The theory of relativity and the Pythagorean theorem∗

285
The theory of relativity
and
the Pythagorean theorem∗
L. B. Okun
Abstract
It is shown that the most important effects of special and general theory of relativity
can be understood in a simple and straightforward way. The system of units in which
the speed of light c is the unit of velocity allows to cast all formulas in a very simple
form.The Pythagorean theorem graphically relates energy, momentum and mass. The
paper is addressed to those who teach and popularize the theory of relativity
1
Introduction
The report ”Energy and mass in the works of Einstein, Landau and Feynman” that I was
preparing for the Session of the Division of Physical Sciences of the Russian Academy
of Sciences (DPS RAS) on the occasion of the 100th anniversary of Lev Davidovich
Landau’s birth was to consist of two parts, one on history and the other on physics.
The history part was absorbed into the article “Einstein’s formula: E0 = mc2. ‘Isn’t the
Lord laughing?”’ that appeared in the May issue of Uspekhi Fizicheskikh Nauk [Physics-
Uspekhi] journal [1]. The physics part is published in the present article. It is devoted to
various, so to speak, technical aspects of the theory, such as the dimensional analysis and
fundamental constants c and ¯
h; the kinematics of a single particle in the entire velocity
range from 0 to c; systems of two or more free particles; and the interactions between
particles: electromagnetic, gravitational, etc. The text uses the slides of the talk at
the session of the Section of Nuclear Physics of the DPS RAS in November 2007 at the
Institute for Theoretical and Experimental Physics (ITEP). My goal was to present the
main formulas of the theory of relativity in the simplest possible way, using mostly the
Pythagorean theorem.
2
Relativity
2.1. The advanced standpoint.
The history of the concept of mass in physics runs to many centuries and is very in-
teresting, but I leave it aside here. Instead, this will be an attempt to look at mass from
an advanced standpoint. I borrowed the words from the famous title of Felix Klein’s
Elementary Mathematics from an Advanced Standpoint (traditionally translated into
∗This is a slightly corrected version of the paper published in Physics - Uspekhi 51 622 (2008). In particular,
the short subsections of the text are numbered as had been suggested by one of the readers.
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Russian incorrectly as Elementary Mathematics from the Standpoint of Higher Math-
ematics. See V.G. Boltyanskii’s foreword to the 4th Russian edition). The advanced
modern standpoint based on principles of symmetry in general and on the theory of rel-
ativity in particular makes it possible to avoid inevitable terminological confusion and
paradoxes.
2.2. The principle of relativity. Ever since the time of Galileo and Newton, the
concept of relativity has been connected with the impossibility of detecting, by means
of any experiment, a translational (uniform and rectilinear) motion of a closed space
(for instance, inside a ship) while remaining within this space. At the turn of XIX and
XX centuries Poincar´e gave to this idea the name ‘the principle of relativity’ . In 1905
Einstein generalized this principle to the case of the existence of the limiting velocity of
propagation of signals. (The finite velocity of propagation of light has been discovered
by R¨omer already in 1676). Planck called the theory constructed in this way ‘Einstein’s
theory of relativity’.
2.3. Mechanics and optics. Newton tried to construct a unified theory uniting
the theory of motion of massive objects (mechanics) and the theory of propagation of
light (optics). In fact, it became possible to create the unified theory of particles of
massive matter and of light only in the XX century. It was established on the road to
the vantage ground of truth (I am using here the ironical wording of Francis Bacon) that
light is also a sort of matter, just like the massive stuff, but that its particles are massless.
This interpretation of particles of light — photons — continues to face resistance from
many students of physics, and even more from physics teachers.
3
Dimensions
3.1. Units in which c = 1. The maximum possible velocity is known as the speed
of light and is denoted by c. When dealing with formulas of the theory of relativity it
is convenient to use a system of units in which c is chosen as a unit of velocity. Since
c/c = 1, using this system means that we set c = 1 in all formulas, thus simplifying them
greatly. If time is measured in seconds, then distance in this system of units should be
measured in light seconds: one light second equals 3 · 1010 cm.
3.2. Poincar´
e and c. One of the creators of the theory of relativity, Henri Poincare,
when discussing in 1904 the fact that c is found in every equation of electrodynamics,
compared the situation with the geocentric theory of Ptolemy’s epicycles in which ev-
ery relation between motions of celestial bodies included the terrestrial year. Poincare
expressed his hope that the future Copernicus would rid electrodynamics of c [3]. How-
ever, Einstein showed already in 1905 that c was to play the key role as the limit for the
velocity of signal propagation.
