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The **mass** (or inertial mass, from Greek: μᾶζα, máza, barley cake, lump of dough) is a physical quantity that represents the amount of matter proper to material bodies that determines their dynamic behavior when subjected to the influence of external forces. In classical mechanics the term mass can refer to three different scalar physical quantities, distinct from each other:

**inertial mass**is proportional to the inertia of a body, which is the resistance to changing its state of motion when a force is applied;**passive gravitational mass**is proportional to the force of a body’s interaction with the force of gravity;**active gravitational mass**, on the other hand, is proportional to the intensity of the gravitational field produced by a body.

Inertial mass and gravitational mass have been experimentally proven to be equivalent, although conceptually they are distinct. The first experiments aimed to establish this equivalence were those of Galileo Galilei.

Throughout the history of physics, in particular classical physics, mass has been considered an intrinsic property of matter, representable by a scalar value and which is conserved in time and space, remaining constant in any isolated system. In addition, the term mass has been used to denote two potentially distinct quantities: the interaction of matter with the gravitational field and the relationship between the force applied to a body and the acceleration induced on it. However, the equivalence of the two masses has been verified in numerous experiments (put in place by Galileo Galilei first).

The mass does not correspond with the amount of substance, the physical quantity for which it has been introduced in the SI a fundamental quantity, the mole (symbol mol). The mass of a body is commonly determined by measuring its inertia which is opposed to a change in its state of motion or the gravitational attraction to other bodies by comparison with a sample (see balance).

Unlike space and time, for which operational definitions can be given in terms of natural phenomena, defining the concept of mass requires explicit reference to the physical theory that describes its meaning and properties. Pre-physical intuitive concepts of quantity of matter (not to be confused with quantity of substance, measured in moles) are too vague for an operational definition, and refer to common properties, inertia and weight, which are considered quite distinct from the first theory to introduce mass in quantitative terms, Newtonian dynamics.

The concept of mass becomes more complex at the level of particle physics where the presence of elementary particles with mass (electrons, quarks, and so on) and without mass (photons, gluons) still has no explanation in fundamental terms. In other words, it is not clear why some particles have mass and others do not. The main theories that try to give an interpretation to mass are: the Higgs mechanism, string theory and loop quantum gravity; of these, as of July 4, 2012 thanks to the LHC particle accelerator, only the Higgs theory has had the first experimental evidence.

In the current International System of Units (SI), mass has been chosen as a fundamental physical quantity, i.e., not expressible solely in terms of other fundamental quantities. Its unit of measurement is the kilogram, indicated by the symbol: kg.

In the CGS system, the unit of mass is the gram. In the United Kingdom and the United States the pound (approximately 454 g) and the stone (14 pounds) are commonly used. Other units of measurement are commonly used in specific fields of physics.

In atomic physics and physics of matter are commonly used the Hartree units, based on the mass of the electron or the atomic mass unit, roughly equivalent to the mass of a proton. In chemistry is frequently used the mole that, even if it is not a mass unit, it is linked by a simple proportionality factor. In nuclear and sub-nuclear physics is common the use of atomic mass unit. However, especially in the field of high energies, it is used to express the mass (at rest or invariant) by its equivalent energy E = mc^{2}. The energy is in turn expressed in: eV.

## Conservation of mass

In classical mechanics there is the fundamental law of conservation of mass, in various formulations. In general, given a control volume, fixed, the variation of mass contained in it is equal to the outgoing flow of mass through the frontier of the system, that is through the closed surface that delimits the volume, changed of sign: in simple words, the variation of mass of a system is equal to the incoming mass minus the outgoing mass; this implies, for example, that the mass can be neither created nor destroyed, but only moved from one place to another. In chemistry, Antoine Lavoisier established in the 18th century that in a chemical reaction the mass of the reactants is equal to the mass of the products.

The principle of conservation of mass is valid with very good approximation in everyday experience, but it ceases to be valid in nuclear reactions and, in general, in phenomena involving relativistic energies: in this case it is incorporated in the principle of conservation of energy.

## Inertial mass

### Newtonian definition

The inertial mass \(m_i\) of a body is defined in the Principia as the amount of matter by tying it to the principle of proportionality as the constant of proportionality between the applied force \(\vec{F}\) and the acceleration undergone \(\vec{a}\):

\[m_i=\dfrac{F}{a}\]

The inertial mass can actually be obtained operationally by measuring the acceleration of the body subjected to a known force, being an index of the resistance of a body to accelerate when subjected to a force, i.e., of the inertia of the body. The problem to use this property as a definition is that it needs the previous concept of force; to avoid the vicious circle generated by Newton who did not specify the instrument to measure it, often the force is then defined by linking it to the elongation of a spring following Hooke’s law, definition clearly unsatisfactory as particular and not general. In addition, this definition has given rise to several problems, particularly related to the reference system in which the measurement is made: the concept of inertia, like that of force, was in fact historically criticized by many thinkers, including Berkeley, Ernst Mach, Percy Williams Bridgman and Max Jammer.

