Relativistic Mass

The term mass in special relativity has two meanings: invariant mass (also called rest mass) is an invariant quantity that is the same for all observers in all reference frames; Whereas relativistic mass depends on the velocity of the observer. According to the concept of mass-energy equivalence , invariant mass is equal to rest energy , while relativistic mass is equal to relativistic energy (also called total energy).

The term “relativistic mass” is not used in particle and nuclear physics and is often avoided by authors on special relativity, in reference to the relativistic energy of a body. [1] In contrast, “invariant mass” is generally preferred over rest energy. The measurable inertia and warp of spacetime by a body in a given frame of reference are determined by its relative mass, not just by its invariant mass. For example, photons have zero rest mass, but contribute to the inertia (and weight in the gravitational field) of any system.

rest mass

The term mass in special relativity usually refers to the object’s rest mass, which is the Newtonian mass measured by an observer moving with the object. Another name for irreversible mass is the rest of the mass of single particles. The more general invariant mass (calculated with a more complex formula) loosely corresponds to the “rest mass” of the “system”. Thus, invariant mass is a natural unit of mass used for systems that have a center of their momentum frame (COM frame)., such as when weighing any closed system (e.g. a bottle of hot gas), which requires the measurement of momentum to be taken at the center of the frame where the system has no net momentum Is. In such circumstances the invariant mass is equal to the relativistic mass (discussed below), which is the total energy of the system divided by c 2 ( speed of light squared ).

However, the concept of immutable mass does not require bound systems of particles. As such, it can also be applied to systems of unbound particles in high-speed relativistic motion. Because of this, it is often employed in particle physics for systems that contain widely separated high-energy particles. If such systems are derived from a single particle, then computing the invariant mass of such systems, which is a never-changing quantity, would give the remaining mass of the original particle (since it is conserved over time).

In calculations it is often convenient that the invariant mass of the system is the total energy ( c divided by 2 ) of the system 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 amount in all inertial frames, this is often a quantity calculated from the total energy in the COM frame, then used to calculate the system energy and momentum in other frames. where the momentum is not zero, and the total energy of the system will essentially be a different quantity than in the COM frame. As with energy and momentum, the irreversible mass of a system cannot be destroyed or changed, and is thus conserved as long as the system is closed to all effects. (technical term is isolated systemWhich means that an ideal boundary is drawn around the system , and no mass/energy is allowed across it.)

relative mass

Relative mass is the total amount of energy in a body or system divided by 2 . Thus, the mass in the formula

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

is relative mass. For a particle of finite rest mass m moving with a velocity relative to the observer, one finds v

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

at the center of the motion frame , and the relativistic mass is equal to the rest mass. In other frames, relativistic mass (a body or system of bodies) includes contributions from the body’s “net” kinetic energy (the kinetic energy of the body ‘s center of mass ), and the faster the body moves. Thus, unlike invariant mass, relativistic mass depends on the observer ‘s frame of reference . However, for a given single frame of reference and isolated systems, relativistic mass is also a conserved quantity. Relativistic mass is also the proportionality factor between velocity and momentum, v=0

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

Newton’s second law remains valid as

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

When a body emits light of frequency and wavelength as a photon of energy , the mass of the body decreases , [2] which some [3] [4] interpret as the relative mass of the emitted photon because It also completes . Although some authors present relativistic mass as a fundamental concept of the theory, it has been argued that this is incorrect because the fundamentals of the theory are related to space-time. There is disagreement over whether the concept is pedagogically useful.

\not\lambda {\displaystyle E = h \ nu = hc / \lambda}{\displaystyle E/c^{2}=h/\lambda c} {\displaystyle p=m_{\text{rel}}c=h/\lambda }

It simply and quantitatively explains why a body subjected to a constant acceleration cannot reach the speed of light, and why the mass of a system emitting a photon is reduced. [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 has no exact relation to the concept in relativity. [9] Relativistic mass is not referenced in atomic and particle physics, [1] and a 2005 survey of introductory textbooks showed that only 5 out of 24 texts used the concept, [10]Although it is still prevalent in being popular.