3.3. Two systems of units: SI and c = 1. The unit of velocity in the In-
ternational System of Units SI, 1 m/s, is forced on us by convenience arguments and
by standardization of manufacturing and commerce but not by the laws of Nature. In
contrast to this, c as a unit of velocity is imposed by Nature itself when we wish to
consider fundamental processes of Nature.
3.4. Dimensional factors. Consider some physical quantity a. Let us denote
by [a] the dimension of the quantity a. The dimension of a definitely changes if it is
multiplied by any power of the universal constant c but its physical meaning remains
unaffected. In what follows I explain why this is so.
3.5. Velocity, momentum, energy, mass. The dimensions of momentum, mass,
and velocity of a particle are usually related by the formula [p] = [m][v] while the
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dimensions of energy, mass, and velocity are related by the formula [E ] = [m][v 2]. Let
us introduce dimensionless velocity v/c and from now on denote this ratio as v. Likewise,
referring to momentum p we actually mean the ratio p/c. When speaking of energy, we
actually mean the ratio e = E/c 2. Obviously, the dimensions of p, e, and m become
identical and therefore, these quantities can be measured in the same units, for example,
in grams or electron-volts, as is customary in elementary particle physics.
3.6. On the letter e denoting energy. Choosing e as the notation for energy
may invite the reader’s ire since this symbol traditionally stands for electron and electric
charge. However, this choice cannot cause confusion and, importantly, it will lead to a
compact form of formulas for a single particle, always reminding us that these formulas
were written using the system of units in which c = 1. On the other hand, it will be
clear a little later that the letter E is a convenient notation for the energy of two or
more particles. I happened to see Einstein’s formula with a lower-case e on a billboard
on Rublevskoye highway in Moscow. I wonder, why should this e irritate physicists?
3.7. On the difference between energy and frequency. Two paragraphs ago I
insisted that e = E/c 2 is energy even though its dimension is that of mass. In that case
it is logical to ask why ω = E/¯
h is not energy but frequency? Indeed, the quantum of
action ¯h, like the speed of light c, is a universal constant. The answer to this question
can be found by considering how e and ω are measured. E and e are measured by the
same procedure, say, using a calorimeter, while frequency is measured in a drastically
different manner, say, using clocks. Therefore, the equality ω = E/¯
h informs us of the
link between two different types of measurement, while the equality e = E/c 2 carries
no such information. Arguments similar to those concerning frequency hold equally well
for wavelength. I have to emphasize that these metrological distinctions are mostly of a
historical nature since in our day atomic clocks operate on the difference between atomic
energy levels.
4
Single particle
4.1. Relative and absolute quantities. The kinetic energy of any body is a relative
quantity: it depends on the reference frame in which it is measured. The same is true
for the momentum of a body and its velocity. In contrast to them, the mass of a body
is an absolute quantity: it characterizes the body as such, irrespective of the observer.
The rest energy of a body (see below) is also an absolute quantity since the frame of
reference is fixed in it once and for all — ‘nailed to it’.
4.2. Invariant mass. The mass of a body is defined in the theory of relativity by
the formula
m2 = e2 − p2.
(1)
Here and in what follows p = |p|. Likewise, v = |v|. Note that energy and momentum of
a given body are not bounded from above while the mass of the body is fixed. Formula
(1) is the simplest relation between energy, momentum, and mass that one could write
‘off the top of one’s head’. (The relation between e, p, and m cannot be linear since p
is a vector while e and m are scalars in three-dimensional space.) We shall see now that
formula (1) has another, much more profound theoretical foundation.
4.3. The 4-momentum. Minkowski was the first to point out that the theory of
relativity gains the simplest form if considered in four-dimensional spacetime [4]. Energy
and momentum in the theory of relativity form a four-dimensional energy-momentum
vector pi(i = 0, a), where p0 = e, pa = p, and a = 1, 2, 3. Mass is the Lorentz scalar that
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characterizes the length of the 4-vector p
2
i: m 2 = pi
= e 2 − p2; four-dimensional space
is pseudo-Euclidean, which explains the minus sign in the formula for length squared.
(The reader will recall that p2 = p2.) Another way to clarify why the sign is negative is
by introducing the imaginary momentum ip. Then m 2 = e 2 + (ip)2 and we are dealing
with the Pythagorean theorem for such a pseudo-Euclidean right triangle in which the
hypotenuse m is shorter than the cathetus e.
4.4. Relation between momentum and velocity. The momentum of a body is
related to its velocity v by the formula
p = ev.
