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The teaching of relativity usually starts with kinematics: The invariance of the speed of light, clock synchronization, time dilatation and length contraction, the relativity of simultaneity, Lorentz transformation and the Minkowski diagram. The change of the reference frame is a central topic. Only afterwards problems of relativistic dynamics are discussed. Such an approach closely follows the historical development of the Special Theory of Relativity. We believe that this access to relativity is unnecessarily complicated, and unsuitable for beginners. We present the basics of a teaching approach in which the initial postulate of relativity is the identity of energy and relativistic mass. Reference frame changes are largely avoided.
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F. Herrmannand M. Pohlig
Institute for Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT),
E-mail: f.herrmann@kit.edu
www.kit.edu
The teaching of relativity usually starts with kinematics: The invariance of the speed
of light, clock synchronization, time dilatation and length contraction, the relativity of
simultaneity, Lorentz transformation and the Minkowski diagram. The change of the
reference frame is a central topic. Only afterwards problems of relativistic dynamics are
discussed. Such an approach closely follows the historical development of the Special
Theory of Relativity.
We believe that this access to relativity is unnecessarily complicated, and unsuitable
for beginners. We present the basics of a teaching approach in which the initial postulate
of relativity is the identity of energy and relativistic mass. Reference frame changes are
largely avoided.
Keywords: Special relativity, additional postulate, reference frame change, relativistic
dynamics.
1. Introduction
Relativity, and we mean for the moment only special relativity, is more than 100
electromagnetism, which is certainly not simpler than special relativity, would still
not be included in the curricula 120 years after its creation, i.e. in 1980. There
are several reasons for this deﬁciency. Here we want to discuss only one of them:
Relativity is still taught today as it has originated historically. One starts with
a very special relativistic eﬀect, and works through with great eﬀort to the more
important and useful general statements. It is as if one enters a splendid palace not
by the beautiful main portal, but by some shabby servants’ entrance.
We describe the basics of a course on special relativity. The concept has been
tested and is used at numerous secondary schools: parts of it already in the lower
secondary school, the complete program in the upper secondary school. The course
is part of the Karlsruhe Physics Course 1. It can be downloaded from the Internet
in various languages. A bilingual English-Chinese version was published recently.2
We do not describe the details of this course. We are merely presenting some
ideas that underlie its development. Some of the topics we address are also discussed
in articles of the column Historical burdens on physics 3.
We justify our paradigm change in Section 2. We choose a diﬀerent “entrance”
to relativity. We substantiate our choice in section 2.1. In sections 2.2 and 2.3 we
begin with a critical discussion of two popular topics: the reference frame change,
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and the role of the observer. Section 2.4 is about naming. How is the word mass
used and how do we want to use it. Section 2.5 deals with the way to write Einstein’s
famous equation E=mc2. We are thus prepared for our main topic, which will be
treated in section 3: relativistic dynamics.
2.1. Additional postulate of the special theory of relativity
The laws of the special theory of relativity, or special relativity for short, are ob-
tained from those of classical mechanics by adding one extra postulate. Tradition-
ally and for historical reasons, the choice was made for the invariance of the speed
of light upon a change of the reference frame.
Once one has become aware that this choice as a starting point is only one of
several possibilities, one discovers that completely new perspectives arise for the
development of a teaching concept. We have decided to introduce the mass-energy
equivalence as an additional postulate instead of the invariance of the speed of light.
With this choice, we arrive more quickly at that part of special relativity that
we consider being the most important one, namely relativistic dynamics.
The traditional choice of the invariance of the speed of light has a rather in-
cidental cause: when special relativity came into being, light was the only known
system that behaved relativistically. Einstein’s work – both his famous publication
of 1905 Zur Elektrodynamik bewegter K¨orper 4and his textbook Grundz¨uge der Rel-
ativit¨atstheorie 5– begins with a detailed, and one can say somewhat tiring part on
relativistic kinematics.
One might imagine what the course of history would have been if the ﬁrst rela-
tivistic observation had been that a cup of hot coﬀee is heavier than a cup of cold
coﬀee (or that the corresponding observation had been made with particles in an
accelerator). The presentation of the theory of relativity in our textbooks would
certainly be very diﬀerent from what it is actually.
Of course, the mass-energy equivalence is not supported by our everyday expe-
rience (neither is the invariance of the speed of light). But one can make it easily
plausible and discuss its consequences even in beginners’ classes. It leads to sur-
prising and at ﬁrst unbelievable statements; but it does not lead to the cognitive
conﬂicts one has to deal with in the traditional approach to relativistic kinematics,
which questions our basic convictions about space and time.
