Mass

# Mass

## Definitions

The mass of a physical system is its intrinsic energy.

It is a somewhat arbitrary choice, made necessary by the theory of relativity, to decide whether the old term ‘mass’ should be used for intrinsic energy or total energy; one may clarify that intrinsic energy is rest mass while total energy is relativistic mass.

Traditionally, mass and energy are measured in different units related by the speed of light: $E = m c^2$. Thus one may call a system's mass its rest energy when it is measured in units of energy; conversely, one may call a systems's total energy its relativistic mass when it is measured in units of mass.

The kinetic energy of a system is the difference between its total energy and its mass (or rest energy). Thus the kinetic energy is the energy due to the motion of the system (or better, due to the relative motion of the system and the observer).

When a system is thought of as the combination of several subsystems, then its mass also may be broken into several pieces:

• for each subsystem, its mass,
• for each subsystem, its kinetic energy due to relative motion of the subsystem,
• for each pair of subsystems, the potential energy of their interaction.

The latter entry may be positive (in which case the system is liable to separate into its component parts, at least at sufficiently high temperature, giving off energy) or negative (in which case the system is bound and can be expected to remain together until additional energy is added).

## Formulas

Mass $m$ may be given in terms of energy $E$ and linear momentum $\mathbf{p}$ as

$m = \sqrt {E^2 - {|\mathbf{p}|}^2} .$

As $\mathbf{p} = E \mathbf{v}$ for $\mathbf{v}$ the linear velocity, we also have

$m = E \sqrt {1 - {|\mathbf{v}|}^2} .$

(For nonrelativistic units, change $E$ to $E/c^2$, $\mathbf{p}$ to $\mathbf{p}/c$, and $\mathbf{v}$ to $\mathbf{v}/c$.)

For a particle travelling at the speed of light, we therefore have $m = 0$. When $\mathbf{v} = 0$, we have $m = E$. When $0 \lt {|\mathbf{v}|} \ll c$, we still have $m \approx E$. However, in nonrelativistic physics, it is much more useful to use kinetic energy $K = E - m$ instead of the total energy $E$; then we have

$m = K \frac { 1 - {|\mathbf{v}|}^2 + \sqrt {1 - {|\mathbf{v}|^2}} } { {|\mathbf{v}|}^2 }$

exactly and

$m \approx 2 \frac K { {|\mathbf{v}|}^2 }$

when $0 \lt {|\mathbf{v}|} \ll c$. Notice that this last expression makes sense already in nonrelativistic units.

For a system made up of subsystems (indexed by $i,j,\ldots$), we have

$m = \sum_i m_i + \sum_i \tilde{K}_i + \sum_{i,j} U_{i,j} ,$

where $m_i$ is the mass of subsystem $i$, $\tilde{K}_i$ is the kinetic energy of subsystem $i$ relative to the whole system, and $U_{i,j}$ is the potential energy of the forces between system $i$ and system $j$. The sum

$- \sum_{i,j} U_{i,j} = \sum_i m_i + \sum_i \tilde{K}_i - m$

is the binding energy? of the system; when it is positive, it gives the minimum energy that must be added to the system to break it into components that do not interact. In many cases, the relative kinetic energies $\tilde{K}_i$ are negligible, so we can calculate binding energy by subtracting masses as

$- \sum_{i,j} U_{i,j} \approx \sum_i m_i - m .$

In nonrelativistic physics, where kinetic energy is always negligible, the $U_{i,j}$ are also negligible compared to the masses, so it is not possible at all to calculate the sign of the binding energy using masses. Instead, we have

$m \approx \sum_i m_i ,$

the statement of nonrelativistic conservation of mass.

Many of the formulas above rely on ${|\mathbf{v}|} \leq c$, or equivalently ${|\mathbf{p}|} \leq E$. For tachyons, where this is violated, the mass becomes an imaginary number. For this reason, it may make more sense to work with $m^2$ (which is always real) than with $m$ itself. We also implicitly assume that $E \gt 0$; for exotic particle?s in which $E \lt 0$, it is convenient also to take $m \lt 0$. (In this case, of course, it is not sufficient to work only with $m^2$.)

fundamental scales (fundamental physical units)

• speed of light$\,$ $c$

• Planck's constant$\,$ $\hbar$

• gravitational constant$\,$ $G_N = \kappa^2/8\pi$

• Planck scale

• Planck length$\,$ $\ell_p = \sqrt{ \hbar G / c^3 }$

• Planck mass$\,$ $m_p = \sqrt{\hbar c / G}$

• depending on a given mass $m$

• Compton wavelength$\,$ $\lambda_m = \hbar / m c$

• Schwarzschild radius$\,$ $2 m G / c^2$

• depending also on a given charge $e$

• Schwinger limit$\,$ $E_{crit} = m^2 c^3 / e \hbar$
• GUT scale

• string scale

• string tension$\,$ $T = 1/(2\pi \alpha^\prime)$

• string length scale$\,$ $\ell_s = \sqrt{\alpha'}$

• string coupling constant$\,$ $g_s = e^\lambda$