The mass of a body is called a physical quantity equal to the ratio of the force acting on the body to the acceleration acquired by it: a. Under normal conditions (velocities of bodies are much less than the speed of light c), this ratio is constant.

Derived on the basis of Newton's second law, the mass determines the inertial properties of the body, and therefore it is called the inertial mass. There is also the concept of gravitational (heavy) mass - a physical quantity that determines the measure of the gravitational interaction of the body in question with other bodies, say the Earth. In the wording of the law gravity stating, in particular, that any object at a distance r from its center of mass creates a gravitational acceleration:

the gravitational mass is present in the above formula as a factor m. The quantity G included in formula (1) is called the gravitational constant, the numerical value of which depends on the choice of the system of units; in the SI system. According to this definition of the law of universal gravitation, it is in principle possible, for example, to measure the gravitational acceleration that causes a standard of mass of 1 kg, and any object that causes the same acceleration at the same distance can be assigned a mass of 1 kg.

The definitions of inertial and gravitational masses are at first glance very different. The inertial mass, which characterizes the body's ability to "resist" external influences, plays a passive role; the gravitational mass generates attraction, i.e., it is an active principle.

For hundreds of years, scientists have been concerned with the question: are these two concepts equivalent? The classic experience of checking the equivalence of inertial and gravitational masses was carried out by I. Newton and described in the "Mathematical Principles of Natural Philosophy":

“I tested gold, silver, lead, glass, sand, table salt, wood, water and wheat. I got two identical boxes. I filled one of them with wood, and in the center of the swings of the other I placed a piece of gold of the same (as accurately as I could) weight. Suspended on threads 11 feet long, the boxes formed a pair of pendulums, exactly the same in weight and shape, and equally subject to air resistance; placing them side by side, I watched them rock back and forth together for a long time with the same oscillation.

And therefore (by virtue of Corollaries I and VI, Proposition XXIV, Book II) the amount of substance in gold was related to the amount of substance in wood, as the action driving force on all gold to the action of the driving force on the whole tree; in other words, as the weight of one to the weight of the other.

And with the help of these experiments in bodies of the same weight, it was possible to detect a difference in the quantities of a substance, constituting one thousandth of the total.

After Newton's experiments, the technique for measuring inertial and gravitational masses improved and their accuracy increased.

At present, in the experiments of Soviet physicists V. B. Braginsky and V. I. Panov, the equivalence of gravitational and inertial masses has been proved with an accuracy of 10-12, which is a billion times higher than the accuracy of the experiment described by Newton.

The unity of the nature of inertial and gravitational masses, and, consequently, the fact of their numerical coincidence established in experiments, was explained by A. Einstein. In his theory of relativity, he gave the concept of mass a new meaning, linking the mass of a body with the energy E contained in it:

From this definition follows: if the rest mass of the body is equal, then this means that it contains energy, which is called the rest energy. According to this definition, the mass of a body turns out to be dependent on its speed:

(at enormous speeds comparable to the speed of light).

As is known, the energy of a photon is determined by its frequency v: E where h is Planck's constant. On the other hand, according to Einstein's formula, . Comparison of these two formulas leads to the conclusion that the photon has an inertial mass equal to (it must be distinguished from the rest mass, which, of course, is equal to zero).

In 1960, American scientists Pound and Rebke performed the most subtle experiment, which showed that the photon also has a gravitational mass, which is equal to the inertial mass. If a photon with frequency v is emitted at a height H above the Earth towards the center of the Earth, then at the level earth's surface his kinetic energy increases by decreasing potential energy. From the law of conservation of energy we have:

It is assumed here that the mass of the photon does not change during the fall. Thus, a photon flew up to the receiver with a frequency v different from that with which it was emitted by the source. At

Such a subtle experiment was carried out using the Mössbauer effect.

, they simply talk about the mass, without specifying which one they mean.

In classical mechanics, the mass of a system of bodies is equal to the sum of the masses of its constituent bodies. AT relativistic mechanics mass is not an additive physical quantity, that is, the mass of the system in the general case is not equal to the sum of the masses of the components, but includes the binding energy and depends on the nature of the movement of particles relative to each other.

Direct generalizations of the concept of mass include such tensor characteristics as the moment of inertia, and such characteristics of the properties of the "body plus medium" system as mass displacement, added mass and effective mass, used in hydrostatics, hydrodynamics and quantum theory.

Principle of equivalence

All phenomena in the gravitational field occur in exactly the same way as in the corresponding field of inertial forces, if the strengths of these fields coincide and the initial conditions for the bodies of the system are the same.

Gravitational mass is a characteristic of bodies in classical mechanics, which is a measure of their gravitational interaction. It differs by definition from inertial mass, which determines the dynamic properties of bodies.