2.2. Reference frames and reference frame changes
In the traditional approach to special relativity, the following topics are dealt with
The invariance of the speed of light
Clock synchronization
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The relativity of simultaneity
Time dilatation and length contraction
Lorentz transformation
Minkowski diagram
The problem with such an approach is that one begins with the most confusing
part of the theory: the relationship between space and time.
Certainly, students can learn a lot of physics by analyzing the same process
in diﬀerent reference frames. But we should not forget that we are dealing with
beginners, and it is better to stick to the old rule: Choose a suitable reference
frame right at the beginning, i.e. a reference frame in which the description of your
problem becomes as simple as possible, and don’t change it anymore.
And above all, don’t change the reference frame in the middle of dealing with
your problem (as is usually done when discussing the twin paradox). By the way, in
classical mechanics and electromagnetism, too, one can create the greatest confusion
if one chooses the reference frame improperly or if one changes it in the middle of
the discussion.
This is why our decision was not to make reference frame changes the main
topic of our lessons and to avoid them as far as possible. Above all, the impression
should not be created that special relativity is essentially a theory of reference frame
changes – an impression that some presentations certainly arouse. Even the name
relativity gives that impression.
2.3. The observer
Closely related to the question of the choice of the reference frame is the problem
of the so-called observer. The observer seems to be particularly important in two
areas of physics: in quantum physics (where the observer always appears as the one
making a “measurement”) and in the theory of relativity.
An observation is always made from a certain perspective. It thereby emphasizes
something that does not play a particular role in the phenomenon to be described.
We believe the observation should not be in the foreground as long as the un-
derstanding of a process is the objective. This is especially true when teaching at
school, i.e. beginners.
It is true that we get all the information about the world by observing and
measuring. But the idea we form of the world is quite diﬀerent from what we
observe. So, if we wanted to explain the shape of the earth to someone, we would
The earth is a sphere.
In our opinion when teaching physics we should primarily give a picture of what
nature is like – not how it is perceived by an observer.
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2.4. The use of the term mass
Mass is a physical quantity that until not so long ago did not cause any problems.
It was known for which properties it is a measure.
With Einstein’s special theory of relativity, this only changed insofar as the mass
of a body became dependent on its velocity, temperature and other variables. It
was no longer a quantity that had a characteristic value for a body or a particle.
Thus, a body, a particle, a ﬁeld, or any other structure, has a mass that depends,
among other things, on its velocity. The value that the mass assumes when the
centre of mass of the particle or body is at rest is called its rest mass (symbol
m0). Even more appropriate would actually be the less common term proper mass,
because when the centre of mass is at rest, this does not mean that the parts or
particles of the system are at rest.
It is that simple, or, unfortunately, one must say: it could be that simple.
For there is an area of physics in which another use of the term mass has estab-
lished itself: Particle physics. A particle has a well-deﬁned rest mass. The rest mass
is characteristic of the particle species. Among the various other parameters, such
as electric charge, spin, lepton number, etc., it is considered the main characteristic.
It seems to constitute the identity of the particle. For this property, a compact,
plausible name was needed, and particle physicists simply called it mass. Thus, in
particle physics, the term mass refers to only part of the quantity that describes
the inertia of a particle.
However, this custom also spread beyond particle physics, and this results in
several misunderstandings and ambiguities. What is to be understood by the mass
of a macroscopic body that is at rest? Is it the mass that would be measured with
a (very accurate) scale, or is it the sum of the (rest) masses of the particles that
constitute the body? This is a question that particle physicists probably don’t ask,
but we teachers do.
We have therefore decided to use the term mass (symbol m) exclusively for
the quantity that measures gravity and inertia, no matter what kind of object is
considered and in what state it is. Thus, a hot cup of coﬀee has a larger mass than
the same coﬀee when it is cold. A photon has a mass and a magnetic ﬁeld has
a mass (a liter of magnetic ﬁeld near a neutron star has a mass of some hundred
grams).
By the way, if one follows this use of the term mass, it makes no sense to say
that mass is a “form of energy” or that mass can be converted into energy.
2.5. The identity of mass and energy
First, let us look at the term mass-energy equivalence. It is a pity that a simple fact
is expressed so unclearly. The word equivalence is certainly not wrong, but why not
say directly: Mass and energy are the same physical quantity.
