Mass is an
intrinsic property of a body. It was traditionally believed to be related to the
quantity
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a u ...
of
matter in a
physical body, until the discovery of the
atom and
particle physics. It was found that different atoms and different
elementary particles, theoretically with the same amount of matter, have nonetheless different masses. Mass in modern physics has multiple
definitions which are conceptually distinct, but physically
equivalent. Mass can be experimentally defined as a
measure of the body's
inertia, meaning the resistance to
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
(change of
velocity) when a
net force is applied. The object's mass also determines the
strength of its
gravitational attraction to other bodies.
The
SI base unit of mass is the
kilogram (kg). In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, mass is
not the same as
weight, even though mass is often determined by measuring the object's weight using a
spring scale, rather than
balance scale comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity, but it would still have the same mass. This is because weight is a force, while mass is the property that (along with gravity) determines the strength of this force.
Phenomena
There are several distinct phenomena that can be used to measure mass. Although some theorists have speculated that some of these phenomena could be independent of each other, current experiments have found no difference in results regardless of how it is measured:
* ''Inertial mass'' measures an object's resistance to being accelerated by a force (represented by the relationship
).
* ''Active gravitational mass'' determines the strength of the gravitational field generated by an object.
* ''Passive gravitational mass'' measures the gravitational force exerted on an object in a known gravitational field.
The mass of an object determines its acceleration in the presence of an applied force. The inertia and the inertial mass describe this property of physical bodies at the qualitative and quantitative level respectively. 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 and is affected by a
gravitational field. If a first body of mass ''m''
A is placed at a distance ''r'' (center of mass to center of mass) from a second body of mass ''m''
B, each body is subject to an attractive force , where is the "universal
gravitational constant". This is sometimes referred to as gravitational mass.
[When a distinction is necessary, the active and passive gravitational masses may be distinguished.] Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are identical; since 1915, this observation has been incorporated ''
a priori
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ex ...
'' in the
equivalence principle of
general relativity.
Units of mass
The
International System of Units (SI) unit of mass is the
kilogram (kg). The kilogram is 1000 grams (g), and was first defined in 1795 as the mass of one cubic decimetre of water at the
melting point of ice. However, because precise measurement of a cubic decimetre of water at the specified temperature and pressure was difficult, in 1889 the kilogram was redefined as the mass of a metal object, and thus became independent of the metre and the properties of water, this being a copper prototype of the
grave in 1793, the platinum
Kilogramme des Archives in 1799, and the platinum-iridium
International Prototype of the Kilogram (IPK) in 1889.
However, the mass of the IPK and its national copies have been found to drift over time. The
re-definition of the kilogram and several other units came into effect on 20 May 2019, following a final vote by the
CGPM in November 2018. The new definition uses only invariant quantities of nature: the
speed of light, the
caesium hyperfine frequency, the
Planck constant and the
elementary charge.
[
]
Non-SI units accepted for use with SI units include:
* the
tonne (t) (or "metric ton"), equal to 1000 kg
* the
electronvolt
In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacu ...
(eV), a unit of
energy, used to express mass in units of eV/''c''
2 through
mass–energy equivalence
* the
dalton (Da), equal to 1/12 of the mass of a free
carbon-12 atom, approximately .
[The dalton is convenient for expressing the masses of atoms and molecules.]
Outside the SI system, other units of mass include:
* the
slug (sl), an
Imperial unit of mass (about 14.6 kg)
* the
pound (lb), a unit of mass (about 0.45 kg), which is used alongside the similarly named
pound (force) (about 4.5 N), a unit of force
[These are used mainly in the United States except in scientific contexts where SI units are usually used instead.]
* the
Planck mass (about ), a quantity derived from fundamental constants
* the
solar mass (), defined as the mass of the
Sun, primarily used in astronomy to compare large masses such as stars or galaxies (≈ )
* the mass of a particle, as identified with its inverse
Compton wavelength ()
* the mass of a star or
black hole, as identified with its
Schwarzschild radius
The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteri ...
