Classical Electrodynamics by Konstantin K. Likharev - HTML preview
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2 R
>> . Use the Huygens principle to calculate the wave’s
z
intensity at a distance z >> R behind the disk’s center –
S
0
0
see the figure on the right. Discuss the result.
8.19. Use the Huygens principle to analyze the Fraunhofer diffraction of a plane wave normally incident on a square-shaped hole, of size a a, in an opaque screen. Sketch the diffraction pattern you would observe at a sufficiently large distance, and quantify the expression “sufficiently large” for this case.
8.20. Use the Huygens principle to analyze the propagation of a monochromatic Gaussian beam described by Eq. (7.181), with the initial characteristic width a 0 >> , in a uniform isotropic medium.
Use the result for a semi-quantitative derivation of the so-called Abbe limit for the spatial resolution of Chapter 8
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Essential Graduate Physics
EM: Classical Electrodynamics
an optical system: w min = /2sin, where is the half-angle of the wave cone propagating from the object and captured by the system.
T
8.21. Within the Fraunhofer approximation,
1
analyze the pattern produced by a diffraction grating
with the 1D-periodic transparency profile shown in
w
the figure on the right, for the normal incidence of a
monochromatic plane wave.
x
d
0
d
q
8.22. N equal point charges are attached, at equal intervals, to a circle rotating with a constant angular velocity about its center – see the figure on the R
right. For what values of N does the system emit:
(i) the electric dipole radiation?
2 / N
(ii) the magnetic dipole radiation?
(iii) the electric quadrupole radiation?
8.23. What general statements can you make about:
(i) the electric dipole radiation, and
(ii) the magnetic dipole radiation,
due to a collision of an arbitrary number of similar non-relativistic classical particles?
8.24. Calculate the angular distribution and the total power radiated by a small planar loop antenna of radius R, fed with ac current with frequency and amplitude I 0, into free space.
8.25. The orientation of a magnetic dipole, with a constant magnitude m of its moment, is rotating about a certain axis with an angular velocity , with the angle between them staying constant.
Calculate the angular distribution and the average power of its radiation into the free space.
8.26. Solve Problem 12 (also in the low-frequency limit kR << 1), for the case when the sphere’s material has a frequency-independent Ohmic conductivity , and opt = 0, in two limits: (i) of a very large skin depth (s >> R), and
(ii) of a very small skin depth (s << R).
8.27. Complete the solution of the problem started in Sec. 9, by calculating the full power of radiation of the system of two charges oscillating in antiphase along the same straight line – see Fig. 16.
Also, calculate the average radiation power for the case of harmonic oscillations, d( t) = acos t, compare it with the case of a single charge performing similar oscillations, and interpret the difference.
8.28. The system of four alternating charges located at the angles of a square, considered in Problem 3.3(i), is now being rotated around the axis normal to their plane and passing through the square’s center, with a constant angular frequency << v/ a. Calculate the time-averaged angular distribution and the total power of the resulting radiation.
Chapter 8
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Chapter 9. Special Relativity
This chapter starts with a review of special relativity’s basics, including its very convenient 4-vector formalism. This background is then used for the analysis of the relation between the electromagnetic field’s values measured in different inertial reference frames moving relative to each other. The results enable us to discuss relativistic particle dynamics in the electric and magnetic fields, and the analytical mechanics of the particles – and of the electromagnetic field as such.
9.1. Einstein postulates and the Lorentz transform
As was emphasized at the derivation of expressions for the dipole and quadrupole radiation in the last chapter, they are only valid for systems of non-relativistic particles moving with velocities u much lower than c. In order to generalize these results to particles moving with arbitrary u, we need help from the relativity theory. Moreover, an analysis of the motion of charged relativistic particles in electric and magnetic fields is also a natural part of electrodynamics. This is why I will follow the tradition of using this course for a (by necessity, brief) introduction to the special relativity theory. This theory is based on the fundamental idea that measurements of physical variables (including the spatial and even temporal intervals between two events) may give different results in different reference frames, in particular in two inertial frames moving relative to each other translationally (i.e. without rotation), with a certain constant velocity v (Fig. 1).
y
y'
r { x, y, z
}
' r { x' , y' , z' }
v
0
'
0
x
x'
Fig. 9.1. Translational mutual motion of
two reference frames.
z
z'
In the non-relativistic (Newtonian) mechanics the problem of transfer between such reference frames has a simple solution at least in the limit v << c, because the basic equation of particle dynamics (the 2nd Newton law) 1
m r
U r
r ,
(9.1)
k k
k
(
)
k
k '
k '
where U is the potential energy of inter-particle interactions, is invariant with respect to the so-called Galilean transformation (or just “transform” for short).2 Choosing the coordinates in both frames so that their axes x and x’ are parallel to the vector v (as in Fig. 1), the transform may be represented as 1 Let me hope that the reader does not need a reminder that for Eq. (1) to be valid, the reference frames 0 and 0 ’
have to be inertial – see, e.g., CM Sec. 1.2.
2 It had been first formulated by Galileo Galilei, if only rather informally, as early as 1638 – four years before Isaac Newton was born! Note also the very unfortunate term “boost” used sometimes to describe such translational transformations. (It is especially unnatural in the special relativity, not describing accelerations.) In my course, this term is avoided, with the equivalent “transform” used instead.
© K. Likharev
Essential Graduate Physics
EM: Classical Electrodynamics
x x' vt' ,
y y' ,
z z' ,
t t' ,
(9.2a) Galilean
transform
and plugging Eq. (2a) into Eq. (1), we get an absolutely similarly looking equation of motion in the
“moving” reference frame 0 ’. Since the reciprocal transform,
x' x vt,
y y' ,
z' z,
t' t ,
(9.2b)
is similar to the direct one, with the replacement of (+ v) with (– v), we may say that the Galilean invariance means that there is no “master” ( absolute) spatial reference frame in classical mechanics, although the spatial and temporal intervals between different instant events are absolute, i.e. reference-frame invariant: x = x’,…, t = t’.
However, it is straightforward to use Eq. (2) to check that the form of the wave equation 2
2
2
1
2
f 0
,
(9.3)
2
2
2
2
2
x
y
z
c t
describing, in particular, the electromagnetic wave propagation in free space,3 is not Galilean-invariant.4
For the “usual” (say, elastic) waves, which obey a similar equation albeit with a different speed,5 this lack of Galilean invariance is natural and is compatible with the invariance of Eq. (1), from which the wave equation originates. This is because the elastic waves are essentially the oscillations of interacting particles of a certain medium (e.g., an elastic solid), making the reference frame connected to this medium, special. So, if the electromagnetic waves were oscillations of a certain special medium (which was first called the “luminiferous aether”6 and later aether – or just “ether”), similar arguments might be applicable to reconcile Eqs. (2) and (3).
The detection of such a medium was the goal of the measurements carried out between 1881 and 1887 (with better and better precision) by Albert Abraham Michelson and Edward Williams Morley, which are sometimes called “the most famous failed experiments in physics”. Figure 2 shows a crude scheme of these experiments.
mirror
v v
R
E
semi-
expt 1
light
transparent
v E
source
mirror
mirror
expt 2
Fig. 9.2. The Michelson-
Earth
Morley experiment.
detector
3 The discussions in this chapter and most of the next chapter will be restricted to the free-space (and hence dispersion-free) case; some media effects on the radiation by relativistic particles will be discussed in Sec.10.4.