(2)
This formula satisfies in the simplest manner the requirement that the momentum 3-
vector be proportional to the velocity 3-vector and that the dimensional proportionality
coefficient not vanish for the massless photon. Conservation of the thus defined momen-
tum in the theory of relativity is implied by the uniformity of 3-space while conservation
of energy is implied by the uniformity of time (Noether’s theorem).
4.5. The Pythagorean theorem. Formula (1) is shown in Fig. 1 by an ordinary
right triangle in which m and p are catheti and e is the hypotenuse.
4.6. Transition from m �= 0 to m = 0. Formula (1) is obviously valid at m = 0
while formula (2) holds for v = 1. This implies that there is a smooth transition from
massless particles to massive, when the energy of the latter particles greatly exceeds
their mass.
4.7. Physics from p = 0 to p = e. Let us consider formulas (1) and (2) first at
zero momentum, then in the limit of very low momenta (when p � m), and then in the
limit of very high momenta when p ∼ e � m, and finally in the case of massless photons.
We will call the case of very small momenta and velocities the Newtonian case, and that
of very high momenta and velocities close to the speed of light, the ultrarelativistic case.
We will start with zero momentum.
e
p
m
Fig. 1
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5
Rest energy
5.1. Zero momentum. If momentum is zero, then in the case of a massive particle
the velocity is also zero and energy e is by definition equal to the rest energy e0. (The
subscript 0 reminds us that here we are dealing not with the energy of a given body in
general but with its energy precisely in the case when its momentum is zero!) Hence
equation (1) implies
e0 = m.
(3)
If, however, the particle is massless, then equation (1) at p = 0 implies that e = e0 = 0
(see 7.6).
5.2. Horizontal ‘biangle’. If m �= 0 and p = 0, then the triangle shown in Fig.1
‘collapses’ to a horizontal ‘biangle’ (Fig. 2).
5.3 Einstein’s great discovery. In units in which c �= 1, equation (3) has the form
E0 = mc 2 .
(4)
The realization that ordinary matter at rest stores an enormous amount of energy in its
mass was Einstein’s great discovery.
5.4. The ‘famous formula’. Equation (4) is very often written (especially in
popular physics literature) in the form of ‘Einstein’s famous equation’ that drops the
subscript 0:
E = mc 2 .
(5)
This simplification, to which Einstein himself sometimes resorted, might seem innocuous
at first glance, but it results in unacceptable confusion in understanding the foundations
of physics. In particular, it generates a totally false idea that ‘according to the theory
of relativity’ the mass of a body is equivalent to its total energy and, as an inevitable
result, depends on its velocity. (‘Wished to make it simpler, got it as always’.1)
e0
m
Fig. 2
5.5. No experiment can disprove the ‘famous formula’. Very clever people
thought up this formula in such a way that it never contradicts experiments. However,
it contradicts the essence of the theory of relativity. In this respect, the situation with
the ‘famous formula’ is unique — I do not know another case that could be compared
with this one.
5.6. This is not a matter of taste but of understanding. You hear time
and again that the introduction of momentum-dependent mass is ‘a matter of taste’.
Of course, one can write the letter m instead of E/c 2 and even call it ‘mass’, although
1A paraphrase of former Russian Prime Minister Chernomyrdin’s ‘statement of the day’: “Wished to make
it better, got as always.” (Note added by the Author in translation.)
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it is no more sensible than writing p instead of E/c and calling it ‘momentum’. Alas,
this ‘dress changing’ introduces unnecessary and bizarre notions — relativistic mass and
rest mass m0— and creates an obstacle to understanding the theory of relativity. A
well-known Russian proverb comes to mind: “Call me a pot if you wish but don’t push
me into the oven.” Unfortunately, people who call E/c 2 ‘mass’ do place this ‘pot’ into
the ‘oven’ of physics teaching.
5.7.
Longitudinal and transverse masses. In addition to relativistic mass,
concepts of intense use at the beginning of the XX century were the transverse and
longitudinal masses: mt and ml. This longitudinal masses increased as (e 3/m 3) m and
‘explained’ — in terms of Newton’s formula F = ma — why a massive body cannot
be accelerated to the speed of light. Then it was forgotten and such popularizers of
the theory of relativity as Stephen Hawking started to persuade their readers that even
much gentler growth of mass with velocity ((e/m) m) could explain why the velocity of
a massive body cannot reach c. I single out Hawking only because, printed on the dust
jacket of the Russian edition of his latest popular science book [5], which advertises the
formula E = mc 2, we see this text: “Translated into 40 languages. More than 10 million
copies sold worldwide.”
