Physics
Relativistic Mass and Energy
Relativistic mass and energy are concepts in physics that arise from Einstein's theory of special relativity. They describe how an object's mass and energy change as its velocity approaches the speed of light. According to this theory, as an object's velocity increases, its mass and energy also increase, leading to the famous equation E=mc^2.
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12 Key excerpts on "Relativistic Mass and Energy"
eBook - PDF- Gary N. Felder, Kenny M. Felder(Authors)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
We implicitly used those assumptions when we solved the elastic collision problem on p. 64 by setting the masses of the final objects equal to their initial masses. When the colliding objects are particles like electrons, protons, or photons, you can assume that they always have the same mass, so particle collisions are somewhat simpler to analyze than macroscopic ones. Summary: Momentum, Energy, and Mass The relativistic concepts of momentum, energy, and mass closely parallel their classical coun- terparts. Momentum is based on an object’s mass and velocity, and is conserved in any isolated system. Energy is also conserved, but can be converted between different forms. Mass is reference-frame-independent, and quantifies how difficult it is to accelerate an object. But different formulas lead to different results and some new concepts. In relativity both momentum and energy approach infinity in the limit v → c. The energy mc 2 of a composite system at rest includes all the rest energies of its constituents, plus their internal kinetic and potential energies. When the whole system is moving, its mass is the same but its energy now also includes a new kinetic energy term. The “external potential energy,” such as the energy of a brick on top of a building, is not included in E = γ mc 2 for the brick. That energy belongs to the Earth–brick system, but not to either object individually. (If you ask very strictly “where is that energy, then?” the answer is “in the gravitational field.” You can write a formula for the energy density of a gravitational or electric field and then integrate to find its total energy.) We’ve summarized some of the key properties of these quantities in Table 2.1. In reading the table, keep in mind the following definitions: • A quantity is “invariant” if it is the same in all reference frames. The number of atoms in a table is invariant; every observer will agree about it.
eBook - PDF- Lev Borisovich Okun(Author)
- 1991(Publication Date)
- World Scientific(Publisher)
THE CONCEPT OF MASS (MASS, ENERGY, RELATIVITY) Reprinted from: Sov. Phys. Usp. 32(7), July 1989, pp. 629 - 638. Present-day ideas concerning the relationship between mass and energy are presented. The history of the origin of archaic terms and concepts that are widely used in the literature in discussing the problem of mass and energy is related, and arguments are presented for the necessity of abandoning these archaic terms and concepts. Contents 1. A small test instead of an introduction 95 I Facts 97 2. Mass in Newtonian mechanics 97 3. Galileo's principle of relativity 99 4. Einstein's principle of relativity 99 5. Energy, momentum, and mass in the theory of relativity 100 6. Limiting cases of the relativistic equations 101 7. Connection between the force and acceleration in the theory of relativity 103 8. Gravitational attraction in the theory of relativity 104 9. Mass of a system of particles 104 10. Examples of transformations into each other of rest energy and kinetic energy 106 11. Comparison of the role played by mass in the theories of Einstein and Newton 107 12. The nature of mass: question no. 1 of modern physics 108 II Artifacts 109 13. At the turn of the century: four masses 109 14. Mass and energy in Einstein's papers in 1905 111 15. Generalized Poincare formula 112 16. A thousand and two books 113 17. Imprinting and mass culture 115 18. Why it is bad to call E/c 2 the mass 116 19. Does mass really depend on velocity, Dad? 118 20. Fizika v Shkole (Physics at School) 119 References 120 1. A Small Test Instead of an Introduction Einstein's relation between the mass of a body and the energy contained in it is undoubtedly the most famous formula of the theory of relativity. It permitted a new and deeper understanding of the world that surrounds us. Its practical consequences are vast and, to a large degree, tragic. In a certain sense this formula has become the symbol of science in the 20th century.
