In the history of scientific thought, physics and philosophy were not always clearly

distinct disciplines. However, even after they have become established as separate

disciplines, knowledge of the interactions between both fields is important in understanding

their development. A clear illustration of this interdependence is given by Einstein's special

theory of relativity, published 1905. Einstein was deeply impressed by Mach's positivistic

philosophy, which led him to regard only measureable quantities as physically meaningfull.

This allowed him to show that such quantities as time, length, velocity, mass, and energy

cannot be defined absolutely, but depend on the state of motion of the observer. This, in

turn, is philosophically significant since it shows that the ordinary notion of these

fundamental quantities as being absolute is wrong. Meter sticks don't really exist per se,

only meter sticks, where v is the velocity of the observer with respect to the stick

(in one dimension), and c is the speed of light. Clocks tick only then at the same rate,

if they are at rest with respect to each other; more generally,

where t0 is the time

of the moving clock. These phenomena are called length contraction and time dilation. Another

interesting consequence of STR is the twin paradox: If a twin leaves her sibling, she will be

younger when she returns.

The basis for STR is the constancy of the speed of light and the assumption that nothing can

move faster than light. This is not an issue in Newtonian mechanics, where anything can have

any speed, in principle (except infinite speed). It was assumed that light was a wave with

the aether as the medium. Newton had assumed that light consisted of particles, but later it

was shown that light is an electro-magnetic wave. Now a wave needs a medium to propagate,

and since it even reaches us from distant stars, it must fill all of space and be otherwise

unnoticeable. Einstein, as a good Machian, never assumed the existence of the aether. As

a consequence, the speed of light must be constant and the same for any observer, regardless

of his state of motion. Based on this assumption, it can then be shown mathematically that

it would take an infinite amount of energy in order to accelerate a body to the speed of light.

In Newtonian mechanics, space and time are independent of each other - there are three spatial

dimensions and one time dimension. Spatial intervals, dr^2 = dx^2+dy^2+dz^2 are measured

idependent of time intervals, dt. However, the fact that length and time are functions of

the state of motion of objects and observers suggests that they cannot be treated independently.

Minkowski was the first to mathematically unify space and time into four-dimensional space-time.

An interval in Minkowski spacetime, ds^2 = dx^2+dy^2+dz^2-c^2dt^2, is a mixed interval, combining

space and time intervals.

Length contraction, time dilation, and the twin paradox can easily be demonstrated

mathematically, but it is very instructive to demonstrate them graphically, using Minkowski

diagrams. A Minkowski diagram is a plot of time versus space. Since it is difficult to

display four dimensions on two-dimensional paper, only one spacial dimension is displayed (fig.1).

Fig.1

The path of a particle in spacetime is called it's world line. If the particle is at rest,

the line is vertical; if it moves with the speed of light, the line makes an angle of 45o

with the x-axis, since the vertical axis is ct instead of just t. A curved line indicates

acceleration.

In order to compare how one observer sees another observer's clock or meter stick,

two coordinate systems are used, one for each observer, in which they are at rest. These

coordinate systems are inertial frames which means that they are not accelerating. Here it

is assumed that observer O' is moving with velocity v in the x-direction. ct' and x' are O''s

coordinates. So how does observer O see O''s coordinates? Clearly, O''s world line (ct'-axis)

makes an angle of less than 45o with the ct-axis. It is less clear how the x'-axis should be

drawn. The Lorenz transformatons can be used to get the exact equations for the primed axes

in terms of the umprimed axes:

where b = v/c. For the ct'-axis, set x'=0. This gives vt = x or ct = x/b. For the x'-axis,

set ct'=0, which gives ct = bx (fig.2).

Fig.2

In the appendix it is shown that q = f. The Lorenz transformations can also be used to

figure out how the scale changes as we go from the unprimed to the primed system. For this

the inverse transformations,

are also needed. To do this, the

hyperbolae c2t2 - x2 = +1 are drawn with the primed and unprimed coordinate systems (fig.3).

