Mass in special relativity
Encyclopedia
Mass
in special relativity
incorporates the general understandings from the concept of mass-energy equivalence
. Added to this concept is an additional complication resulting from the fact that "mass" is defined in two different ways in special relativity: one way defines mass ("rest mass" or "invariant mass") as an invariant quantity which is the same for all observers in all reference frames; in the other definition, the measure of mass ("relativistic mass") is dependent on the velocity of the observer.
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 c (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 c
2) 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. (In this case, "closure" implies that an idealized boundary is drawn around the system, and no mass/energy is allowed across it).
The term relativistic mass is also sometimes used. This is the sum total quantity of energy in a body or system (divided by c2). As seen from the center of momentum frame, the relativistic mass is also the invariant mass, as discussed above (just as the relativistic energy of a single particle is the same as its rest energy, when seen from its rest frame). For 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 closed systems, the relativistic mass is also a conserved quantity.
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. The notion of mass as a property of an object from Newtonian mechanics does not bear a precise relationship to the concept in relativity.
For a discussion of mass in general relativity
, see mass in general relativity
. For a general discussion including mass in Newtonian mechanics
, see the article on 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 weight 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."
in 1881 that a charged body is harder to set in motion than an uncharged body, which was worked out in more detail by Oliver Heaviside
(1889) and George Frederick Charles Searle
(1897). So the electrostatic energy behaves as having some sort of electromagnetic mass
, which can increase the normal mechanical mass of the bodies.
Now, 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 Lorentz ether theory
. He defined mass as the ratio of force to acceleration, not as the ratio of momentum to velocity, so he needed to distinguish between the mass
parallel to the direction of motion and the mass 
perpendicular to the direction of motion (where 
is the Lorentz factor
, v is the relative velocity between the aether and the object, and c is the speed of light
). Only when the force is perpendicular to the velocity, Lorentz's mass is equal to what is now called "relativistic mass". Max Abraham
(1902) called
longitudinal mass and 
transverse mass
(although Abraham used more complicated expressions than Lorentz's relativistic ones). So, according to Lorentz's theory no body can reach the speed of light because the mass becomes infinitely large at this velocity.
The precise relativistic expression (which is equivalent to Lorentz's) relating force and acceleration for a particle with non-zero rest mass
moving in the x direction with velocity v and associated Lorentz factor 
is
In the first years after 1905, following 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 which was first defined by Richard C. Tolman
in 1912, who stated: “the expression m0(1 - v2/c2)-1/2 is best suited for THE mass of a moving body.”
In 1934, Tolman also defined relativistic mass as
which holds for all particles, including those moving at the speed of light.
For a slower than light particle, a particle with a nonzero rest mass, the formula becomes
Where m0 is the rest mass.
Tolman remarked on this relation that "We have, moreover, of course the experimental verification of the expression in the case of moving electrons to which we shall call attention in §29. We shall hence have no hesitation in accepting the expression as correct in general for the mass of a moving particle."
When the relative velocity is zero,
is simply equal to 1, and the relativistic mass is reduced to the rest mass as one can see in the next two equations below. As the velocity increases toward the speed of light c, the denominator of the right side approaches zero, and consequently 
approaches infinity.
In the formula for momentum
the mass that occurs is the relativistic mass. In other words, the relativistic mass is the proportionality constant between the velocity and the momentum.
While Newton's second law remains valid in the form
the derived form
is not valid because 
in 
is generally not a constant (see the section above on transverse and longitudinal mass).
Also Einstein at first used a relativistic mass concept in the form of longitudinal and transverse mass in his 1905 electrodynamics paper (equivalent to those of Lorentz, but with a different
by an unfortunate force definition, which was later corrected), and in another paper in 1906. On the other hand, in his first paper on 
(1905) he treated m as what would now be called the rest mass.
Some claim that (in later years) he did not like the idea of "relativistic mass", by referring to this quote:
is the ratio of four-momentum to four-velocity
:
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:
Many contemporary authors such as Taylor and Wheeler avoid using the concept of relativistic mass altogether:
While space-time has the unbounded geometry of Minkowski-space, the velocity-space is bounded by c and has the geometry of hyperbolic geometry where relativistic-mass plays an analogous role to that of Newtonian-mass in the barycentric-coordinates of Euclidean geometry
. The connection of velocity to hyperbolic-geometry enables the 3-velocity-dependent relativistic-mass to be related to the 4-velocity Minkowski-formalism.
