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, but it is usually reserved for systems which consist of widely separated particles.

The term relativistic mass is also used, and this is the total quantity of energy in a body (divided by c²). The relativistic mass includes a contribution from the kinetic energy of the body, and is bigger the faster the body moves, so unlike the invariant mass, the relativistic mass depends on the observer's frame of reference.

Because the relativistic mass is proportional to the energy, it has gradually fallen into disuse in particle physics. Lev B. Okun makes the case that the concept is no longer even pedagogically useful. However, T.R. Sandin has argued otherwise.

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.

Terminology

If a box contains many particles, it weighs more the faster the particles are moving. Any energy in the box adds to the mass, so that the relative motion of the particles contributes to the mass in the box. But if the box itself 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, while the relativistic mass is calculated including it.

Relativistic mass and rest mass are both traditional concepts in physics, but the relativistic mass is just a redundant name for the total energy. The relativistic mass is the mass of the system as it would be measured on a scale, but if the object is moving at relativistic speeds, it has to be moving around in circles otherwise the scale would need to be very wide. If the object was stopped and weighed, it would not be moving, and the relativistic and rest masses would be the same.

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. This is why the invariant mass is also called the rest mass. This special frame 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 linear 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.

If an object is moving at the speed of light, it is never at rest in any frame. In this case the total energy of the object becomes smaller and smaller in frames which move faster and faster in the same direction. The rest mass of such an object is zero, and the only mass which the object has is relativistic mass-- a quantity which depends on the observer.

The relativistic mass concept

See also: Special relativity § Mass-energy equivalence

Early developments: transverse and longitudinal mass

It was recognized by J. J. Thomson in 1881 that a charged body is harder to set in motion than an uncharged body, which was worked out on more detail by Oliver Heaviside (1889) and George Frederick Charles Searle (1896). So the electrostatic energy behaves as having some sort of electromagnetic mass, which can increase the normal mechanical mass of the bodies. Later Wilhelm Wien (1900), Max Abraham (1902), came to the conclusion that the total mass of the bodies is identical to its electromagnetic mass. And because the em-mass depends on the em-energy, the formula for the energy-mass-relation given by Wien (1900) was $m=(4/3)E/c^{2}$.

It was pointed out by Thomson and Searle, that this electromagnetic mass also increases with velocity. This was also recognized by Hendrik Lorentz (1899, 1904) in the framework of Lorentz's Theory of Electrons. 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 $m_{L}=\gamma ^{3}m_{0}$ parallel to the direction of motion and the mass $m_{T}=\gamma m_{0}$ perpendicular to the direction of motion. Only when the force is perpendicular to the velocity is Lorentz's mass equal to what is now called "relativistic mass". (Where $\gamma =1/{\sqrt {1-v^{2}/c^{2}}}$ is the Lorentz factor, v is the relative velocity between the aether and the object, and c is the speed of light). Abraham (1902) called $m_{L}$ longitudinal mass and $m_{T}$ transverse mass, (whereby Abraham's own expressions were more complicated than Lorentz's relativistic ones). So, according to this 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 $m$ moving in the x direction with velocity v and associated Lorentz factor $\gamma$ is

$f_{x}=m\gamma ^{3}a_{x}=m_{L}a_{x},\,$

$f_{y}=m\gamma a_{y}=m_{T}a_{y},\,$

$f_{z}=m\gamma a_{z}=m_{T}a_{z}.\,$

Einstein calculated the longitudinal and transverse mass (which are equivalent to those of Lorentz, but for a mistake in $m_{T}$, which was later corrected ) in his 1905 electrodynamics paper and in another paper in 1906. However, in his first paper on $E=mc^{2}$ (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":

It is not good to introduce the concept of the mass $M=m/{\sqrt {1-v^{2}/c^{2}}}$ of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the ’rest mass’ m. Instead of introducing M it is better to mention the expression for the momentum and energy of a body in motion.

— Albert Einstein in letter to L Barnett (quote

from L. B. Okun, “The Concept of Mass,” Phys. Today 42,

31, June 1989.)

Modern relativistic concepts

In special relativity, as in Lorentz's ether theory, 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.

The velocity dependent mass of Lorentz and Abraham 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

$M={\frac {E}{c^{2}}}\!$

which holds for all particles, including those moving at the speed of light. Even a photon, a particle which moves at the speed of light, has relativistic mass.

For a slower than light particle, a particle with a nonzero rest mass, the formula becomes

$M=\gamma m\!$

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, γ 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 γ approaches infinity.

In the formula for momentum

$\mathbf {p} =M\mathbf {v}$

the mass that occurs is the relativistic mass. In other words, the relativistic mass is the proportionality constant between the velocity and the momentum.