As established experimentally, these two masses proportional each other. No deviations from this law were found, therefore, new units of measurement for the inertial mass are not introduced (the units of measurement of the gravitational mass are used) and the proportionality coefficient is considered equal to one, which allows us to speak about equality inertial and gravitational masses.

It can be said that the first test of the proportionality of two kinds of mass was performed by Galileo Galilei, who discovered the universality free fall. According to Galileo's experiments on observing the free fall of bodies, all bodies, regardless of their mass and material, fall with the same free fall acceleration. Now these experiments can be interpreted as follows: an increase in the force acting on a more massive body from the Earth's gravitational field is fully compensated by an increase in its inert properties.

Newton drew attention to the equality of inertial and gravitational masses, he was the first to prove that they differ by no more than 0.1% (in other words, they are equal to within 10 −3). To date, this equality has been experimentally verified with a very high degree of accuracy (the sensitivity to the relative difference between the inertial and gravitational masses in the best experiment for 2009 is (0.3±1.8)·10 −13) .

A distinction should be made between the "weak equivalence principle" and the "strong equivalence principle". The strong equivalence principle can be formulated as follows: at each point of space-time in an arbitrary gravitational field, one can choose a locally inertial coordinate system such that in a sufficiently small neighborhood of the considered point Nature laws will have the same form as in non-accelerated Cartesian coordinate systems, where "laws of nature" means all the laws of nature.
The weak principle differs in that the words "laws of nature" are replaced in it by the words "laws of motion of freely falling particles". The weak principle is nothing more than another formulation of the observed equality of gravitational and inertial masses, while the strong principle is a generalization of observations of the effect of gravity on any physical objects.

Determination of mass

M 2 = E 2 c 4 − p 2 c 2 (\displaystyle m^(2)=(\frac (E^(2))(c^(4)))-(\frac (\mathbf (p) ^ (2))(c^(2)))),

where E - total energy free body, p- its momentum, c- the speed of light.

The mass defined above is a relativistic invariant, that is, it is the same in all frames of reference. If we go to the reference frame where the body is at rest, then m = E 0 c 2 (\displaystyle m=(\tfrac (E_(0))(c^(2))))- mass is determined by the rest energy ( Equivalence of mass and energy).

These definitions look especially simple in the system of units in which the speed of light is taken as 1 (for example, in the Planck or in the accepted in physics elementary particles system of units in which mass, momentum and energy are measured in electron volts):

In the service station: m = p i 2 = E 2 − p 2 (\displaystyle m=(\sqrt (p_(i)^(2)))=(\sqrt (E^(2)-\mathbf (p) ^(2)) )). In OTO: m = g i k p i p k (\displaystyle m=(\sqrt (g_(ik)p^(i)p^(k)))).

However, it should be noted that particles with zero mass (photon and hypothetical graviton) move in vacuum at the speed of light ( c≈ 300,000 km/s), and therefore there is no frame of reference in which they would be at rest. In contrast, particles with non-zero mass always move slower than the speed of light.

About "rest mass" and "relativistic mass"

In modern terminology, the term weight used instead of terms invariant mass or rest mass, being completely equivalent to them in meaning. In some situations (especially in the popular literature), however, this is specified explicitly to avoid confusion due to the understanding of the term weight in another - obsolete - sense, described in this paragraph.

In a large number of sources relating to the beginning and middle of the 20th century, as well as in popular science, the concept of mass introduced above was called "rest mass", while the mass itself was introduced on the basis of the classical definition of momentum

In this case m = E c 2 (\displaystyle m=(\tfrac (E)(c^(2)))) and said that body mass increases with increasing speed. With this definition, the concept of mass was equivalent to the concept of energy, and also required to separately introduce the "rest mass", measured in its own CO, and the "relativistic mass" of the moving body. This approach was widespread for a long time, as it allowed to draw numerous analogies with classical physics, however, in modern scientific literature rarely used because it introduces additional confusion in terminology without giving any new results. So-called relativistic mass turns out to be additive (in contrast to the rest mass of the system, which depends on the state of its constituent particles). However, massless particles (for example, photons) in this terminology turn out to have a variable mass; moreover, the relativistic mass does not simplify the formulation of the laws of particle dynamics in the least.

The covariant equality should be considered as a complete analog of the classical definition of momentum in terms of mass and velocity in SRT

positive mass

Particles with positive mass (tardions) include almost all particles of the Standard Model: leptons (including neutrinos, which were considered massless in the original version of the Standard Model), quarks, W- and Z-bosons, Higgs boson. These particles can move at any speed less than the speed of light, including at rest. Tardions also include all known compound particles: baryons (including the proton and neutron) and mesons.

