User:Edgerck/Mass in special relativity

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The terms invariant mass and mass are synonymous in physics and do not depend on the observer or the inertial frame of reference used to observe it. The preferred form in physics is simply mass; when physicists use the term mass they invariably mean invariant mass.

Because the term relativistic mass has also been used in physics, this occasionally leads to confusion.

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.

Summary
The terms invariant mass, intrinsic mass, or the proper mass are synonymous with mass, an observer-independent quantity that is an inherent property of a body.

The term rest mass is applied for the mass of a body that is isolated (free) and at rest relative to the observer. Due to the special relativity theory mass-energy equivalence, the rest mass is equivalent to the energy of the isolated (free) body at rest, divided by the speed of light squared.

The term relativistic mass (also known as the apparent mass) increases with observed speed according to the Lorentz transformation and depends on the observer's frame of reference.

The term relativistic mass has gradually fallen into disuse since the 1950s, when particle physics, the development of Minkowski four-vector notation, and general relativity showed that relativistic mass had no physical existence as mass, while invariant mass was the fundamental quantity also known as mass. Relativistic mass is virtually never used in current scientific research literature.

In the earlier years of relativity, however, relativistic mass was sometimes taken to be the "correct" notion of mass, and the invariant mass was referred to as the rest mass. Physics text books from the early 1920s to the 1980s, written by well-respected physicists, made the term "relativistic mass" common in popular discussions.

Einstein himself always meant invariant mass when he wrote "m" in his equations. Einstein first deduced in 1905 that the mass (inertia) of bodies increases with their internal energy (energy content), and that mass does not depend on the inertial frame used to observe it (see section below on mass in systems).

Today, following Einstein, instead of introducing the concept of relativistic mass when changing frames of reference, one uses the relativistic energy-momentum relation. In this relation, mass is always invariant. Scales and balances operate in the rest frame of objects being measured, measuring mass.

This usage is less confusing because it does not appear to make the increase of energy of an object with velocity or momentum to be connected with some change in internal structure of the object that would increase its mass, which change cannot be observed. In other words, one cannot observe changes in the mass of an object as a function of the speed of an observer relative to the object (to make it clearer, as it does not matter who is considered to be at rest).

In popular science, however, "relativistic mass" may refer to the mass-equivalence of the particle energy in the laboratory frame of reference. This is not the same as what physicists call relativistic mass or rest mass.

Mass in relativity theory
The relativistic energy-momentum relation states


 * $$ E^2 - (p c)^2 = (m c^2)^2 \,\!$$

where c is the speed of light, $$E \;$$ is total energy, $$m \;$$ is the (invariant) mass, and $$p\;$$ is momentum.

The energy $$ E\,$$ and the momentum $$ p\,$$ are observer dependent (vary from frame to frame). The quantity $$ m\,$$ is independent of observer or inertial frame.

If and only if the particle is not massless, a rest frame (also called center of mass frame or COM frame) can be defined for the particle.

For a free body in the rest frame, p = 0. Using the formula above, one obtains Einstein's famous formula Eo = mc² for the rest energy. Using this formula, the mass of a free body in the rest frame is defined by:


 * $$ m = \frac {Eo}{c^2}$$  (in Center of Momentum frame)

When one relates four-force to mass and four-acceleration Newton's second law takes the form


 * $$F^\mu = mA^\mu.\!$$

where m is the mass. In an inertial frame of reference, mass is a constant times the "length" of the momentum 4-vector.

Mass and gravitation
In "Relativity : the Special and General Theory", Part II, Chapter 22, Albert Einstein warns the reader about the limits of special relativity with these words: "We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light)."

Therefore, if the reader wants to study the influence of gravitational fields on phenomena, such as light and moving masses, the reader is advised to look beyond the scope of this article on special relativity.

The relativistic mass concept
According to special relativity, an object with mass cannot travel at the speed of light. As such an object approaches the speed of light, a stationary observer will observe that the object's energy and momentum are increasing toward infinity.

In 1912, Richard C. Tolman coined the concept of relativistic mass, in stating: “the expression m0(1 − v2/c2)−1/2 is best suited for THE mass of a moving body”.

The formula above is frequently written as:


 * $$M = \gamma m \!$$

where


 * M is the relativistic mass
 * m is the mass, and
 * $$\gamma = {1 \over {\sqrt{1 - \frac{|\mathbf{v}|^2}{c^2}}}} \!$$ is the Lorentz factor,
 * v is the relative velocity between the observer and the object, and
 * c is the speed of light.

It is clear that m0(1 − v2/c2)−1/2 is not defined for v = c (for example, for a photon).

When v is zero, γ is simply equal to 1, and the relativistic mass is equal to mass &mdash; the familiar concept in Newtonian mechanics. The use of this formula was justified by the wish to preserve the role of mass in Newtonian mechanics, first of all as a measure of inertia. Another aim was to preserve the additivity: the mass of a system of free particles is equal to the sum of their masses.

The main benefit of using the relativistic mass is also often cited to be that the formula for momentum


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

from Newtonian mechanics would retain its form by simply replacing m by M. However, some relations do not work right by doing so. For example, even though 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 invalid as $$M\,$$ in $${d(M\mathbf{v})}\!$$ is generally not a constant. The correct relativistic expression relating force and acceleration for a particle with non-zero mass moving in the x direction with velocity v and associated Lorentz factor γ is


 * $$f_x = \gamma^3 m a_x = \gamma^2 M a_x, \,$$


 * $$f_y = \gamma m a_y = M a_y, \,$$


 * $$f_z = \gamma m a_z = M a_z. \,$$

These equations, which are almost forgotten today, define $$\gamma^3 m\,$$ as longitudinal mass and γ m as transverse mass.

Another difficulty of this approach is that since γ depends on velocity, observers in different inertial reference frames will calculate different values for relativistic mass, whereas our description of the world should not depend on a arbitrary choice of reference frame.

After 1934, relativistic mass was also defined by Richard C. Tolman as


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

Because the energy $$ E\,$$ is observer dependent (vary from frame to frame), this formula makes m depend on the observer.

Mainly for the reasons noted above, the modern practice in physics is not to use relativistic mass.

Relativistic mass and gravitation
Changes of relativistic mass with velocity in relation to an observer, as defined in special relativity, have no consequence regarding inertia or gravitation.

In "A brief history of time" (p225, 1996 edition) Professor Stephen Hawking says that particle energies of around a hundred GeV, which were the most one could produce in a laboratory at that time, are still far from what is called the Planck energy of 10 million million million Gev (1 followed by 19 zeros), such that its mass would be so concentrated it would form a black hole. Further, Hawking says "We shall not bridge that gap with particle accelerators in the foreseeable future!".

This explanation is sometimes confused with the possibility of manufacturing a black hole by increasing the velocity of a particle until it becomes massive enough. This is incorrect because, according to special relativity, if an object has a mass of 10 Kg in one inertial reference frame then it has a mass of 10 Kg in any other inertial reference frame.