If a stationary box contains many particles that weigh more than it does in its rest frame, the particles are moving faster. Any energy in the box (including the kinetic energy of the particles) is added 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), the question remains whether the kinetic energy of the overall motion should be included in the mass of the system. The kinetic energy of the system as invariant mass is calculated as a whole, (calculating a velocity of the box, which is to say the velocity of the center of mass of the box), while the relativistic mass is calculated including the invariant mass is calculated plus the kinetic energy of the system which is calculated from the velocity of the center of mass.

relativistic vs rest mass

Relativistic mass and rest mass are both traditional concepts in physics, but relativistic mass corresponds to total energy. 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 means, the measurement is at the center of its momentum frame ). For example, if a cyclotronIf an electron is moving in circles with a relativistic velocity, the mass of the cyclotron + electron system increases by the relative mass of the electron, not the rest mass of the electron. But this 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 simply the lack of total momentum in the system (system momenta sum zero) that allows to “weight” the kinetic energy of the electron. If the electron is stoppedand is weighed, or thereafter sent to the scale somehow, it will not be moving with respect to the scale, and again the relativistic and rest mass will be the same (and smaller) for a single electron. In general, relativistic and rest-mass are the same only in systems in which there is no net motion and the system’s center of mass is at rest; Otherwise they may differ.

The invariant mass is proportional to the value of the total energy in a reference frame, the frame where the whole object is at rest (as defined below with reference to the center of mass). This is why the irreversible mass is the same as the rest mass for single particles. However, invariant mass also represents the mass measured when the center of mass is at rest for systems of many particles. This particular frame where this occurs is also called the center of the momentum frame , and is defined as the inertial frame in which the object ‘s center of massoccurs at rest (another way of stating this is that this is the frame in which the momentum is when the parts of the system add up to zero). For compound objects (composed of many smaller objects, some of which may be in motion) and sets of unbound objects (some of which may also be in motion), only the center of mass of the system needed to be at rest, the object For the 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 change that would bring the particle to rest. The total energy of such particles gets smaller and smaller in frames that move faster and faster in the same direction. As such, they have no rest mass, because they can never be measured in the frame where they are at rest. This property of having no rest mass makes these particles “massless”. However, massless particles also have a relative mass, which varies with their observed energy in different frames of reference.

Immutable mass

The ratio of the invariant mass to char velocity (a four-dimensional generalization of classical motion) is char velocities:


And so is the ratio of char-acceleration and char-force when rest mass is constant. The four dimensional form of Newton’s second law is:

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

relativistic energy-motion equation

The relative expressions for E and p obey the relative energy-motion relationship:

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

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

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

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

And so


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

The momentum p of an object at rest is zero, so

E_0 = mc^2 \,\!

[True only for particles or systems with momentum = 0]

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

When the object is in motion, the total energy is given by

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

To find the form of momentum and energy as a function of velocity, it may be noted that the char-velocity, which is , is the only char-vector associated with the motion of the particle, so that if there is a conserved char-momentum , It must be proportional to this vector. This allows the ratio of energy to momentum to be expressed as

{\displaystyle \left(c,{\vec {v}}\right)}{\displaystyle \left(E,{\vec {p}}c\right)}
{\displaystyle pc=E{\frac {v}{c}}},

As a result of which the relation between E and v becomes :

{\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}}}}}}


{\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 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 relativistic equations are simple and the quantities energy, momentum, and mass have the same natural dimension: 

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

The equation is often written as such because the differential energy is the relative length of the momentum four-vector, a length that is associated with the rest mass or invariant mass in the system. Where m > 0 and p = 0 , this equation again expresses the mass-energy equivalence E = m .

{\displaystyle E^{2}-p^{2}}

mass of composite systems

The rest mass of a combined system is not the sum of the rest mass 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 the sum of the energies of its components together. The total motion of a system, a vector quantity, can also be calculated by adding together the momentum of all its components. Given the total energy E and the length (magnitude) p of the total momentum vector , the invariant mass is given by:

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

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

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

where, again, the particle momentum is first expressed as vectors, and then the square of their resultant total magnitude (the Euclidean norm) is used. This results in a scalar number, which is subtracted from the scalar value of the square of the total energy.

