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Special Relativity for Economics Students

Takao Fujimoto, Dept of Econ, Fukuoka University

(submitted to this Journal: 28th, April, 2006 )

1 Introduction

In this note, an elementary account of special relativity is given. The knowledge of basic calculus is enough to understand the theory. In fact we do not use differentiation until we get to the part of composition of velocities. If you can accept the two hypotheses by Einstein, you may skip directly to Section 5. The following three sections are to explain that light is somehow different from sound, and thus compels us to make further assumptions about the nature of matter, if we stick to the existence of static ubiquitous ether as the medium of light. There is nothing original here. I have collected from various sources the parts which

seemwithin the capacity of the average economics students. I heavily draws upon Harrison ([4]). And yet, something new in the way of exposition may be found in the subsections 5.4 and 6.1. Thus the reader can reach an important formula by Einstein that E = mc2

with all the necessary mathematical steps displayed explicitly. Thanks are due to a great friend of mine, Makoto Ogawa, for his comments. He read

this note while he was travelling on the train between Tokyo and Niigata at an average speed of 150km/h.

2 An Experiment Using Sound We use the following symbols:

c = speed of light; s = speed of sound;

V = velocity of another frame; x = position in x-coordinate;

t = time; 0 = value in another frame;

β ≡ r 1− V

2

c2

(Occasionally, some variables observed in another frame(O0) appear without a prime (0) to avoid complicated display of equations.)

1

2

Consider the following experiment depicted in Fig.1. In this experiment, we are on a small stage with two walls, A and B. The distances between our ears and these walls are the same, and written as l = l1 = l2. The stage itself is moving to the right at the velocity V , while the air, i.e., the medium of sound, stands still. So, for the observer on the stage, the wind is blowing to the left at the velocity V . We assume that V < s, where s is the velocity of sound at a given temperature. Now, two sounds are sent out from the same source at the same time, one to the

direction of wall A, the other to the wall B. What are the times required for the two sound waves to return to our ears? The actual paths taken by the two waves are depicted in Fig.2. First, let us calculate the return time, tA, for the sound reflected by Wall A. We have the following equation:

tA = l

s− V + l

s+ V =

2sl

s2 − V 2 . (1)

The first term in the middle of the above eq.(1) stands for the time required to reach wall A, and the second for that time to came back to the ears. Then, for the sound wave directed to wall B, we obtain this:

tB = 2 · r l2 + (

V · tB 2

)2

s . (2)

Pythagoras theorem is here used to get the length of the hypotenuse. From eq.(2), it follows

(s · tB)2 = 4 l2 + (V · tB)2. Thus,

tB = 2l√

s2 − V 2 . (3) From eqs.(1) and (3), we have

tA tB =

s√ s2 − V 2 =

1r 1− V

2

s2

> 1 (∵ 0 < V < s ).

Now two sound waves are displaced from each other because their arrival times are different. (Note that no Doppler effect is involved since the source of sound, the walls, and the ears are on the same stage, keeping the same mutual distance among them.) Two waves may be in phase or out of phase when we change slightly the distance between the ears and the wall A. That is, there can be interference. When the effect of interference is visualized through an optical apparatus after converting it into electric current, we can observe bright and dark stripes or rings called fringes.

3

3 Experiments by Michelson and Morley Michelson invented an interferometer (today called Michelson interferometer), which was to observe interference between two light waves in place of sound in the above experi- ment. (On the Internet, you can see the home pages of companies which produce this device today.) The air surrounding the stage was then the (luminiferous) ether which was supposed to exist ubiquitously in the universe and the medium through which light is transmitted. ( The ether should be like solid matter because light is a transversal wave.) Using his interferometer, Michelson repeated the experiments, later together with Morley, to find out what is the speed of ether wind, more precisely, whether it is in the order of the average orbital speed of the earth, i.e., roughly 30km/s. See Figs. 3, 4, and 5. (Figs.3 and 4 are from Michelson and Morley [9].) In Fig.5( which is adapted from Fig.15-3 in Panofsky and Phillips[11, p.275].), the

experiment is depicted so as to show the similarity to the one in the previous section. The walls, A and B, are now the mirrors, and the light emitted at a, goes through b, c, and here at c , half the light passes through c, and reaches d, reflected and returns to c, half the light reflected at c, and finally to e. At the point c, after going through b, half the light is reflected and proceed through f , g, f , c, and here at c , half the light passes through c, and finally to e. Then, if there is ether everywhere like the air in the first experiment, we have for the same reason the following equations. (Now, the speed of sound s is replaced by that of light c, and in this experiment, the two distances l1 and l2 need not be equal.)

