# Relativistic mass

(Redirected from Rest mass)

The term mass in special relativity can be used in different ways, occasionally leading to a great deal of confusion. Specifically, mass can refer to either the rest mass or the relativistic mass. The rest mass, or invariant mass, is an observer independent quantity, while the relativistic mass, or apparent mass depends on one's frame of reference. In particular, the relativistic mass increases with velocity while the rest mass stays the same.

Both of these terms are somewhat outdated. In the earlier years of relativity, it was the relativistic mass that was taken to be the "correct" notion of mass, and the invariant mass was referred to as the rest mass. Gradually, as special relativity gave way to general relativity and found application in quantum field theory, it was realized that the invariant mass was the more useful quantity and people stopped referring to the relativistic mass altogether.

When particle physicists talk about the mass of an object they always mean the rest mass. They use other terms, like energy, to refer to the relativistic mass (The reason for this is explained in the next section). The terms rest mass and relativistic mass can still be found in elementary textbooks and, especially, in popularizations of physics. There are several arguments, discussed below, as to why this terminology should be dropped. However, the fact that some relativity courses continue to use relativistic mass demonstrates that this is a matter of opinion.

In modern usage the term mass, when unqualified, always refers to the invariant (rest) mass.

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## Definitions

The rest mass of an object is the true invariant mass of the object. That is, all observers in inertial reference frames will agree on what the invariant mass is. The relativistic mass, on the other hand, is observer dependent. For an object traveling with a velocity v relative to some inertial observer, the relativistic mass is given by

[itex]M = {m \over {\sqrt{1 - v^2/c^2}}}[itex]

where m is the invariant (rest) mass and c denotes the speed of light in a vacuum. This is often written as [itex]M = \gamma m[itex] where γ (the Lorentz factor) is the quantity given by

[itex]\gamma = {1 \over {\sqrt{1 - v^2/c^2}}}[itex]

Note that when the object is at rest, v = 0 and γ = 1, and the relativistic mass equals the rest mass (hence the name). At the other extreme, as the velocity approaches the speed of light, γ and the relativistic mass increase without bound.

Given an object with momentum p and energy E we can define the rest mass by the equation

[itex]m = \frac{1}{c}\sqrt{\left(\frac{E}{c}\right)^2 - p^2}[itex]

while the relativistic mass is given by

[itex]M = \frac{E}{c^2}[itex]

Physicists usually work in units where c = 1 so that energy and relativistic mass become identical concepts. Instead of talking about relativistic mass they simply talk about energy.

## Arguments regarding this terminology

The original reasoning for regarding the relativistic mass as the proper notion of mass has to do with Newton's second law of motion:

[itex]F = m a = \frac{dp}{dt}[itex]

whereby the inertial mass of an object measures the "resistance" of that object to undergo acceleration when a given force is applied. In special relativity, this resistance becomes unbounded as v approaches the speed of light. This is another way of saying that it is impossible to accelerate anything with mass to the speed of light: one would have to push "infinitely hard".

By taking the mass to be M = γ m, with m the rest mass, it is not possible to fully retain this notion of inertia as F=ma doesn't hold in general (see below). However, the more general equation for force

[itex] F = \frac{d(M v)}{dt}[itex]

and the equations for energy and momentum

[itex] E = M c^2 \,[itex]
[itex] p = M v \,[itex]

remain valid in any reference frame when M is treated as the relativistic mass. This is particularly appealing as they are the same definitions as used in non-relativistic mechanics. If instead one uses the rest mass, the equations become:

[itex] F = m \frac{d(\gamma v)}{dt}[itex]
[itex] E = \gamma m c^2 \,[itex]
[itex] p = \gamma m v \,[itex]

In this form the velocity dependence of E and p is slightly more transparent (as long as the velocity dependence of γ is understood). In particular, it is clear that momentum is no longer a linear function of velocity.