If one were to ask someone who has never seen the equation E=mc2to express
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this fact in a formula, he would probably write something like this:
E=k·m . (1)
The factor ktells us how the units joule and kilogram are converted into each
other. As the deﬁnitions of the units kilogram and joule are independent of the
choice of the reference frame, kis a universal constant.
Its value is obtained by a measurement. One ﬁnds
k= 9 ·1016 J/kg .(2)
But what is wrong with writing
E=mc2? (3)
Every student learns in mathematics that a linear relationship between the variables
xand yis written as
y=a·x . (4)
On the right side ﬁrst the factor of proportionality a, and second the independent
variable. The unbiased student might interpret the famous equation (3) this way:
The energy is proportional to the square of the speed of light – and not: energy
and mass are the same physical quantity. One might object: This can easily be
explained to the students. Of course it can. But doesn’t the statement become
clearer if one writes E=k·m? Would the iconic character of equation (3) survive
if it were formulated in this way?
3. The laws of dynamics
In the Karlsruhe Physics Course1, the extensive quantities energy, momentum, elec-
tric charge and entropy are introduced as basic quantities. Especially momentum
and entropy have a very direct and vivid interpretation. Momentum is a measure
of the “amount of motion”, that is, what one would colloquially call “impetus” or
“drive”. Entropy measures almost perfectly what would colloquially be called the
amount of heat (not to be confused with the rather diﬃcult concept of heat that
has established itself in physics).
Therefore, in the context of relativity, it is natural to ask in the ﬁrst place for the
dependence of diﬀerent quantities on momentum. Momentum is our independent
variable. We give momentum to a body or particle and ask how it reacts to it: How
does its mass (= energy) behave? What happens to its velocity? In other words:
We ask for the functions E(p) and v(p).
3.1. The energy momentum relationship
To derive E(p), we take over as much as possible from non-relativistic physics. In
addition, we only require the identity of mass and energy, i.e. we assume the validity
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of equation (1). We start with the change dEof the energy, that results from a
change of the momentum dp.
dE=vdp . (5)
With p=m·vwe obtain
dE=p
mdp . (6)
Replacing mwith E/k, and reordering returns
EdE=kpdp . (7)
We thus obtain
dE2=kdp2(8)
and
E2=kp2+C , (9)
where Cis the constant of integration.
The value of Ccan easily be determined, because for p= 0 the energy Eassumes
the value of the rest energy E0. Thus, Cmust be equal to E0
2. We therefore get
E2=E0
2+kp2(10)
and for the sought-after relationship between energy and momentum we get:
E(p) = qE0
2+kp2.(11)
The red line in Figure 1 shows the graphic representation of relation (11). Two
limiting cases are of particular interest.
Fig. 1. Relationship between mass/energy and momentum (red line). For large values of the mo-
mentum the curve approaches the asymptote (dashed line), for small values the classical quadratic
relation (grey line).
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For small momentum values, equation (11) changes to
E(p) = E0+kp2
2E0
=E0+p2
2m0
.(12)
We obtain the classical kinetic energy, increased by the rest energy (grey line in
Figure 1).
If the momentum is very large, so that E0
2can be neglected in comparison with
kp2, equation (11) turns into
E(p) = kp , (13)
see the dashed line in Figure 1. For bodies whose rest mass is 0 kg, equation (13)
applies for all values of the momentum, not only for large values (Figure 2). Thus,
in the highly relativistic limiting case, energy and momentum are proportional to
each other. This shows that there is a similarity between these quantities, which
becomes even clearer when we solve equation (11) according to E0
2
E0
2=E2kp2.(14)
Fig. 2. Energy momentum relationship for four diﬀerent rest masses. For photons (rest mass
zero) the relation is linear.
We thus have the rule: If the momentum of a body changes, its energy changes
in such a way that the diﬀerence E2kp2retains its value. This value is the square
of the rest energy E0.
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3.2. The velocity momentum relationship
Now our second question: How does the velocity of a body depend on its momen-
tum? We solve p=mv for v, then apply equations (1) and (11) and obtain
v(p) = p
m=kp
E=kp
pE0
2+kp2.(15)
If we replace the rest energy with the rest mass we get
v(p) = kp
pk2m02+kp2.(16)
Figure 3 shows the dependence of the velocity on the momentum for diﬀerent rest-
masses. From equation (16) follows that the velocity of a body approaches a termi-
nal value as the momentum increases. It is
lim
p→∞ v(p) = lim
p→∞
kp
pk2m02+kp2=k . (17)
Fig. 3. Dependence of the velocity on the momentum for diﬀerent rest masses.