The relativistic expressions for E and p obey the relativistic energy-momentum equation:
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:
and therefore
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
[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
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
, is the only four-dimensional arrow associated with the particle's motion, so that if there is a conserved four-momentum 
, it must be proportional to this vector. This allows expressing the ratio of energy to momentum as
,
resulting in a relation between E and v:
This results in
and
these expressions can be written as
,
,
and
When working in units where c = 1, known as the natural unit system, all relativistic equations simplify. In particular, all three quantities E, p, m have the same dimension:
.
The equation is often written this way because the difference
is the relativistic length of the energy momentum four-vector, a length which is associated with rest mass or invariant mass in systems. If m > 0, then there is the rest frame, where p = 0, this equation states that E = m, revealing once more that invariant mass is the same as the energy in the rest frame.
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
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 
, the invariant mass is given by:
In a mathematical system where c = 1, for systems of particles (whether bound or unbound) the total system invariant mass is given equivalently by the following:
Where, again, the particle momenta
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 momenta 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
.
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 mass
es 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 over-all velocity, or else (equivalently) it may be viewed from an inertial frame in which it has an over-all 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 momenta 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.
Conservation of invariant mass also requires the system to be enclosed 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 these merely act to change the inertial frame of the system or the observer. Though such actions may change the total energy or momentum of the bound system, these two changes cancel, so that there is no change in the system's invariant mass. This is just the same result as with single particles: their calculated rest mass also remains constant no matter how fast they move, or how fast an observer sees them move.
On the other hand, for systems which are unbound, the "closure" of the system may be enforced by an idealized surface, inasmuch as no mass-energy can be allowed into or out of the test-volume over time, if conservation of system invariant mass
is to hold during that time. If a force is allowed to act on (do work on) only one part of such an unbound system, this is equivalent to allowing energy into or out of the system, and the condition of "closure" to mass-energy (total isolation) is violated. In this case, conservation of invariant mass of the system also will no longer hold. Such a loss of rest mass in systems when energy is removed, according to E=mc2 where E is the energy removed, and m is the change in rest mass, reflect changes of mass associated with movement of energy, not "conversion" of mass to energy.
, and divided by c2, is the system's invariant mass.
In special relativity, mass is not "converted" to energy, for all types of energy still retain their associated mass. Neither energy nor invariant mass can be destroyed in special relativity, and each is separately conserved over time in closed systems. Thus, a system's invariant mass
may change only because invariant mass
is allowed to escape, perhaps as light or heat. Thus, when reactions (whether chemical or nuclear) release energy in the form of heat and light, if the heat and light is not allowed to escape (the system is closed and isolated), the energy will continue to contribute to the system rest mass, and the system mass will not change. Only if the energy is released to the environment will the mass be lost; this is because the associated mass has been allowed out of the system, where it contributes to the mass of the surroundings.
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
in special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
incorporates the general understandings from the concept of mass-energy equivalence
Mass-energy equivalence
In physics, mass–energy equivalence is the concept that the mass of a body is a measure of its energy content. In this concept, mass is a property of all energy, and energy is a property of all mass, and the two properties are connected by a constant...
. Added to this concept is an additional complication resulting from the fact that "mass" is defined in two different ways in special relativity: one way defines mass ("rest mass" or "invariant mass") as an invariant quantity which is the same for all observers in all reference frames; in the other definition, the measure of mass ("relativistic mass") is dependent on the velocity of the observer.
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
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
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 c (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 c
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
2) 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. (In this case, "closure" implies that an idealized boundary is drawn around the system, and no mass/energy is allowed across it).
The term relativistic mass is also sometimes used. This is the sum total quantity of energy in a body or system (divided by c2). As seen from the center of momentum frame, the relativistic mass is also the invariant mass, as discussed above (just as the relativistic energy of a single particle is the same as its rest energy, when seen from its rest frame). For 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
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
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
Frame of reference
A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
. However, for given single frames of reference and for closed systems, the relativistic mass is also a conserved quantity.
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. The notion of mass as a property of an object from Newtonian mechanics does not bear a precise relationship to the concept in relativity.