4 It is interesting that the usual (non-relativistic) Schrödinger equation, whose fundamental solution for a free particle is a similar monochromatic wave (albeit with a different dispersion law), is Galilean-invariant, with a certain change of the wavefunction’s phase – see, e.g., QM Chapter 1.
5 See, e.g., CM Secs. 6.5 and 7.7.
6 In ancient Greek mythology, aether is the clean air breathed by the gods residing on Mount Olympus.
Chapter 9
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A nearly monochromatic wave from a light source is split into two parts (optimally, of equal intensity), using a semi-transparent mirror tilted by the angle /4 to the incident wave direction. These two partial waves are reflected back by two fully-reflecting mirrors and arrive at the same semi-transparent mirror again. Here half of each wave is directed toward the light source (they vanish there without affecting the source), but another half is passed toward an intensity detector, forming, with its counterpart, an interference pattern similar to that in the Young experiment. Thus each of the interfering waves has traveled twice (back and forth) each of two mutually perpendicular “arms” of the interferometer. Assuming that the aether, in which light propagates with speed c, moves with speed v < c along one of the arms, of length ll, it is straightforward (and hence left for the reader’s exercise :-) to get the following expression for the difference between the light roundtrip times:
2
2
l
l
t
l
l v
t
,
(9.4)
c
2
2
1 v / c 1/2
2
2
1 v / c c c
where lt is the length of the second, “transverse” arm of the interferometer (perpendicular to v), and the last, approximate expression is valid at lt ll l and v << c.
Since the Earth moves around the Sun with a speed v E 30 km/s 10-4 c, the arm positions relative to this motion alternate, due to the Earth’s rotation about its axis, every 6 hours – see the right panel of Fig. 2. Hence if we assume that the aether rests in the Sun’s reference frame, then t (and the corresponding shift of the interference fringes), has to change its sign with this half-period as well. The same alternation may be achieved, at a smaller time scale, by a deliberate rotation of the instrument by
/2. In the most precise version of the Michelson-Morley experiment (circa 1887), this shift was expected to be close to 0.4 of the interference pattern period. The results of the search for such a shift were negative, with the error bar about 0.01 of the period.7
The most prominent immediate explanation for this zero result8 was suggested in 1889 by George Francis FitzGerald and (independently and more qualitatively) by H. Lorentz in 1892: as evident from Eq. (4), if the longitudinal arm of the interferometer itself experiences the so-called length contraction:
1/ 2
2
v
l ( v) l ( )
0 1
,
(9.5)
l
l
2
c
while the transverse arm’s length is not affected by its motion through the aether, this effect kills the shift t. This radical idea received strong support from the proof, in 1887-1905, that the Maxwell equations, and hence the wave equation (3), are form-invariant under the so-called Lorentz transform,9
which in particular describes Eq. (5). For the choice of coordinates shown in Fig. 1, the transform reads 7 Through the 20th century, the Michelson-Morley-type experiments were repeated using more and more refined experimental techniques, always with zero results for the apparent aether motion speed. For example, recent experiments using cryogenically cooled optical resonators have reduced the upper limit for such speed to just 310-15 c –see H. Müller et al., Phys. Rev. Lett. 91, 020401 (2003).
8 The zero result of a slightly later experiment, namely a precise measurement of the torque that should be exerted by the moving aether on a charged capacitor, carried out in 1903 by F. Trouton and H. Noble (following G.
FitzGerald’s suggestion), seconded the Michelson and Morley’s conclusions.
9 The theoretical work toward this result included important contributions by Woldemart Voigt (in 1887), Hendrik Lorentz (in 1892-1904), Joseph Larmor (in 1897 and 1900), and Henri Poincaré (in 1900 and 1905).
Chapter 9
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2
x' vt'
t' ( v / c ) x'
x
Lorentz
, y y' , z z' , t
.
(9.6a) transform
2
2
1 v / c 1/2
2 2
1 v / c 1/2
It is elementary to solve these equations for the primed coordinates to get the reciprocal transform x vt
t ( v / 2
c ) x
x'
y' y
z' z
t'
(9.6b)
1
2
v / 2
c ,
,
,
1/ 2
1 2 2
v c .
/
1/ 2
(I will soon represent Eqs. (6) in a more elegant form – see Eqs. (19) below.)
The Lorentz transform relations (6) are evidently reduced to the Galilean transform formulas (2) at v 2 << c 2. However, all attempts to give a reasonable interpretation of these equalities while keeping the notion of the aether have failed, in particular because of the restrictions imposed by results of earlier experiments carried out in 1851 and 1853 by Hippolyte Fizeau – which were repeated with higher accuracy by the same Michelson and Morley in 1886. These experiments have shown that if one sticks to the aether concept, this hypothetical medium has to be partially “dragged” by any moving dielectric material with a speed proportional to ( – 1). Such local drag would be irreconcilable with the assumed continuity of the aether.
In his famous 1905 paper, Albert Einstein suggested a bold resolution of this contradiction, essentially removing the concept of the aether altogether.10 Moreover, he argued that the Lorentz transform is the general property of time and space, rather than of the electromagnetic field alone. He started with two postulates, the first one essentially repeating the relativity principle formulated a bit earlier (in 1904) by H. Poincaré in the following form:
“… the laws of physical phenomena should be the same, whether for an observer fixed or for an observer carried along in a uniform movement of translation; so that we have not and could not have any means of discerning whether or not we are carried along in such a motion.”11
The second Einstein postulate was that the speed of light c, in free space, should be constant in all reference frames. (This is essentially a denial of the aether’s existence.)
Then, Einstein showed that the Lorenz transform relations (6) naturally follow from his postulates, with a few (very natural) additional assumptions. Let a point source emit a short flash of light, at the moment t = t’ = 0 when the origins of the reference frames shown in Fig. 1 coincide. Then, according to the second of Einstein’s postulates, in each of the frames, the spherical wave propagates with the same speed c, i.e. the coordinates of points of its front, measured in the two frames, have to obey the following equalities:
( ct)2 ( 2
2
2
x y z ) ,
0
(9.7)
( ct' )2 ( 2
2
2
x' y' z' ) 0.
10 In hindsight, this was much relief, because the aether had been a very awkward construct to start with. In particular, according to the basic theory of elasticity (see, e.g., CM Ch. 7), in order to carry such transverse waves as the electromagnetic ones, this medium would need to have a non-zero shear modulus, i.e. behave as an elastic solid – rather than as a rarified gas hypothesized initially by C. Huygens.
11 Note that though the relativity principle excludes the notion of the special (“absolute”) spatial reference frame, its quoted verbal formulation still leaves the possibility of the Galilean “absolute time” t = t’ open. The quantitative relativity theory kills this option – see Eqs. (6) and their discussion below.
Chapter 9
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What may be the general relation between the combinations in the left-hand side of these equations –
not for this particular wave’s front, but in general? A very natural (essentially, the only justifiable) choice is
( ct)2 ( 2
2
2
x y z ) f ( 2
v
) ( ct' )2 ( 2
2
2
x' y' z' ).
(9.8)
Now, according to the first postulate, the same relation should be valid if we swap the reference frames ( x x’, etc.) and replace v with (– v). This is only possible if f 2 = 1, so excluding the option f = –1
(which is incompatible with the Galilean transform in the limit v/ c 0), we are left with f = +1, i.e.
( ct)2 ( 2
2
2
x y z ) ( ct' )2 ( 2
2
2
x' y' z' ) .
(9.9)
For the line with y = y’ = 0 and z = z’ = 0, Eq. (9) is reduced to 2
2
2
2
( ct) x ( ct' ) x' .
