Physics
Relativistic Kinematics
Relativistic kinematics is a branch of physics that deals with the motion of objects at speeds close to the speed of light. It involves the application of Einstein's theory of special relativity to describe how time, length, and mass change for objects in motion. Relativistic kinematics provides a framework for understanding the behavior of particles at high velocities.
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11 Key excerpts on "Relativistic Kinematics"
eBook - PDF- Palash B. Pal(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
Chapter 2 Relativistic Kinematics Einstein published his ideas on relativity in two major instalments. The first instalment 1905 examined how the perception of time, space and energy might differ for observers in uniform relative motion. This constituted the basis of the special theory of relativity . When the speed of any particle is very high, close to the speed of light, the kinematical consequences of this theory were found to be drastically different from those derived from Newton’s laws of motion and associated ideas of space and time. Since the study of particle physics is intimately connected to high energies and therefore large speeds, we must take relativistic effects into account in discussing kinematics of particle motion, which is what we do in this chapter. We will be brief, because we will assume that the reader is familiar with the basic tenets of the special theory of relativity. For fuller treatments, the reader is advised to consult textbooks on the subject. Einstein’s second instalment of relativity came around 1915, where he discussed observers in accelerated motion with respect to one another. This general theory of relativity turned out to be a theory of gravitation. Since we decided to neglect all gravitational effects, the ramifications of this theory will be ignored completely. 2.1 Lorentz transformation equations The special theory of relativity is based on two axioms. One of them is the principle of relativity, which states that all physical laws should have the same form to all observers who might be moving in uniform relative motion with respect to one another. This principle was first advocated by Galileo, and was inherent in the Newtonian formulation of dynamics. Later, when Maxwell formulated his theory of electromagnetism, it was found that the equations involve a constant which has the dimension of ve-locity. Further, it was realized that the constant equalled the speed of prop-agation of electromagnetic waves in the vacuum.
eBook - PDF- David Agmon, Paul Gluck;;;(Authors)
- 2009(Publication Date)
- WSPC(Publisher)
Chapter 12 The Special Theory of Relativity — Kinematics We must do science because it reveals the beauty of nature. 12.1 Introduction Einstein Newtonian mechanics is concerned with the motion of bodies acted upon by forces. Each chapter in its development was a link in a chain of elements with a common logic. The special theory of relativity developed by Albert Einstein in the year 1905 was a fundamental break in this tradition and a point of departure from its predecessors. It challenged the fundamental assumptions of Newtonian theory and replaced them with bold new axioms. It forced us to review and revise our most fundamental notions concerning the nature of time, space, force, mass, momentum and energy. The major difficulty in understanding relativity theory is not mathematical, but rather in coming to terms with its basic abstract ideas. The mathematical equipment required of the reader of the present chapter comprises only basic algebra and differential calculus. The theory of relativity rests on two basic postulates: 1. The velocity of light is independent of the frame of reference of any observer. Therefore, it does not depend on the velocity of the source or that of the observer. 2. The principle of relativity: All fundamental laws of physics are identical for all observers moving with constant velocities with respect to each other. Albert Einstein (1879-1955) Comments: (a) The velocity of light waves in vacuum is c = (299772.5+0.1) km/s. In this book this will be rounded to 3-10 5 km/s = 3-10 8 m/s. (b) Space is assumed to be homogeneous and isotropic, so that the velocity of light is the same in all directions. (c) An observer moving at constant velocity is called an inertial observer. (d) The second postulate embraces all laws of physics, not just mechanics. Its content is the following: All the equations describing physical laws in terms of 432
eBook - PDF- David Griffiths(Author)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
Introduction to Elementary Particles David Griffiths © 2004 WILEY-VCH Verlag GmbH & Co. Chapter 3 Relativistic Kinematics In this chapter I summarize the basic principles, notation, and terminology of Relativistic Kinematics. This is material you must know cold in order to understand Chapters 6 through 11 (it is not needed for Chapters 4 and 5, however, and if you prefer you can read themfirst).Although the treatment is reasonably self-contained, I do assume that you have encountered special relativity before if not, you should pause here and read the appropriate chapter in any introductory physics text before proceeding. If you are already quite familiar with relativity, this chapter will be an easy review but read through it anyway because some of the notation may be new to you. 3.1 LORENTZ TRANSFORMATIONS According to the special theory of relativity, 1 the laws of physics apply just as well in a reference system moving at constant velocity as they do in one at rest. An embarrassing implication of this is that there's no way of telling which system (if any) is at rest, and hence there is no way of knowing what "the" velocity of any other system might be. So perhaps I had better start over. Ahem. According to the special theory of relativity, 1 the laws of physics are equally valid in all inertial reference systems. An inertial system is one in which Newton's first law (the law of inertia) is obeyed: objects keep moving in straight lines at constant speeds unless acted upon by some force.