Content uploaded by Friedrich Herrmann

Author content

All content in this area was uploaded by Friedrich Herrmann on Oct 06, 2021

Content may be subject to copyright.

September 30, 2021 16:5 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ”Teaching Relativity A paradigm

change” page 1

1

Teaching relativity: A paradigm change

F. Herrmann∗and M. Pohlig

Institute for Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT),

Karlsruhe, Baden-W¨urttemberg, Germany

∗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

years old, but still has not found its place in school. Just compare: Faraday-Maxwell

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,

September 30, 2021 16:5 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ”Teaching Relativity A paradigm

change” page 2

2

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. Paradigm change

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

before relativistic dynamics is addressed:

The invariance of the speed of light

Clock synchronization

September 30, 2021 16:5 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ”Teaching Relativity A paradigm

change” page 3

3

The relativity of simultaneity

Time dilatation and length contraction

Velocity addition

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

certainly not start with the shadow of the obelisk in Alexandria, but simply say:

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.

change” page 4

4

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

change” page 5

5

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

change” page 6

6

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).

change” page 7

7

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=E2−kp2.(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 E2−kp2retains its value. This value is the square

of the rest energy E0.

change” page 8

8

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

change” page 9

9

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 p−v

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

q1−v2

c2

(24)

and

T(v) = dp

dv=m0

1−v2

c2

3

2

.(25)

change” page 10

10

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 U−Icharacteristic. 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).