CONTENTS

Introduction

Chapter 01. Classical Mechanics
Chapter 02. Relativistic Mechanics
Chapter 03. Electromagnetic Field Theory
Chapter 04. Gravitational Field Theory
Chapter 05. Quantum Mechanics (Nonrelativistic Theory)
Chapter 06. Relativistic Quantum Theory
Chapter 07. Physics of Elementary Particles
Chapter 08. Nuclear Physics
Chapter 09. Statistical Physics
Chapter 10. Physical Kinetics

Chapter 02. Relativistic Mechanics

Galileo’s relativity principle, we saw, leads to the conclusion that interactions between bodies propagate through space instantaneously. Experience shows, however,

The constant c
that there is no such things as instantaneous interactions in nature. Hence, classical mechanics, which proceeds from the concept of instantaneous propagation of interaction, carries an inherent error. For, if there is a change in one of two interacting bodies, it will tell on the other body only after a certain time interval. Only after this necessary time has elapsed will the processes engendered by the change begin to take place in the second body. Division of the distance between the two bodies by this time interval yields the velocity of propagation of the interaction.
           From the relativity principle(1) it follows, in particular, that the limiting velocity of propagation of interactions is the same in all inertial frames of reference. In other words, it is a universal constant, denoted by the symbol c. The latest measurements set its value at
c = 2.99776 x  cm/sec.
           The tremendous magnitude of this velocity explains why classical mechanics is sufficiently accurate for real situations. Most of the velocities with which we have to deal are so small in comparison with c that in practice the proposition concerning the infinity of the latter has no effect on the accuracy of computations.

Relativistic relativity principle
           The relativity principle combined with the postulate of the finality of the propagation speed of interactions gives Einstein’s principle of relativity, as opposed to Galileo’s which presumed that interactions can propagate with infinite velocity.
           The mechanics based on Einstein’s relativity principle is called relativistic. In the limiting case of small velocities in comparison with c the effects of a finite propagation speed can be neglected, and relativistic mechanics turns into conventional classical mechanics based on the assumption of instantaneous propagation of interaction. The limiting transition from relativistic to classical mechanics may be formally expressed as going over to the limit  in the equations of relativistic mechanics.

Space-time separation
           We saw before that the limiting propagation speed c is the same in all inertial reference frames. Let us express this mathematically. Consider two coordinate systems O and O’ moving with uniform velocity relative to each other. Let the coordinate axes be so directed that axis Ox coincides with axis O’x’ and the y and z axes are parallel to the respective primed axes; the time is t and t’ in the unprimed and primed systems, respectively.
          Consider an event(2) viewed in system O as taking place at a point  at time  in that system. Let the event be the dispatch of a signal(3) propagating with the velocity c. Let the second event be the arrival of the signal at a point  at time . The velocity of the signal being c, the distance traveled by it is . On the other hand, that same distance is equal to

.

We can thus write the following relationship between the coordinates of both events in system O :

.             (10)

           The propagation of the signal and the two events can also be observed from system O’. Let the space-time coordinates of the first event in the primed system be  and of the second event . As the velocity of the signal is the same in both systems, we have, by analogy with (10),

.

The coordinates of any two events being  and , the quantity



is called the space-time separation between the two events.

           It follows thus from the constancy of the velocity c that if the separation between the two events is zero in one inertial frame it is zero in any other. For the space-time separation(4) between two infinitely close events we get

,

which is known also as the squared interval.

Invariance of space-time separation
           As shown above, if dS=0 in some inertial frame, then dS’=0 in any other. On the other hand, dS and dS’ are quantities of the same order of smallness. It follows from these two considerations that dS and dS’ must be proportionate:

,

the coefficientbeing dependent only on the absolute value of the relative velocity of both inertial systems. It cannot depend on the space-time coordinates, as otherwise different points of space and moments of time would not be homologous, which contradicts the notion of space and time homogeneity. Neither can it be depended on the direction of the relative velocity, as this would contradict the notion of space isotropy. Hence, we can write


with the same justification as we wrote , as the velocity of the first system relative to the second is equal to the velocity of the second relative to the first. Substituting  into , we find that , i.e., . For selecting one of these values note that  can be either always +1 or always –1. For if  could be +1 for some velocities and –1 for others, for yet others it would have to have values lying between +1 and –1, which is impossible. But if that is so then a must always be equal to +1, since a special case of the transformation of  is the identity , where . From dS’=dS it follows that for all finite separation S’=S.

           Thus we arrive at a very important result: the space-time separation between events is the same in all inertial frames of reference, i.e., it is invariant in relation to transformation from one coordinate system to another. This invariance is the mathematical expression of the constancy of the velocity c.

