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.”
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."
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. In later years Einstein expressed his dislike of the idea of "relativistic mass":
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.
— Albert Einstein in letter to Lincoln Barnett, 19 June 1948 (quote from L. B. Okun (1989), p. 42)
Read more about this topic: Relativistic Mass Distortion, The Relativistic Mass Concept
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