(9.10)
It is very illuminating to interpret this relation as the one resulting from a mutual rotation of the reference frames (that now have to include clocks to measure time) on the plane of the coordinate x and the so-called imaginary time ict – see Fig. 3.
'
x'
Fig. 9.3. The Lorentz transform as a mutual rotation
0
x
of two reference frames on the [ x, ] plane.
Indeed, rewriting Eq. (10) as
2
2
2
2
x ' x' ,
(9.11)
we may consider it as the invariance of the squared radius at the rotation shown in Fig. 3 and described by the following geometric relations:
x x' cos ' sin ,
(9.12a)
x' sin ' cos ,
with the reciprocal relations
x' x cos sin ,
(9.12b)
' x sin cos .
So far, the angle has been arbitrary. In the spirit of Eq. (8), a natural choice is = ( v), with the requirement (0) = 0. To find this function, let us write the definition of the velocity v of frame 0 ’, as measured in frame 0 (which was implied above): for x’ = 0, x = vt. In the variables x and , this means x
x
v
.
(9.13)
x'0
ict x'0
ic
On the other hand, for the same point x’ = 0, Eqs. (12a) yield
Chapter 9
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x
tan .
(9.14)
x'0
These two expressions are compatible only if
iv
tan
,
(9.15)
c
so
tan
iv / c
1
1
sin
i
(9.16)
1 tan 2 1/2
1 2
v / 2
c
,
cos
1/ 2
1 tan2 1/2 1 2 2
v c
,
/
1/ 2
where and are two very convenient and commonly used dimensionless parameters defined as v
1
1
Relativistic
β ,
.
(9.17) parameters
c
2
2
1 v / c 1/2
2
1 1/2
and
(The vector is called the normalized velocity, while the scalar is the Lorentz factor.)12
Using the above relations for , Eqs. (12) become
x x' i ' ,
i x' ' ,
(9.18a)
x' x i ,
' i x .
(9.18b)
Now returning to the real variables [ x, ct], we get the Lorentz transform relations (6), in a more compact form:
x x' ct' , y y' , z z' , ct ct' x' , (9.19a) Lorentz
transform
x' x ct, y' y, z' z, ct' ct x.
(9.19b) – again
An immediate corollary of Eqs. (19) is that for to stay real, we need v 2 c 2, i.e. that the speed of any physical body (to which we could connect a meaningful reference frame) cannot exceed the speed of light, as measured in any other meaningful reference frame.13
9.2. Relativistic kinematic effects
Before proceeding to other corollaries of Eqs. (19), let us spend a few minutes discussing what these relations actually mean. Evidently, they are trying to tell us that the spatial and temporal intervals are not absolute (as they are in the Newtonian space), but do depend on the reference frame they are measured in. So, we have to understand very clearly what exactly may be measured – and thus may be discussed in a meaningful physics theory. Recognizing this necessity, A. Einstein introduced the notion of numerous imaginary observers that may be distributed all over each reference frame. Each observer 12 Note the following identities: 2 1/(1- 2) and (2 – 1) 2/(1- 2) 22, which are frequently handy in relativity-related algebra. One more function of , the rapidity tanh–1 (so that = i), is also useful for some calculations.
13 All attempts to rationally conjecture particles moving with v > c (called tachyons) have failed – so far, at least.
Possibly the strongest objection against their existence is the fact that the tachyons could be used to communicate back in time, thus violating the causality principle – see, e.g., G. Benford et al., Phys. Rev. D 2, 263 (1970).
Chapter 9
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has a clock and may use it to measure the instants of local events, taking place at the observer’s location. He also conjectured, very reasonably, that:
(i) all observers within the same reference frame may agree on a common length measure (“a scale”), i.e. on their relative positions in that frame, and synchronize their clocks,14 and (ii) the observers belonging to different reference frames may agree on the nomenclature of world events (e.g., short flashes of light) to which their respective measurements refer.
Actually, these additional postulates have been already implied in our “derivation” of the Lorentz transform in Sec. 1. For example, by the set { x, y, z, t} we mean the results of space and time measurements of a certain world event, about that all observers belonging to frame 0 agree. Similarly, all observers of frame 0 ’ have to agree about the results { x’, y’, z’, t’}. Finally, when the origin of frame 0 ’ passes by some sequential points xk of frame 0, the observers in the latter frame may measure its passage times tk without a fundamental error, and know that all these times belong to x’ = 0.
Now we can analyze the major corollaries of the Lorentz transform, which are rather striking from the point of view of our everyday (rather non-relativistic) experience.
(i) Length contraction. Let us consider a thin rigid rod oriented along the x-axis, with its length l
x 2 – x 1, where x 1,2 are the coordinates of the rod’s ends, as measured in its rest frame 0, at any instant t (Fig. 4). What would be the rod’s length l’ measured by the Einstein observers in the moving frame 0 ’?
y
y'
x
l
x
1
2
x
x'
0
'
0
v
Fig. 9.4. The relativistic length contraction.
z
z'
At a time instant t’ agreed upon in advance, the observers who find themselves exactly at the rod’s ends, may register that fact, and then subtract their coordinates x’ 1,2 to calculate the apparent rod length l’ x 2 ’ – x 1 ’ in the moving frame. According to Eq. (19a), l may be expressed via this l’ as l x x ( x ' ct' ) ( x ' ct' ) ( x ' x ' ) l' .
(9.20a)
2
1
2
1
2
1
Hence, the rod’s length, as measured in the moving reference frame is
1/ 2
Length
l
v 2
l'
l 1
l
contraction
,
(9.20b)
c 2
in accordance with the FitzGerald-Lorentz hypothesis (5). This is the relativistic length contraction effect: an object is always the longest (has the so-called proper length l) if measured in its rest frame.
14 A posteriori, the Lorenz transform may be used to show that consensus-creating procedures (such as clock synchronization) are indeed possible. The basic idea of the proof is that since at v << c, the relativistic corrections to space and time intervals are of the order of ( v/ c)2, they have negligible effects on clocks being brought together into the same point for synchronization slowly, with a speed u << c. The reader interested in a detailed discussion of this and other fine points of special relativity may be referred to, e.g., either H. Arzeliès, Relativistic Kinematics, Pergamon, 1966, or W. Rindler, Introduction to Special Relativity, 2nd ed., Oxford U. Press, 1991.
Chapter 9
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Note that according to Eqs. (19), the length contraction takes place only in the direction of the relative motion of two reference frames. As was noted in Sec. 1, this result immediately explains the zero result of the Michelson-Morley-type experiments, so they give very convincing evidence (if not irrefutable proof) of Eqs. (18)-(19).
(ii) Time dilation. Now let us use Eqs. (19a) to find the time interval t, as measured in some reference frame 0, between two world events – say, two ticks of a clock moving with another frame 0 ’
(Fig. 5), i.e. having fixed values of x’, y’, and z’.
y
y'
x
x'
0
v
Fig. 9.5. The relativistic time dilation.
z
z'
Let the time interval between these two events, measured in the clock’s rest frame 0’, be t’ t 2 ’
– t 1 ’. At these two moments, the clock would fly by two Einstein’s observers at rest in frame 0, so they can record the corresponding moments t 1,2 shown by their clocks, and then calculate t as their difference. According to the last of Eqs. (19a),
c Δ t ct ct
,
(9.21a)
2
1
( ct ' x' ) ( ct ' x' )
2
1
c t'
Δ
so, finally,
t'
Δ
t
Δ t'
Δ
Time
Δ .
