Physics
Relativistic Dynamics
Relativistic dynamics is a branch of physics that describes the motion of objects at speeds approaching the speed of light. It incorporates the principles of special relativity, which include the concept of time dilation, length contraction, and the equivalence of mass and energy. Relativistic dynamics provides a more accurate description of motion at high speeds than classical mechanics.
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11 Key excerpts on "Relativistic Dynamics"
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.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
In addition, particle accelerators routinely accelerate and measure the properties of particles moving at near the speed of light, where their behavior is completely consistent with relativity theory and inconsistent with the earlier Newtonian mechanics. These machines would simply not work if they were not engineered according to relativistic principles. General relativity A simulated black hole of ten solar masses as seen from a distance of 600 kilometers with the Milky Way in the background. ________________________ WORLD TECHNOLOGIES ________________________ General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1915. It is the current description of gravitation in modern physics. It generalises special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the four-momentum (mass-energy and linear momentum) of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations. Many predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time dilation, the gravitational redshift of light, and the gravitational time delay. General relativity's predictions have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data.
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.
eBook - PDF- A R Prasanna(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
Chapter 3 Special Relativity 3.1 Introduction Towards the end of the nineteenth century, before the beginnings of the new era of physics, the two long range forces, gravitation (mechanics of masses) and electromagnetism (mechanics of charges and currents), the two funda-mental attributes of matter, held the center stage for understanding Nature. Encompassing them were two major theories, the concept of inertia and the laws of mechanics as propounded by Galileo and Newton, and the laws of elec-trodynamics as given by Faraday and Maxwell. By then it was also confirmed that light is an electromagnetic wave, having a finite velocity, albeit very large. Galilean relativity, which was the basis of Newtonian mechanics, was ac-cepted totally, with its consequential ‘action at a distance’ theory for grav-ity, supported by the Newtonian idea of ‘absolute time’. However, there were physicists who were not happy with the fact, whereas there was a theory of rel-ativity for mechanics, there was none for electrodynamics. Prominent among them were Lorentz and Poincare. This discomfort was further strengthened by the fact that the Michelson–Morley experiments did not give any positive result regarding the nature of aether and its role as a medium for light waves[7]. It is at this juncture that Einstein appeared on the scientific scene to create the theory of relativity that held good both for mechanics and elec-trodynamics. Before taking up the Einsteinian approach, let us briefly review the background on the theoretical front during that time.
eBook - PDF- Ashok Das(Author)
- 2011(Publication Date)
- World Scientific(Publisher)
Chapter 2 Relativistic Dynamics 2.1 Relativistic point particle It is clear that Newton’s equation which is covariant (form invari-ant) under Galilean transformations, is not covariant under Lorentz transformations. So if we require Lorentz invariance to be the invari-ance group of the physical world, we must modify Newton’s laws. Let us note, however, that since Newton’s laws have been tested again and again in the laboratory for slowly moving particles, it is only for highly relativistic particles ( | v | similarequal c = 1) that the equation may require any modification. We have already seen that in Minkowski space we can write the line element as d s 2 = d τ 2 = η μν d x μ d x ν = d t 2 -d x 2 , (2.1) where d τ is known as the “proper time” of the particle and is related to the proper length through the speed of light (d s = c d τ ) which we have set to unity. The proper time, like the proper length, is an invariant quantity (since the speed of light is an invariant) and hence can be used to characterize the trajectory of a particle. To describe a Lorentz covariant generalization of Newton’s equation, let us define a force four vector f μ such that m d 2 x μ d τ 2 = f μ , (2.2) where m represents the rest mass of the particle. Clearly since the proper time does not change under a Lorentz transformation, equa-tion (2.2) takes the same form in all Lorentz frames. Furthermore, if 35 36 2 Relativistic Dynamics we know the form of f μ , we can calculate the trajectory of the parti-cle. To understand the nature of this force four vector, we note that the form of the relativistic force four vector can be obtained from the observation that since (2.2) holds in all Lorentz frames, it must be valid in the rest frame as well. In the rest frame of the particle ( d x d t = 0) d τ = d t, (2.3) and hence (2.2) takes the form m d 2 x i d t 2 = f i (rest) , f 0 (rest) = 0 .
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...
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...
- F N H Robinson(Author)
- 1996(Publication Date)
- WSPC(Publisher)
This is readily integ-rated to give ct = (x 2 +2xmc 2 /F)^, and from this we get x = [mc 2 /F] [ (1 + F 2 t 2 /m 2 c 2 )* - 1] . (6.31) For Ft«mc this gives the classical result x = HFt 2 /m, and for Ft » mc, x -> ct as we might expect. If we identify F/m with the constant acceleration a, it is identical with Eq.(3.10f), x = (c 2 /a) [ (1+a 2 1 2 / c 2 )* -1] and this confirms the idea that we introduced in Chapter 3: that constant acceleration in the instantaneous, comoving, rest frame of a particle corresponds to a constant force. 6.11 Summary We started our development of special relativity by add-ing the universal constancy of the velocity of light to a few rather general assumptions about the nature of space and time. The resulting Lorentz transformation specifies the relation between the coordinates in time and space ascribed to the same event, when we use two different inertial systems in uniform relative motion. We then explored how this new relation modifies the kinematic description of physical phenomena when speeds comparable with c are involved, the most obvious effects being time dilation and the relativistic Doppler effect. Next, in Chapter 5, we looked at the consequences of requir-ing conservation of linear momentum in a simple collision to be valid in any inertial frame, if it were valid in one particular frame. This led to replacing p=mv by p=my(v)v, and thus to new relations such as F 2 =p 2 c 2 +m 2 c 4 and the new form of dynamics used in this chapter. A theory which started by describing how descriptions of events in two different frames are related has thus generated new dynami-cal laws operating within a single reference frame. 80 Relativity The behaviour of an equation expressing a physical law, when the coordinate system is subjected to a Lorentz transformation, determines whether that law and the princi-ples of special relativity are compatible.