For a discussion of mass in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, see mass in general relativity
Mass in General Relativity
The concept of mass in general relativity is more complex than the concept of mass in special relativity. In fact, general relativity does not offer a single definition for the term mass, but offers several different definitions which are applicable under different circumstances...
. For a general discussion including mass in Newtonian mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, see the article on mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
.
Terminology
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 massCenter of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
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
Cyclotron
In technology, a cyclotron is a type of particle accelerator. In physics, the cyclotron frequency or gyrofrequency is the frequency of a charged particle moving perpendicularly to the direction of a uniform magnetic field, i.e. a magnetic field of constant magnitude and direction...
is moving in circles with a relativistic velocity, the weight 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
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
is at rest for systems of many particles. This special frame where this occurs is also called the center of momentum frame
Center of momentum frame
A center-of-momentum frame of a system is any inertial frame in which the center of mass is at rest . Note that the center of momentum of a system is not a location, but rather defines a particular inertial frame...
, and is defined as the inertial frame in which the center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
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."
Transverse and longitudinal mass
Concepts that were similar to what nowadays is called "relativistic mass", were already developed before the advent of special relativity. For example, it was recognized by J. J. ThomsonJ. J. Thomson
Sir Joseph John "J. J." Thomson, OM, FRS was a British physicist and Nobel laureate. He is credited for the discovery of the electron and of isotopes, and the invention of the mass spectrometer...
in 1881 that a charged body is harder to set in motion than an uncharged body, which was worked out in more detail by Oliver Heaviside
Oliver Heaviside
Oliver Heaviside was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's field equations in terms of electric and...
(1889) and George Frederick Charles Searle
George Frederick Charles Searle
George Frederick Charles Searle was a British physicist and teacher, and a Fellow of the Royal Society.-Biography:Searle was born in Oakington, Cambridgeshire, England....
(1897). So the electrostatic energy behaves as having some sort of electromagnetic mass
, which can increase the normal mechanical mass of the bodies.Now, it was pointed out by Thomson and Searle that this electromagnetic mass also increases with velocity. This was further elaborated by Hendrik Lorentz
Hendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
(1899, 1904) in the framework of Lorentz ether theory
Lorentz ether theory
What is now often called Lorentz Ether theory has its roots in Hendrik Lorentz's "Theory of electrons", which was the final point in the development of the classical aether theories at the end of the 19th and at the beginning of the 20th century....
. He defined mass as the ratio of force to acceleration, not as the ratio of momentum to velocity, so he needed to distinguish between the mass
parallel to the direction of motion and the mass perpendicular to the direction of motion (where is the Lorentz factorLorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
, v is the relative velocity between the aether and the object, and c is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
). Only when the force is perpendicular to the velocity, Lorentz's mass is equal to what is now called "relativistic mass". Max Abraham
Max Abraham
Max Abraham was a German physicist.Abraham was born in Danzig, Imperial Germany to a family of Jewish merchants. His father was Moritz Abraham and his mother was Selma Moritzsohn. Attending the University of Berlin, he studied under Max Planck. He graduated in 1897...
(1902) called
longitudinal mass and transverse massTransverse mass
The transverse mass is a useful quantity to define for use in particle physics as it is invariant under Lorentz boost along the z direction. In natural units it is:Hadron collider physicists use another definition of transverse mass, in the case of a decay into two particles:-References: - See...
(although Abraham used more complicated expressions than Lorentz's relativistic ones). So, according to Lorentz's theory no body can reach the speed of light because the mass becomes infinitely large at this velocity.
The precise relativistic expression (which is equivalent to Lorentz's) relating force and acceleration for a particle with non-zero rest mass
moving in the x direction with velocity v and associated Lorentz factor isRelativistic mass
In special relativity, an object that has a mass cannot travel at the speed of light. As the object approaches the speed of light, the object's energy and momentum increase without bound.In the first years after 1905, following 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 which was first defined by Richard C. Tolman
Richard C. Tolman
Richard Chace Tolman was an American mathematical physicist and physical chemist who was an authority on statistical mechanics. He also made important contributions to theoretical cosmology in the years soon after Einstein's discovery of general relativity...
in 1912, who stated: “the expression m0(1 - v2/c2)-1/2 is best suited for THE mass of a moving body.”