Newton's second law remains valid in the form

$\mathbf {f} ={\frac {d(M\mathbf {v} )}{dt}},\!$

the derived form $\mathbf {f} =M\mathbf {a}$ is not valid because $M\,$ in ${d(M\mathbf {v} )}\!$ is generally not a constant [1] (see the section above on transverse and longitudinal mass).

The rest mass is the ratio of four-momentum to four-velocity:

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

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:

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

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 ${\vec {p}}$ 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 p of the total momentum vector ${\vec {p}}$, the invariant mass is given by:

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

This is the four-dimensional length of the four-dimensional vector composed of E and ${\vec {p}}$ together, calculated using a pythagorean theorem with minus signs.

Note that the invariant mass of a closed system is also independent of observer or inertial frame, and is a constant, conserved quantity for closed systems and single observers, even during chemical and nuclear reactions. It 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.

The relativistic energy-momentum equation

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

The relativistic expressions for E and p obey the relativistic energy-momentum equation:

$E^{2}-(pc)^{2}=(mc^{2})^{2}\,\!$

the m is the rest mass.

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

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

$E=pc\,\!$

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,

$E=mc^{2}\,\!$

And the rest mass is only equal to the total energy in the rest frame of the object.

If the object is moving, the total energy is

$E={\sqrt {(mc^{2})^{2}+(pc)^{2}}}\,\!$

Which has both positive and negative solutions. In classical physics, the negative energy solutions are spurious, and as the momentum increases with the increase of the velocity v, so does the total energy.

To find the form of the momentum and energy as a function of velocity, note that the four-velocity, which is propotional to $(c,{\vec {v}})$, is the only four-dimensional arrow associated to the particle's motion, so that if there is a conserved four-momentum $(E,{\vec {p}}c)$, it must be proportional to this vector. This gives the ratio of energy and momentum:

$pc=E{v \over c}$

Which makes the energy-momentum equation a relation between E and v.

$E^{2}=(mc^{2})^{2}+E^{2}{v^{2} \over c^{2}}$

Which gives E

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

and P.

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

The relativistic mass equation is the formula for E divided by c²

$m_{\mathrm {rel} }={m \over {\sqrt {1-{v^{2} \over c^{2}}}}}$

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 dimensions.

$m^{2}=E^{2}-p^{2}\,\!$

The equation is often written in this way because the difference $E^{2}-p^{2}$ is the relativistic length of the energy momentum four-vector. In the rest frame, the equation above just states that E=m, again revealing that the rest mass is the energy in the rest frame.

Conservation of mass in special relativity

Energy is an additive conserved quantity but rest mass is not. This means that rest mass is only conserved under those conditions where it can be identified as the total energy of an isolated system. The relativistic mass is synonymous with the energy, so conservation of energy means that relativistic mass is conserved.

If a system is closed, then the total momentum is also conserved, so that the rest mass of the entire system, which is determined by the total energy-momentum, is constant. Note that the rest mass of a system is not equal to the sum of the rest masses of the parts--- a massive particle can decay into photons.

For a system to keep the same total mass, it must be enclosed so that no heat and radiation can escape. It does not need to be completely isolated from external forces, because although these can change the magnitude of the momentum and the energy, if the forces only do work on the whole system and not on the individual parts, the changes in momentum and energy keep the rest mass constant. When reactions release energy in the form of heat and light, and if the heat and light is not allowed to escape, the energy will continue to contribute to the rest mass. Only if the energy is released to the environment will the mass be lost..

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:

$p={m_{0}v \over {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\!$

rather than through relativistic mass....

On the other hand, T. R. Sandin has written:

The concept of relativistic mass brings a consistency and simplicity to the teaching of special relativity to introductory students. For example, $E=mc^{2}$ then expresses the beautifully simplifying equivalence of mass and energy. Those who claim not to use relativistic mass actually do so—if not by name—when considering systems of particles or photons. Relativistic mass does not depend on the angle between force and velocity—this supposed dependence results from incorrect use of Newton's second law of motion.

It's important to notice that a relationship between speed and mass such as

$m={m_{0} \over {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\!$

implies that the velocity is measured relative to a frame of reference.

External links

• Gary Oas On the Abuse and Use of the Relativistic Mass, 2005, arXiv.org:physics/0504110.
• Usenet Physics FAQ
  • "Does light have mass?" by Philip Gibbs, 1997, retrieved Aug 10,2006
  • "Does mass change with velocity?" by Philip Gibbs et al., 2002, retrieved Aug 10 2006
  • "What is the mass of a photon?" by Matt Austern et al., 1998, retrieved Jun 27, 2007
  • Max Jammer (1997). Concepts of Mass in Classical and Modern Physics. Courier Dover Publications. p. pp. 177-178. ISBN 0486299988. {{cite book}}: |page= has extra text (help)