A New Way of Teaching the Special Theory of Relativity

-1-
A New Way of Teaching the Special Theory of Relativity

by Michael Pohlig and Hans Michael Strauch



1 What changes Newtonian Mechanics into the Special Theory of Relativity?

Usually, in introducing in the Special Theory of Relativity we begin by discussing the failure
of the MICHELSON-MORLEY experiment. FITZGERALD’s hypothesis, elaborated by LORENTZ
stated that all bodies in motion should be shortened in the direction of their velocity. He be-
lieved this contraction to be caused by special molecular forces. Very different to this explana-
tion was the assumption made by EINSTEIN:

Axiom: The velocity of light in empty space is the same in all reference frames and is inde-
pendent of the motion of the emitting body.

This axiom turns NEWTONIAN mechanics into EINSTEIN’s Special Theory of Relativity. Using
this axiom, EINSTEIN could prove the very important theorem:

Theorem: The mass of a body and the energy of a body are just different words for the same
physical quantity.1


The well-known formula for this theorem is:

E = m ⋅ c2 (i)

In the Karlsruhe Physics Course2 EINSTEIN’s axiom and theorem change places. The sentence
- mass and energy are the same physical quantity - is now the axiom which changes
NEWTONIAN mechanics into EINSTEIN’s Special Theory of Relativity and the sentence - the
velocity of light in empty space is the same in all reference frames and is independent of the
motion of the emitting body - becomes a theorem. For mathematicians such an exchange is not
unusual. One reason for exchanging an axiom with a theorem is that it actually aids compre-
hension of the problem. This paper will show, that many important results from the Special
Theory of Relativity can be developed from the axiom: Mass and energy are the same physi-
cal quantity.

1 EINSTEIN writes in his article: ‘Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?’ : „Die
Masse eines Körpers ist ein Maß für seinen Energieinhalt; ändert sich seine Energie L, so ändert sich die
Masse in dem selben Sinne um L/9.1020, wenn die Energie in Erg und die Masse in Gramm gemessen wird.“
(quoted from [1]). Translation: The mass of a body is a measurement of its energy, if its energy L changes, so
the mass will change in the same way by L/9.1020 (if the energy is measured in erg and the mass in gramme).
In [2] EINSTEIN writes: „Nach der Relativitätstheorie gibt es keinen prinzipiellen Unterschied zwischen Masse
und Energie,...“ and „Masse ist Energie..“ Translated: "According to the Special Theory of Relativity there is,
principally, no difference between mass and energy,... " and "Mass is energy...")
2 The Karlsruhe physics course was developed by F. HERRMANN, university Karlsruhe, Germany.

E-mail: didaktik@physik.uni-karlsruhe.de.
Further information (publications etc.) Internet: http://www.physik.uni-karlruhe.de/˜didaktik/

A new Way of Teaching the Special Theory of Relativity - Pohlig-Strauch

---

-2-
Formula (i) is often misunderstood. People think the formula means that energy could be
changed into mass or vice versa - mass could be changed into energy. Such an interpretation
of the formula leads to error if it is used in the same physical system. It would mean that in a
system energy could increase at the cost of mass and vice versa - the mass could increase at
the cost of energy. This, however, is not correct. In fact, mass and energy are different words
for the same physical quantity. The factor c2 in formula (i) converts the unit Kilogram (kg)
into Joule (J), nothing more. It would be better to write formula (i) as

E = k ⋅ m (i’)

The fact that c = k is a velocity, with a certain role in Einstein’s Special Theory of Relativ-
ity, will become evident later. In school formula (i’) should be used; to save time and space
formula (i) is used in this paper.



2 The role of momentum and energy in the description of bodies in motion

In Newtonian mechanics the primary concepts are trajectory, velocity, mass and force. Mo-
mentum and energy are nothing more than tools for easier calculations. In contrast to this, the
Karlsruhe Physics Course is based on quantities which are primary quantities of quantum me-
chanics. These quantities have something in common, namely, each can be pictured as a kind
of "stuff". This is the reason why they are called "substance-like" [3]. Like any substance these
quantities can be thought to be brought into a physical system, they can flow from one system
to another and there exists currents of such quantities. Momentum p and energy E, respec-
tively, are such "substance-like" quantities. Their currents are traditionally called force F and
power P. We prefer to use the names "momentum current" instead of force and "energy cur-
rent" instead of power. The pictures these names create in our minds are more practical than
the ones created by the traditional names.