{\vec {p}}

For such a system, at the particular center of the momentum frame where the momentum is zero, again the system mass (called the invariant mass) corresponds to the total system energy or, in units where c = 1 . For a system this inertial mass remains the same amount in any inertial frame, although the total energy and total momentum of the system are functions of the particular inertial frame that is chosen, and will vary between inertial frames such that Keeping the invariant mass the same for all observers. Invariant mass thus acts in the same capacity for systems of particles as “rest mass” does for single particles.

Note that the invariant mass of an isolated system (that is, closed to both mass and energy) is also independent of the 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 irreversible mass is widely used in particle physics, because the irreversible mass of a particle’s decay products is equal to its rest mass. It is used to measure the mass of particles such as the Z boson or top quark.

Invariance vs Conservation of Mass in Special Relativity

The total energy is an additively conserved quantity (for single observers) in systems and in reactions between particles, but the rest mass (in the sense of the particle being the sum of the rest mass) cannot be conserved through the phenomenon in which the particles The rest of the mass is converted into other forms of energy, such as kinetic energy. Finding the sum of individual particle rest masses would require several observers, one for each particle at rest inertial frame, and these observers disregard the individual particle kinetic energy. Conservation laws require an observer and an inertial frame.

In general, for isolated systems and single observers, relativistic mass is conserved (each observer sees it as 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 with time).

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

In contrast, the rest mass and the invariant mass of the system and particles are both conserved and irreversible . For example: a system in a closed container of gas (also closed to energy) has “rest mass”, in the sense that it can be weighed on a rest scale, even though it has 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) when it is measured at the center of the motion frame. As is the case with single particles, the calculated “rest mass” of such a container of gas does not change when it is in motion, although its “relative mass” does change.

The container can also be subjected to a force that gives it an overall velocity, or (equally) it can be viewed from an inertial frame in which it has an overall velocity (that is, technically, a frame which has its center of mass) is velocity). In this case, its total relative mass and energy increase. However, in such a situation, although the total relative energy of the container and the total momentum increase, the increase in these energies and momentum decreases in the invariant mass definition, so that the invariant mass of the moving container will be calculated as the same value as it was. was measured at rest, on the scale.

closed (meaning completely isolated) systems

All conservation laws in special relativity (for energy, mass, and momentum) require isolated systems, meaning that systems that are completely isolated do not allow any mass-energy in or out of time. . If a system is isolated, both the total energy and the total momentum in the system are conserved over time for any observer in any one inertial frame, although different observers in different inertial frames have their Absolute prices will vary. The invariant mass of the system is also conserved, but does not change with different observers. This is a familiar situation even with single particles: all observers calculate the same particle rest mass (a special case of invariant mass) ,It doesn’t matter how they move (which inertial frame they choose), but different observers see different total energies and speeds of the same particle.

Conservation of invariant mass also requires enclosing the system so that no heat and radiation (and thus invariant mass) can escape. As in the example above, a physically enclosed or bound system does not need to be completely isolated from external forces for its mass to remain constant, because for bound systems it is only the inertial frame of the system or observer. work to change. Although such actions can change the total energy or momentum of the bound system, these two changes cancel out, so that there is no change in the system’s irreversible mass. This is the same consequence as for single particles: their calculated rest mass remains constant, no matter how fast they move, or how fast an observer observes them moving.

On the other hand, for systems that are unbound, the “closure” of the system can be enforced by an ideal surface, since no mass-energy can be allowed into or out of the test-volume over time, If the conservation of the system is to hold invariant mass during that time. If a force is allowed to act (do work) on only one part of such an unbound system, it is equivalent to allowing energy to move in or out of the system, and to mass-energy (total separation). “Closed” status. is violated. In this case, the conservation of irreversible mass of the system also no longer holds. According to E = mc 2 there is such a loss of rest mass in the system when energy is removed, where E energy is removed, and MChanges in rest mass refer to changes of mass associated with the movement of energy, not “conversions” from mass to energy.