### Machian definition

The concept of inertial mass was revolutionized by Mach’s work. He was able to eliminate metaphysical elements that persisted in classical mechanics, reformulating the definition of mass in an operationally precise way and without logical contradictions. From this redefinition, general relativity started, even if Einstein himself was not able to include Mach principle within general relativity.

The Machian definition is based on the principle of action-reaction, leaving the principle of proportionality to define the force later. Consider an isolated system consisting of two interacting (point-like) bodies. Whatever is the force acting between the two bodies, it is observed experimentally that the accelerations undergone by the two bodies are always proportional and in constant ratio between them:

\[\vec{a}_2=-\mu_{12}\vec{a}_1\]

What is particularly relevant is that the ratio \(\mu_{12}\) between the two instantaneous accelerations is not only constant over time, but does not depend on the initial state of the system: it is therefore associated with an intrinsic physical property of the two bodies under consideration. By changing one of the two bodies, the constant of proportionality also varies. Let’s suppose then to use three bodies, and carry out separately three experiments with the three possible pairs (it is always assumed the absence of external forces). In this way we can measure the constants \(\mu_{12},\mu_{23},\mu_{31}\). Note that by definition:

\[\mu_{ab}=\dfrac{1}{\mu_{ba}}\]

Comparing the values of the observed constants, one will invariably find that they satisfy the relation \(\mu_{12}\cdot \mu_{23}\cdot \mu_{31}=1\). Thus the product \(\mu_{12}\cdot \mu_{31}\) does not depend on the nature of body 1, since it is equal to the inverse of \(\mu_{23}\), namely \(\mu_{32}\), which is independent of it due to the independence of \(\mu_{23}\). From this it follows that any coefficient \(\mu_{ij}\) must be able to be expressed as a product of two constants, each dependent on only one of the two bodies.

\[\mu_{ab}=\nu_{b}\cdot m_{a}=\dfrac{1}{\nu_a m_b}=\dfrac{1}{\mu_{ba}} \quad \Rightarrow\begin{cases} \nu_a=\dfrac{1}{m_a} \\ \nu_b=\dfrac{1}{m_b} \end{cases}\]

\[\mu_{ij}=\dfrac{m_i}{m_j}\quad\Rightarrow\quad m_i\vec{a}_i=-m_j\vec{a}_j\]

at any instant of time, for any pair of bodies. The quantity \(m\) that results so defined (unless a constant factor, which corresponds to the choice of the unit of measurement) is called inertial mass of the body: it is therefore possible to measure the mass of a body by measuring the accelerations due to interactions between this and another body of known mass, without needing to know what are the forces acting between the two points (provided that the system formed by the two bodies can be considered isolated, ie not subject to external forces). The link between the masses is given by:

\[m_2=\dfrac{a_1}{a_2}m_1\]

## Gravitational mass

If a body, such as a basket ball, is left free in the air, it is attracted downwards by a force, in first approximation constant, called weight force. By means of a plate scale, it can be seen that different bodies, in general, are attracted differently by the weight force, i.e. they weigh differently. The plate balance can be used to give an operational definition of gravitational mass: a unit mass is assigned to a sample object and the other objects have a mass equal to the number of samples needed to balance the plates.

Passive gravitational mass is a physical quantity proportional to the interaction of each body with the gravitational field. Within the same gravitational field, a body with small gravitational mass experiences a smaller force than a body with large gravitational mass: the gravitational mass is proportional to the weight, but while the latter varies with the gravitational field, the mass remains constant. By definition, the weight force \(F_w\) is expressed as the product of the gravitational mass \(m_g\) by a vector \(\vec{g}\), called gravity acceleration, which depends on the place where the measurement is made and whose units depend on that of the gravitational mass.

The active gravitational mass of a body is proportional to the intensity of the gravitational field generated by it. The greater is the active gravitational mass of a body, the more intense is the gravitational field generated by it, and therefore the force exerted by the field on another body; to make an example, the gravitational field generated by the Moon is less (at the same distance from the center of the two celestial bodies) than that generated by the Earth because its mass is less. Measurements of active gravitational masses can be performed, for example, with torsion balances like the one used by Henry Cavendish in the determination of the constant of universal gravitation.

