Mass in special relativity

The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The invariant mass is another name for the rest mass of single particles. The more general invariant mass (calculated with a more complicated formula) loosely corresponds to the "rest mass" of a "system". Thus, invariant mass is a natural unit of mass used for systems which are being viewed from their center of momentum frame (COM frame), as when any closed system (for example a bottle of hot gas) is weighed, which requires that the measurement be taken in the center of momentum frame where the system has no net momentum. Under such circumstances the invariant mass is equal to the relativistic mass (discussed below), which is the total energy of the system divided by c² (the speed of light squared).

The concept of invariant mass does not require bound systems of particles, however. As such, it may also be applied to systems of unbound particles in high-speed relative motion. Because of this, it is often employed in particle physics for systems which consist of widely separated high-energy particles. If such systems were derived from a single particle, then the calculation of the invariant mass of such systems, which is a never-changing quantity, will provide the rest mass of the parent particle (because it is conserved over time).

It is often convenient in calculation that the invariant mass of a system is the total energy of the system (divided by c²) in the COM frame (where, by definition, the momentum of the system is zero). However, since the invariant mass of any system is also the same quantity in all inertial frames, it is a quantity often calculated from the total energy in the COM frame, then used to calculate system energies and momenta in other frames where the momenta are not zero, and the system total energy will necessarily be a different quantity than in the COM frame. As with energy and momentum, the invariant mass of a system cannot be destroyed or changed, and it is thus conserved, so long as the system is closed to all influences. (The technical term is isolated system meaning that an idealized boundary is drawn around the system, and no mass/energy is allowed across it.)

The relativistic mass is the sum total quantity of energy in a body or system (divided by c²). Thus, the mass in the formula

${\displaystyle E=m_{\text{rel}}c^{2}\,}$

is the relativistic mass. For a particle of finite rest mass m moving at a speed ${\displaystyle v}$ relative to the observer, one finds

${\displaystyle m_{\text{rel}}={m \over {\sqrt {1-\displaystyle {v^{2} \over c^{2}}}}}}$ (see below).

In the center of momentum frame, ${\displaystyle v=0}$ and the relativistic mass equals the rest mass. In other frames, the relativistic mass (of a body or system of bodies) includes a contribution from the "net" kinetic energy of the body (the kinetic energy of the center of mass of the body), and is larger the faster the body moves. Thus, unlike the invariant mass, the relativistic mass depends on the observer's frame of reference. However, for given single frames of reference and for isolated systems, the relativistic mass is also a conserved quantity. The relativistic mass is also the proportionality factor between velocity and momentum,

${\displaystyle \mathbf {p} =m_{\text{rel}}\mathbf {v} }$.

Newton's second law remains valid in the form

${\displaystyle \mathbf {f} ={\frac {d(m_{\text{rel}}\mathbf {v} )}{dt}}.}$

When a body emits light of frequency ${\displaystyle \nu }$ and wavelength ${\displaystyle \lambda }$ as a photon of energy ${\displaystyle E=h\nu =hc/\lambda }$, the mass of the body decreases by ${\displaystyle E/c^{2}=h/\lambda c}$,[2] which some[3][4] interpret as the relativistic mass of the emitted photon since it also fulfills ${\displaystyle p=m_{\text{rel}}c=h/\lambda }$. Although some authors present relativistic mass as a fundamental concept of the theory, it has been argued that this is wrong as the fundamentals of the theory relate to space–time. There is disagreement over whether the concept is pedagogically useful.[5][3][6] It explains simply and quantitatively why a body subject to a constant acceleration cannot reach the speed of light, and why the mass of a system emitting a photon decreases.[3] In relativistic quantum chemistry, relativistic mass is used to explain electron orbital contraction in heavy elements.[7][8] The notion of mass as a property of an object from Newtonian mechanics does not bear a precise relationship to the concept in relativity.[9] Relativistic mass is not referenced in nuclear and particle physics,[1] and a survey of introductory textbooks in 2005 showed that only 5 of 24 texts used the concept,[10] although it is still prevalent in popularizations.

If a stationary box contains many particles, it weighs more in its rest frame, the faster the particles are moving. Any energy in the box (including the kinetic energy of the particles) adds to the mass, so that the relative motion of the particles contributes to the mass of the box. But if the box itself is moving (its center of mass is moving), there remains the question of whether the kinetic energy of the overall motion should be included in the mass of the system. The invariant mass is calculated excluding the kinetic energy of the system as a whole (calculated using the single velocity of the box, which is to say the velocity of the box's center of mass), while the relativistic mass is calculated including invariant mass plus the kinetic energy of the system which is calculated from the velocity of the center of mass.