Here is a short summary of the most important rules pupils should know when they follow the
Karlsruhe Physics Course:

(1) A body in motion has momentum. If the mass and the velocity of the body are known,
we can calculate its momentum by the formula:
p = m⋅ v (ii)3
A body with the mass 1kg and the velocity 1m/s has a momentum of 1Hy4

(2) Momentum can flow from one body to another, providing that there is a conducting
connection between both bodies.
If in the time interval ∆t the momentum ∆p flows into a body, we say, the momentum cur-
rent into the body is:

3 We discuss motions in only one direction, hence we consider only one component of the vector p=(px,py,pz).
If there is only the x-component, we omit the index ‘x’. In this paper this is valid for all vectors.
4 Hy is the abriviation for Huygens.

A new Way of Teaching the Special Theory of Relativity - Pohlig-Strauch

---

-3-
p
∆
I = F =
p
t
∆ (iii) 5
Hy
The unit of the momentum current is 1Newton ( 1N = 1
)
s
(3) If momentum carries energy, the energy current is given by:
P = v ⋅ F (iv)

3 Model-building and the Karlsruhe physics course [4]

In this paper the model-building software Powersim™ [5] is used. A great advantage of a
model-building system like Powersim and others like Stella [6] is that models can be edited by
using graphic symbols. Another advantage is that these flow diagram symbols, used by Pow-
ersim6, allow us to build a picture of the substance-like quantities in our minds. In the model
(Fig. 1) a level symbol is shown. It represents the quantity X, the momentum p for example.
An arrow leads from a cloud into the level symbol. If X is momentum p, then there is a current
of momentum and momentum will be accumulated. The fact that the arrow starts in a cloud
shows us that it is not specified where the momentum current comes from.

?
X
Ix
rate symbol
level symbol
Figure 1: Some symbols of Powersim

The model shown in fig 1. represents the following iteration loop: the rate of change ∆X of X
in a region is equal to the current into or out of the region, and the quantity X obeys a general
conservation law.

X
:= X
+ X
∆ = X + I (t) ⋅ ∆t
new
old
old
X

The loop will be executed from t1 to t2 with a time step ∆t . The initial value of the quantity X,
that is Xold for the first run of the loop, and the time step ∆t can be chosen by the user of the
program. The iteration loop will be executed automatically. This is programmed by editing the
model using the graphic symbols, as shown in fig 1. Another advantage of the model building
program is that I (t) must not be constant. When pupils first start to work with a model
X
building system, very simple models should be chosen to familiarise pupils with such systems.

5 Currents are often designated by the symbol ‘IX ‘,while the index X indicates the quantity which is flowing. If
X is the momentum p, traditionally the momentum current is named F. Each component of momentum obeys
a general conservation law, hence the flow of momentum into a region is equal to the rate of change of mo-
mentum inside the region.
6 MIT - standard

A new Way of Teaching the Special Theory of Relativity - Pohlig-Strauch

---

-4-
3.1 Free falling of a body - the non relativistic view

One simple example is free falling of a body. Fig 2 shows a stone hanging on gallows. The
spring is stretched. Thus momentum is leaving the stone through the spring. Since nothing is
in motion, no momentum is accumulated. If momentum is leaving the stone but the stone re-
mains in rest, a momentum current of the same amount must enter it. Its value is:
F = m ⋅ g (v)
Here g is the acceleration due to gravity. If we cut the connection between the stone and the
spring, momentum cannot leave the stone anymore, thus the stone will fall to earth. Using the
equations (ii), (iii), and (v) we find v = g ⋅ t , the velocity increases linearly with time. This is
shown by several well-known experiments.


Figure 2: Momentum flows through the gravitation field into the stone and through the bars back to earth


Figure 3: View of a model in a Powersim application window: diagram (left) equations (right)
Fig. 3 shows the Powersim application window with two views of the model ‘free falling’ in
the workspace, each in separate smaller windows. The left one, the diagram view, displays the
model’s structure using the flow diagram symbols. The mass m and acceleration g are gener-
ated as constants, the velocity as a variable. Constants are displayed as rhombuses, variables
as circles. The curved lines from the mass symbol to the velocity symbol and from the mo-

A new Way of Teaching the Special Theory of Relativity - Pohlig-Strauch

---

-5-
mentum symbol to the velocity symbol represent the links between these quantities. The ve-
locity v is defined as:
p
v =

m
The momentum must be initialized. Here the initial value is taken to be 0 Hy, this means that
at the beginning of the simulation the body is not in motion. The momentum current is set to
constant 10N and the mass to 2kg. Fig. 4 shows the simulation setup, that is: start time, stop
time and time step. The iteration loop is calculated as a numerical integration. In the model we
use the integration method RUNGE-KUTTA with variable step size7.


