intrinsic property
In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject. An extrinsic property is not essential or inherent to the subject that is being characterized. For example, mass ...
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 unit ...
of
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
in a
physical body
In common usage and classical mechanics, a physical object or physical body (or simply an object or body) is a collection of matter within a defined contiguous boundary in three-dimensional space. The boundary must be defined and identified by t ...
, until the discovery of the
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, ...
and
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
. It was found that different atoms and different
elementary particles
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, anti ...
, theoretically with the same amount of matter, have nonetheless different masses. Mass in modern physics has multiple
definitions
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definiti ...
which are conceptually distinct, but physically
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
. Mass can be experimentally defined as a
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
of the body's
inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
, 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 the ...
(change of
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
) when a
net force Net Force may refer to:
* Net force, the overall force acting on an object
* ''NetForce'' (film), a 1999 American television film
* Tom Clancy's Net Force, a novel series
* Tom Clancy's Net Force Explorers, a young adult novel series
{{disam ...
is applied. The object's mass also determines the
strength
Strength may refer to:
Physical strength
*Physical strength, as in people or animals
* Hysterical strength, extreme strength occurring when people are in life-and-death situations
*Superhuman strength, great physical strength far above human c ...
of its
gravitation
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 stron ...
al attraction to other bodies.
The
SI base unit
The SI base units are the standard units of measurement defined by the International System of Units (SI) for the seven base quantities of what is now known as the International System of Quantities: they are notably a basic set from which all ...
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 r ...
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
, even though mass is often determined by measuring the object's weight using a spring scale, rather than
balance scale
A scale or balance is a device used to measure weight or mass. These are also known as mass scales, weight scales, mass balances, and weight balances.
The traditional scale consists of two plates or bowls suspended at equal distances from a ...
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 ...
'' in the
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 ...
of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
.
Units of mass
The
International System of Units
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
(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
The melting point (or, rarely, liquefaction point) of a substance is the temperature at which it changes state from solid to liquid. At the melting point the solid and liquid phase exist in equilibrium. The melting point of a substance depends ...
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
A grave is a location where a dead body (typically that of a human, although sometimes that of an animal) is buried or interred after a funeral. Graves are usually located in special areas set aside for the purpose of burial, such as grave ...
in 1793, the platinum
Kilogramme des Archives
The grave, abbreviated ''gv'', is the unit of mass used in the first metric system which was implemented in France in 1793. In 1795, the grave was renamed as the kilogram.
Origin
The modern kilogram has its origins in the Age of Enlightenment an ...
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 speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
and the elementary charge.
Non-SI units accepted for use with SI units include:
* the
tonne
The tonne ( or ; symbol: t) is a unit of mass equal to 1000 kilograms. It is a non-SI unit accepted for use with SI. It is also referred to as a metric ton to distinguish it from the non-metric units of the short ton ( United State ...
(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
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defi ...
(eV), a unit of
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
, used to express mass in units of eV/''c''2 through
mass–energy equivalence
In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physici ...
* the
dalton
Dalton may refer to:
Science
* Dalton (crater), a lunar crater
* Dalton (program), chemistry software
* Dalton (unit) (Da), the atomic mass unit
* John Dalton, chemist, physicist and meteorologist
Entertainment
* Dalton (Buffyverse), minor ch ...
(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
The imperial system of units, imperial system or imperial units (also known as British Imperial or Exchequer Standards of 1826) is the system of units first defined in the British Weights and Measures Act 1824 and continued to be developed thr ...
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)
The pound of force or pound-force (symbol: lbf, sometimes lbf,) is a unit of force used in some systems of measurement, including English Engineering units and the foot–pound–second system.
Pound-force should not be confused with pound-m ...
(about 4.5 N), a unit of forceThese 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
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
, 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 Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It was ...
()
* 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 characteris ...
().
Definitions
In
physical science
Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical sciences".
Definition
Phy ...
, 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
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
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
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
.