eBook - PDF- Peter E. Hodgson(Author)
- 2017(Publication Date)
- Routledge(Publisher)
We can therefore define m g as the relativistic energy, with E ¼ mc 2 as the rest energy and 1 = 2 m v 2 as the kinetic energy. This shows that every material particle of mass m has an energy equivalent mc 2 . Thus if in a nuclear reaction the sum of the masses of the products is less than that of the particles before the reaction, then energy is released. This has been verified experimentally to high precision. It is very often said (e.g. Planck, 1933) that experiments with beams of electrons show that their mass increases with their velocity according to the relation m ¼ m 0 g , where now m 0 is invariant. The mass thus tends to infinity as the velocity approaches that of light. This seems very strange, as we are used to thinking of mass as corresponding to quantity of matter, and this should be invariant. The apparent increase of mass with velocity is a consequence of retaining the Newtonian definition of velocity in a relativistic context. This may been seen by recalling that the relativistic momentum is m v g . We then have to choose between defining the mass as m and the velocity as v g , or the mass as m g and the velocity as v . Since we know that relativistic velocities behave in a non-Newtonian way, it is very natural to make the former choice, and this ensures that velocities transform according to the Lorentz transformation. Thus by defining velocity as v g corresponding to a derivative with respect to the proper time, mass remains an invariant quantity. The apparent variation of mass with 100 Theology and Modern Physics velocity is thus not a new and profound property of matter, but simply the result of a failure to use relativistic dynamics in a consistent way (Leighton, 1959, pp. 35, 36).
eBook - PDF- Daniel Kleppner, Robert Kolenkow(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
RELATIVISTIC DYNAMICS 13 13.1 Introduction 478 13.2 Relativistic Momentum 478 13.3 Relativistic Energy 481 13.4 How Relativistic Energy and Momentum are Related 487 13.5 The Photon: A Massless Particle 488 13.6 How Einstein Derived E = mc 2 498 Problems 499 478 RELATIVISTIC DYNAMICS 13.1 Introduction In Chapter 12 we saw how the postulates of special relativity lead to new kinematical relations for space and time. These relations can naturally be expected to have important implications for dynamics, particularly for the meaning of momentum and energy. In this chapter we examine the modifications to the Newtonian concepts of momentum and energy required by special relativity. The underlying strategy is to ensure that momentum and energy in an isolated system continue to be conserved. This approach is often used in extending the frontiers of physics: by reformulating conservation laws so that they are preserved in new situa-tions, we are led to generalizations of familiar concepts. We can also be led to the discovery of unfamiliar concepts, for instance the concept of massless particles that can nevertheless carry energy and momentum. x y A B 13.2 Relativistic Momentum To investigate the nature of momentum in special relativity, consider a glancing elastic collision between two identical particles A and B in an isolated system. We want the total momentum of the system to be con-served, as it is in non-relativistic physics. We shall view the collision in two frames: A ’s frame, the frame moving along the x axis with A so that A is at rest while B approaches along the x direction with speed V , and then in B ’s frame, which is moving with B in the opposite direction so that B is at rest and A is approaching. (The term “frame” is used synony-mously with “reference system.”) We take the collisions to be completely symmetrical. Each particle has the same y speed u 0 in its own frame be-fore the collision, as shown in the sketches.
- Martinus J G Veltman(Author)
- 2003(Publication Date)
- World Scientific(Publisher)
In fact, the energy increases less sharply with momentum, and for very high values of the momentum the energy becomes proportional to it. (Energy approximately equals momen-tum times c , the speed of light.) A typical case is shown in the next figure, with the dashed line showing the non-relativistic case, the solid curve the relativistically correct relation. kinetic energy 0 Momentum Relativ. The quantitatively minded reader may be reminded of the equa-tions quoted in Chapter 1. In particular there is the relation between energy and momentum, plotted in the next figure: 2 2 2 c m p c E + = . or, using the choice of units such that c = 1: 2 2 2 m p E + = . 129 E N E R G Y, M O M E N T U M A N D M A S S -S H E L L Another important fact is the Einstein equation E = mc 2 . This very famous equation can be deduced in a number of ways, none of which is intuitively appealing. This equation tells us that even for a particle at rest the energy is not zero, but equal to its mass multiplied with the square of the speed of light. In particle physics this equation is a fact of daily life, because in inelastic processes, where the set of secondary particles is different from the primary one, there is no energy conservation unless one includes these rest-mass energies in the calculation. As the final particles have generally masses different from the primary ones, the mass-energy of the initial state is in general different from that of the final state. In fact, the first example that has already been discussed extensively is neutron decay; this decay is a beautiful and direct demonstration of Einstein’s law, E = mc 2 . Indeed it is in particle physics that some very remarkable aspects of the theory of relativ-ity are most clearly demonstrated, not just the energy-mass equa-tion. Another example is the lifetime of unstable particles, in particular the muon.