().
Definitions
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''.
Every experiment to date has shown these seven values to be
proportional, and in some cases equal, and this proportionality gives rise to the abstract concept of mass. There are a number of ways mass can be measured or
operationally defined:
* Inertial mass is a measure of an object's resistance to acceleration 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
[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 for many purposes. In most practical applications, Newtonian gravity is assumed 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". 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 an identical result. The property, observed by Galileo, that objects of different mass fall with the same rate of acceleration (ignoring air resistance), shows that inertial and gravitational mass are the same.] 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 fall freely and measuring its
free-fall acceleration. For example, an object in free-fall near the
Moon is subject to a smaller gravitational field, and hence accelerates more slowly, 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.
* According to
relativity, mass is nothing else than the
rest energy
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
of a system of particles, meaning the energy of that system in a
reference frame
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
where it has zero
momentum. Mass can be converted into other forms of energy according to the principle of
mass–energy equivalence. This equivalence is exemplified in a large number of physical processes including
pair production,
beta decay and
nuclear fusion. Pair production and nuclear fusion are processes in which measurable amounts of mass are converted to
kinetic energy or vice versa.
* Curvature of
spacetime is a relativistic manifestation of the existence of mass. Such
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.
* Quantum mass manifests itself as a difference between an object's quantum
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
and its
wave number. The quantum mass of a particle is proportional to the inverse
Compton wavelength and can be determined through various forms of
spectroscopy. In relativistic quantum mechanics, mass is one of the irreducible representation labels of the Poincaré group.
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 a constant gravitational field, the weight of an object is proportional to its mass, and it is unproblematic to use the same unit for both concepts. But because of slight differences in the strength of the
Earth's gravitational field at different places, the
distinction becomes important for measurements with a precision better than a few percent, and for places far from the surface of the Earth, such as in space or on other planets. Conceptually, "mass" (measured in
kilograms) refers to an intrinsic property of an object, whereas "weight" (measured in
newtons) measures an object's resistance to deviating from its current 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.
The force known as "weight" is proportional to mass and
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
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
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
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 , where is the acceleration due to
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 , 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).
Inertial vs. 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
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
developed his
general theory of relativity starting with the assumption that the inertial and passive gravitational masses are the same. This is known as the
equivalence principle.
The particular equivalence often referred to as the "Galilean equivalence principle" or the "
weak equivalence principle
In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (suc ...
" has the most important consequence for freely falling objects. Suppose an object has inertial and gravitational masses ''m'' and ''M'', respectively. If the only force acting on the object comes from a gravitational field ''g'', the force on the object is:
:
Given this force, the acceleration of the object can be determined by Newton's second law:
:
Putting these together, the gravitational acceleration is given by:
:
This says that the ratio of gravitational to inertial mass of any object is equal to some constant ''K''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
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 as 1 by defining our units appropriately.
The first experiments demonstrating the universality of free-fall were—according to scientific 'folklore'—conducted by
Galileo obtained by dropping objects from the
Leaning Tower of Pisa. This is most likely apocryphal: he is more likely to have performed his experiments with balls rolling down nearly frictionless
inclined plane
An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle from the vertical direction, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six cla ...
s to slow the motion and increase the timing accuracy. Increasingly precise experiments have been performed, such as those performed by
Loránd Eötvös, using the
torsion balance pendulum, in 1889. , no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the precision 10
−6. 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
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
, 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.
Origin
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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
. To date, a number of different models have been proposed which advocate different views of 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.
Pre-Newtonian concepts
Weight as an amount
The concept of
amount
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a un ...
is very old and
predates recorded history. 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:
:
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:
:
, or equivalently
An early use of this relationship 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.
Consequently, historical weight standards were often defined in terms of amounts. The Romans, for example, used the
carob
The carob ( ; ''Ceratonia siliqua'') is a flowering evergreen tree or shrub in the Caesalpinioideae sub-family of the legume family, Fabaceae. It is widely cultivated for its edible fruit pods, and as an ornamental tree in gardens and lands ...
seed (
carat or
siliqua) as a measurement standard. If an object's weight was equivalent t
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:
:
Planetary motion
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 orbit 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
The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting pla ...