The Lorentz transformations
           Now let us develop the formulas for going over from one inertial reference frame to another, i.e., the formulas by which, knowing the coordinates x,y,z,t of an event in some reference system O one can find the coordinates x’,y’,z’,t’ of the same event in another inertial reference system O’.
           In classical non-relativistic mechanics this problem is easily solved. As time is absolute, t=t’. Then, if the coordinate axes are so chosen that axes x and x’ coincide and axes y and z are respectively parallel to axes y’ and z’, the motion being along the axes x and x’, the coordinates y and z will, evidently, be equal to the corresponding coordinates y’ and z’, while the difference between x and x’ will be given by the distance traveled by one system with respect to the other. If the time count has been chosen to start when the origins of the two systems coincided, and the velocity of the primed system relative to the unprimed one is V, the distance traveled is Vt. Thus,

, , , .                   (11)

These are the transformation formulas in classical mechanics (the Galilean transformations). It can easily be proved that they do not satisfy relativity theory, as could be expected: they do not leave the separation between events invariant.

The relativistic transformation formulas must be sought on the basis of the requirement of separation invariance. Using in the following discourse the more convenient quantity , the space-time separation, as we have seen (cf. footnotes on …) can be regarded as the distance between two corresponding points in a four-dimensional coordinate system. We can say, therefore, that the required transformation must leave unchanged all the lengths x, y, z, in the four-dimensional space. But such transformations represent either parallel translations or rotations of coordinate systems. Of these, translation of coordinate system parallel to itself is of no interest as it simply means a transfer of the origin of the spatial coordinates and a new beginning of the time count. Thus the required transformation must be mathematically expressed as a rotation of a four-dimensional coordinate system x, y, z,.
           Any rotation in four-dimensional space can be resolved into six rotations, viz., rotations in the planes xy, zy, xz, ,, (just as any rotation in ordinary space can be resolved into three rotations in the planes xy, zy, and zx). The first three of these rotations transform only the spatial coordinates; they correspond to ordinary spatial rotations.
           Let us investigate the rotation of plane ; the coordinates y and z remain unchanged. If  is the angle of rotation, the connection between the old and new coordinates is given by the formulas

,
,                            (12)

           We are seeking the formulas for going over from one inertial reference system O to another system O’ moving relative to the former with a velocity V along the x axis. It is evident that only the x coordinates and the time t undergo a transformation, hence the transformation must be of the form (12). All that is left is to determine the angle , which can be dependent only on the relative velocity V(5).

           Consider the motion of the origin of system O’ in system O. In this case x’=0, and the equations (12) take the form

,   ,

or, dividing the one by the other,

.
But , and x/t is, evidently, the velocity V of system O’ relative to O. Thus,

,

whence

,   .

Substituting the expressions for  and  into (12), we find that

,   ,   ,   .

Substituting, furthermore,  and , we obtain finally

,   ,   ,   .               (13)

These expressions are known as the Lorentz transformations, and they are of fundamental importance in contemporary physics.

Corollaries of the Lorentz transformations
           It will be readily observed that in the limiting case of classical mechanics () the Lorentz transformations change into the old Galilean transformations (11).
           At  the coordinates x and t in equations (13) become imaginary; this corresponds to the statement that motion faster than c is impossible. It is also impossible to use a reference system traveling with the velocity c, as the denominators in the formulas (13) would vanish.

Length in relativity theory
           Now let there be a measuring rod at rest in system O  lying parallel to the x axis. Its length measured in the system is , where  and  are the coordinates of its ends. What is the length of the rod in system O’  ? To determine it we must locate the coordinates  and  of both ends of the rod in the primed system at precisely the same instant . From equations (13) we have

,   
The length of the rod in the primed system is . Subtraction of  from  yields

,                               (14)

           The proper length of the measuring rod is its length in the reference system where it is at rest. Denoting it by the symbol , and its length in some other reference system O’  by l, we obtain

.

           Thus the rod is longest in the system in which it is at rest. Its length in a reference system relative to which it is moving with velocity V is shorter by the factor . This consequence of theory is known as the Lorentz contraction.

           For the limiting case of  (classical mechanics) we obtain from equation (14) , which is completely in accord with classical notions concerning the absolute nature of length.

Time interval in relativity theory
           Now consider two events taking place at the same point in space, whose coordinates are  in system O’. According to a clock at rest in the system the time between the two events in the primed system is . Let us find the time interval  between the two events as measure in system O.
           From (13), we have

,    .

Subtracting, we obtain

,

which can be rewritten

.                             (15)

Time measured by a clock moving together with a given object is called the proper time of that object. The above relationship establishes the connection between proper time and time in the reference system relative to which the motion is being considered.

It is evident from equation (15) that the proper time of a moving object is always less that a corresponding time interval in a stationary system. In other words, the rhythm of a traveling cock is slower that that of a stationary clock.

In the limiting case of classical mechanics () the equality  develops, which is in accord with classical notions concerning the absolute nature of time.
           Thus relativity theory introduces important changes in the fundamental physical concepts of space and time. Our notions based on daily experience, we find, are approximations due to the fact that in everyday life we deal with velocities far below c.
           Another consequence of the Lorentz transformations arises from the formulas which relate the velocity of a material particle in one system to its velocity in another system.