Relativistic mass and rest mass are both traditional concepts in physics, but the relativistic mass corresponds to the total energy. The relativistic mass is the mass of the system as it would be measured on a scale, but in some cases (such as the box above) this fact remains true only because the system on average must be at rest to be weighed (it must have zero net momentum, which is to say, the measurement is in its center of momentum frame). For example, if an electron in a cyclotron is moving in circles with a relativistic velocity, the mass of the cyclotron+electron system is increased by the relativistic mass of the electron, not by the electron's rest mass. But the same is also true of any closed system, such as an electron-and-box, if the electron bounces at high speed inside the box. It is only the lack of total momentum in the system (the system momenta sum to zero) which allows the kinetic energy of the electron to be "weighed". If the electron is stopped and weighed, or the scale were somehow sent after it, it would not be moving with respect to the scale, and again the relativistic and rest masses would be the same for the single electron (and would be smaller). In general, relativistic and rest masses are equal only in systems which have no net momentum and the system center of mass is at rest; otherwise they may be different.

The invariant mass is proportional to the value of the total energy in one reference frame, the frame where the object as a whole is at rest (as defined below in terms of center of mass). This is why the invariant mass is the same as the rest mass for single particles. However, the invariant mass also represents the measured mass when the center of mass is at rest for systems of many particles. This special frame where this occurs is also called the center of momentum frame, and is defined as the inertial frame in which the center of mass of the object is at rest (another way of stating this is that it is the frame in which the momenta of the system's parts add to zero). For compound objects (made of many smaller objects, some of which may be moving) and sets of unbound objects (some of which may also be moving), only the center of mass of the system is required to be at rest, for the object's relativistic mass to be equal to its rest mass.

A so-called massless particle (such as a photon, or a theoretical graviton) moves at the speed of light in every frame of reference. In this case there is no transformation that will bring the particle to rest. The total energy of such particles becomes smaller and smaller in frames which move faster and faster in the same direction. As such, they have no rest mass, because they can never be measured in a frame where they are at rest. This property of having no rest mass is what causes these particles to be termed "massless". However, even massless particles have a relativistic mass, which varies with their observed energy in various frames of reference.

The invariant mass is the ratio of four-momentum (the four-dimensional generalization of classical momentum) to four-velocity:[11]

${\displaystyle p^{\mu }=mv^{\mu }\,}$

and is also the ratio of four-acceleration to four-force when the rest mass is constant. The four-dimensional form of Newton's second law is:

${\displaystyle F^{\mu }=mA^{\mu }.}$

Dependency between the rest mass and E, given in 4-momentum (p₀, p₁) coordinates, where p₀c = E

The relativistic expressions for E and p obey the relativistic energy–momentum relation:[12]

${\displaystyle E^{2}-(pc)^{2}=\left(mc^{2}\right)^{2}}$

where the m is the rest mass, or the invariant mass for systems, and E is the total energy.

The equation is also valid for photons, which have m = 0:

${\displaystyle E^{2}-(pc)^{2}=0}$

and therefore

${\displaystyle E=pc}$

A photon's momentum is a function of its energy, but it is not proportional to the velocity, which is always c.

For an object at rest, the momentum p is zero, therefore

${\displaystyle E_{0}=mc^{2}\,\!}$ [true only for particles or systems with momentum = 0]

The rest mass is only proportional to the total energy in the rest frame of the object.

When the object is moving, the total energy is given by

${\displaystyle E={\sqrt {\left(mc^{2}\right)^{2}+(pc)^{2}}}}$

To find the form of the momentum and energy as a function of velocity, it can be noted that the four-velocity, which is proportional to ${\displaystyle \left(c,{\vec {v}}\right)}$, is the only four-vector associated with the particle's motion, so that if there is a conserved four-momentum ${\displaystyle \left(E,{\vec {p}}c\right)}$, it must be proportional to this vector. This allows expressing the ratio of energy to momentum as

${\displaystyle pc=E{\frac {v}{c}}}$,

resulting in a relation between E and v:

${\displaystyle E^{2}=\left(mc^{2}\right)^{2}+E^{2}{\frac {v^{2}}{c^{2}}},}$

This results in

${\displaystyle E={mc^{2} \over {\sqrt {1-\displaystyle {v^{2} \over c^{2}}}}}}$

and

${\displaystyle p={mv \over {\sqrt {1-\displaystyle {v^{2} \over c^{2}}}}}.}$

these expressions can be written as

${\displaystyle E_{0}=mc^{2},}$

${\displaystyle E=\gamma mc^{2},}$ and

${\displaystyle p=mv\gamma .}$

where the factor ${\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.}$

When working in units where c = 1, known as the natural unit system, all the relativistic equations are simplified and the quantities energy, momentum, and mass have the same natural dimension:[13]

${\displaystyle m^{2}=E^{2}-p^{2}}$.

The equation is often written this way because the difference ${\displaystyle E^{2}-p^{2}}$ is the relativistic length of the energy momentum four-vector, a length which is associated with rest mass or invariant mass in systems. Where m > 0 and p = 0, this equation again expresses the mass–energy equivalence E = m.

