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Energy and momentum in special relativity
Abstract
The special relativistic expressions for momentum and energy are
obtained by requiring their conservation in a totally inelastic variant
of the Lewis-Tolman symmetric collision. The resulting analysis is
simpler and more straightforward than the usual textbook treatments of
relativistic dynamics.
Mass-Energy Equivalence (MEE) has become a basis of modern physics. In spite of the current educational trends highlighting modern physics education, it has been pointed out that interpretations of MEE are still not in general agreement. In addition, derivations of MEE appeared in textbooks gloss over some logical oversights. MEE also is often introduced only with a declarative knowledge that represents the rest energy of a particle, making students experience difficulty in learning it. To resolve the instructional challenges, distinguished papers on MEE were analyzed. By specifying common features of derivations in each paper, it was found that there were at least three types of MEE. By identifying the entire hierarchical structure of them, a potential type of MEE was suggested that can be useful to establish the connection between the particle and field.
Relativity requires that a particle's momentum and energy are the same functions of the particle's velocity in all inertial frames. Using the fact that momentum and energy must transform linearly between reference frames, we present a novel derivation of the mass-energy equivalence, namely, the relation that the energy is proportional to the moving mass, with no postulate about the existence of light or its properties. We further prove the mass-velocity relation without relying on momentum and energy conservation or on the Lorentz transformation. It is demonstrated that neither conservation laws nor the Lorentz transformation are necessary to establish those relations, and that those relations have a wider scope of validity than that of the conservation laws and the invariance of the speed of light.
We obtain the expressions for the energy and momentum of a relativistic particle by incorporating the equivalence of mass and energy into Newtonian mechanics.
Based on relativistic velocity addition and the conservation of momentum and energy, I present simple derivations of the expressions for the relativistic momentum and kinetic energy of a particle, and for the formula E = mc2.
We consider a relativistic elastic collision between a projectile of
momentum p with a target atom of momentum k in a general inertial frame.
We employ one space plus one time Minkowski geometry and calculate the
momentum transfer vector qμ suffered by the projectile
calculated via a Lorentz transformation to the barycentric frame and
then eliminate k. The resulting expression for qμ
reproduces several known cases, its algebraic behavior can be
interpreted physically, and it leads to a simple understanding of the
relativistic equipartition law.
We show that the relativistic expressions for momentum and energy as well as the way in which they transform could be derived without involving collisions and conservation laws. Our approach involves relativistic kinematics via the addition law of relativistic velocities.
Mass is one of the most fundamental concepts of physics. Understanding and calculating the masses of the elementary particles is the central problem of modern physics, and is intimately connected with other fundamental problems such as the origin of CP violation, the mystery of the energy scales that determine the properties of the weak and gravitational interactions, the compositeness of particles, supersymmetrytheory and the properties of the not‐yet‐discovered Higgs bosons.
In the modern language of relativity theory there is only one mass, the Newtonian mass m, which does not vary with velocity; hence the famous formula E = mc 2 has to be taken with a large grain of salt.
The concept of relativistic mass brings a consistency and simplicity to
the teaching of special relativity to introductory students. For
example, E=mc2 then expresses the beautifully simplifying equivalence of
mass and energy. Those who claim not to use relativistic mass actually
do so-if not by name-when considering systems of particles or photons.
Relativistic mass does not depend on the angle between force and
velocity-this supposed dependence results from incorrect use of Newton's
second law of motion.
Relativistic mass is discussed in both a pedagogical and historical context. It is pointed out that its introduction into the theory of special relativity was much in the way of a historical accident. Gaining widespread use initially in instruction, the use of relativistic mass is showing signs of progressive disfavor. An analysis and criticism of the various ways relativistic mass is used in relativity is detailed and special attention is given to the frequent misuse of relativistic mass as an inertia.
An alternate derivation of the expression for relativistic momentum is given which does not rely on the symmetric glancing collision first introduced by Lewis and Tolman in 1909 and used by most authors today. The collision in the alternate derivation involves a non-head-on elastic collision of one body with an identical one initially at rest, in which the two bodies after the collision move symmetrically with respect to the initial axis of the collision. Newtonian momentum is found not to be conserved in this collision and the expression for relativistic momentum emerges when momentum conservation is imposed. In addition, kinetic energy conservation can be verified in the collision. Alternatively, the collision can be used to derive the expression for relativistic kinetic energy without resorting to a work-energy calculation. Some consequences of a totally inelastic collision between these two bodies are also explored.
A purely mechanical version is given of Einstein’s 1905 argument that the mass of a body depends on its energy content. The Gedankenexperiment described here is the same as Einstein’s, except that the body loses energy not through electromagnetic radiation, but through the emission of massive particles. The concept of mass is not assumed, but is extracted as one of two constants of integration. Viewing mass in this way emphasizes a sometimes obscured distinction between mass as a proportionality constant in the kinetic energy (for which E=mc2 has profound physical content), and mass as it appears in the conventional relativistic rest energy (for which E=mc2 is merely a convenient convention).
From the relativity principle and the conservation of energy in particle collisions we deduce the form of the energy function, and the conservation of inertial mass and three-momentum. We show that the arguments are parallel under Einsteinian and Galilean kinematics.
DOI:https://doi.org/10.1103/PhysRev.1.161
The relativistic theory of collisions of macroscopic particles is developed from the two axioms of energy conservation and relativity, by use of standard relativistic kinematics (without, of course, assuming the mathematical expression for relativistic energy). We derive, in turn, the equivalence of rest-mass and rest-energy, the usual mathematical expression for the total energy in terms of the rest-energy, the conservation of 3-momentum, and the equivalence of total energy with inertial mass. We incidentally find all possible conservation laws of quantities described by the product of a scalar with a function of the speed.
A discussion of what E=mc2 means and other issues associated with the equation are presented. The differences between matter, mass, and energy, a derivation of the equation, the history of the word mass and examples of how it is used, misconceptions surrounding the equation, and a discussion of uranium fission are included. (KR)
The derivation of the functional form of the relativistic momentum of a particle has a history going back to Lewis and Tolman's paper of 1909, yet satisfactory presentations seem to be few in number. Careful examination of the several types of derivation shows that their shortcomings are avoidable and allows the presentation of exact and improved analyses.
Modern Physics ͑McGraw-Hill
- F J Blatt
F. J. Blatt, Modern Physics ͑McGraw-Hill, New York, 1992͒, Sec. 2.2.
1 is also a consequence of symmetry and the principle of relativity, because if ͉V x ͉ ͉V x Ј͉, say ͉V x ͉ Ͼ ͉V x Ј͉, then we could distinguish frame S from frame SЈ in a way that would violate the principle of relativity
- The
- Fig
The equality of ͉V x ͉ and ͉V x Ј͉ in Fig. 1 is also a consequence of symmetry
and the principle of relativity, because if ͉V x ͉ ͉V x Ј͉, say ͉V x ͉ Ͼ ͉V x Ј͉, then
we could distinguish frame S from frame SЈ in a way that would violate
the principle of relativity. The velocity transformation Eq. ͑6a͒ with vʈ
= V x = U and v ʈ
Ј= V x Ј=−U gives Eq. ͑7͒.
- J G Taylor
J. G. Taylor, Special Relativity ͑Clarendon, Oxford, 1975͒, Chap. 5.