(9.21b) dilation
1 v 2 / 2
t'
c 1/ 2
This is the famous relativistic time dilation (or “dilatation”) effect: a time interval is longer if measured in a frame (in our case, frame 0) moving relative to the clock, while that in the clock’s rest frame is the shortest possible – the so-called proper time interval.
This rather counter-intuitive effect is the everyday reality in experiments with high-energy elementary particles. For example, in a typical (and by no means record-breaking) experiment carried out in Fermilab, a beam of charged 200 GeV pions with 1,400 traveled a distance of l = 300 m with the measured loss of only 3% of the initial beam intensity due to the pion decay (mostly, into muon-neutrino pairs) with the proper lifetime t 0 2.5610-8 s. Without the time dilation, only an exp{- l/ ct 0}
~10-17 fraction of the initial pions would survive, while the relativity-corrected number, exp{- l/ ct} =
exp{- l/ c t 0} 0.97, was in full accordance with experimental measurements.
As another example, the global positioning systems (say, the GPS) are designed with the account of the time dilation due to the velocity of their satellites (and also some gravity-induced, i.e. general-relativity corrections, which I would not have time to discuss) and would give large errors without such corrections. So, there is no doubt that time dilation (21) is a reality, though the precision of its experimental tests I am aware of15 has been limited to a few percent, because of the almost unavoidable involvement of less controllable gravity effects – which provide a time interval change of the opposite sign in most experiments near the Earth’s surface.
15 See, e.g., J. Hafele and R. Keating, Science 177, 166 (1972).
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Before the first reliable observation of time dilation (by B. Rossi and D. Hall in 1940), there had been serious doubts about the reality of this effect, the most famous being the twin paradox first posed (together with an immediate suggestion of its resolution) by P. Langevin in 1911. Let us send one of two twins on a long space roundtrip with the maximum speed approaching c. Upon his return to Earth, who of the twins would be older? The naïve approach is to say that due to the relativity principle, not one can be (and hence there is no time dilation) because each twin could claim that their counterpart rather than them, was moving, with the same speed but in the opposite direction. The resolution of the paradox is that one of the twins had to be accelerated to be brought back, and hence the reference frames have to be dissimilar: only one of them may stay inertial all the time. As a result, the twin who had been accelerated (“actually traveling”) would be younger than their sibling when they finally came together.
Constructive proof of this conclusion for the particular case of straight-line travel with a piecewise-constant acceleration, is simple and hence left for the reader’s exercise.
(iii) Velocity transformation. Now let us calculate the velocity u of a moving point, as observed in reference frame 0, provided that its velocity, as measured in frame 0 ’, is u ’ (Fig. 6).
y'
u '
y
u
0
'
0
v
x
x'
Fig. 9.6. The relativistic velocity addition.
z
z'
Keeping the usual definition of velocity, but with due attention to the relativity of not only spatial but also temporal intervals, we may write
dr
d '
r
u
,
u '
.
(9.22)
dt
dt'
Plugging in the differentials of the Lorentz transform relations (6a) into these definitions, we get dx
dx' vdt'
u' v
x
dy
1
dy'
1
u'
u
,
u
y
, (9.23)
x
dt
dt' vdx' / 2
c
1 u' v / 2
c
y
dt
dt' vdx' / 2
c
1 u' v / 2
c
x
x
with a similar formula for uz. In the classical limit v/ c 0, these relations are reduced to u u' v,
u u' ,
u u' ,
(9.24a)
x
x
y
y
z
z
and may be merged into the familiar Galilean form
u u ' v,
v
for c .
(9.24b)
In order to see how unusual the full relativistic rules (23) are at u ~ c, let us first consider a purely longitudinal motion, uy = uz = 0; then16
16 With an account of the identity tanh( a + b) = (tanh a + tanh b)/(1 + tanh a tanh b), which readily follows from MA Eq. (3.5), Eq. (25) shows that rapidities tanh-1 add up exactly as longitudinal velocities at non-relativistic motion, making that notion very convenient for the analysis of transfer between several frames.
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u' v
Longitudinal
u
,
(9.25) velocity
2
1 u'v / c
addition
where u ux and u’ u’x. Figure 7 shows this u as the function of u’, for several values of the reference frames’ relative velocity v.
1
v / c 0.9
0.5
0
u
0.5
0
c
0.9
Fig. 9.7. The addition of longitudinal velocities.
1 1
0
1
u' / c
The first sanity check is that if v = 0, i.e. if the reference frames are at rest relative to each other, then u = u’, as it should be – see the diagonal straight line in Fig. 7. Next, if magnitudes of u’ and v are both below c, so is the magnitude of u. (Also good, because otherwise, ordinary particles in one frame would be tachyons in the other one, and the theory would be in big trouble.) Now strange things begin: even as u’ and v are both approaching c, then u is also close to c, but does not exceed it. As an example, if we fired forward a bullet with the relative speed of 0.9 c, from a spaceship moving from the Earth also at 0.9 c, Eq. (25) predicts the speed of the bullet relative to the Earth to be just [(0.9 + 0.9)/(1 +
0.90.9)] c 0.994 c < c, rather than (0.9 + 0.9) c = 1.8 c > c as in the Galilean kinematics. Actually, we could expect this strangeness, because it is necessary to fulfill the 2nd Einstein’s postulate: the independence of the speed of light in any reference frame. Indeed, for u’ = c, Eq. (25) yields u = c, regardless of v.
In the opposite case of a purely transverse motion, when a point moves across the relative motion of the frames (for example, at our choice of coordinates, u’ x = u’ z = 0), Eqs. (23) yield a much less spectacular result
1
u u' u' .
(9.26)
y
y
y
This effect comes purely from the time dilation because the transverse spatial intervals are Lorentz-invariant.
In the case when both u x’ and uy’ are substantial (but uz’ is still zero), we may divide Eqs. (23) by each other to relate the angles of the point’s propagation, as observed in the two reference frames: u
u'
Stellar
y
y
sin '
tan
.
(9.27) aberration
u
cos /
effect
x
u' v
x
' v u'
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This expression describes, in particular, the so-called stellar aberration effect: the dependence of the observed direction toward a star on the speed v of the telescope’s motion relative to the star – see Fig.
8. (The effect is readily observable experimentally as the annual aberration due to the periodic change of speed v by 2 v E 60 km/s because of the Earth’s rotation about the Sun. Since the aberration’s main part is of the first order in v E/ c ~ 10-4, this effect is very significant and has been known since the early 1700s.)
( u' c)
u '
'
v
Fig. 9.8. The stellar aberration.
For the analysis of this effect, it is sufficient to take, in Eq. (27), u’ = c, i.e. v/ u’ = , and interpret ’ as the “proper” direction to the star, which would be measured at v = 0.17 At << 1, both Eq. (27) and the Galilean result (which the reader is invited to derive directly from Fig. 8), sin '
tan
,
(9.28)
cos '
may be well approximated by the first-order term
' sin '
.
(9.29)
Unfortunately, it is not easy to use the difference between Eqs. (27) and (28), of the second order in , for special relativity’s confirmation, because other components of the Earth’s motion, such as its rotation, nutation, and torque-induced precession,18 give masking first-order contributions to the aberration.
Finally, for a completely arbitrary direction of the vector u ’, Eqs. (22) may be readily used to calculate the velocity’s magnitude. The most popular form of the resulting expression is the following expression for the square of the relative velocity (or rather the reduced relative velocity ) of two points,
β β β β
2
1
2 2
1
2
.
(9.30)
β β
1
1
1
2
2
where 1,2 v1,2/ c are their normalized velocities as measured in the same reference frame.