- Sergey Siparov(Author)
- 2011(Publication Date)
- World Scientific(Publisher)
This is an example of science fiction for scientists. Classical relativity: Scope and beyond 35 physical world, but there was one that is suited. Einstein’s principle of relativity accounts for electrodynamics alongside with mechanics and is the natural extension of Galileo’s principle of relativ-ity, but the way it was obtained by Einstein is different from that of Galileo. The problem of relative motion was realized and discussed throughout all the preceding three centuries but mainly in the philosophical qualitative sense. Einstein gave it an operational definition, that is, in order to con-struct a model of space and time, he started not with the concept but with the procedure that was expected to test it, and attributed the character of a postulate to the result of this procedure. Large variety of experiments with electricity and magnetism are described by one and the same set of Maxwell equations. The equations predicted the existence of electromag-netic waves that were subsequently discovered by Hertz, and Faraday’s experiments proved that the light and electromagnetic waves had the same nature. As one can test in an experiment, the speed of light propagation is a) finite and b) appears to be the same when measured in any inertial frame. As it was underlined by Poincar´ e already in 1898, item a) ruins the absolute simultaneity of events, i.e. the absolute character of time. Item b) is equivalent to the following statement: with Galileo transformations of coordinates, eq. (1.10), that preserve the intuitive model of absolute space with or without ether, the results of the solution of Maxwell equations and even the form of these equations start to depend on the chosen IRF, that is on the state of an observer; with Lorentz transformations eq. (1.33) that preserve causality and the form of the testable Maxwell equations in every IRF, the intuitive model of space requires modification.
eBook - PDFForces in Physics
A Historical Perspective
- Steven N. Shore(Author)
- 2008(Publication Date)
- Greenwood(Publisher)
8 THE RELATIVITY OF MOTION In the last few days I have completed one of the finest papers of my life. When you are older, I will tell you about it. —Albert Einstein to his son Hans Albert, November 4, 1915 Electromagnetic theory achieved a unification of two forces but at a price. It requi- red a medium, the ether, to support the waves that transmit the force. But this takes time and there is a delay in arrival of a signal from a source when something is changing or moving during this time interval. How do you know something is mov- ing? Remember that Aristotle used the notion of place within space or memory to recognize first change and then motion. So there the motion is known by compari- son to a previous state. But this isn’t the same question that recurs in the seven- teenth century. According to Descartes, we have to look instead at the disposition of bodies relative to which motion is known. These aren’t the same statement, although that would again divert us into metaphysics. Instead, we can look at what this means for the force concept. As Newton was at pains to explain, inertia requires two quantities, velocity and mass, to specify the quantity, momentum that is conserved. While mass is both a scalar quantity and a primitive in the dynamical principles, the momentum is not. It requires a direction and therefore, if we identify the inertial state as one moving at constant velocity, we need to know relative to what we take this motion. This is where we begin our discussion of the revolutionary developments at the start of the twentieth century: time, not only space and motion, is relative to the observer and dependent on the choice of reference frame and, consequently, has the same status as a coordinate because of its dependence on the motion of the observers. It follows that the notion of force as defined to this point in our discussions must be completely re-examined.
Our intuitions about the addition of velocities could be wrong, in which case the universe might be a very strange place indeed. Einstein focused on the third of these possibilities. The idea that the laws of nature are the same in all frames of reference is called the principle of relativity, and can be stated as follows: • • • Every observer must experience the same natural laws. • • • This statement is the central assumption of Einstein’s theory of relativity. Hid- den beneath this seemingly simple statement lies a view of the universe that is both strange and wonderful. The extraordinary theoretical effort required to understand the consequences of this one simple assumption occupied Einstein during much of the first decades of the twentieth century. Stop and Think! It may seem obvious that the laws of nature are the same everywhere in the universe, but how can we know for sure? How might you test this statement? We can begin to understand Einstein’s work by recalling what Isaac Newton had demonstrated three centuries earlier—that all motions fall into one of two categories: uniform motion or acceleration (Chapter 2). Einstein therefore divided his theory of relativity into two parts—one dealing with each of these kinds of motion. The easier part, first published by Einstein in 1905, is called special relativity and deals with all frames of reference in uniform motion relative to one another—reference frames that do not accelerate. It took Einstein another decade to complete his treatment of general relativity, mathematically a much more complex theory, which applies to any reference frame whether or not it is accelerating relative to another. At first glance, the underlying principle of relativity seems obvious, perhaps almost too simple. Of course the laws of nature are the same everywhere—that’s the only way that scientists can explain how the universe behaves in an ordered way.
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