In 1934, Tolman also defined relativistic mass as
which holds for all particles, including those moving at the speed of light.
For a slower than light particle, a particle with a nonzero rest mass, the formula becomes
Where m0 is the rest mass.
Tolman remarked on this relation that "We have, moreover, of course the experimental verification of the expression in the case of moving electrons to which we shall call attention in §29. We shall hence have no hesitation in accepting the expression as correct in general for the mass of a moving particle."
When the relative velocity is zero,
is simply equal to 1, and the relativistic mass is reduced to the rest mass as one can see in the next two equations below. As the velocity increases toward the speed of light c, the denominator of the right side approaches zero, and consequently approaches infinity.In the formula for momentum
the mass that occurs is the relativistic mass. In other words, the relativistic mass is the proportionality constant between the velocity and the momentum.
While Newton's second law remains valid in the form
the derived form
is not valid because in is generally not a constant (see the section above on transverse and longitudinal mass).Also Einstein at first used a relativistic mass concept in the form of longitudinal and transverse mass in his 1905 electrodynamics paper (equivalent to those of Lorentz, but with a different
by an unfortunate force definition, which was later corrected), and in another paper in 1906. On the other hand, in his first paper on (1905) he treated m as what would now be called the rest mass.Some claim that (in later years) he did not like the idea of "relativistic mass", by referring to this quote:
Modern view
The invariant massInvariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
is the ratio of four-momentum to four-velocity
Four-velocity
In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector that replaces classicalvelocity...
:
and is also the ratio of four-acceleration
Four-acceleration
In special relativity, four-acceleration is a four-vector and is defined as the change in four-velocity over the particle's proper time:whereandand \gamma_u is the Lorentz factor for the speed u...
to four-force when the rest mass is constant. The four-dimensional form of Newton's second law is:
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, it applies the name mass - belonging to the magnitude of a 4-vector - to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself."
While space-time has the unbounded geometry of Minkowski-space, the velocity-space is bounded by c and has the geometry of hyperbolic geometry where relativistic-mass plays an analogous role to that of Newtonian-mass in the barycentric-coordinates of Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. The connection of velocity to hyperbolic-geometry enables the 3-velocity-dependent relativistic-mass to be related to the 4-velocity Minkowski-formalism.
The relativistic energy-momentum equation
where the m is the rest mass, or the invariant mass
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
for systems, and E is the total energy.
The equation is also valid for photons, which have m = 0:
and therefore
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
[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
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
, is the only four-dimensional arrow associated with the particle's motion, so that if there is a conserved four-momentum , it must be proportional to this vector. This allows expressing the ratio of energy to momentum as,resulting in a relation between E and v:
This results in
and
these expressions can be written as
,,and
When working in units where c = 1, known as the natural unit system, all relativistic equations simplify. In particular, all three quantities E, p, m have the same dimension:
.The equation is often written this way because the difference
is the relativistic length of the energy momentum four-vector, a length which is associated with rest mass or invariant mass in systems. If m > 0, then there is the rest frame, where p = 0, this equation states that E = m, revealing once more that invariant mass is the same as the energy in the rest frame.The mass of composite systems
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
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 , the invariant mass is given by:In a mathematical system where c = 1, for systems of particles (whether bound or unbound) the total system invariant mass is given equivalently by the following:
Where, again, the particle momenta
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
Center of momentum frame
A center-of-momentum frame of a system is any inertial frame in which the center of mass is at rest . Note that the center of momentum of a system is not a location, but rather defines a particular inertial frame...
where momenta sum to zero, again the system mass (called the invariant mass
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
) 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 momenta 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
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
of an isolated system
Isolated system
In the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy - mass stays constant...
(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
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, because the invariant mass
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
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
Top quark
The top quark, also known as the t quark or truth quark, is an elementary particle and a fundamental constituent of matter. Like all quarks, the top quark is an elementary fermion with spin-, and experiences all four fundamental interactions: gravitation, electromagnetism, weak interactions, and...
.
