the equality of inertial and active gravitational mass [...] remains as puzzling as ever".[13]

### Inertial mass

Inertial mass is the mass of an object measured by its resistance to acceleration.

To understand what the inertial mass of a body is, one begins with classical mechanics and Newton's Laws of Motion. Later on, we will see how our classical definition of mass must be altered if we take into consideration the theory of special relativity, which is more accurate than classical mechanics. However, the implications of special relativity will not change the meaning of "mass" in any essential way.

According to Newton's second law, we say that a body has a mass m if, at any instant of time, it obeys the equation of motion

$\boldsymbol{F}=m \boldsymbol{a},$

where F is the resultant force acting on the body and a is the acceleration of the body's centre of mass.[note 7] For the moment, we will put aside the question of what "force acting on the body" actually means.

This equation illustrates how mass relates to the inertia of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. We might say that the larger mass exerts a greater "resistance" to changing its state of motion in response to the force.

However, this notion of applying "identical" forces to different objects brings us back to the fact that we have not really defined what a force is. We can sidestep this difficulty with the help of Newton's third law, which states that if one object exerts a force on a second object, it will experience an equal and opposite force. To be precise, suppose we have two objects X and Y, with constant inertial masses mX and mY. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on X by Y, which we denote FXY, and the force exerted on Y by X, which we denote FYX. Newton's second law states that

\begin{align} \boldsymbol{F_{XY}} & =m_X\boldsymbol{a_X},\\ \boldsymbol{F_{YX}} & =m_Y\boldsymbol{a_Y}, \end{align}

where aX and aY are the accelerations of X and Y, respectively. Suppose that these accelerations are non-zero, so that the forces between the two objects are non-zero. This occurs, for example, if the two objects are in the process of colliding with one another. Newton's third law then states that

$\boldsymbol{F_{XY}}=-\boldsymbol{F_{YX}};$

and thus

$\frac{m_X}{m_Y}=-\frac{\boldsymbol{a_Y}}{\boldsymbol{a_X}}\!.$

Note that our requirement that aX be non-zero ensures that the fraction is well-defined.

This is, in principle, how we would measure the inertial mass of an object. We choose a "reference" object and define its mass mY as (say) 1 kilogram. Then we can measure the mass of any other object in the universe by colliding it with the reference object and measuring the accelerations.

### Newtonian gravitational mass

The Newtonian concept of gravitational mass rests on Newton's law of gravitation. Let us suppose we have two objects A and B, separated by a distance RAB. The law of gravitation states that if A and B have gravitational masses MA and MB respectively, then each object exerts a gravitational force on the other, of magnitude

$\boldsymbol{F_{AB}}=-GM_AM_B\frac{\widehat{\boldsymbol{R_{AB}}}}{|\boldsymbol{R_{AB}}|^2}\$,

where G is the universal gravitational constant. The above statement may be reformulated in the following way: if g is the magnitude at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is

$\boldsymbol{F}=M\boldsymbol{g}\!$.

This is the basis by which masses are determined by weighing. In simple spring scales, for example, the force F is proportional to the displacement of the spring beneath the weighing pan, as per Hooke's law, and the scales are calibrated to take g into account, allowing the mass M to be read off. A balance measures gravitational mass; only the spring scale measures weight.

### Equivalence of inertial and gravitational masses

The equivalence of inertial and gravitational masses is sometimes referred to as the "Galilean equivalence principle" or the "weak equivalence principle". The most important consequence of this equivalence principle applies to freely falling objects. Suppose we have an object with inertial and gravitational masses m and M, respectively. If the only force acting on the object comes from a gravitational field g, combining Newton's second law and the gravitational law yields the acceleration

$\boldsymbol{a}=\frac{M}{m}\boldsymbol{g}.$

This says that the ratio of gravitational to inertial mass of any object is equal to some constant K if and only if all objects fall at the same rate in a given gravitational field. This phenomenon is referred to as the "universality of free-fall". (In addition, the constant K can be taken to be 1 by defining our units appropriately.)

The first experiments demonstrating the universality of free-fall were conducted by Galileo. It is commonly stated that Galileo obtained his results by dropping objects from the Leaning Tower of Pisa, but this is most likely apocryphal; actually, he performed his experiments with balls rolling down nearly frictionless inclined planes to slow the motion and increase the timing accuracy. Increasingly precise experiments have been performed, such as those performed by Loránd Eötvös,[14] using the torsion balance pendulum, in 1889. As of 2008, no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the precision 10−12. More precise experimental efforts are still being carried out.

The universality of free-fall only applies to systems in which gravity is the only acting force. All other forces, especially friction and air resistance, must be absent or at least negligible. For example, if a hammer and a feather are dropped from the same height through the air on Earth, the feather will take much longer to reach the ground; the feather is not really in free-fall because the force of air resistance upwards against the feather is comparable to the downward force of gravity. On the other hand, if the experiment is performed in a vacuum, in which there is no air resistance, the hammer and the feather should hit the ground at exactly the same time (assuming the acceleration of both objects towards each other, and of the ground towards both objects, for its own part, is negligible). This can easily be done in a high school laboratory by dropping the objects in transparent tubes that have the air removed with a vacuum pump. It is even more dramatic when done in an environment that naturally has a vacuum, as David Scott did on the surface of the Moon during Apollo 15.