Up to now, konly played the role of a conversion factor, but now it gets a
physical meaning. Its value is the square of the terminal speed. Since kis a universal
constant, its square root, i.e. the terminal speed, is also a universal constant. The
terminal speed is the same for all bodies and particles and is independent of the
reference frame.
This can be seen in Figure 3. The diagram also shows: the smaller the rest mass
of a body is, the “faster” it approaches the terminal speed.
Let us come back to equation (16). We see: If one supplies momentum to a
body, its velocity initially increases linearly with the momentum, while its mass
almost does not change. This is the Newtonian limiting case. When its momentum
has become very large, its velocity no longer changes, but its mass increases.
But what is the value of kand thus the value of the terminal speed? So far,
nothing has been said about it. The answer to this question can only be obtained
by a measurement. There are several ways to do that: Either one increases the
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momentum of a particle until its velocity no longer changes (in a particle accelerator)
and then measures its velocity, or one measures the velocity of photons, i.e. particles
of rest mass zero. Photons always move with the terminal speed.
Because of the great importance of the terminal speed, one gives it its own
symbol
c:= k . (18)
The measurement results in
c= 3 ·108m/s (19)
and therefore
k= 9 ·1016 J/kg .(20)
The constant cis also called speed of light. But our derivation shows that light
does not play a particular role in special relativity. That is why we prefer to call c
terminal speed.
3.3. Mass and inertia
From classical physics we are used to consider mass as a measure of inertia. Let us
ﬁrst clarify what is meaningfully understood by inertia.
To determine the inertia of an object, we supply a certain amount of momentum
to the object and we look at the resulting change in velocity. The more momentum
dpis needed to achieve a desired change in velocity dv, the greater the inertia.
Therefore we can deﬁne the inertia as
T:= dp
dv.(21)
We ﬁrst consider a classical motion, i.e. a motion with vc. We know the pv
relationship to be
p=m·v . (22)
This results in
T=m , (23)
which is no surprise.
If however the movement is relativistic, i.e. if no longer vc, things become
more complicated. From equation (15) we obtain
p(v) = m0v
q1v2
c2
(24)
and
T(v) = dp
dv=m0
1v2
c2
3
2
.(25)
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The inertia now depends on the velocity. It can no longer be described by a single
number. By the way, it is also not identical with the so-called relativistic mass.
We know a similar behavior from other contexts. The current-voltage relation-
ship of an ohmic resistor can be characterized by a single number, its resistance.
In general, however, the resistive behavior of an electrical component cannot be
characterized by a single number. What we need is the UIcharacteristic. The
situation is like that of inertia. In general, one cannot say that the mass is a measure
for the inertia of a body. Rather, the inertial behavior of a body is characterized by
a characteristic curve, equation (25). Sometimes the quantity deﬁned by equation
(25) is called the longitudinal mass. We think this is rather clumsy. The simple
facts are thereby somewhat obscured.
4. Conclusion
The development of a teaching concept for the school, in our case for the secondary
school, is a balancing act.
On one hand, teaching at school diﬀers fundamentally from popular science
presentations. The latter can limit themselves to showcasing the spectacular, the
impressive and the surprising of the scientiﬁc results – one can almost say: to exhibit
them like objects in a museum.
School teaching has to meet other requirements. The statements must be logi-
cally coherent. They have to ﬁt into the previous teaching and form a foundation
for the future teaching, for example, at the university.
On the other hand, we must make sure that we do not treat high-school students
like university students, that we do not overburden them. Let us not forget: One
can calculate and prove without generating understanding.
It should also be borne in mind that most high school students a priori have no
particular interest in physics.
We have tried to develop a course under these constraints. We would like to
emphasize once again that the above remarks do not represent the content of our
course. We have presented only what we believe is diﬀerent in our approach from
that of other textbooks.
References
1. F. Herrmann et al., The Karlsruhe Physics Course (2016), http://www.
physikdidaktik.uni-karlsruhe.de/.
2. F. Herrmann et al., The Karlsruhe Physics Course (Guangzhou: Guangdong Educa-
tion Publishing House , 2018), http://www.physikdidaktik.uni-karlsruhe.de/.
3. F. Herrmann, G. Job, Historical burdens on physics (2019), http://www.
physikdidaktik.uni-karlsruhe.de/.
4. A. Einstein, Zur Elektrodynamik bewegter K¨orper, Vol 10 (322) , (Annalen der
Physik, 1905), pp. 891–921.
5. A. Einstein, Grundz¨uge der Relativit¨atstheorie, 5th edn. (Berlin: Akademie-Verlag,
1970).