tA = l1

c− V + l1

c+ V =

2cl1 c2 − V 2 =

2l1

c · β2 . (4)

tB = 2l2√ c2 − V 2 =

2l2 c · β . (5)

tA tB =

l1 l2 β

. (6)

The time difference is:

∆t ≡ tA − tB = 2l1 c(1− (V

c )2) − 2l2 c

r 1− (V

c )2 .

4

5

Michelson rotated the whole interferometer by 90 degrees, and the time difference would be then (with l1 and l2 interchanged):

∆t0 = 2l2

c(1− (V c )2) − 2l1 c

r 1− (V

c )2

.

The whole time difference between the two cases before and after the rotation by 90 degrees becomes

∆t+∆t0 = 2

c · ((l1 + l2) · (1− (V

c )2)−1)− (l1 + l2) · (1− (V

c )2)−

1 2 )

6

= 2

c · ((l1 + l2) · (1 + (V

c )2 + · · ·) − (l1 + l2) · (1 + 1

2 ( V

c )2 + · · ·))

+ 2 c · ((l1 + l2) · (1 + (V

c )2)− (l1 + l2) · (1 + 1

2 ( V

c )2))

= l1 + l2 c

· (V c )2.

(The formula (1+x)n = 1+nx+ · · · is used, and the terms of orders not lower than two are dropped. Why then addition, ∆t + ∆t0 , ? When the mirror for adjustment is slid in order to annul the effect of interference, we simply add the adjustment shifts whether they are forward or backward.) Hence, the difference in light path lengths between the two cases is

c · (∆t+∆t0) = (l1 + l2) · (V c )2,

where the speed of light c = 3 × 105 km/s= 3 × 108m/s, and the orbital speed of the earth V = 30 km/s= 3 × 104m/s, and l1 + l2 + 11m. (See Fig.4 to note that the light goes along the diagonal 15 times.) We can calculate

c · (∆t+∆t0) = (l1 + l2) · (V c )2 + 22× 10−8 = 2.2× 10−7.

On the other hand, the light used in the experiments was the yellow light from heated sodium whose wave length λ = 5.9× 10−7m, and so we have

c · (∆t+∆t0) λ

+ 2.2× 10 −7

5.9× 10−7 + 3.7× 10 −1 + 0.4.

Michelson wished to observe the displacements of interference fringes, sometimes and somewhere larger than 0.4 The results up to 1887 was reported in Michelson and Morley[9]. The results were to

their disappointment, and the speed of ether wind was estimated less than one sixth of the orbital speed of the earth. Note that the experiments were conducted day and night, each of four seasons, and on the top of mountains so that they could avoid the off-setting of the orbital velocity of the earth by some other movements such as the solar system, which is at present estimated as large as 250 km/s. (The rotational speed of the earth near the equator is 500 m/s.)

4 Lorentz Contraction

In 1892, H.A.Lorentz published a paper which explains the negative results of Michelson- Morley experiments. (See Janssen[5] for the information on the papers written by Lorentz, and on how Lorentz improved on his ideas for a considerable period.) The idea was very simple: the things shrink or contract in the direction of motion, and the size of

7

contraction depends on the speed of the movement, and is just enough to make the above two time durations tA (eq.(4)) and tB (eq.(5)) equal. (G.F.FitzGerald arrived at the same contraction theory in 1894.) That is, when a certain body is moving at a velocity v, its length l is contracted in the direction of motion as

l1 =

r 1− (V

c )2 · l2 = β · l2 < l2 . (7)

It is obvious from eq.(6) that no ether wind could be observed. Along with his theory of contraction, he had to prepare many formulas for conversion

or transformation concerning the time and the motion of matter on a moving frame. A set of these formulas were named by Poincaré Lorentz transformation, and most of them re-emerge in Einstein’s theory of relativity as a necessary consequence of his two hypotheses.

5 Special Relativity

5.1 Einstein’s Two Hypotheses

First, we define inertial re

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