* Active gravitational massThe distinction between "active" and "passive" gravitational mass does not exist in the Newtonian view of gravity as found in
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, 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
In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on i ...
acceleration. For example, an object in free-fall near the
Moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
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 mathe ...
where it has zero
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
. Mass can be converted into other forms of energy according to the principle of
mass–energy equivalence
In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physici ...
. This equivalence is exemplified in a large number of physical processes including
pair production
Pair production is the creation of a subatomic particle and its antiparticle from a neutral boson. Examples include creating an electron and a positron, a muon and an antimuon, or a proton and an antiproton. Pair production often refers specifi ...
,
beta decay
In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar of that nuclide. For ...
and
nuclear fusion
Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is manifest ...
. Pair production and nuclear fusion are processes in which measurable amounts of mass are converted to
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
or vice versa.
* Curvature of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
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
An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwee ...
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
Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass. The lower the gravitational potential (the closer ...
. Other forms of curvature have been measured using the
Gravity Probe B
Gravity Probe B (GP-B) was a satellite-based experiment to test two unverified predictions of general relativity: the geodetic effect and frame-dragging. This was to be accomplished by measuring, very precisely, tiny changes in the direction of ...
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
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
. The quantum mass of a particle is proportional to the inverse
Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It was ...
and can be determined through various forms of
spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
. 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
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
" 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
The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially ...
) 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
In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on ...
, 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 the ...
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 surfa ...
or the
Moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
. 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
In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at ...
. 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
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, 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 theory ...
developed his
general theory of relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
starting with the assumption that the inertial and passive gravitational masses are the same. This is known as the
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 ...
.
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 b ...
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
The Leaning Tower of Pisa ( it, torre pendente di Pisa), or simply, the Tower of Pisa (''torre di Pisa'' ), is the ''campanile'', or freestanding bell tower, of Pisa Cathedral. It is known for its nearly four-degree lean, the result of an unst ...
. 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 clas ...
s to slow the motion and increase the timing accuracy. Increasingly precise experiments have been performed, such as those performed by
Loránd Eötvös
Baron Loránd Eötvös de Vásárosnamény (or Loránd Eötvös, , '' hu, vásárosnaményi báró Eötvös Loránd Ágoston''; 27 July 1848 – 8 April 1919), also called Baron Roland von Eötvös in English literature, was a Hungarian physicist ...
, using the
torsion balance
A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. When it is twisted, it exerts a torque in the opposite direction, proportional ...
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
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
and
air resistance
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
, 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 dis ...
, 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
David Randolph Scott (born June 6, 1932) is an American retired test pilot and NASA astronaut who was the seventh person to walk on the Moon. Selected as part of the third group of astronauts in 1963, Scott flew to space three times and ...
did on the surface of the
Moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
during
Apollo 15
Apollo 15 (July 26August 7, 1971) was the ninth crewed mission in the United States' Apollo program and the fourth to Moon landing, land on the Moon. It was the first List of Apollo missions#Alphabetical mission types, J mission, with a ...
.
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
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
. 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
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, a
mass generation mechanism
In theoretical physics, a mass generation mechanism is a theory that describes the origin of mass from the most fundamental laws of physics. Physicists have proposed a number of models that advocate different views of the origin of mass. The probl ...
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 r ...
. 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
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 stron ...
but a theory of the latter has not been yet reconciled with the currently popular model of
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
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 unit ...
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
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
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
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
:
: , or equivalently
An early use of this relationship is a
balance scale
A scale or balance is a device used to measure weight or mass. These are also known as mass scales, weight scales, mass balances, and weight balances.
The traditional scale consists of two plates or bowls suspended at equal distances from a ...
, 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 ...
siliqua
The siliqua (plural ''siliquae'') is the modern name given (without any ancient evidence to confirm the designation) to small, thin, Roman silver coins produced in the 4th century A.D. and later. When the coins were in circulation, the Latin wo ...
) 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:
:
Tycho Brahe
Tycho Brahe ( ; born Tyge Ottesen Brahe; generally called Tycho (14 December 154624 October 1601) was a Danish astronomer, known for his comprehensive astronomical observations, generally considered to be the most accurate of his time. He was ...
, 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
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
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 planets ...