Zero mass

The currently known particles of zero mass (massless, luxons) include photons and gluons, as well as hypothetical gravitons. Such particles in a free state can only move at the speed of light. But since it follows from quantum chromodynamics that gluons in a free state do not exist, only photons can be directly observed moving at the speed of light (in fact, that is why it is called the speed of light). For a long time it was believed that neutrinos also have zero mass, but the discovery of vacuum neutrino oscillations indicates that the neutrino mass, although very small, is not equal to zero.

It should be noted that a combination of several particles of zero mass can (and in the case of, for example, linked particles, must) have a non-zero mass.

negative mass

imaginary mass

Within the framework of the special theory of relativity, the existence of particles with an imaginary mass, the so-called tachyons, is mathematically possible. Such particles will have real values ​​of energy and momentum, and their speed must always be higher than the speed of light. However, the assumption of the possibility of observing single tachyons causes a number of methodological difficulties (for example, violation of the principle of causality), therefore, in most modern theories, single tachyons are not introduced. However, in quantum field theory, an imaginary mass can be introduced to consider tachyon condensation, which does not violate the principle of causality.

Mass units

The mass of very small particles can be determined using the reciprocal of the Compton wavelength: 1 cm -1 ≈ 3.52 × 10 -41 kg. The mass of a very large star or black hole can be identified with its gravitational radius: 1 cm ≈ 6.73 × 10 24 kg.

Mass measurement

Most instruments for measuring mass are based on the use of the principle of equivalence between inertial and gravitational mass. With the help of such instruments, called scales, the mass of bodies is determined by their weight. In spring scales, weight is measured by the degree of deformation of a flexible spring. In lever - the weight is determined by comparing the weight of the body of interest with the weight of standards (weights) of known mass.

The masses of charged elementary particles are determined by their traces in the cloud chamber. The masses of short-lived elementary particles that do not leave traces in a cloud chamber are determined by estimating the total energy of their decay products.

The mass of the Earth is determined on the basis of Newton's law of universal gravitation, based on the known values ​​of the gravitational constant and the radius of the Earth. The mass of the Sun is determined, also on the basis of Newton's law of universal gravitation, based on the known values ​​of the gravitational constant, the distance between the Earth and the Sun, and the period of the Earth's revolution around the Sun. The mass of our Galaxy is determined based on the period of revolution of the neighborhood of the Sun around the center of the Galaxy and the distance to the center of the Galaxy.

The masses of the nearest binary stars are determined from the distance between them and their period of revolution. If a star has no satellite and belongs to the main sequence, then its mass can be determined based on its luminosity or surface temperature.

Etymology and history of the concept

Mass as a scientific term was introduced by Newton as a measure of the amount of matter, before that, natural scientists operated with the concept of weight. In The Mathematical Principles of Natural Philosophy (1687), Newton first defined the "amount of matter" in a physical body as the product of its density and volume. He further indicated that in the same sense he would use the term weight. Finally, Newton introduces mass into the laws of physics: first into Newton's second law (through the momentum), and then into the law of gravity, from which it immediately follows that weight is proportional to mass. Newton clearly pointed out this proportionality and even tested it experimentally with all the accuracy possible in those years: “The mass is determined by the weight of the body, because it is proportional to the weight, which I found by experiments on pendulums, produced in the most accurate way” (Newton described these experiments in detail in III volume of his "Beginnings").

In fact, Newton uses only two understandings of mass: as a measure of inertia and as a source of gravity. Its interpretation as a measure of the "amount of matter" is nothing more than a clear illustration, and it was criticized as early as the 19th century as non-physical and meaningless.

For a long time, the law of conservation of mass was considered one of the main laws of nature. However, in the 20th century it turned out that this law is a limited version of the law of conservation of energy, and in many situations it is not observed.

Mass in the Universe

– physical quantity, which is one of the main characteristics of matter, which determines its inertial, energy and gravitational properties.
Mass is usually denoted by a Latin letter m.
Mass units
The CI unit for mass is kilogram. In the Gaussian system, mass is measured in grams. AT atomic physics it is customary to equate mass to the atomic mass unit in physics solid body- to the mass of an electron, in high-energy physics, the mass is measured in electron volts. In addition to these units used in science, there is a wide variety of historical units of mass that have retained their separate scope of use: the pound, ounce, carat, ton, etc. In astronomy, the unit for comparing masses celestial bodies is the mass of the sun.
Mass types
Strictly speaking, there are two different quantities that have common name"weight":

inertial mass characterizes the ability of a body to resist a change in its state of motion under the action of a force. Provided that the force is the same, an object with less mass changes its state of motion more easily than an object with more mass. Inertial mass appears in Newton's second law.

gravitational mass characterizes the intensity of interaction of the body with the gravitational field. It appears in Newton's law of universal gravitation.