* It's easy to see that any two inertial systems must be moving at constant velocity with respect to one another, and conversely, that any system moving at constant velocity with respect to an inertial system is itself inertial. * If you are wondering whether a freely falling system in a uniform gravitationalfieldis "inertial," you know more than is good for you. Let's just keep gravity out of it. 81
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
The predictions of special relativity agree well with Newtonian mechanics in their common realm of applicability, specifically in experiments in which all velocities are small compared with the speed of light. Special relativity reveals that c is not just the velocity of a certain phenomenon—namely the propagation of electromagnetic radiation (light)—but rather a fundamental feature of the way space and time are unified as spacetime. One of the consequences of the theory is that it is impossible for any particle that has rest mass to be accelerated to the speed of light. The theory is termed special because it applies the principle of relativity only to the special case of inertial reference frames, i.e. frames of reference in uniform relative motion with respect to each other. Einstein developed general relativity to apply the principle in the more general case, that is, to any frame so as to handle general coordinate transformations, and that theory includes the effects of gravity. From the theory of general relativity it follows that special relativity will still apply locally (i.e., to first order), and hence to any relativistic situation where gravity is not a significant factor. Inertial frames should be identified with non-rotating Cartesian coordinate systems constructed around any free falling trajectory as a time axis. Postulates “ Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the ” ________________________ WORLD TECHNOLOGIES ________________________ possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results...
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
The predictions of special relativity agree well with Newtonian mechanics in their common realm of applicability, specifically in experiments in which all velocities are small compared with the speed of light. Special relativity reveals that c is not just the velocity of a certain phenomenon—namely the propagation of electromagnetic radiation (light)—but rather a fundamental feature of the way space and time are unified as spacetime. One of the consequences of the theory is that it is impossible for any particle that has rest mass to be accelerated to the speed of light. The theory is termed special because it applies the principle of relativity only to the special case of inertial reference frames, i.e. frames of reference in uniform relative motion with respect to each other. Einstein developed general relativity to apply the principle in the more general case, that is, to any frame so as to handle general coordinate transformations, and that theory includes the effects of gravity. From the theory of general relativity it follows that special relativity will still apply locally (i.e., to first order), and hence to any relativistic situation where gravity is not a significant factor. Inertial frames should be identified with non-rotating Cartesian coordinate systems constructed around any free falling trajectory as a time axis. Postulates “ Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of ” ________________________ WORLD TECHNOLOGIES ________________________ discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results...
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
The predictions of special relativity agree well with Newtonian mechanics in their common realm of applicability, specifically in experiments in which all velocities are small compared with the speed of light. Special relativity reveals that c is not just the velocity of a certain phenomenon—namely the propagation of electromagnetic radiation (light)—but rather a fundamental feature of the way space and time are unified as spacetime. One of the consequences of the theory is that it is impossible for any particle that has rest mass to be accelerated to the speed of light. The theory is termed special because it applies the principle of relativity only to the special case of inertial reference frames, i.e. frames of reference in uniform relative motion with respect to each other. Einstein developed general relativity to apply the principle in the more general case, that is, to any frame so as to handle general coordinate transformations, and that theory includes the effects of gravity. From the theory of general relativity it follows that special relativity will still apply locally (i.e., to first order), and hence to any relativistic situation where gravity is not a significant factor. Inertial frames should be identified with non-rotating Cartesian coordinate systems constructed around any free falling trajectory as a time axis. Postulates “ Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results...