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of these two values is constant for all planets in the
Solar System
The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...
.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 was ...
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
Vincenzo Viviani (April 5, 1622 – September 22, 1703) was an Italian mathematician and scientist. He was a pupil of Torricelli and a disciple of Galileo.balls of the same material, but different masses, from the
Leaning Tower of Pisa
The Leaning Tower of Pisa ( it, torre pendente di Pisa), or simply, the Tower of Pisa (''torre di Pisa'' ), is the ''campanile'', or freestanding bell tower, of Pisa Cathedral. It is known for its nearly four-degree lean, the result of an unst ...
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
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* Groove 101.7FM, a former Perth, Australia, radio station
...
. 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.
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, 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 Newton's law of universal gravitation, 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 (vector), displacement RAB, 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, for example, the force ''F'' is proportional to the displacement of the spring (device), spring beneath the weighing pan, as per Hooke's law, and the scales are calibration, 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 Beam balance, 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 MachOri 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 Operationalization, operationalism by Percy W. Bridgman. The simple
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
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
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
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 the ...
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
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
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 F12, and the force exerted on ''m''2 by ''m''1, which we denote F21. Newton's second law states that
:
where a1 and a2 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
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
p to its linear
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
v:
: ,
and the body's
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
''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
Dalton may refer to:
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* Dalton (unit) (Da), the atomic mass unit
* John Dalton, chemist, physicist and meteorologist
Entertainment
* Dalton (Buffyverse), minor ch ...
(also known as the unified atomic mass unit). By definition, 1 Da (one
dalton
Dalton may refer to:
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) 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 (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 speed of light, ''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 system, closed 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 Mass–energy equivalence, ''E'' = ''mc''2, yielding rest energy 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 Mass–energy equivalence, proportional to the energy, it has gradually fallen into disuse among physicists. There is disagreement over whether the concept remains useful Pedagogy, pedagogically.
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
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the
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 ...
is the equivalence of Gravitational mass, gravitational and Inertial Mass, inertial mass. At the core of this assertion is Albert Einstein, 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 ''Fictitious force, pseudo-force'' experienced by an observer in a non-Inertial frame of reference, 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 Nonlinear system, non-linearity 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 quantum physics
In
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, 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
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
operator:
: .
In the ostensibly Covariance and contravariance of vectors#Informal usage, covariant (relativistically invariant) Dirac equation, and in natural units, this becomes:
:
where the "Invariant mass, 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
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
as developed in the 1960s, this term arises from the coupling of the field ψ to an additional field Φ, the Higgs field. In the case of fermions, the Higgs mechanism results in the replacement of the term ''m''ψ in the Lagrangian with . This shifts the explanandum 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, or simply tachyon, is a quantum field with an imaginary number, imaginary mass. Although tachyons (particles that move faster-than-light, 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." field (physics), fields with imaginary mass have come to play an important Higgs boson, 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 Standard Model, current models of
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
.
The term "tachyon" was coined by Gerald Feinberg in a 1967 paper, but it was soon realized that Feinberg's model in fact did not allow for superluminal 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 Higgs mechanism, condensation of the Higgs boson in
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, and ferromagnetism in condensed matter physics.
Although the notion of a tachyonic imaginary number, 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 field theory, quantum fields, the field operators at Minkowski space, spacelike separated points still Canonical commutation relation, 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 number, complex mass, with the real part being their mass in the usual sense, and the imaginary part being the Particle decay#Decay rate, 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 of the Hamiltonian (quantum mechanics), Hamiltonian. An Particle decay, unstable particle 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 "bradyons" in discussions of tachyons) must also apply to tachyons. In particular the energy–momentum relation:
:
(where p is the relativistic
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
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 number, "imaginary", as the value under the square root, radical is negative. Because the total
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
must be real number, 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)
* International System of Quantities
* 2019 redefinition of SI base units
Notes
References
External links
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* Jim Baggott (27 September 2017) The Concept of Mass (video) published by the Royal Institution on YouTube.
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Mass,
Physical quantities
SI base quantities
Moment (physics)