The gravitational mass is to all intents and purposes the charge of the gravitational field, exactly in the same sense in which the electric charge is the charge of the electric field: it simultaneously generates and suffers the effects of the gravitational field. Objects with zero gravitational mass (e.g. photons) would not suffer the effects of the field: in fact a result of general relativity is that any body follows a trajectory due to gravitational field.

## Electromagnetic mass

Charged objects have a greater inertia than the same uncharged bodies. This is explained by an interaction of electric charges in motion with the field generated by them, called field reaction, the effect can be interpreted as an increase in the inertial mass of the body and is derived from Maxwell’s equations. The interaction of electric charges with the field depends on the geometry of the system: the inertia of a charged body assumes a tensor character, in contradiction with classical mechanics, and therefore we must distinguish between a component parallel to the motion and two transverse components. It is shown that we can divide the inertial mass of a charged body in two components, the electromagnetic mass and the non-electromagnetic mass. While the electromagnetic mass depends on the geometry of the system, the non-electromagnetic mass has the same “standard” invariance characteristics of the inertial mass, and the inertial mass is related to it if the body is uncharged.

The concept of electromagnetic mass also exists in the theory of special relativity and quantum field theory. The electromagnetic mass had a great importance in the history of physics at the turn of the nineteenth and twentieth centuries because of the attempt, carried out mainly by Max Abraham and Wilhelm Wien, initially supported by the experimental work of Walter Kaufmann, to derive the inertial mass only from the electromagnetic inertia; this interpretation of inertia was later abandoned with the acceptance of the theory of relativity; more precise experiments, performed for the first time by A. H. Bucherer in 1908, showed that the correct relationships for longitudinal mass and transverse mass were not those provided by Abraham, but those of Hendrik Antoon Lorentz.

## Negative mass

In theoretical physics, **negative mass** is a hypothetical concept of matter whose mass has a negative sign relative to ordinary (positive) matter, e.g.: -2 kg. Such type of matter violates one or more conditions of energy and exhibits special properties, arising from the ambiguity of how attraction should refer to force or acceleration oriented opposite to negative mass. It is used in certain speculative theories, for the construction of Einstein-Rosen bridges. Originally, the most realistic known representation of this type of exotic matter was the density with pseudo negative pressure produced by the Casimir effect.

General relativity describes gravity and Newton’s laws of motion as both positive and negative particles and therefore also with negative mass but not including the other fundamental interactions. On the other hand, the standard model describes elementary particles and other interactions but does not include gravity. A new unifying theory that can make the concept of negative mass better understood would be appropriate.

The concept of negative mass arises in the first instance by analogy with electric charges, of which there are both positive and negative varieties. Just as a positive electric charge can be canceled by a negative charge, thus giving rise to the possibility of screening against electric forces, so we can envisage the possibility of “gravity screens,“ if negative mass existed to neutralize ordinary, positive mass.

Several scientists have speculated on its properties. Among these are Hermann Bondi in the 1950s, Banesh Hoffman (1906-1986), of the City University, New York, in the 1960s and ’70s, and Robert Forward, in the context of spacecraft propulsion, in the 1980s. In both Newton’s and Einstein’s theories of gravity, negative mass is a requirement for antigravity to exist. Since April 2017, researchers at Washington State University in the United States have experimentally demonstrated the existence of negative mass by cooling Rubidium atoms with lasers. Professor Peter Engels and his team at Washington State University say they observed a negative mass on April 10, 2017 by reducing the temperature of Rubidium atoms near absolute zero and generating a Bose-Einstein condensate with these atoms. Using a laser trap, the team was able to reverse the spin of some of the atoms in the condensate and observed that once released from the trap the atoms expanded and exhibited negative mass properties, and in particular accelerating in the direction from which the force was coming instead of away from it. See: Negative-Mass Hydrodynamics in a Spin-Orbit–Coupled Bose-Einstein Condensate. M. A. Khamehchi, Khalid Hossain, M. E. Mossman, Yongping Zhang, Th. Busch, Michael McNeil Forbes, and P. Engels. Phys. Rev. Lett. 118, 155301 – Published 10 April 2017 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.155301

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## References

- Bondi, H. “Negative Mass in General Relativity,”
*Reviews of Modern Physics*, Vol. 29, No.3, July 1957, pp. 423-428. - Forward, R. L. “Negative Matter Propulsion,”
*Journal of Propulsion and Power*(AIAA), Vol. 6, No. 1, Jan.-Feb. 1990, pp. 28-37.