- Raymond Serway, John Jewett(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Finally, because the mass m of a particle is independent of its motion, m must have the same value in all reference frames. For this reason, m is often called the invariant mass. On the other hand, because the total energy and linear momen- tum of a particle both depend on velocity, these quantities depend on the reference frame in which they are measured. When dealing with subatomic particles, it is convenient to express their energy in electron volts (Section 24.1) because the particles are usually given this energy by acceleration through a potential difference. The conversion factor, as you recall from Equation 24.5, is 1 eV 5 1.602 3 10 219 J For example, the mass of an electron is 9.109 3 10 231 kg. Hence, the rest energy of the electron is m e c 2 5 (9.109 3 10 231 kg)(2.998 3 10 8 m/s) 2 5 8.187 3 10 214 J 5 (8.187 3 10 214 J)(1 eV/1.602 3 10 219 J) 5 0.511 MeV Another way to represent this same idea is to express the mass in units of MeV/ c 2 by dividing both sides of the previous equation by c 2 : m e 5 0.511 MeV c 2 Total energy of a relativistic particle Energy–momentum relationship for a relativistic particle Energy-momentum relationship for a photon 4 One way to remember this relationship is to draw a right triangle having a hypotenuse of length E and legs of lengths pc and mc 2 . PITFALL PREVENTION 38.6 Watch Out for “Relativistic Mass” Some older treatments of relativ- ity maintained the conservation of momentum principle at high speeds by using a model in which a particle’s mass increases with speed. You might still encounter this notion of “relativistic mass” in your outside reading, especially in older books. Be aware that this notion is no longer widely accepted; today, mass is consid- ered as invariant, independent of speed. The mass of an object in all frames is considered to be the mass as measured by an observer at rest with respect to the object. Copyright 2019 Cengage Learning. All Rights Reserved.
eBook - PDF- Norman Gray(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
As you will recall, in natural units the speed = 1, and this extra factor of 2 becomes redundant. Thus in natural units the units of mass are the same as the units of energy, and we habitually write e = 0.511 MeV. 7.6 Relativistic Force Another way of thinking about this is through the idea of relativistic force. Possibly surprisingly, force is a less useful concept in relativistic dynamics, than it is in newtonian dynamics – it’s more often useful to talk of momen- tum and energy, and indeed energy-momentum. However, there are a few useful things we can discover. We can define force in the usual way, in terms of the rate of change of momentum. Defining the 4-vector = d d , we can promptly discover that = . Referring back to Eq. (6.16), and writing for the spatial components of the 4-vector, , we discover that = ( + ) (observing that in the non-relativistic limit = reassures us that this quantity does have at least something to do with the thing we are familiar with as ‘force’, and that we are justified in calling the ‘relativistic 3-force’). But this tells us that is not parallel to , as we (and Newton) might expect. What else might we expect? In newtonian dynamics, ⋅ d is the work done on a particle by a force which displaces it by an amount d – what is the analogue in relativity? We can write ⋅ d = ⋅ d, but ⋅ = d d ⋅ = ⋅ d d = 1 2 d d ( ⋅ ) = 0, since we know that ⋅ is conserved. So, although in newtonian dynamics the effect of a force on a particle is to do ‘work’ on it, and so change its energy, the effect of a relativistic force is not to change the particle’s energy- momentum, but instead to simply ‘rotate’ the particle in Minkowski space – it changes the direction of the particle’s velocity vector, , but not its (invariant) length. 7.7 An Example: Compton Scattering 139 Q 1 E, p Q 2 θ φ Figure 7.3 Compton scattering: a photon being scattered from a charged par- ticle.