eBook - PDF- Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
451 20-1 TROUBLES WITH CLASSICAL PHYSICS The kinematics developed by Galileo and the mechanics developed by Newton, which form the basis of what we call classical physics, had many triumphs. Particularly notewor- thy are the understanding of the motion of the planets and the use of kinetic theory to explain certain observed proper- ties of gases. However, a number of experimental phenom- ena cannot be understood with these otherwise successful classical theories. Let us consider a few of these difficulties. We will consider examples of experiments specifically de- signed to reveal the limitations of classical physics and — as we shall see — to serve as tests of Einstein’s special the- ory of relativity. Troubles with Our Ideas about Time The pion ( or ) is a particle that can be created in a high-energy particle accelerator. It is a very unstable parti- cle; pions created at rest are observed to decay (to other particles) with an average lifetime of only 26.0 ns (26.0 10 9 s). In one particular experiment, pions were created in motion at a speed of v 0.913c (where c is the speed of light, 3.00 10 8 m/s). In this case they were observed to travel in the laboratory an average distance of D 17.4 m before decaying, from which we conclude that they decay in a time given by D/v 63.7 ns, much larger than the life- time measured for pions at rest (26.0 ns). This effect, called time dilation, suggests that something about the relative motion between the pion and the laboratory has stretched the measured time interval by a factor of about 2.5. Such an effect cannot be explained by Newtonian physics, in which time is a universal coordinate having identical values for all observers. CHAPTER 20 CHAPTER 20 THE SPECIAL THEORY OF RELATIVITY* T he special theory of relativity has an undeserved rep- utation as a difficult subject. It is not mathematically complicated; most of its details can be understood us- ing techniques well known to readers of this text.
eBook - PDF- A R Prasanna(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
The name is connected with the fact that motion from the point of view of possible experience always appears as the relative motion of one object with respect to another. Motion is never observ-able as ‘motion with respect to space’ or, as it has been expressed as ‘absolute Special Relativity 87 motion’. The principle of relativity in its widest sense is contained in the state-ment: ‘The totality of physical phenomena is of such a character that it gives no basis for the introduction of the concept of, absolute motion’. 3.2 Postulates and kinematics Einstein’s famous paper of 1905 [14], ‘On the Electrodynamics of Moving Bodies’ starts thus: It is known that Maxwell‘s electrodynamics, when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. For example in the case of the reciprocal electrodynamic action of a magnet and a conductor, the observable depends only on the relative motion of the conductor and the magnet. Examples of this sort, together with the unsuccess-ful attempts to discover any motion of the earth relative to the ‘light medium’, suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest; rather the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (hereafter be called the ‘Principle of relativity’) to the status of a postulate. Physics deals with observations of events which occur in space (location) at a given time. Thus one requires four numbers to identify an event . While dealing with Newtonian physics, since ‘time’ was absolute, it was considered to be the same for all observers and thus one could do with the three– dimensional Euclidean manifold for describing the physical laws associated with ‘events’ in space and time.
eBook - PDFRelativity: The Theory and Its Philosophy
Foundations & Philosophy of Science & Technology
- Roger B. Angel, Mario Bunge(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
The reader may wonder 94 Relativity: The Theory and its Philosophy how it is possible that a photon, a particle of light, is capable of travelling at the speed of light. The answer is that although a photon possesses relativistic inertia, it has a vanishing rest-mass. We may now give formal recognition to the relativistic counterpart of classical momentum, which was implicit in (3.39) and (3.40). It is a four-vector which is defined by P» = df. moil. (3.43) Clearly, the first three of its components are m 0 v l τ n , tt t m 0 c r. Its fourth component will be , . X4.CJ IV^V*» 111 V V i l i l U V l l V H l YT 111 <^W . . jl-v 2 /c 2 J-v 2 /c 2 Consequently, the relativistic law of conservation telescopes the conservation of momentum and the conservation of mass. The reader should take note that Ρ μ as defined by (3.43) may be multiplied by the factor c. The fourth component then has the dimensions of energy. It then turns out that (cP 1 ) 2 -E 2 is an invariant. Hence, the possibility of constructing the four-vector Ρ μ leads to the combining of the principles of conservation of momentum and conservation of energy. The reason why the time component of Ρ μ may be variously treated as pertaining to both mass and energy will shortly emerge. Our final step is to construct a relativistic law of motion involving the four-force or Minkowski force f. The relativistic law of motion is simply άΡ μ άι μ It has already been remarked that the relativistic increase in mass or inertia is a highly confirmed experimental fact. We shall now establish another highly confirmed result for which special relativity is perhaps unjustly famous. Our derivation will be somewhat heuristic since the rigorous proof is a trifle complicated. We begin by making use of the purely formal result, which the more ambitious reader may derive for himself, that F ß and Ρ μ are orthogonal vectors in spacetime.