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.
[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:
: ]
On 25 August 1609,
Galileo Galilei
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He ...
demonstrated his first telescope to a group of Venetian merchants, and in early January 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 free fall
Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects in free fall, 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. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo, 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.
[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 that predicted that bodies ''of the same material'' falling through the same medium would fall at the same speed. See ] 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.
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
Parchment is a writing material made from specially prepared untanned skins of animals—primarily sheep, calves, and goats. It has been used as a writing medium for over two millennia. Vellum is a finer quality parchment made from the skins ...
, 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
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
s 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.
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:
:
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, Galileo's free fall motions and Kepler's planetary motions remained distinct during Galileo's lifetime.
Newtonian mass
Robert Hooke had published his concept of gravitational forces in 1674, stating that all
celestial bodies have an attraction or gravitating power towards their own centers, and also attract all the other celestial bodies that are within the sphere of their activity. He further stated that gravitational attraction increases by how much nearer the body wrought upon is to its own center. In correspondence with
Isaac Newton from 1679 and 1680, Hooke conjectured that gravitational forces might decrease according to the double of the distance between the two bodies. Hooke urged Newton, who was a pioneer in the development of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, 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. 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"). 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 1685–86; the second on 2 March 1686–87; and the third on 6 April 1686–87. The Royal Society published Newton's entire collection at their own expense in May 1686–87.
Isaac Newton had bridged the gap between Kepler's gravitational mass and Galileo's gravitational acceleration, resulting in the discovery of the following relationship which governed both of these:
:
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.
Newton's cannonball
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."
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."
Universal gravitational mass
In contrast to earlier theories (e.g.
celestial spheres) which stated that the heavens were made of entirely different material, Newton's theory of mass was groundbreaking partly because it introduced
universal gravitational mass: 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. If a large collection of small objects were formed into a giant spherical body such as the Earth or Sun, Newton calculated the collection would create a gravitational field proportional to the total mass of the body,
and inversely proportional to the square of the distance to the body's center.
[These two properties are very useful, as they allow spherical collections of objects to be treated exactly like large individual objects.]
For example, according to Newton's theory of universal gravitation, each carob seed produces a gravitational field. 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. In fact, by
unit conversion 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. 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.
Henry 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.
Given two objects A and B, of masses ''M''
A and ''M''
B, separated by a
displacement R
AB, Newton's law of gravitation states that each object exerts a gravitational force on the other, of magnitude
:
,
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
:
.
This is the basis by which masses are determined by
weighing. In simple
spring scales
A spring scale, spring balance or newton meter is a type of mechanical force gauge or weighing scale. It consists of a spring fixed at one end with a hook to attach an object at the other. It works in accordance with Hooke's Law, which states ...
, for example, the force ''F'' is proportional to the displacement of the
spring
Spring(s) may refer to:
Common uses
* Spring (season), a season of the year
* Spring (device), a mechanical device that stores energy
* Spring (hydrology), a natural source of water
* Spring (mathematics), a geometric surface in the shape of a h ...
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. Assuming the gravitational field is equivalent on both sides of the balance, a
balance measures relative weight, giving the relative gravitation mass of each object.
Inertial mass
''Inertial mass'' is the mass of an object measured by its resistance to acceleration. This definition has been championed by
Ernst Mach[Ori Belkind, "Physical Systems: Conceptual Pathways between Flat Space-time and Matter" (2012) Springer (''Chapter 5.3'')] and has since been developed into the notion of
operationalism by
Percy W. Bridgman. The simple
classical mechanics definition of mass differs slightly from the definition in the theory of
special relativity, but the essential meaning is the same.
In classical mechanics, 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
:
where F is the resultant
force acting on the body and a is the
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
of the body's centre of mass.