17 Strictly speaking, to reconcile the geometries shown in Fig. 1 (for which all our formulas, including Eq. (27), are valid) and Fig. 8 (giving the traditional scheme of the stellar aberration), it is necessary to invert the signs of u (and hence of sin ’ and cos ’) and v, but as it is evident from Eq. (27), all the minus signs cancel, and the formula is valid “as is”.
18 See, e.g., CM Secs. 4.4-4.5.
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(iv) The Doppler effect. Let us consider a monochromatic plane wave of some physical nature, traveling along the x- axis:
f
Re f
i kx t
f
kx
exp (
cos
t arg f f cos
.
(9.31)
Its total phase, kx – t + arg f (in contrast to its amplitude f– see Sec. 5 below) cannot depend on the observer’s reference frame, because the variable f vanishes completely at = ( n + ½) (for all integer n), and such “world events” should be observable in all reference frames. The only way to keep
= ’ at all times is to have19
kx t
k'x' 't'
.
(9.32)
First, let us use this general relation to consider the Doppler effect in the usual non-relativistic mechanical waves, e.g., oscillations of particles of a certain medium. Using the Galilean transform (2), we may rewrite Eq. (32) as
k( x' vt) t k'x' 't
.
(9.33)
Since this transform leaves all space intervals (including the wavelength = 2/ k) intact, we can take k
= k’, so Eq. (33) yields
'
kv .
(9.34)
For a dispersion-free medium, the wave number k is the ratio of its frequency , as measured in the reference frame bound to the medium, and the wave velocity v w. In particular, if the wave source rests in the medium, we may bind the reference frame 0 to the medium as well, and frame 0 ’ to the wave’s receiver (i.e. v = v r), so
k
,
(9.35)
v w
and for the frequency perceived by the receiver, Eq. (34) yields
v v
w
r
'
.
(9.36)
v w
On the other hand, if the receiver and the medium are at rest in the reference frame 0 ’, while the wave source is bound to the frame 0 (so v = – v s), Eq. (35) should be replaced with
'
k k'
,
(9.37)
v w
and Eq. (34) yields a different result:
v
w
'
,
(9.38)
v v
w
s
Finally, if both the source and detector are moving, it is straightforward to combine these two results to get the general relation
v v
w
r
'
.
(9.39)
v v
w
s
19 Strictly speaking, Eq. (32) is valid to an additive constant, but for notation simplicity, it may be always made equal to zero by selecting (as has already been done in all relations of Sec. 1) the reference frame origins and/or clock turn-on times so that at t = 0 and x = 0, t’ = 0 and x’ = 0 as well.
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At low speeds of both the source and the receiver, this result simplifies,
v v
' 1
r
s
,
with
,
(9.40)
v w
but at speeds comparable to v w we have to use the more general Eq. (39). Thus, the usual Doppler effect is generally affected not only by the relative speed ( v r – v s) of the wave’s source and detector but also by their speeds relative to the medium in which the waves propagate.
Somewhat counter-intuitively, for the electromagnetic waves the calculations are simpler because for them the propagation medium (aether) does not exist, the wave velocity equals c in any reference frame, and there are no two separate cases: we can always take k = / c and k’ = ’/ c.
Plugging these relations, together with the Lorentz transform (19a), into the phase-invariance condition (32), we get
ct' x'
'
( x' ct' )
x' 't' .
(9.41)
c
c
c
This relation has to hold for any x’ and t’, so we may require that the net coefficients before these variables vanish. These two requirements yield the same equality:
' 1
( ) .
(9.42)
This result is already quite simple, but may be transformed further to be even more illuminating: 1
1 1 1/2
'
.
(9.43)
2
1 1/2
1 1
At any sign before , one pair of parentheses cancels, so20
1/ 2
Longitudinal
1
Doppler
'
effect
.
(9.44)
1
Thus the Doppler effect for electromagnetic waves depends only on the relative velocity v = c between the wave source and detector – as it should be, given the aether’s absence. At velocities much lower than c, Eq. (44) may be approximated as
1 / 2
'
1 ,
(9.45)
1 / 2
i.e. in the first approximation in v/ c, it tends to the corresponding limit (40) of the usual Doppler effect.
If the wave vector k is tilted by angle to the vector v (as measured in frame 0), then we have to repeat the calculations, with k replaced by kx, and components ky and kz left intact at the Lorentz transform. As a result, Eq. (42) is generalized as
20 It may look like the reciprocal expression of via ’ is different, violating the relativity principle. However, in this case, we have to change the sign of , because the relative velocity of the system is opposite, so we return to Eq. (44) again.
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' 1 cos .
(9.46)
For the case cos = 1, Eq. (46) reduces to our previous result (42). However, at = /2 (i.e. cos = 0), the relation is rather different:
Transverse
'
.
(9.47) Doppler
2
1 1/2
effect
This is the transverse Doppler effect – which is absent in non-relativistic physics. Its first experimental evidence was obtained using electron beams (as had been suggested in 1906 by J. Stark), by H. Ives and G. Stilwell in 1938 and 1941. Later, similar experiments were repeated several times, but the first unambiguous measurements were performed only in 1979 by D. Hasselkamp et al. who confirmed Eq. (47) with a relative accuracy of about 10%. This precision may not look too spectacular, but besides the special tests discussed above, the Lorentz transform formulas have been also confirmed, less directly, by a huge body of other experimental data, especially in high energy physics, agreeing with calculations incorporating this transform as their part. This is why, with due respect to the spirit of challenging authority, I should warn the reader: if you decide to challenge the relativity theory (called
“theory” by tradition only), you would also need to explain all these data. Best luck with that! 21
9.3. 4-vectors, momentum, mass, and energy
Before proceeding to the relativistic dynamics, let us discuss the mathematical formalism that makes all calculations more compact – and more beautiful. We have already seen that the three spatial coordinates { x, y, z} and the product ct are Lorentz-transformed similarly – see Eqs. (18)-(19) again. So it is natural to consider them as components of a single four-component vector (or, for short, 4-vector),
{ x , x , x , x } ct
,
(9.48)
0
1
2
3
, r
with components
Space
x ct, x x, x y, x z .
(9.49) -time
0
1
2
3
4-vector
According to Eqs. (19), its components are Lorentz-transformed as
3
Lorentz
x
L x' ,
(9.50) transform:
j
jj' j'
j'0
4-form
where Ljj’ are the elements of the following 44 Lorentz transform matrix
0 0
0 0
Lorentz
.
(9.51) transform
0
0
1 0
matrix
0
0
0 1
Since such 4-vectors are a new notion for this course and will be used for many more purposes than just the space-time transform, we need to discuss the general mathematical rules they obey. Indeed, 21 The same fact, ignored by crackpots, is also valid for other favorite directions of their attacks, including the Universe expansion, quantum measurement uncertainty, and entropy growth in physics, and the evolution theory in biology.
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as was already mentioned in Sec. 8.9, the usual (three-component) vector is not just any ordered set ( string) of three scalars { Ax, Ay, Az}; if we want it to represent a reference-frame-independent physical reality, the vector’s components have to obey certain rules at the transfer from one reference frame to another. In particular, in the non-relativistic limit the vector’s norm (its magnitude squared), 2
2
2
2
A A A A ,
(9.52)
x
y
z
should be invariant with respect to the transfer between different reference frames. However, a naïve extension of this approach to 4-vectors would not work, because, according to the calculations of Sec. 1, the Lorentz transform keeps intact the combinations of the type (7), with one sign negative, rather than the sum of all components squared. Hence for the 4-vectors, all the rules of the game have to be reviewed and adjusted – or rather redefined from the very beginning, for example as follows.22
An arbitrary 4-vector is a string of 4 scalars,23
General
4-vector
A , A , A , A ,
(9.53)
0
1
2
3
whose components Aj, as measured in the reference frames 0 and 0 ’ shown in Fig. 1, obey the Lorentz transform relations similar to Eq. (50):
Lorentz
3
transform:
A
L A' .
