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= Mass =
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Introduction
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Mass is an intrinsic property of a body. It was traditionally believed
to be related to the quantity of matter in a 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 (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, 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.
In the Standard Model of physics, the mass of elementary particles is
believed to be a result of their coupling with the Higgs boson in what
is known as the Brout-Englert-Higgs mechanism.
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 Newton's laws of motion).
* '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.
Repeated experiments since the 17th century have demonstrated that
inertial and gravitational mass are identical; since 1915, this
observation has been incorporated 'a priori' 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 (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 .
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
* 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 ().
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 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
of a system of particles, meaning the energy of that system in a
reference frame 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 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 all situations where the mass is accelerated away from free fall.
For example, when a body is at rest in a gravitational field (rather
than in free fall), it must be accelerated by a force from a scale or
the surface of a planetary body such as the Earth or the Moon. This
force keeps the object from going into free fall. Weight is the
opposing force in such circumstances and is thus determined by the
acceleration of free fall. On the surface of the Earth, for example,
an object with a mass of 50 kilograms weighs 491 newtons, which means
that 491 newtons is being applied to keep the object from going into
free fall. By contrast, on the surface of the Moon, the same object
still has a mass of 50 kilograms but weighs only 81.5 newtons, because
only 81.5 newtons is required to keep this object from going into a
free fall on the moon. Restated in mathematical terms, on the surface
of the Earth, the weight 'W' of an object is related to its mass 'm'
by , 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 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" 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 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
planes 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, 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 spacetime, 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. 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.
Weight as an amount
=====================
The concept of amount is very old and predates recorded history. The
concept of "weight" would incorporate "amount" and acquire a double
meaning that was not clearly recognized as such.
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 seed (carat or
siliqua) as a measurement standard. If an object's weight was
equivalent to [
http://std.dkuug.dk/JTC1/SC2/WG2/docs/n3138.pdf 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 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 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. 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, 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.
Mass as distinct from weight
==============================
According to K. M. Browne: "Kepler formed a [distinct] concept of mass
('amount of matter' ('copia materiae')), but called it 'weight' as did
everyone at that time." Finally, in 1686, Newton gave this distinct
concept its own name. In the first paragraph of 'Principia', Newton
defined quantity of matter as “density and bulk conjunctly”, and mass
as quantity of matter.
Newtonian mass
======================================================================
colspan=2| Earth's Moon !! rowspan=2| Mass of Earth
Semi-major axis !! Sidereal orbital period
|0.002 569 AU 0.074 802 sidereal year |rowspan=3|
Earth's gravity !! Earth's radius
|9.806 65 m/s2 6 375 km
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' (English: '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:
: 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 **F** is the resultant force acting on the body and **a** is
the acceleration of the body's centre of mass. For the moment, we will
put aside the question of what "force acting on the body" actually
means.
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
Newton's cannonball
=====================
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.
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), 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
'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 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.
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
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.
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.
:English Wikisource translation: On the Dynamics of Moving Systems
('See paragraph 16.') 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 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 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, 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. 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 mass. Although tachyons (particles that move faster than
light) are a purely hypothetical concept not generally believed to
exist, fields with imaginary mass have come to play an important role
in modern physics and are discussed in popular books on physics. 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" 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
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, 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 of the Hamiltonian. An 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 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)
* International System of Quantities
* 2019 revision of the SI base
External links
======================================================================
*
*
*
*
*
*
* Jim Baggott (27 September 2017).
[
https://www.youtube.com/watch?v=HfHjzomqbZc The Concept of Mass]
(video) published by the Royal Institution on YouTube.
License
=========
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License URL:
http://creativecommons.org/licenses/by-sa/3.0/
Original Article:
http://en.wikipedia.org/wiki/Mass