[In its original form, Newton's second law is valid only for bodies of constant mass.] 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 of constant inertial masses ''m''
1 and ''m''
2. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on ''m''
1 by ''m''
2, which we denote F
12, and the force exerted on ''m''
2 by ''m''
1, which we denote F
21. Newton's second law states that
:
where a
1 and a
2 are the accelerations of ''m''
1 and ''m''
2, 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
:
and thus
:
If is non-zero, the fraction is well-defined, which allows us to measure the inertial mass of ''m''
1. In this case, ''m''
2 is our "reference" object, and we can define its mass ''m'' 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.
Additionally, mass relates a body's
momentum p to its linear
velocity v:
:
,
and the body's
kinetic energy ''K'' to its velocity:
:
.
The primary difficulty with Mach's definition of mass is that it fails to take into account the
potential energy (or
binding energy) needed to bring two masses sufficiently close to one another to perform the measurement of mass.
[ This is most vividly demonstrated by comparing the mass of the proton in the nucleus of deuterium, to the mass of the proton in free space (which is greater by about 0.239%—this is due to the binding energy of deuterium). Thus, for example, if the reference weight ''m''2 is taken to be the mass of the neutron in free space, and the relative accelerations for the proton and neutron in deuterium are computed, then the above formula over-estimates the mass ''m''1 (by 0.239%) for the proton in deuterium. At best, Mach's formula can only be used to obtain ratios of masses, that is, as ''m''1 / ''m''2 = / . An additional difficulty was pointed out by Henri Poincaré, which is that the measurement of instantaneous acceleration is impossible: unlike the measurement of time or distance, there is no way to measure acceleration with a single measurement; one must make multiple measurements (of position, time, etc.) and perform a computation to obtain the acceleration. Poincaré termed this to be an "insurmountable flaw" in the Mach definition of mass.
]
Atomic masses
Typically, the mass of objects is measured in terms of the kilogram, which since 2019 is defined in terms of fundamental constants of nature. The mass of an atom or other particle can be compared more precisely and more conveniently to that of another atom, and thus scientists developed the dalton (also known as the unified atomic mass unit). By definition, 1 Da (one dalton) is exactly one-twelfth of the mass of a carbon-12 atom, and thus, a carbon-12 atom has a mass of exactly 12 Da.
In relativity
Special relativity
In some frameworks of special relativity, physicists have used different definitions of the term. In these frameworks, two kinds of mass are defined: rest mass (invariant mass),[It is possible to make a slight distinction between "rest mass" and "invariant mass". For a system of two or more particles, none of the particles are required be at rest with respect to the observer for the system as a whole to be at rest with respect to the observer. To avoid this confusion, some sources will use "rest mass" only for individual particles, and "invariant mass" for systems.] and relativistic mass
The word " mass" has two meanings in special relativity: '' invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity ...
(which increases with velocity). Rest mass is the Newtonian mass as measured by an observer moving along with the object. ''Relativistic mass'' is the total quantity of energy in a body or system divided by ''c''2. The two are related by the following equation:
:
where is the Lorentz factor:
:
The invariant mass of systems is the same for observers in all inertial frames, while the relativistic mass depends on the observer's frame of reference. In order to formulate the equations of physics such that mass values do not change between observers, it is convenient to use rest mass. The rest mass of a body is also related to its energy ''E'' and the magnitude of its momentum p by the relativistic energy-momentum equation:
:
So long as the system is closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
with respect to mass and energy, both kinds of mass are conserved in any given frame of reference. The conservation of mass holds even as some types of particles are converted to others. Matter particles (such as atoms) may be converted to non-matter particles (such as photons of light), but this does not affect the total amount of mass or energy. Although things like heat may not be matter, all types of energy still continue to exhibit mass.[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 or radiation 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.] 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.
Both rest and relativistic mass can be expressed as an energy by applying the well-known relationship ''E'' = ''mc''2, yielding rest energy
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
and "relativistic energy" (total system energy) respectively:
:
:
The "relativistic" mass and energy concepts 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. Because the relativistic mass is proportional to the energy, it has gradually fallen into disuse among physicists. There is disagreement over whether the concept remains useful pedagogically
Pedagogy (), most commonly understood as the approach to teaching, is the theory and practice of learning, and how this process influences, and is influenced by, the social, political and psychological development of learners. Pedagogy, taken a ...