The rest mass of a composite system is not the sum of the rest masses of the parts, unless all the parts are at rest. The total mass of a composite system includes the kinetic energy and field energy in the system.

The total energy E of a composite system can be determined by adding together the sum of the energies of its components. The total momentum ${\displaystyle {\vec {p}}}$ of the system, a vector quantity, can also be computed by adding together the momenta of all its components. Given the total energy E and the length (magnitude) p of the total momentum vector ${\displaystyle {\vec {p}}}$, the invariant mass is given by:

${\displaystyle m={\frac {\sqrt {E^{2}-(pc)^{2}}}{c^{2}}}}$

In the system of natural units where c = 1, for systems of particles (whether bound or unbound) the total system invariant mass is given equivalently by the following:

${\displaystyle m^{2}=\left(\sum E\right)^{2}-\left\|\sum {\vec {p}}\ \right\|^{2}}$

Where, again, the particle momenta ${\displaystyle {\vec {p}}}$ are first summed as vectors, and then the square of their resulting total magnitude (Euclidean norm) is used. This results in a scalar number, which is subtracted from the scalar value of the square of the total energy.

For such a system, in the special center of momentum frame where momenta sum to zero, again the system mass (called the invariant mass) corresponds to the total system energy or, in units where c = 1, is identical to it. This invariant mass for a system remains the same quantity in any inertial frame, although the system total energy and total momentum are functions of the particular inertial frame which is chosen, and will vary in such a way between inertial frames as to keep the invariant mass the same for all observers. Invariant mass thus functions for systems of particles in the same capacity as "rest mass" does for single particles.

Note that the invariant mass of an isolated system (i.e., one closed to both mass and energy) is also independent of observer or inertial frame, and is a constant, conserved quantity for isolated systems and single observers, even during chemical and nuclear reactions. The concept of invariant mass is widely used in particle physics, because the invariant mass of a particle's decay products is equal to its rest mass. This is used to make measurements of the mass of particles like the Z boson or the top quark.

Total energy is an additive conserved quantity (for single observers) in systems and in reactions between particles, but rest mass (in the sense of being a sum of particle rest masses) may not be conserved through an event in which rest masses of particles are converted to other types of energy, such as kinetic energy. Finding the sum of individual particle rest masses would require multiple observers, one for each particle rest inertial frame, and these observers ignore individual particle kinetic energy. Conservation laws require a single observer and a single inertial frame.

In general, for isolated systems and single observers, relativistic mass is conserved (each observer sees it constant over time), but is not invariant (that is, different observers see different values). Invariant mass, however, is both conserved and invariant (all single observers see the same value, which does not change over time).

The relativistic mass corresponds to the energy, so conservation of energy automatically means that relativistic mass is conserved for any given observer and inertial frame. However, this quantity, like the total energy of a particle, is not invariant. This means that, even though it is conserved for any observer during a reaction, its absolute value will change with the frame of the observer, and for different observers in different frames.

By contrast, the rest mass and invariant masses of systems and particles are both conserved and also invariant. For example: A closed container of gas (closed to energy as well) has a system "rest mass" in the sense that it can be weighed on a resting scale, even while it contains moving components. This mass is the invariant mass, which is equal to the total relativistic energy of the container (including the kinetic energy of the gas) only when it is measured in the center of momentum frame. Just as is the case for single particles, the calculated "rest mass" of such a container of gas does not change when it is in motion, although its "relativistic mass" does change.

The container may even be subjected to a force which gives it an overall velocity, or else (equivalently) it may be viewed from an inertial frame in which it has an overall velocity (that is, technically, a frame in which its center of mass has a velocity). In this case, its total relativistic mass and energy increase. However, in such a situation, although the container's total relativistic energy and total momentum increase, these energy and momentum increases subtract out in the invariant mass definition, so that the moving container's invariant mass will be calculated as the same value as if it were measured at rest, on a scale.

• Mass
• Special relativity
• Tests of relativistic energy and momentum

• Silagadze, Z. K. (2008), "Relativity without tears", Acta Physica Polonica B, 39 (4): 811–885, arXiv:0708.0929, Bibcode:2008AcPPB..39..811S
• Oas, Gary (2005), "On the Abuse and Use of Relativistic Mass", arXiv:physics/0504110
• Usenet Physics FAQ
  • "Does mass change with velocity?" by Philip Gibbs et al., 2002, retrieved August 10, 2006
  • "What is the mass of a photon?" by Matt Austern et al., 1998, retrieved June 27, 2007
• Max Jammer (1997), Concepts of Mass in Classical and Modern Physics, Courier Dover Publications, pp. 177–178, ISBN 978-0-486-29998-3
• Mass as a Variable Quantity