System invariant mass versus individual rest mass of parts of the system

Again, in special relativity, the rest mass of a system does not need to be equal to the sum of the rest mass of the parts (a condition that would be analogous to gross-mass-conservation in chemistry). For example, a massive particle may decay into photons that have no mass individually, but which (as a system) preserve the invariant mass of the particle that generated them. Furthermore a box of moving non-interacting particles (e.g., photons, or an ideal gas) will have a larger invariant mass than the sum of the rest masses of the particles that make up it. This is because the total energy of all particles and fields in a system must be summed up, and this quantity, as seen at the center of the motion frame, and c2 splits, the system has an invariant mass.

In special relativity, mass is not “converted” into energy, because all forms of energy still retain their respective mass. Neither energy nor immutable mass can be destroyed in special relativity, and each is conserved in closed systems with varying amounts of time. Thus, the invariant mass of a system can change only because the invariant mass is allowed to exit, perhaps in the form of light or heat. Thus, when reactions (whether chemical or nuclear) release energy in the form of heat and light, if heat and light are not allowed to escape(the system is closed and isolated), then the energy will continue to contribute to the rest mass of the system, and the system mass will not change. Mass will only be destroyed when energy is released to the environment; This is because the associated mass has been allowed out of the system, where it contributes to the mass of the surroundings.

History of Relativistic Mass Concept

transverse and longitudinal mass

Concepts similar to what is today called “relativistic mass” had already been developed before the advent of special relativity. For example, it was recognized in 1881 by JJ Thomson that a charged body is harder to set in motion than an uncharged body, which was described in more detail by Oliver Heaviside (1889) and George Frederick Charles Searle (1897). was worked from. So electrostatic energy behaves as some kind of electromagnetic mass , which can increase the normal mechanical mass of bodies.

{\displaystyle m_{\text{em}}=(4/3)E_{\text{em}}/c^{2}}

Then, it was pointed out by Thomson and Searle that this electromagnetic mass also increases with velocity. This was further elaborated by Hendrik Lorentz (1899, 1904) in the framework of the Lorentz ether theory. He defined mass as the ratio of force to acceleration, not velocity to velocity, so he needed to differentiate between mass parallel to the direction of motion and mass perpendicular to the direction of motion (where Lorentz factor, v is the relative velocity between the ether and the object, and c is the speed of light). Only when the force is perpendicular to velocity, does Lorentz’s mass equal what is now called “relative mass”. Max Abraham (1902) said) longitudinal mass and transverse mass

{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}

(Although Abraham used more complex expressions than Lorentz’s relativistic expressions). So, according to Lorentz’s principle no body can reach the speed of light because at this velocity the mass becomes infinitely large.

Albert Einstein also initially used the concepts of longitudinal and transverse mass in his 1905 paper Electrodynamics (equivalent to Lorentz, but with a different one by an unfortunate force definition, which was later corrected), and in 1906 In another paper. [19] [20] However, he later abandoned the velocity-dependent mass concepts (see citation at the end of the next section).

{\displaystyle m_{\text{T}}}

The exact relativistic expression (which is equivalent to Lorentz) relates the force and acceleration for a particle with non-zero rest mass moving in the direction X with velocity V and the associated Lorentz factor .

{\displaystyle {\begin{aligned}f_{\text{x}}&=m\gamma ^{3}a_{\text{x}}&=m_{\text{L}}a_{\text{x}},\\f_{\text{y}}&=m\gamma a_{\text{y}}&=m_{\text{T}}a_{\text{y}},\\f_{\text{z}}&=m\gamma a_{\text{z}}&=m_{\text{T}}a_{\text{z}}.\end{aligned}}}

relative mass

In special relativity, an object that has a non-zero rest mass cannot travel at the speed of light. As an object approaches the speed of light, the object’s energy and speed increase without limit.

In the first years after 1905, after Lorentz and Einstein, the terms longitudinal and transverse mass were still in use. However, those expressions were replaced by the concept of relativistic mass , an expression first introduced in 1909 by Gilbert N. Lewis and Richard C. was defined by Tolman. [21] He defined the total energy and mass of a body as

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

and of a body at rest

m_0 = \frac{E_0}{c^2}\!,

with the ratio

{\displaystyle {\frac {m_{\text{rel}}}{m_{0}}}=\gamma \!}.

