Relativistic mass

Mass in special relativity

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, 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.

Relativistic 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 in 1912, who stated: “the expression m₀(1 - v²/c²)^-1/2 is best suited for THE mass of a moving body.”^[10]

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

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."^[11]

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

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

the derived form is not valid because in is generally not a constant [1] (see the section above on transverse and longitudinal mass).

Also Einstein at first used the 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 m_T by an unfortunate force definition, which was later corrected), and in another paper in 1906. On the other hand, in his first paper on E = mc² (1905) he treated m as what would now be called the rest mass.^{[12] [13] [14]}

Some claim that (in later years) he did not like the idea of "relativistic mass", by referring to this quote:^[15]

It is not good to introduce the concept of the mass 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.