eBook - ePub- J. Kenneth Shultis, Richard E. Faw(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
If the particle was also in some potential field, for example, an electric field, the total energy would also include the potential energy. Thus one has E = m c 2. (2.11) This well known equation is the cornerstone of nuclear energy analyses. It shows the equivalence of energy and mass. One can be converted into the other in precisely the amount specified by E = mc 2. When we later study various nuclear reactions, many examples of energy being converted into mass and mass being converted into energy will be seen. Example 2.2: What is the energy equivalent in MeV of the electron rest mass? From data in Table 1.5 and Eq. 1.1 one finds E = m e c 2 = (9.109 × 10 − 31 kg) × (2.998 × 10 8 m / s) 2 × (1 J / (kg m 2 s − 2) / (1.602 × 10 − 13 J / MeV) = 0.5110 MeV When dealing with masses on the atomic scale, it is often easier to use masses measured in atomic mass units (u) and the conversion factor of 931.49 MeV/u. With this important conversion factor one obtains E = m e c 2 = (5.486 × 10 − 4 u) × (931.49 MeV/u) = 0.5110 MeV. Reduction to Classical Mechanics For slowly moving particles, that is, v ≪ c, Eq. (2.10) yields the usual classical result. Since, 1 1 − v 2 / c 2 ≡ (1 − v 2 / c 2) − 1 / 2 = 1 + v 2 2 c 2 + 3 v 2 8 c 2 + 3 v 2 8 c 2 + ⋯ ≃ 1 + v 2 2 c 2, (2.12) the kinetic energy of a slowly moving particle. is T = m o c 2 (1 1 − v 2 / c 2 = 1) = m o c 2 ([ 1 + v 2 2 c 2 + ⋯ ] − 1) ≃ 1 2 m o v 2. (2.13) Thus the relativistic kinetic energy reduces to the classical expression for kinetic energy if v ≪ c, a reassuring result since the validity of classical mechanics is well established in the macroscopic world. Relation Between Kinetic Energy and Momentum Both classically and relativistically, the momentum p of a particle is given by p = m v. (2.14) In classical physics, a particle’s kinetic energy T is given. by T = m v 2 2 = p 2 2 m, which yields p = 2 m T. (2.15) For relativistic particles, the relationship between momentum and kinetic energy is not as simple
eBook - ePubThe Nature of Temporal (t > 0) Science
A Physically Realizable Principle
- Francis T.S. Yu(Author)
- 2022(Publication Date)
- CRC Press(Publisher)
where E is energy, M is mass, and c is the velocity of light. From this we see that our current science is limited by the speed of light. The fact is that Einstein's energy equation had given us the fundamental limit of energy; that is, mass and energy are equivalent. In this one of most famous and important equations that more than haft the world's population may have known it but many of them may not actually understand its physical significance. Anyway, this equation strictly speaking it is not consistent with a physically realizable axiom of our temporal (t > 0) universe since the special theory was developed from a non-physically realizable paradigm as I had shown. Yet the physical significance of Einstein's energy equation remains; energy and mass are equivalent.5.5 A New Mass Energy Equation
It is trivial that Einstein's relativity equation can be written in a relativistic mass formula as given by:M =(5.14)M 0(1 −)v 2/c 2− 1 / 2where M is the effective mass (or mass in motion), Mois the rest mass, v is the velocity of the moving M, and c is the speed of light. In other words, the effective mass of a moving particle increases at the same amount with respect to the relativistic time window Δt′ (i.e., time dilation). Since Einstein's special theory was developed within an empty subspace, it has no induced gravitational field, as can be seen in Figure 5.11(a) . Besides, it is not a physically realizable subspace paradigm, and it is trivial that empty space cannot support an induced gravitational field. In contrast, if it is situated within a temporal (t > 0) subspace shown in Figure 5.11(b), an induced gravitational field F is inherently attached with a temporal (t > 0) mass M(t) in motion.Figure 5.11
eBook - ePubClassical Mechanics
From Newton to Einstein: A Modern Introduction
- Martin W. McCall(Author)
- 2011(Publication Date)
- Wiley(Publisher)
are already accounted for once it is established experimentally that the rest mass and momentum of an isolated system are conserved. The conservation of energy has a higher objective status in relativity.Why doesn’t the rest energy matter in Newtonian physics? For a particle moving with speed v we could include a rest energy term and write for a free particle(6.19)Applying energy conservation to the elastic collision of two particles would then lead to the equation(6.20)which is of course the same as Equation (4.19 ) since the rest energy terms cancel. The rest terms still cancel in calculating the loss of kinetic energy in inelastic collisions. However, in both cases there is no implication for the total system mass, which is given by m 1 + m 2 . In Newtonian physics there is no coupling between the loss of energy and the total system mass, and we simply say that the energy is converted to heat. Relativistically, however, the total rest mass of two moving particles is always greater than the individual rest masses. When two particles collide