Tolman elaborated this concept further in 1912, saying: “The expression 0 (1 – 2 / 2 ) -1/2 is best suited for the mass of a moving body.” [22] [23] [24]

In 1934, Tolman argued that the relativistic mass formula holds for all particles, including those moving at the speed of light, while the formula only holds for a slower-than-light particle (a particle with a non-zero rest mass). applies to. Tolman remarked on this relation that “in addition, we have, of course, experimental verification of the expression in the case of moving electrons … so we have little to take the expression for the mass of a moving particle to be true in general.” Won’t hesitate.”

{\displaystyle m_{\text{rel}}=E/c^{2}\!}{\displaystyle m_{\text{rel}}=\gamma m_{0}\!}

When the relative velocity is zero, just equals 1, and the relativistic mass is reduced to the rest mass as can be seen in the next two equations below. As the velocity increases towards the speed of light c , the denominator on the right approaches zero, and as a result approaches infinity. While Newton’s second law remains valid as

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

The derived form is not valid because in is generally not constant [26] (see the section above on transverse and longitudinal masses).

{\displaystyle \mathbf {f} =m_{\text{rel}}\mathbf {a} }{\displaystyle m_{\text{rel}}}{\displaystyle {d(m_{\text{rel}}\mathbf {v} )}\!}

Even though Einstein initially used “longitudinal” and “transverse” mass in two papers (see the previous section), in his first paper (1905) he treated m as what would now be called rest mass . [2] Einstein never worked out an equation for “relativistic mass”, and in later years he expressed his dislike for the idea: 

E = mc^2

It is not good to introduce the concept of mass of a moving body for which no clear definition can be given. It is better not to introduce any other mass concept than ‘rest mass’ m . It is better to refer to expressions for the momentum and energy of a body in motion than to introduce M.

M = m/\sqrt{1 - v^2/c^2}

-  Albert Einstein in a letter to Lincoln Barnett, 19 June 1948 (quote from LB Okun (1989), p 42  )

Popular Science and Textbooks

The concept of relativistic mass is widely used in popular science writings and in high school and graduate textbooks. Writers such as Okun and Abby Arons have made archaic and confusing arguments against it, not in line with modern relativistic theory. [5] [28] Aarons wrote: [28]

For many years it was traditional to enter the discussion of dynamics through the derivation of relativistic mass, which is the mass–velocity relationship, and it is probably still the dominant mode in textbooks. Recently, however, it has been increasingly recognized that relativistic mass is a troublesome and questionable concept. [See, for example, Okun (1989). [5] ]… The sonic and rigorous approach to relativistic dynamics is through the direct development of an expression for momentum that ensures the conservation of momentum in all frames:

p = {m_0 v \over {\sqrt{1 - \frac{v^2}{c^2}}}} \!

rather than through relativistic mass.

C. Elder takes a similarly rejected stance on mass in relativity. Writing on the said subject, he states that “its introduction into the theory of special relativity was much in the way of a historical accident”, noting the widespread knowledge of E = mc2 and the public’s interpretation of the equation has largely informed on. How is it taught in higher education? [29] Instead he believes that the distinction between rest and relativistic mass should be explicitly taught, so that students know why mass should be considered invariant “in most discussions of inertia”.

Many contemporary authors such as Taylor and Wheeler avoid using the concept of relativistic mass altogether:

The concept of “relativistic mass” is subject to misunderstanding. That’s why we don’t use it. First, the name applies the mass – related to the magnitude of the 4-vector – to a very different concept, the time component of the 4-vector. Second, it appears to be associated with some change in the internal structure of the object with velocity or momentum as an increase in an object’s energy. In fact, the increase of energy with velocity is not in the object itself but in the geometric properties of spacetime. [12]

Whereas spacetime has the infinite geometry of Minkowski space, velocity-space is bounded by c and has the geometry of hyperbolic geometry where relativistic mass plays a role similar to Newtonian mass in the barycentric coordinates of Euclidean geometry. [30] The connection of velocity to hyperbolic geometry enables 3-velocity-dependent relativistic mass to be related to the 4-velocity Minkowski formalism.