Mass

Prerequisites: Newtonian Mechanics. Full path (5 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ

Intuitive

Mass shows up in two roles in everyday physics. It measures how hard it is to change the motion of a body: pushing a loaded shopping trolley takes more effort than pushing an empty one. It also measures how strongly a body is pulled by gravity: the loaded trolley is also heavier. That the same number appears in both roles is an experimental fact. It is the reason why all bodies fall equally fast in a vacuum, and Einstein took it as the starting point of general relativity.

Relativity changed the picture. Mass turned out to be the energy a body has when it is at rest, divided by the square of the speed of light. The sum of the masses of the parts of a system is therefore not conserved: when a nucleus splits or a particle and its antiparticle annihilate, rest mass turns into the energy of motion and radiation, and the total energy stays the same. In modern physics the word "mass" refers to this rest mass, which is the same for all observers.

Most of the mass around us does not come from the masses of the elementary particles. A proton consists of quarks and gluons. The quarks it contains are very light and the gluons are massless, yet the proton is heavy. Most of its mass is the energy of the quarks and gluons moving and interacting inside it, through the strong force. The Higgs field gives the elementary particles, such as the electron and the quarks, their masses, but it accounts for only a small fraction of the mass of ordinary matter.

• The Concept of Mass by Lev Okun
• The equivalence of mass and energy: the “center of energy” | Gravity and Levity
• http://nautil.us/issue/54/the-unspoken/physics-has-demoted-mass
• Origins of Mass by Wilzeck

Concrete

Inertial and gravitational mass. In Newtonian mechanics the inertial mass $m_i$ appears in Newton's second law, $\vec F = m_i \vec a$, and the gravitational mass $m_g$ appears in the law of gravitation, $\vec F = -G M m_g \hat r/r^2$. Combining the two gives the acceleration in a gravitational field,

$$\vec a = -\frac{m_g}{m_i}\,\frac{GM}{r^2}\,\hat r .$$

Torsion balance experiments (starting with Eötvös) find that $m_g/m_i$ is the same for all materials to high precision, so with suitable units $m_g = m_i$ and all bodies fall with the same acceleration.

Mass in special relativity. A free particle has four-momentum $p^\mu = (E/c, \vec p)$. With metric signature $(+,-,-,-)$ its square is the same in every inertial frame and defines the mass:

$$p_\mu p^\mu = \frac{E^2}{c^2} - \vec p^{\,2} = m^2c^2, \qquad E = \sqrt{m^2c^4 + \vec p^{\,2}c^2} .$$

At rest $E = mc^2$. For a massless particle such as the photon, $E = |\vec p|c$.

Mass is not additive. For a system of particles the mass is defined from the total four-momentum, $M^2c^2 = P_\mu P^\mu$ with $P^\mu = \sum_a p_a^\mu$. Two photons with energies $E$ moving in opposite directions have $\vec P = 0$ and total energy $2E$, so the system has mass $M = 2E/c^2$ although each photon is massless. In the other direction, a bound system weighs less than its constituents. The deuteron is lighter than a free proton plus a free neutron by its binding energy, about $2.2\ \mathrm{MeV}/c^2$.

Some older texts use a velocity-dependent "relativistic mass" $\gamma m$. This is the energy divided by $c^2$ and carries no new information. The invariant mass $m$ is the quantity used in modern physics.

Mass in quantum theory. In the Schrödinger equation the mass enters through the kinetic term $-\frac{\hbar^2}{2m}\nabla^2$. In relativistic field theory a free field of mass $m$ obeys the Klein-Gordon equation

$$\left(\frac{1}{c^2}\partial_t^2 - \nabla^2 + \frac{m^2c^2}{\hbar^2}\right)\phi = 0,$$

whose plane-wave solutions satisfy exactly the relation $E^2 = m^2c^4 + \vec p^{\,2}c^2$ with $E = \hbar\omega$ and $\vec p = \hbar\vec k$. In an interacting theory the physical mass of a particle is the position of the pole of its propagator, $p^2 = m^2c^2$, which in general differs from the mass parameter in the Lagrangian.

See: The Concept of Mass by Okun

Abstract

• For a nice description see Mass in Quantum Yang-Mills Theory by L. D. Faddeev

In classical physics, mass is a conserved quantity, as has been experimentally demonstrated by Lavoisier (around 1790). In SR mass conservation is no longer valid—as has been shown in the 1930s by more accurate experimental techniques—and is superseded by energy-momentum conservation, as has been most vividly demonstrated in Alamogordo in 1945.Gauge Theory of Gravity and Spacetime by Friedrich W. Hehl

Why is it interesting?

A lower limit to the amount of energy that a particle can have is a property of a particle with a mass.An object may have any amount of kinetic energy, the amount getting smaller as the body slows toward rest. At this point the particle has its mini-mum energy; the amount of which, E, would then correspond to a mass m given by Einstein’s famous equation $E=mc^2$. For a massless body, the minimum energy in principle can be zero. From "The Infinity Puzzle", by Frank Close

FAQ

Path to this page

1. Calculus
2. Linear Algebra
3. Newton's Second Law
4. Vector Calculus
5. Newtonian Mechanics