A stronger version of the equivalence principle, known as the Einstein equivalence principle or the strong equivalence principle, lies at the heart of the general theory of relativity. Einstein's equivalence principle states that within sufficiently small regions of space-time, it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that the force acting on a massive object caused by a gravitational field is a result of the object's tendency to move in a straight line (in other words its inertia) and should therefore be a function of its inertial mass and the strength of the gravitational field.

## Mass in relativity

### Special relativity

Special relativity shows that rest mass (or invariant mass) and rest energy are essentially equivalent, via the well-known relationship E = mc2. This same equation also connects relativistic mass and "relativistic energy" (total system energy). The latter two "relativistic" mass and energy are concepts that are related to their "rest" counterparts, but they do not have the same value as their rest counterparts in systems where there is a net momentum. In order to deduce any of these four quantities from any of the others, in any system which has a net momentum, an equation that takes momentum into account is needed. Mass (so long as the type and definition of mass is agreed upon) is a conserved quantity over time. From the viewpoint of any single unaccelerated observer, mass can neither be created or destroyed, and special relativity does not change this understanding. All unaccelerated observers agree on the amount of invariant mass in closed systems at all times, and although different observers may not agree with each other on how much relativistic mass is present in any such system, all agree that the amount does not change over time.

The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The invariant mass is another name for the rest mass of single particles. However, the more general invariant mass (calculated with a more complicated formula) may also be applied to systems of particles in relative motion, and because of this, is usually reserved for systems which consist of widely separated high-energy particles. The invariant mass of systems is the same for all observers and inertial frames, and cannot be destroyed, and is thus conserved, so long as the system is closed. In this case, "closure" implies that an idealized boundary is drawn around the system, and no mass/energy is allowed across it.

In as much as energy is conserved in closed systems in relativity, the mass of a system is also a quantity which is conserved: this means it does not change over time, even as some types of particles are converted to others. For any given observer, the mass of any system is separately conserved and cannot change over time, just as energy is separately conserved and cannot change over time. The incorrect popular idea that mass may be converted to (massless) energy in relativity is because some matter particles may in some cases be converted to types of energy which are not matter (such as light, kinetic energy, and the potential energy in magnetic, electric, and other fields). However, this confuses "matter" (a non-conserved and ill-defined thing) with mass (which is well-defined and is conserved). Even if not considered "matter," all types of energy still continue to exhibit mass in relativity[citation needed]. Thus, mass and energy do not change into one another in relativity; rather, both are names for the same thing, and neither mass nor energy appear without the other. "Matter" particles may not be conserved in reactions in relativity, but closed-system mass always is.

For example, a nuclear bomb in an idealized super-strong box, sitting on a scale, would in theory show no change in mass when detonated (although the inside of the box would become much hotter). In such a system, the mass of the box would change only if energy were allowed to escape from the box as light or heat. However, in that case, the removed energy would take its associated mass with it. Letting heat out of such a system is simply a way to remove mass. Thus, mass, like energy, cannot be destroyed, but only moved from one place to another.[15]

In bound systems, the binding energy must (often) be subtracted from the mass of the unbound system, simply because this energy has mass, and this mass is subtracted from the system when it is given off, at the time it is bound. Mass is not conserved in this process because the system is not closed during the binding process. A familiar example is the binding energy of atomic nuclei, which appears as other types of energy (such as gamma rays) when the nuclei are formed, and (after being given off) results in nuclides which have less mass than the free particles (nucleons) of which they are composed.

The term relativistic mass is also used, and this is the total quantity of energy in a body or system (divided by c2). The relativistic mass (of a body or system of bodies) includes a contribution from the kinetic energy of the body, and is larger the faster the body moves, so unlike the invariant mass, the relativistic mass depends on the observer's frame of reference. However, for given single frames of reference and for closed systems, the relativistic mass is also a conserved quantity.

Because the relativistic mass is proportional to the energy, it has gradually fallen into disuse among physicists.[16] There is disagreement over whether the concept remains pedagogically useful.[17][18][19]

### General relativity

In general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass. At the core of this assertion is Albert Einstein's idea that the gravitational force as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

However, it turns out that it is impossible to find an objective general definition for the concept of invariant mass in general relativity. At the core of the problem is the non-linearity of the Einstein field equations, which makes it impossible to write the gravitational field energy as part of the Stress–energy tensor in a way that is invariant for all observers. For a given observer, this can be achieved by the Stress–energy–momentum pseudotensor.[20]

## Mass in quantum physics

In classical mechanics, the inert mass of a particle appears in the Euler–Lagrange equation as a parameter m,

$\frac{\mathrm{d}}{\mathrm{d}t} \ \left( \, \frac{\partial L}{\partial \dot{x}_i} \, \right) \ = \ m \, \ddot{x}_i$.