5.8.
False intuition. After my talk at the ITEP A N Skrinsky told me that
the notion of relativistic mass hampered a well-known physicist’s understanding that a
relativistic electron colliding with an electron at rest can transfer all its energy to the
latter. Indeed, how could a heavy baseball bat transfer all its energy to the lightest ping-
pong ball? In physics, as in daily life, people very often rely on intuition. This is why
it is so important, when studying the theory of relativity, to work out the relativistic
intuition and mistrust nonrelativistic intuition. (In order to ‘feel’ how an electron at
rest can receive the entire energy of a moving electron it is sufficient to use their center-
of-inertia frame to consider scattering by 180 degrees, and then return back to the
laboratory frame.)
6
Newtonian mechanics
6.1. Momentum in Newtonian mechanics. Newtonian mechanics describes with
high accuracy the motion of macroscopic bodies in a terrestrial environment and of
massive celestial bodies because their velocities are much smaller than the speed of
light. For instance, the velocity of a bullet is of the order of 1 km/s, which corresponds
to v = 1/300000 and v 2 = 10−11. In this situation equation (2) reduces to
p = mv .
(6)
Equation (1) is schematically shown in the Newtonian limit in Fig. 3. The side of the
triangle representing p in Fig. 3 is far too long. Scaled correctly, it should be a few
microns.
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e
p
m
Fig. 3
6.2. Kinetic energy ek. It is reasonable to rewrite formula (1) for low velocities
so as to isolate the contribution of the short cathetus:
e 2 − m2 = p2
(7)
and then to present it in the form
(e − m)(e + m) = p2 .
(8)
This allows us to obtain a nonrelativistic expression for kinetic energy without resorting
to the conventional series expansion of the square root. We take into account that the
total energy e is the sum of rest energy e0 and kinetic energy ek and therefore e = m+ek.
6.3. Energy in Newtonian mechanics. In the Newtonian limit we have ek � m
(e.g. for a bullet ek/m = 10−11). Energy can therefore be replaced with high accuracy
by mass m in formula (2) for momentum and in the factor (e + m) in equation (8).
This last equation immediately implies an expression for kinetic energy ek in Newtonian
mechanics:
p 2
mv 2
ek =
=
.
(9)
2m
2
6.4. Potential energy. In addition to velocity-dependent kinetic energy, an impor-
tant role in nonrelativistic mechanics is played by potential energy, which depends only
on the position (coordinate) of the body. The sum of kinetic and potential energy is
conserved at any instance of time. The potential energy of a body placed in an external
field of force is defined to within an arbitrary additive constant because the force acting
on the body equals the gradient of potential energy. In a similar manner, the potential
energy of interaction of several bodies depends only on their positions at the moment of
interaction. However, in the theory of relativity any interaction propagates at a finite
velocity. Hence, potential energy is an essentially nonrelativistic concept.
6.5. Newton and modern physics. Newton’s flash of genius marked the birth
of modern science. The post-Newtonian progress of science is fantastic. Today’s under-
standing of the structure of matter is radically different from Newton’s. Nevertheless,
even in the XXI century many physics textbooks continue to use Newton’s equations
at energies ek � e0, which exceed the limits of applicability of Newton’s mechanics
(ek � e0) by many orders of magnitude. If some professors prefer to insist on keeping
up with this tradition of velocity-dependent mass, they ought to at least familiarize
their students with the fundamental concepts of mass and rest energy, and with the true
Einstein equation E0 = mc 2.
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7
Ultrarelativism
7.1. High energy physics. Let us now consider in some detail the limiting case in
which e/m � 1. The ratio of energy and mass characteristic for high energy physics is
precisely this. For example, this ratio for electrons in the LEP (Large Electron-Positron)
Collider at CERN was e/m = 105, since m = 0.5 MeV and e = 50 GeV. For protons in
the LHC (Large Hadron Collider), which is located in the same tunnel where the LEP
was in previous years, we find e/m ∼ 104. (Here, m ∼ 938 MeV, e ∼ 7 TeV.)
7.2. A vertical triangle. The triangle for protons in the LHC is drawn highly
schematically in Fig. 4. Its base is in fact shorter than its hypotenuse by four orders of
magnitude.