Although inertial mass and gravitational mass are conceptually different concepts, all experiments known to date show that all two masses are proportional to each other. This allows you to build a system of units so that the unit of measurement of all three masses would be the same and all of them were equal to each other. Almost all systems of units are built on this principle.
In general relativity inert and gravitational the masses are considered to be completely equivalent.
Equations
As a measure of body inertia, mass is included in Newton's second law, written as

Where - Acceleration, and - Force acting on the body.
In a corresponding way, mass also enters into the quantum equations of motion: the Schrödinger equation, the Dirac equation, and so on.
As a quantity that determines the gravitational interaction of bodies, mass is included in the formulation of the law of universal gravitation

,

Where G is the gravitational constant, m 1 and m 2 – masses of two bodies interacting with each other, – Force acting from the side of the second body on the first one, – Distance vector between the bodies. So the mass m 2 determines the magnitude of the gravitational field created by the second body, and the mass m 1 force with which this field acts on the body. Both masses enter the law of universal gravitation symmetrically.
Connection with energy
Mass is an invariant quantity. That is, the energy and momentum components are converted through each other when passing to another inertial system coordinates, while the mass remains constant.

Conservation laws

Read more in the article Law of Conservation of Mass

In the 18th century, chemical experiments established the law of conservation of mass for chemical transformations. The total mass of substances entering into chemical reaction, is equal to the total mass of substances that are settled as a result of the reaction. However, in relativistic physics, the law of conservation of mass does not apply.
Mass of elementary particles
Mass, or rather rest mass, is an important characteristic of elementary particles. The question of what causes those values ​​of the mass of particles observed in experiment is important issue physics of elementary particles. So, for example, the mass of a neutron is somewhat larger than the mass of a proton, which is due to the difference in the interaction of quarks that make up these particles. The approximate equality of the masses of some particles allows us to combine them into groups, interpreting them as different states of one common particle with different values ​​of the isotopic spin.
Generalization of the concept of mass
For small values ​​of the momentum of a free particle, i.e. such that no forces act on it, the energy of the particle is determined by the formula

Where p is the momentum of the particle. This dependence of energy on momentum is called parabolic dispersion law.
In many cases, the energy dependence of the complex physical system on the mass has a similar quadratic form. For example, such a dependence is typical for the law of dispersion of energy bands in a solid. For such systems, one can introduce a quantity similar to the mass, which is called the effective mass.

Weight- physical quantity corresponding to the ability physical bodies keep your forward movement(inertia), as well as characterizing the amount of substance.

Mass refers to two different properties of matter:

* inertial mass, which characterizes the measure of inertia of bodies and appears in Newton's second law;
* gravitational mass, which determines with what force the body interacts with external gravitational fields (passive gravitational mass) and what gravitational field this body itself creates (active gravitational mass).

As established experimentally, these two masses are proportional to each other. No deviations from this law have been found, so the proportionality factor is usually chosen equal to unity and one speaks of the equality of the inertial and gravitational masses. The equality of inertial and gravitational masses is the content of the weak equivalence principle - an integral part of the Einstein equivalence principle, which is one of the main provisions of the general theory of relativity. Newton drew attention to the equality of inertial and gravitational masses, he was the first to verify this law with an accuracy of the order of 10 ^ -3. On the other hand, we can say that the first test of the equivalence principle was carried out by Galileo, who discovered the universality of free fall - as it became clear later, the independence of the acceleration of free fall from the material of which the body consists is a consequence of the equality of inertial and gravitational masses. To date, the weak equivalence principle has been experimentally verified with a very high degree of accuracy (3 * 10^-13).

Mass in classical mechanics is an additive quantity (the mass of a system is equal to the sum of the masses of its constituent bodies) and invariant with respect to a change in the frame of reference. Mass is also invariant in relativistic mechanics, although here mass is understood as the rest mass - the length of the 4-vector of momentum of a given body, a Lorentz-invariant quantity. The introduction of the so-called relativistic mass, which depends on the speed of the body, was used in early work on the theory of relativity. At present, the terms "relativistic mass" and "rest mass" are considered obsolete (see, for example, the discussion in "Advances in the Physical Sciences", issue 12, 2000). In the relativistic case, the masses are non-additive.

There are objects with zero mass. So, photon, graviton, gluon are massless particles (in vacuum). All of them must move at the speed of light (c ~ 300,000 km/sec). At the same time, for example, a system of two photons with energy E moving in opposite directions has a non-zero mass m = 2E / c ^ 2. Bodies with non-zero mass always move at a speed less than the speed of light.

The existence of particle mass in the Standard Model of particle physics is explained by the interaction of particles with the field of Higgs bosons.

In the SI system, mass is measured in kilograms. The CGS system uses grams. Sometimes other units of mass are also used.