.
In bound systems, the binding energy must often be subtracted from the mass of the unbound system, because binding energy commonly leaves the system at the time it is bound. The mass of the system changes in this process merely because the system was not closed during the binding process, so the energy escaped. For example, the binding energy of atomic nuclei is often lost in the form of gamma rays when the nuclei are formed, leaving nuclides which have less mass than the free particles ( nucleons) of which they are composed.
Mass–energy equivalence also holds in macroscopic systems. For example, if one takes exactly one kilogram of ice, and applies heat, the mass of the resulting melt-water will be more than a kilogram: it will include the mass from the thermal energy ( latent heat) used to melt the ice; this follows from the conservation of energy. This number is small but not negligible: about 3.7 nanograms. It is given by the latent heat of melting ice (334 kJ/kg) divided by the speed of light squared (''c''2 ≈ ).
General relativity
In general relativity, the equivalence principle is the equivalence of gravitational
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...
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 (i.e. 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
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many ot ...
of the Einstein field equations, making 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 In the theory of general relativity, a stress–energy–momentum pseudotensor, such as the Landau–Lifshitz pseudotensor, is an extension of the non-gravitational stress–energy tensor that incorporates the energy–momentum of gravity. It allo ...
.
In quantum physics
In classical mechanics, the inert mass of a particle appears in the Euler–Lagrange equation as a parameter ''m'':
: .
After quantization, replacing the position vector ''x'' with a wave function, the parameter ''m'' appears in the kinetic energy operator:
: .
In the ostensibly covariant (relativistically invariant) Dirac equation, and in natural units, this becomes:
:
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, this term arises from the coupling of the field ψ to an additional field Φ, the Higgs field
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field,
one of the fields in particle physics theory. In the St ...
. In the case of fermions, the Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property " mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other b ...
results in the replacement of the term ''m''ψ in the Lagrangian with . This shifts the explanandum An explanandum (a Latin term) is a sentence describing a phenomenon that is to be explained, and the explanans are the sentences adduced as explanations of that phenomenon. For example, one person may pose an ''explanandum'' by asking "Why is there ...
of the value for the mass of each elementary particle to the value of the unknown coupling constant ''G''ψ.
Tachyonic particles and imaginary (complex) mass
A tachyonic field
In physics, a tachyonic field, or simply tachyon, is a quantum field with an imaginary mass. Although tachyonic particles (particles that move faster than light) are a purely hypothetical concept that violate a number of essential physical princ ...
, or simply tachyon
A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...
, is a quantum field
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
with an imaginary mass.[ Although ]tachyon
A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...
s ( particles that move faster than light) are a purely hypothetical concept not generally believed to exist,[Lisa Randall, ''Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions'', p.286: "People initially thought of tachyons as particles travelling faster than the speed of light...But we now know that a tachyon indicates an instability in a theory that contains it. Regrettably for science fiction fans, tachyons are not real physical particles that appear in nature."] fields with imaginary mass have come to play an important role in modern physics and are discussed in popular books on physics.[Brian Greene, ''The Elegant Universe'', Vintage Books (2000)] Under no circumstances do any excitations ever propagate faster than light in such theories—the presence or absence of a tachyonic mass has no effect whatsoever on the maximum velocity of signals (there is no violation of causality). While the ''field'' may have imaginary mass, any physical particles do not; the "imaginary mass" shows that the system becomes unstable, and sheds the instability by undergoing a type of phase transition called tachyon condensation (closely related to second order phase transitions) that results in symmetry breaking in current models of particle physics.
The term "tachyon
A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...