(9.54)
j
general
jj' j'
j'0
4-vector
As we have already seen in the example of the space-time 4-vector (48), this means in particular that 3
3
Lorentz
2
A
A
A'
A'
.
(9.55)
0
invariance
2
j
02 j2
j1
j1
This is the so-called Lorentz invariance condition for the 4-vector’s norm. (The difference between this relation and Eq. (52), pertaining to Euclidian geometry, is the reason why the Minkowski space is called pseudo-Euclidian.) It is also straightforward to use Eqs. (51) and (54) to check that the evident generalization of the norm, the scalar product of two arbitrary 4-vectors, 3
Scalar
4-product
A B
A B ,
(9.56)
0
0
j j
j1
is also Lorentz-invariant.
Now consider the 4-vector corresponding to a small interval between two close world events: dx
{
, dx , dx , dx }
;
(9.57)
0
1
2
3
cdt, r
d
its norm,
3
Interval
2
2
2
2
2
2
( ds) dx dx c ( dt) ( dr) ,
(9.58)
0
j
j 1
22 The most prominent alternative, which has both advantages and drawbacks, is to use 4-vectors with one imaginary component – for example, the imaginary time ict instead of the real product ct in Eq. (48).
23 Such vectors are said to reside in so-called 4D Minkowski spaces – called after Hermann Minkowski who was the first one to recast (in 1907) the special relativity relations in a form in which the spatial coordinates and time (or rather ct) are treated on an equal footing.
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is of course also Lorentz-invariant. Since the speed of any particle (or signal) cannot be larger than c, for any pair of world events that are in a causal relation with each other, ( dr)2 cannot be larger than ( cdt)2, i.e. such time-like interval ( ds)2 cannot be negative. The 4D surface separating such intervals from space-like intervals ( ds)2 < 0 is called the light cone (Fig. 9).
time-like interval ds 2 0
t
(causal relation possible)
space-like interval ds 2 < 0
x 2
(causal relation impossible)
0
r ct
x
Fig. 9.9. A 2+1 dimensional image of
1
the light cone – which is actually 3+1
dimensional.
Now let us consider two close world events that happen with the same point moving with velocity u. Then in the frame moving with the point (v = u), the last term on the right-hand side of Eq.
(58) equals zero, while the involved time is the proper one, so
ds
cd ,
(9.59)
where d is the proper time interval. But according to Eq. (21), this means that we can write
dt
d
,
(9.60)
where dt is the time interval in an arbitrary (besides being inertial) reference frame, while u
1
1
β
and
(9.61)
c
2
1 1/2
2 2
1 u / c 1/2
are the parameters (17) corresponding to the point’s velocity (u) in that frame, so ds = cdt/.24
Let us use Eq. (60) to explore whether a 4-vector may be formed using the spatial Cartesian components of the point’s velocity
dx dy dz
u ,
,
.
(9.62)
dt dt dt
Here we have a problem: per Eqs. (22), these components do not obey the Lorentz transform. However, let us use d dt/, the proper time interval of the point, to form the following string:
dx dx dx dx
dx dy dz
0
,
1 ,
2 ,
3 c
,
,
,
c,
u .
(9.63) 4-velocity
d d d d
dt dt dt
24 I have opted against using special indices (e.g., u and u) to distinguish Eqs. (17) and (61) here and below, in a hope that the suitable velocity (of either a reference frame or a particle) will be always clear from the context.
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As it follows from the comparison of the middle form of this expression with Eq. (48), since the time-space vector obeys the Lorentz transform, and is Lorentz-invariant, the string (63) is a legitimate 4-vector; it is called the 4-velocity of a point – or of a point particle.
Now we are well equipped to proceed to relativistic dynamics. Let us start with such basic notions as the momentum p and the energy E – so far, for a free particle.25 Perhaps the most elegant way to “derive” (or rather guess26) the expressions for p and E as functions of the particle’s velocity u, is based on analytical mechanics. Due to the conservation of v, the trajectory of a free particle in the 4D
Minkowski space { ct, r} is always a straight line. Hence, from the Hamilton principle,27 we may expect its action S, between points 1 and 2, to be a linear function of the space-time interval (59): 2
2
t 2
Free
dt
particle:
S
action
ds c d c ,
(9.64)
1
1
t
1
where is some constant. On the other hand, in analytical mechanics, the action is defined as t 2
S L dt ,
(9.65)
t 1
where L is the particle’s Lagrangian function.28 Comparing these two expressions, we get 1/ 2
2
c
u
L
c 1
.
(9.66)
2
c
In the non-relativistic limit ( u << c), this function tends to
u 2
u 2
L c 1
c
.
(9.67)
2 c 2
2 c
In order to correspond to the Newtonian mechanics,29 the last (velocity-dependent) term should equal mu 2/2. From here we find = – mc, so, finally,
Free
1/ 2
2
2
particle:
u
mc
2
Lagrangian
L mc 1
.
(9.68)
2
function
c
Now we can find the Cartesian components pj of the particle’s momentum as the generalized momenta corresponding to the corresponding components rj ( j = 1, 2, 3) of the 3D radius-vector r:30
25 I am sorry for using, just as in Sec. 6.3, the same traditional notation (p) for the particle’s momentum as had been used earlier for the electric dipole moment. However, since the latter notion will be virtually unused in the balance of this course, this may hardly lead to confusion.
26 Indeed, such a derivation uses additional assumptions, however natural (such as the Lorentz-invariance of S), i.e. it can hardly be considered as a real proof of the final results, so they require experimental confirmation.
Fortunately, such confirmations have been numerous – see below.
27 See, e.g., CM Sec. 10.3.
28 See, e.g., CM Sec. 2.1.
29 See, e.g., CM Eq. (2.19b).
30 See, e.g., CM Sec. 2.3, in particular Eq. (2.31).
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L
L
2
1/ 2
u 2 u 2 u 2
mu
1
2
3
j
p
mc
1
m u .
(9.69)
j
r
u
u
c 2
1/ 2
2
2
j
j
j
1 u / c
j
Thus for the 3D vector of momentum, we can write the result in the same form as in non-relativistic mechanics,
p m u u
M ,
(9.70) Relativistic
momentum
using the reference-frame-dependent scalar M (called the relativistic mass) defined as m
M m
,
(9.71) Relativistic
1 u 2
2
/
m
c 1/ 2
mass
m being the non-relativistic mass of the particle. (More often, m is called the rest mass, because in the reference frame in which the particle rests, Eq. (71) yields M = m.) Next, let us return to analytical mechanics to calculate the particle’s energy E (which for a free particle coincides with its Hamiltonian function H):31
1/ 2
3
2
2
2
mu
u
mc
2
E H p u L p u L
mc 1
. (9.72)
j
j
j 1
2 2
1 u / c 1/2
2
c
2 2
1 u / c 1/2
Thus, we have arrived at the most famous of Einstein’s formulas – and probably of physics as a whole: 2
2
E m c Mc ,
(9.73) E = Mc 2
which expresses the relation between the free particle’s mass and its energy.32 In the non-relativistic limit, it reduces to
2
2
2
mc
u
mu
2
2
E
mc
mc
(9.74)
1
2
u / 2
c
1
,
1/ 2
2 2
c
2
the first term mc 2 being called the rest energy of a particle.