Figure 4: Simulation Setup
400
80
300
2
1 2
2
60
1 2
200
p1
1
v1
2
1
40
1
1 2
1
p2
100
v2
2
2
2
2
1
20
1
1
01 2
01 2
0
2
4
6
8
10
0
2
4
6
8
10
TIME
TIME

Figure 5: p-t-diagram (left) and v-t-diagram (right)

Powersim is able to display all kinds of time graphs, phase-diagrams and time tables for all
quantities used in the model. In fig. 5 we see the p-t- and the v-t-diagrams for two different
masses. As expected, the curves are straight lines and the v-t-diagrams are the same. We see,
that the accumulation of momentum in a body during free falling leads to an increase in veloc-
ity. In other words: the increase in velocity indicates that momentum is accumulating. This is
different to what is stated in EINSTEIN’s Special Theory of Relativity. As we will see, momen-
tum can be accumulated in a body without the velocity increasing.

7 See further information in [7] or standard books on numerical integration.

A new Way of Teaching the Special Theory of Relativity - Pohlig-Strauch

---

-6-
Fig. 6 shows the modified flow diagram. Energy is added as a variable. The result of the simu-
lation is shown in fig. 7. The energy E is initialized with 0J and the energy current is given by
P = F ⋅ v. Therefore E is not the total energy of the body but only part of it. This part is tradi-
tionally called kinetic energy.

The result of the simulation is displayed in a phase-diagram window (fig.7). Here we can see
how energy is dependent upon momentum. It is clear that the curve, a parabola, represents the
well-known formula:
p2
E
=
.
kin
2m
This can be checked with the help of a table graph, generated by Powersim. (see fig. 7).
m
g
F
p
E_kin
I_E
v

Figure 6: Flow diagram - free falling of a body on the surface of earth. Powersim does not distinguish between normal letters and capital
letters, so energy current is written as I_E instead of P.
TIME
p
m
check
E
9,0
176,58
2,00
7.795,12
7.795,12
8.000
9,1
178,54
2,00
7.969,31
7.969,31
9,2
180,50
2,00
8.145,42
8.145,42
6.000
9,3
182,47
2,00
8.323,46
8.323,46
E
9,4
184,43
2,00
8.503,42
8.503,42
4.000
9,5
186,39
2,00
8.685,31
8.685,31
9,6
188,35
2,00
8.869,12
8.869,12
2.000
9,7
190,31
2,00
9.054,85
9.054,85
9,8
192,28
2,00
9.242,52
9.242,52
0
9,9
194,24
2,00
9.432,10
9.432,10
0
50
100
150
200
10,0
196,20
2,00
9.623,61
9.623,61
p

Figure 7: E-p-diagram (left). Table graph showing the last ten time steps of the iteration loop. The check column contains p2/(2m) (right).





A new Way of Teaching the Special Theory of Relativity - Pohlig-Strauch

---

-7-
3.2 Free falling of a body on a neutron star - the relativistic view

When the Special Theory of Relativity is taught in schools, it is not possible to carry out any
experiments. Therefore, computer aided model building provides us with a real alternative.
Here, free falling of a body on the surface of a neutron star is chosen. There are two reasons
for this: firstly, the basic model - free falling on the surface of the earth - is already known
thus the model only has to be modified slightly. Secondly, from movies and science fiction,
pupils are familiar with neutron stars at least by name. Fig. 8 shows the modified model. The
mass of the body is no longer independent, rather it depends on the energy E due to
E
m =
.
c2
E is no longer the kinetic energy, now it is the total energy of the body. The initial energy of
the body is the energy contained in the body when it rests, that is, when its momentum is 0Hy.
This energy is named E . In our model the value of E is set to E
13
= 9 ⋅10 J and m = 1g.
0
0
0
0
The value of the acceleration due to gravity on a neutron star is set to g = 1012 N / kg .
m
g
c
F
p
E
I_E
v

Figure 8: Flow diagram: relativistic falling on the surface of a neutron star.

The results of the simulation8 (fig. 9) show that c is the highest velocity a body can reach, or,
better: c is the border-velocity for all momentum-energy transport. In fig. 9 v-t- and m-t-
diagrams are displayed for the rest energy E
13
= 9⋅10 J (= 1
m ) and E
13
= 18⋅10 J (= m2)
0
0
respectively. At the beginning of free falling, accumulation of momentum causes the increase
in velocity while the mass remains almost constant. The linear increase in velocity in the be-
ginning represents NEWTONIAN mechanics. Later, when the border velocity is almost reached,
energy and momentum are still flowing into the body. Due to p = m⋅ v , accumulation of mo-
mentum causes an increase in mass (=energy) only. When the velocity is nearly c the increase

8
The time of falling is 0.001s, therefore, as a result of the high velocity, the falling distance is so great, that a
homogeneous gravitation field can not be assumed. However all results are valid for any homogeneous ac-
celeration field, for example an homogeneous electric field. The gravitation field of a neutron star was cho-
sen for motivation.