head on and stick together and are then at rest in the lab frame, their total mass is still greater than m 1 + m 2 , the excess accounting for the loss of kinetic energy. All of this works provided the rest mass of the particles is included in the calculation .The argument works in reverse. A stationary massive particle which disintegrates into two smaller particles which fly apart must liberate some of the energy which held it together to provide kinetic energy to the fragments . This is the basis of nuclear fission, in which the binding energy of a large nucleus is released as kinetic energy of the fragments after it has disintegrated. Every body has the potential to provide energy to fragments into which it might disintegrate, and this potentiality is the rest energy, E rest = mc 2
- Martinus J G Veltman(Author)
- 2018(Publication Date)
- WSPC(Publisher)
When asked: “What is your nationality?”, Einstein answered: “That will be decided only after my death. If my theories are borne out by experiment, the Germans will say that I was a German and the French will say that I was a Jew. If they are not confirmed, the Germans will say that I was a Jew and the French will say that I was a German.” In actual fact, Einstein kept his Swiss nationality until his death, in addition to his US citizenship.In the figure the dashed curve shows the energy versus the velocity in the pre-relativistic theory, the solid curve shows the same relationship in today’s theory. In experimental particle physics one practically always deals with ultra-relativistic particles, with speeds within a fraction of a percent (such as 1/100%) from the speed of light. It is clearly better to work directly with momentum rather than with velocity.The relation between energy and momentum changes much less dramatically when passing from the pre-relativistic formulation to the relativistic theory. In fact, the energy increases less sharply with momentum, and for very high values of the momentum the energy becomes proportional to it. (Energy approximately equals momentum times c, the speed of light.) A typical case is shown in the next figure, with the dashed line showing the non-relativistic case, the solid curve the relativistically correct relation.The quantitatively minded reader may be reminded of the equations quoted in Chapter 1 . In particular there is the relation between energy and momentum, plotted in the next figure:or, using the choice of units such that c = 1:Another important fact is the Einstein equation E = mc2 . This very famous equation can be deduced in a number of ways, none of which is intuitively appealing. This equation tells us that even for a particle at rest the energy is not zero, but equal to its mass multiplied with the square of the speed of light. In particle physics this equation is a fact of daily life, because in inelastic processes, where the set of secondary particles is different from the primary one, there is no energy conservation unless one includes these rest-mass energies in the calculation. As the final particles have generally masses different from the primary ones, the mass-energy of the initial state is in general different from that of the final state. In fact, the first example that has already been discussed extensively is neutron decay; this decay is a beautiful and direct demonstration of Einstein’s law, E = mc2
eBook - PDF- Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
This situation violates the relativity postu- late; the type of collision (elastic versus inelastic) should de- pend on the properties of the colliding objects and not on the particular reference frame from which we happen to be view- ing the collision. As was the case with momentum, we require a new definition of kinetic energy if we are to preserve the law of conservation of energy and the relativity postulate. The classical expression for kinetic energy also violates the second relativity postulate by allowing speeds in excess of the speed of light. There is no limit (in either classical or rela- tivistic dynamics) to the energy we can give to a particle. Yet, if we allow the kinetic energy to increase without limit, the classical expression implies that the velocity must correspondingly increase without limit, thereby violating the second postulate. We must therefore find a way to redefine ki- netic energy, so that the kinetic energy of a particle can be in- creased without limit while its speed remains less than c. The relativistic expression for the kinetic energy of a particle can be derived using essentially the same procedure we used to derive the classical expression, starting with the particle form of the work – energy theorem (see Problem 16). The result of this calculation is (20-27) Equation 20-27 looks very different from the classical ex- pression but we can show (see Exercise 35) that Eq. 20-27 does reduce to the classical expression in the limit of low speeds (v c). You can also see from Eq. 20- 27 that the relativistic expression for kinetic energy allows a particle to have unlimited energy even though its speed remains less than the speed of light.
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