Now let us consider the following string of 4 scalars:
E
E
4-vector of
, p , p , p
.
(9.75)
energy-
1
2
3
, p
c
c
momentum
Using Eqs. (70) and (73) to represent this expression as
E
, p m c,
u ,
(9.76)
c
31 See, e.g., CM Eq. (2.32).
32 Let me hope that the reader understands that all the layman talk about the “mass to energy conversion” is only valid in a very limited sense of the word. While the Einstein relation (73) does allow the conversion of “massive”
particles (with m 0) into particles with m = 0, such as photons, each of the latter particles also has a non-zero relativistic mass M, and simultaneously the energy E related to this M by Eq. (73).
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and comparing the result with Eq. (63), we immediately see that, since m is a Lorentz-invariant constant, this string is a legitimate 4-vector of energy-momentum. As a result, its norm, 2
E
2
p ,
(9.77a)
c
is Lorentz-invariant, and in particular, has to be equal to the norm in the particle-bound frame. But in that frame, p = 0, and according to Eq. (73), E = mc 2, and the norm is just 2
2
2
E
mc
mc2 ,
(9.77b)
c
c
so in an arbitrary frame
2
E
2
2
p ( mc) .
(9.78a)
c
This very important relation33 between the relativistic energy and momentum (valid for free particles only!) is usually represented in the form34
Free
particle:
2
E mc 2
2
pc2 .
(9.78b)
energy
According to Eq. (70), in the so-called ultra-relativistic limit u c, p tends to infinity, while mc 2 stays constant, so pc/ mc 2 . As follows from Eq. (78), in this limit E pc. Though the above discussion was for particles with finite m, the 4-vector formalism allows us to consider compact objects with zero rest mass as ultra-relativistic particles for which the above energy-to-moment relation, E pc,
for m 0 ,
(9.79)
is exact. Quantum electrodynamics35 tells us that under certain conditions, the electromagnetic field quanta (photons) may be also considered as such massless particles with momentum p = k. Plugging (the modulus of) the last relation into Eq. (78), for the photon’s energy we get E = pc = kc = . Please note again that according to Eq. (73), the relativistic mass of a photon is not equal to zero: M = E/ c 2 =
/c 2 , so the term “massless particle” has a limited meaning: m = 0. For example, the relativistic mass of an optical phonon is of the order of 10–36 kg. On the human scale, this is not too much, but still, a noticeable (approximately one-millionth) part of the rest mass m e of an electron.
The fundamental relations (70) and (73) have been repeatedly verified in numerous particle collision experiments, in which the total energy and momentum of a system of particles are conserved –
at the same conditions as in non-relativistic dynamics. (For the momentum, this is the absence of external forces, and for the energy, the elasticity of particle interactions – in other words, the absence of alternative channels of energy escape.) Of course, generally only the total energy of the system is conserved, including the potential energy of particle interactions. However, at typical high-energy 33 Please note one more simple and useful relation following from Eqs. (70) and (73): p = (E/ c 2)u.
34 It may be tempting to interpret this relation as the perpendicular-vector-like addition of the rest energy mc 2 and the “kinetic energy” pc, but from the point of view of the total energy conservation (see below), a better definition of the kinetic energy is T( u) E (u) – E (0).
35 It is briefly reviewed in QM Chapter 9.
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particle collisions, the potential energy vanishes so rapidly with the distance between them that we can use the momentum and energy conservation laws using Eq. (73).
As an example, let us calculate the minimum energy Emin of a proton (p a), necessary for the well-known high-energy reaction that generates a new proton-antiproton pair, p a + p b p + p + p + p , provided that before the collision, proton p b had been at rest in the lab frame. This minimum corresponds to the vanishing relative velocity of the reaction products, i.e. their motion with virtually the same velocity (ufin), as seen from the lab frame – see Fig. 10.
frame
lab
frame
c.o.m.
min
E
u
u
Fig. 9.10. A high-energy proton
p
min
p
a
b
fin
reaction at E Emin – schematically.
Due to the momentum conservation, this velocity should have the same direction as the initial velocity (umin) of proton pa. This is why two scalar equations: for energy conservation, 2
2
mc
4 mc
2
mc
,
(9.80a)
2
2
1 u
/ c
1 u / c
min
1/2
2 2
fin
1/2
and for momentum conservation,
mu
4 mu
0
,
(9.80b)
2
2
1 u
/ c
1 u / c
min
1/
fin
2
2
2
fin
1/2
are sufficient to find both u min and u fin. After a rather tedious solution of this system of two nonlinear equations, we get
4 3
3
u
c 990
.
0
c,
u
c 866
.
0
c .
(9.81)
min
7
fin
2
Finally, we can use Eq. (72) to calculate the required energy; the result is Emin = 7 mc 2. (Note that at this threshold, only a minor 2 mc 2 part of the kinetic energy T min = Emin – mc 2 = 6 mc 2 of the initially moving particle, goes into the “useful” proton-antiproton pair production.) The proton’s rest mass, m p 1.6710-27 kg, corresponds to m p c 2 1.50210-10 J 0.938 GeV, so Emin 6.57 GeV.
The second, more intelligent way to solve the same problem is to use the center-of-mass ( c.o.m.) reference frame that, in relativity, is defined as the frame in which the total momentum of the system vanishes.36 In this frame, at E = Emin, the velocity and momenta of all reaction products are vanishing, while the velocities of the protons p a and p b before the collision are equal and opposite, with an initially unknown magnitude u’. Hence the energy conservation law becomes
2
2 mc
2
4 mc ,
(9.82)
2
2
1 u' / c 1/2
36 Note that according to this definition, the c.o.m.’s radius-vector is R = kMkr k/ kMk k kmkr k/ k kmk, i.e. is generally different from the well-known non-relativistic expression R = kmkr k/ kmk .
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readily giving u’/ c = 3/2. (This is of course the same result as Eq. (81) gives for u fin.) Now we can use the fact that the velocity of the proton p a in the c.o.m. frame is (– u’), to find its lab-frame speed, using the velocity transform (25):
2 u'
u
.
(9.83)
min
2
2
1 u' / c
With the above result for u’, this relation gives the same result as the first method, u min/ c = 43/7, but in a simpler way.
9.4. More on 4-vectors and 4-tensors
This is a good moment to introduce a formalism that will allow us, in particular, to solve the same proton collision problem in one more (and arguably, the most elegant) way. Much more importantly, this formalism will be virtually necessary for the description of the Lorentz transform of the electromagnetic field, and its interaction with relativistic particles – otherwise the formulas would be too cumbersome.
Let us call the 4-vectors we have used before,
Contravariant
A A ,
,
(9.84)
0
A
4-vectors
contravariant, and denote them with top indices, and introduce also covariant vectors, Covariant
A
4-vectors
A , ,
(9.85)
0
A
marked by bottom indices. Now if we form a scalar product of these two vectors using the standard (3D-like) rule, just as a sum of the products of the corresponding components, we immediately get
2
2
A A A A A A
.
(9.86)
0
Note that the first and the second expressions may be understood as sums over four components of the product, with the summation sign dropped.37 The scalar product (86) is just the norm of the 4-vector in our former definition, and as we already know, is Lorentz-invariant. Moreover, the scalar product of two different vectors (also a Lorentz invariant), may be rewritten in any of two similar forms:38
Scalar
product's
A B A B
;
(9.87)
0
0
A B
A
B
forms
again, the only caveat is to take one vector in the covariant, and the other one in the contravariant form.