A new Way of Teaching the Special Theory of Relativity - Pohlig-Strauch

---

-8-
in mass represents ‘high relativistic mechanics’. Fig. 9 (right) verifies this. While v tends to-
wards c, the mass grows infinitely9 (fig. 10 left). This singularity shows that c has the same
value in all reference frames. If this was not the case a body could have ‘nearly infinite’ mass
and it would not be possible to accelerate this body anymore, whereas the same body could be
accelerated if it was seen in another reference frame.

300.000.000
250.000.000
0,025
200.000.000
0,020
150.000.000
v1
0,015
m1
100.000.000
0,010
v2
m2
50.000.000
0,005
0
0,000
0,0000 0,0004
0,0010
0,0000
0,0004
0,0010
TIME
TIME

Figure 9: v-t-diagram (left) and m-t-diagrams for free falling of bodies on the surface of a neutron star.

Very interesting is the comparison of the E-p- phase-diagram shown in fig. 10 (right) and fig.
7 (left). Even when there is little momentum, the relativistic E-p-diagram can be approximated
by the non-relativistic E-p-diagram, which is shifted by the rest energy (fig. 11).
0,015
6e14
5e14
0,010
4e14
m
E 3e14
0,005
2e14
1e14
0,000
0
0
150000000
300000000
0
1000000
2000000
v
p

m2
Figure 10: m-v- (left) and E-p-diagram (right)

9
The rate of acceleration of a body falling in a homogenous gravity field is contrasted to that of a body in a
homogeneous electric field; as a result of the increase in mass while falling the momentum current into the
body must increase in the same way. In a homogeneous electric field, the momentum current remains con-
stant because the electric charge, by which the body is coupled to the electric field does not change its value
while falling.

A new Way of Teaching the Special Theory of Relativity - Pohlig-Strauch

---

-9-

figure 11: non-relativistic and relativistic E-p-diagram

With increasing momentum all curves approach the asymptote E = c ⋅ p [8]. For particles
which only exist at the velocity c, the formula E = c ⋅ p is exact. Photons are such particles.
Energy and momentum of a photon are fixed by its frequency10, so the border velocity is
found to be the speed of light. As c, the border velocity, is an universal constant, the speed of
light must have the same value in all reference frames.


Figure 12: E-p-diagrams for the rest masses 1g and 2g. The asymptote is added


4. Proving the new way

A momentum current
dp
F = I =

p
dt
is always coupled to an energy current according to
I = v ⋅ I ,
E
p

10
Let a photon fall in a gravity field, its energy and momentum will increase, this is only possible by in-
creasing its frequency (Doppler effect)

A new Way of Teaching the Special Theory of Relativity - Pohlig-Strauch

---

-10-
or
dE
dp
= v ⋅
,
dt
dt
or
dE = vdp.
This is better known as the GIBBS fundamental form of a system, all of whose independent
extensive variables except p are held constant. According to E = m ⋅ c2 and p = m(v) ⋅ v the
GIBBS fundamental form changes to
d (c2 ⋅ m) = v ⋅ d (m(v) ⋅ v)
Due to the evaluation of the differentiation and separation of the variables we get
c2 ⋅ dm = v2 ⋅ dm + v ⋅ m(v) ⋅ dv
(c2 − v2 )dm = v ⋅ m(v) ⋅ dv

1
v
⋅dm =
⋅dv.
m(v)
c2 − v2
After integration we have
*
m(v)
1
v
v
∫
dm* = ∫
⋅dv*
m(v=0
*
2
2
)
0
m
c − v*

1
*
v
−2v
= − ∫
⋅dv*
2
2
2
0 c − v*
[
m(v)
v
1
2
ln *]
2
*
m
= − ⋅
c − v
.
m(v=0)
[ln(
]0
2
With m(v = )
0 = m we get
0
(
m v)
ln
=
m0
1
c2 − v2
c2
1

− ⋅ ln
= ln
= ln
2
c2
c2 − v2
v2
1− c2
and at least we have
m
m(v) =
0
.
(I)
v2
1 − c2

A new Way of Teaching the Special Theory of Relativity - Pohlig-Strauch

---