Mass in special relativity
For example, photons have zero rest mass but contribute to the inertia (and weight in a gravitational field) of any system containing them.
The relativistic mass is the sum total quantity of energy in a body or system (divided by c2).
However, for given single frames of reference and for isolated systems, the relativistic mass is also a conserved quantity.
[5][3][6] It explains simply and quantitatively why a body subject to a constant acceleration cannot reach the speed of light, and why the mass of a system emitting a photon decreases.
[7][8] The notion of mass as a property of an object from Newtonian mechanics does not bear a precise relationship to the concept in relativity.
[9] Relativistic mass is not referenced in nuclear and particle physics,[1] and a survey of introductory textbooks in 2005 showed that only 5 of 24 texts used the concept,[10] although it is still prevalent in popularizations.
The relativistic mass is the mass of the system as it would be measured on a scale, but in some cases (such as the box above) this fact remains true only because the system on average must be at rest to be weighed (it must have zero net momentum, which is to say, the measurement is in its center of momentum frame).
But the same is also true of any closed system, such as an electron-and-box, if the electron bounces at high speed inside the box.
A so-called massless particle (such as a photon, or a theoretical graviton) moves at the speed of light in every frame of reference.
However, even massless particles have a relativistic mass, which varies with their observed energy in various frames of reference.
The invariant mass is the ratio of four-momentum (the four-dimensional generalization of classical momentum) to four-velocity:[11]
To find the form of the momentum and energy as a function of velocity, it can be noted that the four-velocity, which is proportional to
Note that the invariant mass of an isolated system (i.e., one closed to both mass and energy) is also independent of observer or inertial frame, and is a constant, conserved quantity for isolated systems and single observers, even during chemical and nuclear reactions.
However, in such a situation, although the container's total relativistic energy and total momentum increase, these energy and momentum increases subtract out in the invariant mass definition, so that the moving container's invariant mass will be calculated as the same value as if it were measured at rest, on a scale.
If a system is isolated, then both total energy and total momentum in the system are conserved over time for any observer in any single inertial frame, though their absolute values will vary, according to different observers in different inertial frames.
Neither energy nor invariant mass can be destroyed in special relativity, and each is separately conserved over time in closed systems.
[12] Concepts that were similar to what nowadays is called "relativistic mass", were already developed before the advent of special relativity.
is the Lorentz factor, v is the relative velocity between the ether and the object, and c is the speed of light).
So, according to Lorentz's theory no body can reach the speed of light because the mass becomes infinitely large at this velocity.
[16][17][18] Albert Einstein also initially used the concepts of longitudinal and transverse mass in his 1905 electrodynamics paper (equivalent to those of Lorentz, but with a different
[19][20] However, he later abandoned velocity dependent mass concepts (see quote at the end of next section).
The precise relativistic expression (which is equivalent to Lorentz's) relating force and acceleration for a particle with non-zero rest mass
In special relativity, an object that has nonzero rest mass cannot travel at the speed of light.
Tolman in 1912 further elaborated on this concept, and stated: "the expression m0(1 − v2/c2)−1/2 is best suited for the mass of a moving body.
The concept of relativistic mass is widely used in popular science writing and in high school and undergraduate textbooks.
B. Arons have argued against this as archaic and confusing, and not in accord with modern relativistic theory.
[5][28] Arons wrote:[28] For many years it was conventional to enter the discussion of dynamics through derivation of the relativistic mass, that is the mass–velocity relation, and this is probably still the dominant mode in textbooks.
Writing on said subject matter, he says that "its introduction into the theory of special relativity was much in the way of a historical accident", noting towards the widespread knowledge of E = mc2 and how the public's interpretation of the equation has largely informed how it is taught in higher education.
In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself.
[30] The connection of velocity to hyperbolic geometry enables the 3-velocity-dependent relativistic mass to be related to the 4-velocity Minkowski formalism.