After quantization, replacing the position vector x with a wave function, the parameter m appears in the kinetic energy operator,

$i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},\,t) = \left(-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\right)\Psi(\mathbf{r},\,t)$.

In the ostensibly covariant (relativistically invariant) Dirac equation, and in natural units, this becomes

$(-i\gamma^\mu\partial_\mu + m) \psi = 0\,$

Where the "mass" parameter m is now simply a constant associated with the quantum described by the wave function ψ.

In the Standard Model of particle physics as developed in the 1960s, there is the proposal that this term arises from the coupling of the field ψ to an additional field Φ, the so-called Higgs field. In the case of fermions, the Higgs mechanism results in the replacement of the term mψ in the Lagrangian with $G_{\psi} \overline{\psi} \phi \psi$. This shifts the explanandum of the value for the mass of each elementary particle to the value of the unknown couplings Gψ. The tentatively confirmed discovery of a massive Higgs boson is regarded as a strong confirmation of this theory. But there is indirect evidence for the reality of the Electroweak symmetry breaking as described by the Higgs mechanism, and the non-existence of Higgs bosons would indicate a "Higgsless" description of this mechanism.

## Origin of mass

In theoretical physics, a mass generation mechanism is a theory which attempts to explain the origin of mass from the most fundamental laws of physics. To date, a number of different models have been proposed which advocate different views at the origin of mass. The problem is complicated by the fact that the notion of mass is strongly related to the gravitational interaction but a theory of the latter has not been yet reconciled with the currently popular model of particle physics, known as the Standard Model.

## Notes

1. ^ When a distinction is necessary, M is used to denote the active gravitational mass and m the passive gravitational mass.
2. ^ Since the Avogadro constant NA is defined as the number of atoms in 12 g of carbon-12, it follows that 1 u is exactly 1/(103NA) kg.
3. ^ The distinction between "active" and "passive" gravitational mass does not exist in the Newtonian view of gravity as found in classical mechanics, and can safely be ignored by laypersons. In most practical applications, Newtonian gravity is used because it is usually sufficiently accurate, and is simpler than General Relativity; for example, NASA uses primarily Newtonian gravity to design space missions, although "accuracies are routinely enhanced by accounting for tiny relativistic effects".www2.jpl.nasa.gov/basics/bsf3-2.php The distinction between "active" and "passive" is very abstract, and applies to post-graduate level applications of General Relativity to certain problems in cosmology, and is otherwise not used. There is, nevertheless, an important conceptual distinction in Newtonian physics between "inertial mass" and "gravitational mass", although these quantities are identical; the conceptual distinction between these two fundamental definitions of mass is maintained for teaching purposes because they involve two distinct methods of measurement. It was long considered anomalous that the two distinct measurements of mass (inertial and gravitational) gave the identical result. The observed property, noted by Galileo, according to which objects of different mass fall with the same rate of acceleration (ignoring air resistance), is an expression of the fact that inertial and gravitational mass are the same.
4. ^ The origin of the discrepancy was later explained by Einstein's theory of relativity, which says that when protons and neutrons come together to form an atomic nucleus, some of the mass of the nucleus is released in the form of binding energy. The more tightly bound the nucleus, the more energy is lost during formation and this binding energy loss (along with the slight difference in masses between protons and neutrons) causes the elemental masses not to be related through simple fractions.
5. ^ This constant ratio was later shown to be a direct measure of the Sun's active gravitational mass; it has units of distance cubed per time squared, and is known as the standard gravitational parameter:
$\mu=4\pi^2\frac{\text{distance}^3}{\text{time}^2}\propto\text{gravitational mass}$
6. ^ At the time when Viviani asserts that the experiment took place, Galileo had not yet formulated the final version of his law of free fall. He had, however, formulated an earlier version which predicted that bodies of the same material falling through the same medium would fall at the same speed. See Drake, S. (1978). Galileo at Work. University of Chicago Press. pp. 19–20. ISBN 0-226-16226-5.
7. ^ In its original form, Newton's second law is valid only for bodies of constant mass.

## References

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4. ^ Drake, S. (1979). "Galileo's Discovery of the Law of Free Fall". Scientific American 228 (5): 84–92. Bibcode:1973SciAm.228e..84D. doi:10.1038/scientificamerican0573-84.
5. ^ Galileo, G. (1632). Dialogue Concerning the Two Chief World Systems.
6. ^ Galileo, G. (1638). Discorsi e Dimostrazioni Matematiche, Intorno à Due Nuove Scienze 213. Louis Elsevier.
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10. ^ Whiteside, D. T., ed. (2008). The Mathematical Papers of Isaac Newton, Volume VI (1684–1691). Cambridge University Press. ISBN 978-0-521-04585-8. Retrieved 12 March 2011.
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13. ^ Rindler, W. (2006). Relativity: Special, General, And Cosmological. Oxford University Press. p. 22. ISBN 0-19-856731-6.
14. ^ Eötvös, R. V.; Pekár, D.; Fekete, E. (1922). "Beiträge zum Gesetz der Proportionalität von Trägheit und Gravität". Annalen der Physik 68: 11.
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