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Relativistic mass

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The term mass in special relativity is used in a couple of 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Both of these terms are somewhat outdated.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.

Today, when 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.

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

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. Which is precisely why it is a good thing to talk about. 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

$M = {m \over {\sqrt{1 - v^2/c^2}}}$

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

$\gamma = {1 \over {\sqrt{1 - v^2/c^2}}}$

Note that when the object is at rest, v = 0 and γ = 1, and the relativistic mass equals the rest mass (whence 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

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

while the relativistic mass is given by

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

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:

$F = m a = \frac{dp}{dt}$

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 possible to retain this notion of inertia. Moreover, the equations for energy and momentum

$E = M c^2 \,$

$p = M v \,$

remain valid in any reference frame when M is treated as the relativistic mass. The second equation is particularly appealing as it is the same definition of momentum that is used in non-relativistic mechanics. If instead one uses the rest mass, the equations become

$E = \gamma m c^2 \,$

$p = \gamma m v \,$

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:

$E^2 = m^2 c^4 + p^2 c^2 \,$

Some reasons for abandoning the notion relativistic mass:

One is forced to make statements like "The rest mass of a photon is zero", which sounds slightly odd because a photon can never be at rest, it always travels at the speed of light.

The idea of mass increasing with velocity leads many to believe that the internal structure of a quickly moving object is somehow altered. This is not true. The increase in energy is a result of the geometric properties of spacetime.

A related misconception is that the idea of relativistic mass combined with the notion of Lorentz contractions leads some people to the absurd 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 truly 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 physicists chose to abandon relativistic mass in favor of the rest mass has to do with the idea of Lorentz covariance. Roughly, this states that in order for an equation to make sense in any reference frame it should be built out of pieces that transform in a well-defined manner under Lorentz transformations. The rest mass satisfies this requirement (in fact, it is actually a Lorentz invariant quantity — it is the same in every reference frame). The relativistic mass does not. Strictly speaking, it is the time-like component of a four-vector (the energy-momentum four-vector). A four-vector is a Lorentz covariant 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, the fact that the latter terms are virtually unused 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.

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Relativistic mass

Definitions

Arguments regarding this terminology

See also

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