7.3. The neutrino. Neutrinos are even more ultrarelativistic particles: their masses
are a fraction of one electron-volt and their energies reach several MeV for neutrinos
emerging from the Sun and nuclear reactors, and several GeV for neutrinos generated
in particle decays in cosmic rays and in accelerators. The base of the triangle shown
schematically in Fig. 4 is much shorter at these energies than its vertical cathetus and
its hypotenuse.
e
p
m
Fig. 4
7.4. Neutrino oscillations and m2/2e. Equation (e−p)(e+p)= m2 immediately
implies that e − p � m2/2e. The differences between the masses of three neutrinos
ν1, ν2, ν3 possessing definite masses in a vacuum result in oscillations between neutri-
nos having no well-defined masses but possessing certain flavors: νe, νµ, ντ . (This phe-
nomenon is similar to well-known beats that occur when several frequencies interfere.)
The neutrino oscillation data give
∆m 2
21 = (0.77 ± 0.04) × 10−4 eV2
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|∆m232| = (24 ± 3) × 10−4 eV2 .
7.5. The photon. The photon mass is so small that no experiment has been able
to detect it. Hence, it is usually assumed that the photon mass equals zero. This means
that for a photon e = p, where p = |p|, and the triangle shown in Fig. 4 collapses to a
vertical biangle (Fig. 5).
p
e
Fig. 5
7.6 The photon and rest energy? It is logical to conclude the discussion of
single-particle mechanics by returning to the question: is the concept of rest energy e0
applicable to massless photon? It may seem at first glance that it is not, since a photon
propagates at the speed c, however small its energy is, so that ‘a rest for it is but a
dream’ 2. This being so, how can we use the equality e0 = 0 if the photon is never at
rest? We can because our e0 is defined as the energy corresponding to zero momentum,
not velocity. Obviously this energy is zero for the photon with p = 0: this is implied
by equation (1). If a particle has m = 0, p = 0, e = 0 and biangle of Fig. 5 collapses
to a point, we can say that it ‘passed away to the state of eternal rest’. Looking at
the limiting transition to zero mass, we can show that the reference frame in which a
photon is ‘eternally at rest’ has to be rigidly connected to another ‘eternally resting’
photon. Consequently, the value e0 = 0 at m = 0 is in perfect agreement with the
limiting transition.
8
Two free particles
8.1. Collision of two particles. Colliders. If two particles collide at relativistic
2This is a paraphrase of the famous line from Alexander Block. (Note added in translation).
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energies, a comparison of the reference frame in which one of them is at rest with
a reference frame in which their common center of inertia is at rest demonstrates the
advantages of the latter. We already saw this in the case commented on by A N Skrinsky.
If the momenta of the colliding particles are equal and oppositely directed, as for example
in the LHC or LEP collider, then practically the entire energy of the colliding particles
may be spent on the creation of new particles.
8.2. Mass of a system of particles. The total energy E and the total momentum
P of an isolated system of particles are conserved. Energy and momentum being additive,
for two free particles we have
E = e1 + e2
(10)
P = p1 + p2.
(11)
We now define the quantity M by the formula
M 2 = E 2 − P2.
(12)
8.3. Masses are additive at v = 0. Equation (12) is invariant under Lorentz
transformations, as is equation (1). Therefore, it is logical to refer to M as the mass of
a system of two particles. In the static limit, when p1 and p2 equal zero, equation (12)
implies that
M = e01 + e02 = m1 + m2.
(13)
In the Newtonian limit, M equals the sum of the masses of the two particles with an
accuracy of (v/c)2, i.e. the masses are practically additive.
8.4. Masses are not additive at v �= 0. However, M and the masses m1 and
m2 are practically unrelated at high velocities. For instance, M exceeds the electron
mass in the LEP collider or the proton mass in the LHC by four orders of magnitude
(see section 7). The value of M is crucially dependent on the relative directions of the
momenta of two particles, since the sum of two vectors is a function of the angle between
them. Thus, we have for two photons moving in the same direction
P = |P| = |p1 + p2| = p1 + p2.
(14)
8.5. Collinear photons. For photons p1 = e1, and p2 = e2. Therefore, for two
photons moving in the same direction we can write
P = p1 + p2 = e1 + e2 = E.
(15)
Equation (12) then implies that in this case the mass of a pair of photons M = 0. And
this means that the mass of a ‘needle’ light beam is zero.
8.6. What if photons fly away from each other? However, if photons fly away
in opposite directions with equal energies, then p1 = −p2 and P = 0. In that case, the
rest energy of two photons simply equals the sum of their energies and the mass of this
system is
M = E0 = 2e.
(16)
8.7. Shock. Of course, the statement that a pair of two massless or very light
particles might have an enormous mass may shock the unprepared reader. Is there any
sense in speaking of the rest energy of two photons if ‘rest is but a dream’ to either of
them? What is at rest in this case?
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