The downside to this is that the famous Einstein equation E = m c2 is only valid in the rest frame of a particle. Be that as it may, this is precisely how the equation is understood today. In a general reference frame, one should use the equation E = γ m c2 or the full energy-momentum relation:

[itex]E^2 = m^2 c^4 + p^2 c^2 \,[itex]

Some reasons for abandoning the notion relativistic mass:

• One is forced to make statements like "A photon has no rest mass", which sounds slightly odd because a photon can never be at rest.
• The idea of mass increasing with velocity leads many to believe that the internal structure of a quickly moving object is somehow altered. However, the standard view of relativity is that the internal structure of an object is always unchanged, and that the different quantities measured by different observers for energy and velocity are simply the same reality seen from different points of view.
• The idea of relativistic mass combined with the notion of Lorentz contractions leads some people to the incorrect conclusion that an object traveling fast enough will form a black hole. However, by the very principle of relativity, if an object is not a black hole in one frame (its rest frame) it cannot be a black hole in any other frame either.
• If one wishes to retain the notion that mass measures the "resistance" to acceleration, then mass can no longer be treated as a scalar quantity. This is because it is easier to accelerate something perpendicular to the direction of motion than parallel to the direction of motion. In effect, an object would have more mass in one direction than other.
• The primary reason that most physicists chose to abandon relativistic mass in favor of the rest mass is that rest mass is a Lorentz invariant quantity — it is the same in every reference frame. Strictly speaking, it is the time-like component of a four-vector (the energy-momentum four-vector). A four-vector is a Lorentz invariant quantity, but its individual components are not.

In the end, the usage of mass, energy, and momentum in place of terms like rest mass and relativistic mass is a matter of semantics. Neither usage is technically wrong. However, that the latter terms are little used in the scientific community is a strong argument in favor of abandoning them altogether. Einstein himself wrote:

“It is not good to introduce the concept of the mass M = m/(1 - v2/c2)1/2 of a body for which no clear definition can be given. It is better to introduce no other mass 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.” – Einstein, in a 1948 letter to Lincoln Barnett

This precisely echoes the modern sentiment.

However, in the “Autobiographical notes” by Einstein published in 1949,

“…the theory had to combine the following things:

1. From general considerations of special relativity theory it was clear that the inert mass of a physical system increases with the total energy (therefore, e.g., with the kinetic energy).

2. From the very accurate experiments… it was empirically known with very high accuracy that the gravitational mass of a body is exactly equal to its inert mass.”

## Usage

"Relativistic mass" or velocity-dependent mass is still prevalent in popular books and some research papers:

• The God Particle: If the Universe Is the Answer, What Is the Question? by Leon Lederman, Nobel Laureate, Houghton Mifflin (1993) Pg 205
• A Different Universe: Reinventing Physics from the Bottom Down by Robert Laughlin, Nobel Laureate, Basic Books, (2005) Pg 125
• Perfectly Reasonable Deviations from the Beaten Track by Richard Feynman, Nobel Laureate, Basic Books, (2005) "It was successful, the necessary consequential phenomena (like mass changing with velocity) were ultimately observed experimentally." Pg 283.
• Facts and Mysteries in Elementary Particle Physics by Martinus J.G. Veltman, Nobel Laureate, World Scientific Publishing Company (2003) Pg 137
• Einstein's Legacy: The unity of space and time by Julian Schwinger, Nobel Laureate, Dover, (2002) Pg 84
• The Road to Reality by Roger Penrose, Jonathan Cape, London (2004)
• A Brief History of Time by Stephen Hawking (1988) "Because of the equivalence of energy and mass, the energy which an object has due to its motion will add to its mass. In other words, it will make it harder to increase its speed."
• Einstein's Cosmos by Michio Kaku (2004) "For example, Einstein could show that the mass of an object increased the faster it moved. (Its mass would in fact become infinite if you hit the speed of light—which is impossible, which proves the unattainability of the speed of light.) This meant that the energy of motion was somehow being transformed into increasing the mass of the object. Thus, matter and energy are interchangeable!"
• Understanding Relativity by Leo Sartori (1996) "Whether or not to speak of velocity-dependent mass is largely a matter of taste. Although it is currently unfashionable to do so, Einstein did and we shall as well."
• John Roche (2005) "What is mass?" European Journal of Physics 26 (2), 225–242.
• Gerald Gabrielse (1995) "Relativistic mass increase at slow speeds", American Journal of Physics, 63(6), 568.

"... the general trend, especially in the literature on elementary particle physics, is toward the elimination of relativistic mass..."

Max Jammer's "Concepts of MASS in contemporary Physics and Philosophy", Pg 55

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