Conservation versus invariance of mass in special relativity
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
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
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 mass
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
es 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
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
, 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
Center of momentum frame
A center-of-momentum frame of a system is any inertial frame in which the center of mass is at rest . Note that the center of momentum of a system is not a location, but rather defines a particular inertial 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 over-all velocity, or else (equivalently) it may be viewed from an inertial frame in which it has an over-all velocity (that is, technically, a frame in which its center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
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 momenta increase, these energy and momentum increases subtract out in the invariant mass definition, so that the moving container's invariant mass
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
will be calculated as the same value as if it were measured at rest, on a scale.
Closed (meaning totally isolated) systems
All conservation laws in special relativity (for energy, mass, and momentum) require isolated systems, meaning systems that are totally isolated, with no mass-energy allowed in or out, over time. If a system is isolated, then both total energy and total momentum in the system are conserved over time for any observer in any single inertial frame, though their absolute values will vary, according to different observers in different inertial frames. The invariant mass of the system is also conserved, but does not change with different observers. This is also the familiar situation with single particles: all observers calculate the same particle rest mass (a special case of the invariant mass) no matter how they move (what inertial frame they choose), but different observers see different total energies and momenta for the same particle.Conservation of invariant mass also requires the system to be enclosed 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 these merely act to change the inertial frame of the system or the observer. Though such actions may change the total energy or momentum of the bound system, these two changes cancel, so that there is no change in the system's invariant mass. This is just the same result as with single particles: their calculated rest mass also remains constant no matter how fast they move, or how fast an observer sees them move.
On the other hand, for systems which are unbound, the "closure" of the system may be enforced by an idealized surface, inasmuch as no mass-energy can be allowed into or out of the test-volume over time, if conservation of system invariant mass
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
is to hold during that time. If a force is allowed to act on (do work on) only one part of such an unbound system, this is equivalent to allowing energy into or out of the system, and the condition of "closure" to mass-energy (total isolation) is violated. In this case, conservation of invariant mass of the system also will no longer hold. Such a loss of rest mass in systems when energy is removed, according to E=mc2 where E is the energy removed, and m is the change in rest mass, reflect changes of mass associated with movement of energy, not "conversion" of mass to energy.
The system invariant mass vs. the individual rest masses of parts of the system
Again, in special relativity, the rest mass of a system is not required to be equal to the sum of the rest masses of the parts (a situation which would be analogous to gross mass-conservation in chemistry). For example, a massive particle can decay into photons which individually have no mass, but which (as a system) preserve the invariant mass of the particle which produced them. Also 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 which compose it. This is because the total energy of all particles and fields in a system must be summed, and this quantity, as seen in the center of momentum frameCenter of momentum frame
A center-of-momentum frame of a system is any inertial frame in which the center of mass is at rest . Note that the center of momentum of a system is not a location, but rather defines a particular inertial frame...
, and divided by c2, is the system's invariant mass.
In special relativity, mass is not "converted" to energy, for all types of energy still retain their associated mass. Neither energy nor invariant mass can be destroyed in special relativity, and each is separately conserved over time in closed systems. Thus, a system's invariant mass
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
may change only because invariant mass
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
is allowed to escape, perhaps as light or heat. Thus, when reactions (whether chemical or nuclear) release energy in the form of heat and light, if the heat and light is not allowed to escape (the system is closed and isolated), the energy will continue to contribute to the system rest mass, and the system mass will not change. Only if the energy is released to the environment will the mass be lost; this is because the associated mass has been allowed out of the system, where it contributes to the mass of the surroundings.
Controversy
According to Lev Okun, Einstein himself always meant the invariant mass when he wrote "m" in his equations, and never used an unqualified "m" symbol for any other kind of mass. Okun and followers reject the concept of relativistic mass. Arnold B. Arons has argued against teaching the concept of relativistic mass:For many years it was conventional to enter the discussion of dynamics through derivation of the relativistic mass, that is the mass–velocity relation, and this is probably still the dominant mode in textbooks. More recently, however, it has been increasingly recognized that relativistic mass is a troublesome and dubious concept. [See, for example, Okun (1989).]... The sound and rigorous approach to relativistic dynamics is through direct development of that expression for momentum that ensures conservation of momentum in all frames:
rather than through relativistic mass....
External links
- Z.K. Silagadze Relativity without tears (2007)
- Gary Oas On the Abuse and Use of the Relativistic Mass, 2005, arXiv.org: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