Now let us return to our sample problem (Fig. 10). Since all components (E/ c and p) of the total 4-momentum of our system are conserved at the collision, its norm is conserved as well:
p p
.
(9.88)
a
b p
p
a
b
(4 p) (4 p)
37 This compact notation may take some time to be accustomed to, but is very convenient (compact) and can hardly lead to any confusion, due to the following rule: the summation is implied when, and only when the same index is repeated twice, once on the top and another at the bottom. (It is frequently called dummy index, because its notation may be replaced with any other letter not used in the same formula.) In this course, this shorthand notation will be used only for 4-vectors, but not for the usual (3D spatial) vectors.
38 Note also that, by definition, for any two 4-vectors, A B = B A.
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Since now the vector product is the usual math construct, we know that the parentheses on the left-hand side of this equation may be multiplied as usual. We may also swap the operands and move constant factors through products as convenient. As a result, we get
p
2
16
.
(9.89)
a pa
pb pb pa pb
p p
Thanks to the Lorentz invariance of each of the terms, we may calculate it in the reference frame we like. For the first two terms on the left-hand side, as well as for the right-hand side term, it is beneficial to use the frames in which that particular proton is at rest; as a result, according to Eq. (77b), each of the two left-hand-side terms equals ( mc)2, while the right-hand side equals 16( mc)2. On the contrary, the last term on the left-hand side is more easily evaluated in the lab frame, because in it, the three spatial components of the 4-momentum pb vanish, and the scalar product is just the product of the scalars E/ c for protons a and b. For the latter proton, being at rest, this ratio is just mc so we get a simple equation,
2
2
min
E
2
( mc) ( mc) 2
mc
(
16 mc) ,
(9.90)
c
immediately giving the final result Emin = 7 mc 2, already obtained earlier in two more complex ways.
Let me hope that this example was a convincing demonstration of the convenience of representing 4-vectors in the contravariant (84) and covariant (85) forms,39 with Lorentz-invariant norms (86). To be useful for more complex tasks, this formalism should be developed a little bit further.
In particular, it is crucial to know how the 4-vectors change under the Lorentz transform. For contravariant vectors, we already know the answer (54); let us rewrite it in our new notation: Lorentz
transform:
A L A' .
(9.91) contravariant
vectors
where
L is the matrix (51), generally called the mixed Lorentz tensor:40
0 0
0 0
Mixed
L
,
(9.92) Lorentz
0
0
1 0
tensor
0
0
0 1
Note that though the position of the indices and in the Lorentz tensor notation is not crucial, because this tensor is symmetric, it is convenient to place them using the general index balance rule: the difference of the numbers of the upper and lower indices should be the same in both parts of any 4-vector/tensor equality. (You may check that all the formulas above do satisfy this rule.) 39 These forms are 4-vector extensions of the notions of contravariance and covariance, introduced in the 1850s by J. Sylvester (who also introduced the term “matrix” in its mathematical sense) for the description of the change of the usual 3-component spatial vectors at the transfer between different reference frames – e.g., resulting from the frame rotation. In this case, the contravariance or covariance of a vector is uniquely determined by its nature: if the Cartesian coordinates of a vector (such as the non-relativistic velocity v = dr/ dt) are transformed similarly to the radius-vector r, it is called contravariant, while the vectors (such as f ) that require the reciprocal transform, are called covariant. In the 4D Minkowski space, both forms may be used for any 4-vector.
40 Just as the 4-vectors, 4-tensors with two top indices are called contravariant, and those with two bottom indices, are covariant. The tensors with one top and one bottom index are called mixed.
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In order to rewrite Eq. (91) in a more general form that would not depend on the particular orientation of the coordinate axes (Fig. 1), let us use the contravariant and covariant forms of the 4-vector of the time-space interval (57),
dx cdt, d
r ,
dx
cdt, d
r ;
(9.93)
then its norm (58) may be represented as41
2
2
2
( ds) ( cdt) ( dr) dx
dx
dx dx .
(9.94)
Applying Eq. (91) to the first, contravariant form of the 4-vector (93), we get
dx L dx' .
(9.95)
But with our new shorthand notation, we can also write the usual rule of differentiation of each component x, considering it a function (in our case, linear) of four arguments x’, as follows:42
x
dx
dx' .
(9.96)
x'
Comparing Eqs. (95) and (96), we can rewrite the general Lorentz transform rule (92) in a new form, Lorentz
x
transform:
A
A' .
(9.97a)
general form
x'
which does not depend on the coordinate axes’ orientation.
It is straightforward to verify that the reciprocal transform may be represented as Reciprocal
Lorentz
x'
transform
A'
A .
(9.97b)
x
However, the reciprocal transform has to differ from the direct one only by the sign of the relative velocity of the frames, so for the coordinate choice shown in Fig. 1, its matrix is 41 Another way to write this relation is ( ds)2 = g dx dx = g dx dx, where double summation over indices
and is implied, and g is the so-called metric tensor,
1 0
0
0
0 1 0
0
g
g
,
0
0
1 0
0 0
0
1
which may be used, in particular, to transfer a covariant vector into the corresponding contravariant one and back: A = g A, A = g A. The metric tensor plays a key role in general relativity, in which it is affected by gravity – “curved” by particles’ masses.
42 Note that in the index balance rule, the top index in the denominator of a fraction is counted as a bottom index in the numerator, and vice versa.
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0 0
x'
0 0
,
(9.98)
x
0
0
1 0
0
0
0 1
Since according to Eqs. (84)-(85), covariant 4-vectors differ from the contravariant ones by the sign of their spatial components, their direct transform is given by matrix (98). Hence their direct and reciprocal transforms may be represented, respectively, as
x'
x
Lorentz
,
transform:
A
A'
A'
A ,
(9.99)
x
x'
covariant
vectors
evidently satisfying the index balance rule. (Note that primed quantities are now multiplied, rather than divided as in the contravariant case.) As a sanity check, let us apply this formalism to the scalar product A A. As Eq. (96) shows, the implicit-sum notation allows us to multiply and divide any equality by the same partial differential of a coordinate, so we can write:
x'
x
x'
A A
A' A'
A' A'
A' A'
A' A' ,
(9.100)
x
x'
x'
i.e. the scalar product A A (as well as A A) is Lorentz-invariant, as it should be.
Now, let us consider the 4-vectors of derivatives. Here we should be very careful. Consider, for example, the following 4-vector operator
, ,
(9.101)
x
( ct)
As was discussed above, the operator is not changed by its multiplication and division by another differential, e.g., x’ (with the corresponding implied summation over all four values of ), so
x'
.
(9.102)
x
x
x'
But, according to the first of Eqs. (99), this is exactly how the covariant vectors are Lorentz-transformed! Hence, we have to consider the derivative over a contravariant space-time interval as a covariant 4-vector, and vice versa.43 (This result might be also expected from the index balance rule.) In particular, this means that the scalar product
A
A
0
A
(9.103)
x
( ct)
should be Lorentz-invariant for any legitimate 4-vector. A convenient shorthand for the covariant derivative, which complies with the index balance rule, is
,
(9.104)
x
43 As was mentioned above, this is also a property of the reference-frame transform of the “usual” 3D vectors.
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so the invariant scalar product may be written just as A. A similar definition of the contravariant derivative,
,,
(9.105)
x ( ct)
allows us to write the Lorentz-invariant scalar product (103) in any of the following two forms: A