Velocity composition in relativity theory
           Once again, let system O’  be moving with velocity V  along the x axis of system O, let  be the velocity component of a particle in the unprimed system and  its velocity component in the primed system. Differentiating equations (13), we obtain

,    ,  ,   .

Dividing the first three equations by the fourth, we get

,      ,      .

Dividing further the numerator and denominator of the right-hand sides of these equations by , we obtain

,       ,      .

These formulas give the law for the change in velocity in going over from one reference system to another and the rule for velocity composition in relativity theory. In the limiting case of  they turn into the formulas

,    ,   

of classical mechanics.
           In the special case of a particle moving parallel to the x axis, , . Then  and , and
.

It is readily apparent from this formula that the sum of two velocities less than, or equal to, c  cannot be greater than c.

Relativistic mechanics
           Let us examine some quantities the very existence of which follows from the most general properties of space-time symmetry. We shall begin with a single free particle.

Momentum of a relativistic particle
           The conservation of the momentum of a free particle, as we know, is a consequence of spatial homogeneity. In relativistic mechanics the momentum of a particle is expressed by the formula

.                              (16)
When velocity is small (), or in the limit when , this expression turns into the classical .

           In classical mechanics every particle is characterized by its mass m. In relativistic mechanics a particle is also characterized by its mass, insofar as in going over from one inertial reference frame to another its value does not change; as they say in relativity theory, the mass m of a particle is a relativistic invariant.

Energy of a relativistic particle
           The conservation of the energy of a free particle is a consequence of time homogeneity. In relativistic mechanics the energy of a particle is expressed by the formula

.                            (17)

It is seen from this expression that in relativistic mechanics the energy of a particle does not vanish even when the velocity is zero. This “rest energy”, i.e., the energy when v= 0, is

.

           At low velocities () the expression (17) takes the form

,

i.e., the classical expression for the kinetic energy of a particle less the rest energy.
           Formulas (16) and (17) yield the following relationship between the energy and momentum of a free material particle:

.                                (18)

           At v = c the momentum p (16) and the energy E (17) of a particle become infinite. This means that a particle whose mass is not zero cannot move with the speed c.

Particle with zero mass
           The momentum of a particle with zero mass written in the form (16) gives, at v = c, an indeterminancy of the form 0/0 and may remain finite. In this case the velocity of the particle must always be c. From equation (18) we have for such particles

.

Thus relativistic mechanics allows for the existence of particles with zero mass moving with the velocity c. Later on we will see that light phenomena can be interpreted in terms of such particles.

Systems of relativistic particles
           The foregoing formulas are equally valid for the motion of a whole composite body consisting of many particles. In this case by mass is always meant the aggregate mass of the body, and by velocity, its velocity as an entity.

           Consider a body at rest (as a whole). Then its energy, which we call internal, is equal to , where M is the mass of the body. This energy comprises, besides the rest energy of its component particles, their kinetic energy and the energy of their interactions. In other words,  is not simply the sum , where is the mass of a particle belonging to the body, and therefore M is not equal to .

           Thus, in relativistic mechanics the law of conservation of mass is no longer valid; the mass of a composite body is not equal to the sum of the masses of its component parts. Instead, there is only the law of conservation of energy, which also includes the rest energy of the particles. The difference  between the mass of a composite body and the net mass of its component parts is called the mass defect. The quantity  is called the binding energy.

           Consider a body consisting of two parts of masses  and  in a frame of reference in which it is at rest and suppose that it is spontaneously splitting into the two parts, the respective velocities of which are  and . Then according to the principle of conservation of energy, we have

.

This equation is satisfied only if , i.e., if the mass defect  is positive. Thus, spontaneous decay is possible only if the mass defect of a body is positive with respect to the parts into which it is disintegrating. Conversely, if the mass defect is negative, the body is stable and cannot decay spontaneously. In this case, evidently, disintegration can be achieved only by injecting energy from outside, this energy being at least equal to the binding energy .

Elementary particles in relativistic theory
           Let some rigid body(6) be put into motion by an external force applied at one of its points. If the body is absolutely rigid, all its points must start moving at precisely the same instant as the one to which the force is applied. This is quite possible in classical mechanics by virtue of the instantaneous propagation of pressure. In relativity theory, however, this is impossible, as a pressure exerted at one point of a body is transmitted to the other points with a finite velocity; hence all points of the body cannot start moving at once, which means that the body suffers a deformation. Thus relativity theory leads to the conclusion that there is no such thing as an absolutely rigid body.
           The foregoing leads to certain conclusions regarding elementary particles. By an elementary particle is meant a body which participates in all physical phenomena as an entity, i.e., it is meaningless to speak of its parts. In other words, the state of an elementary particle is completely defined by its location and velocity as a whole.

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