" was coined by Gerald Feinberg
Gerald Feinberg (27 May 1933 – 21 April 1992) was a Columbia University physicist, futurist and populist author. He spent a year as a Member of the Institute for Advanced Study, and two years at the Brookhaven Laboratories. Feinberg went to B ...
in a 1967 paper, but it was soon realized that Feinberg's model in fact did not allow for superluminal
Faster-than-light (also FTL, superluminal or supercausal) travel and communication are the conjectural propagation of matter or information faster than the speed of light (). The special theory of relativity implies that only particles with z ...
speeds. Instead, the imaginary mass creates an instability in the configuration:- any configuration in which one or more field excitations are tachyonic will spontaneously decay, and the resulting configuration contains no physical tachyons. This process is known as tachyon condensation. Well known examples include the condensation of the Higgs boson in particle physics, and ferromagnetism in condensed matter physics.
Although the notion of a tachyonic imaginary mass might seem troubling because there is no classical interpretation of an imaginary mass, the mass is not quantized. Rather, the scalar field is; even for tachyonic quantum fields
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
, the field operators at spacelike separated points still commute (or anticommute), thus preserving causality. Therefore, information still does not propagate faster than light, and solutions grow exponentially, but not superluminally (there is no violation of causality). Tachyon condensation drives a physical system that has reached a local limit and might naively be expected to produce physical tachyons, to an alternate stable state where no physical tachyons exist. Once the tachyonic field reaches the minimum of the potential, its quanta are not tachyons any more but rather are ordinary particles with a positive mass-squared.
This is a special case of the general rule, where unstable massive particles are formally described as having a complex mass, with the real part being their mass in the usual sense, and the imaginary part being the decay rate in natural units.[
] However, in quantum field theory, a particle (a "one-particle state") is roughly defined as a state which is constant over time; i.e., an eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
of the Hamiltonian. An unstable particle In particle physics, a resonance is the peak located around a certain energy found in differential cross sections of scattering experiments. These peaks are associated with subatomic particles, which include a variety of bosons, quarks and hadrons ...
is a state which is only approximately constant over time; If it exists long enough to be measured, it can be formally described as having a complex mass, with the real part of the mass greater than its imaginary part. If both parts are of the same magnitude, this is interpreted as a resonance appearing in a scattering process rather than a particle, as it is considered not to exist long enough to be measured independently of the scattering process. In the case of a tachyon, the real part of the mass is zero, and hence no concept of a particle can be attributed to it.
In a Lorentz invariant theory, the same formulas that apply to ordinary slower-than-light particles (sometimes called "bradyon
The physics technical term massive particle refers to a massful particle which has real non-zero rest mass (such as baryonic matter), the counter-part to the term massless particle. According to special relativity, the velocity of a massive particl ...
s" in discussions of tachyons) must also apply to tachyons. In particular the energy–momentum relation:
:
(where p is the relativistic momentum of the bradyon and m is its rest mass) should still apply, along with the formula for the total energy of a particle:
:
This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass–energy") and a contribution from its motion, the kinetic energy.
When ''v'' is larger than ''c'', the denominator in the equation for the energy is "imaginary", as the value under the radical is negative. Because the total energy must be real, the numerator must ''also'' be imaginary: i.e. the rest mass m must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number.
See also
* Mass versus weight
* Effective mass (spring–mass system)
* Effective mass (solid-state physics)
* Extension (metaphysics) In metaphysics, extension signifies both 'stretching out' (Latin: ''extensio'') as well as later 'taking up space', and most recently, spreading one's internal mental cognition into the external world.
The history of thinking about ''extension'' ca ...
* International System of Quantities
* 2019 redefinition of SI base units
Notes
References
External links
*
*
*
*
*
*
* Jim Baggott
James Edward Baggott (born 2 March 1957) is a British science writer living in Reading, Berkshire, England who writes about science, philosophy and science history. Baggott is the author of nine books, including ''Farewell to Reality: How Modern ...
(27 September 2017)
The Concept of Mass
(video) published by the Royal Institution on YouTube
YouTube is a global online video sharing and social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by Google, and is the second mo ...
.
{{Authority control
Physical quantities
SI base quantities
Moment (physics)