- Moses Fayngold, Roland Wengenmayr(Authors)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
4.2 Relativistic dynamics The change in our concepts of space and time causes similar changes in our con- cepts of motion and force. We have already seen that the velocity of a moving object behaves “paradoxically” when we consider it from different reference frames. In par- ticular, it does not obey the “obvious” law of addition of velocities. We can rephrase this in the language of transformations by stating that the velocity behaves differ- ently with respect to different transformations: so far as we are confined to one refer- ence frame (for instance, we rotate the axes in our three-dimensional space, but re- main within the same system K), time t behaves as a scalar quantity, and the velocity v transforms like a regular three-dimensional vector. However, when we perform ro- tations in four-dimensional space–time involving the time axis (which physically cor- responds to a transition to another reference frame), the velocity does not behave as a four-dimensional vector. This role is taken up by the “representative” – the 4-velo- city. We have defined the 4-velocity in Equation (1) as u i = dx i /ds, i = 0, 1, 2, 3, where index 0 corresponds to the time axis, and 1, 2, and 3 correspond to the three spatial axes, whereas the interval ds is a scalar under 4-rotations. The same can be done to describe the dynamic characteristics – momentum, energy, and force. By analogy with the definition of 3-momentum as a vector with three com- 66 4 Relativistic Mechanics of a Point Mass ponents, p a = m 0 v a = m 0 dx a /dt, we have defined 4-momentum as a vector with four components: P i m 0 u i m 0 dx i ds 15 How do these changes affect the concept of force and accelerated motion? The basic definition of force is f dP dt 16 Formally it is similar to definition of velocity as v = dr/dt. Like the velocity, force be- haves as a three-dimensional vector under restricted Lorentz transformations not in- volving time (pure spatial rotations).
eBook - PDFForces in Physics
A Historical Perspective
- Steven N. Shore(Author)
- 2008(Publication Date)
- Greenwood(Publisher)
A striking consequence is the change in the addition law for relative motion. Instead of the Galilean form, he found that if a speed v is seen in a frame moving with a velocity u, instead of the relative motion being v + V it becomes v = (v + u)/(1 + uv/c 2 ). In all cases, this is a direct consequence of keeping the constant speed of light for all observers—no signal can propagate causally faster than c and two observers will always, regardless of their relative frame motion V , see the same value for the speed of light. It then follows that no body with a finite mass can travel at a speed c. Let me digress for just a moment. The fundamental picture of the universe containing an absolute space and time relative to which, independent of the observer, all motion can be measured had not changed for two thousand years. It had been elaborated differently through the centuries but had even underpinned Newton’s development of mechanics. The idea that all physical experience is only relative and that there is no longer a separation between the motion of an observer and the observation was a rupture in the foundations of physics. Every change in ideas about motion has provoked a change in how force is understood but Einstein had now replaced space and time with a new entity, spacetime, so designated by Minkowki. To pass from kinematics to dynamics, however, required another postulate, again related to the speed of light. The two observers must be able to agree on the measurements they’re making. That requires some way to establish what it means to do anything simultaneously. At some moment in space and time the two observers must be able to coordinate their measurement device together. Further, their measurements should be independent of which frame they’re in. This is the principle of covariance, that the physical laws are independent of the state of relative motion of the observers.
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