Fig.3

The hyperbolae c2t2 - x2 = 1 (H1) and x2 - c2t2 = 1 (H2) are used because they give unit

spacetime distance to the origin O (s = 1). This is why the hyperbolae are also called

calibration curves. To see how the calibration works, consider event P1, which is the

intersection of H2 with the x'-axis given by ct = bx. Combining these equations gives the

coordinates of P1,

Comparing equations 5 and 6 with equations 3 and 4 shows that equations 5 and 6 represent

unit length (x' = 1) and zero time (ct' = 0) in the O' frame. In other words, the interval

OP1 gives unit length along the x'-axis. Similarily it can be shown that the interval OP2

represents unit time along the ct'-axis. The result of this calibration is shown in figure 4.

Fig.4

In figure 4, b = 0.5, which means that the primed system is moving away from the unprimed

system at half the speed of light. The calibration shows that the unit intervals of the

primed system are longer than the unit intervals of the unprimed system. It is easy to see

how the coordinates of a spacetime event P differ for the two systems. If the coordinates of

P in O are (3, 2.3), they will be (2, 1.2) in O'. Mathematically speaking, the Lorenz

transformations transform an orthogonal system to a nonorthogonal one.

To demonstrate length contraction, a meter stick is placed at rest in O with the

ends of the stick at x=3 and x=4 (fig.5). As time goes on, the ends of the stick trace out

vertical lines parallel to the ct-axis. The stick is parallel to the x-axis because a length

measurement means that the positions of the two ends of the stick are determined simultaneously.

All lines parallel to the x-axis are lines of simultaneity (ct = const.) in O. Lines parallel

to the x'-axis are lines of simultaneity in O'. It is thus clear that events that are

simultaneous in O are not simultaneous in O'. As a matter of fact, length measurements in

O' are different spacetime events from length measurements in O. For example, the events

E1 and E2 constitute a length measurement of the meter stick in O. However, these events

are not simultaneous in O' and cannot, therefore, be used for a length measurement of the

meter stick in O'. A possible pair of events

Fig.5

that can be used is E2 and E3. The length measured in O' can then be read off the x'-axis

and is not one meter, but 0.87 meters (for b = 0.5). Therefore, the meter stick appears

to be contracted to an observer in O'.

It has been assumed that O is at rest and O' is moving away from O at half the speed

of light. What this means is that we are pretending to be observers in O. Actually, it is

impossible to tell which frame is the moving one, and which one is at rest. What counts is

the relative motion. The symmetry of the situation suggests that if the meter stick were at

rest in O', it should appear contracted in O. This is indeed the case, as shown in figure 6.

The ends of the meter stick are at x' = 3 and x' = 4. The world lines of the ends of the

stick trace out lines parallel to the ct'-axis, as time moves on. The length as measured in

O is the intersection of the stick's world lines with the x-axis, and again comes out to be

0.87 meters.

Fig. 6

Time dilation can also be demonstrated geometrically. A clock at rest in O appears to

tick slower to an observer in O' as shown in figure 7. Also shown is the reciprical nature

of time

Fig. 7

dilation. In order for an observer in O' to measure the interval between two ticks of the

clock at rest in O, two clocks are needed, one at x'1 and the other at x'2. The time

interval that they record between ticks t1 and t2 can be read off the ct' axis, and is

larger than one. A clock at rest in O' ticks one unit of time in O', as can be read off

the ct'-axis, but the two clocks in O that record the time of the ticks t'1 and t'2, measure

the interval between the ticks to be more than one unit, as can be read off the ct-axis.

The spacetime geometry of the twin paradox is shown in figure 8. Let's assume that

twin e remains at home while twin d travels with a speed of 0.8c away from e. If d's clock

records 3 years for

Fig.8

the journey, e will have aged 5 years. Now e sees d's clock tick at a slower rate, but

this cannot explain the age difference upon reunion since d also sees e's clock tick slower

while she recedes from and approaches e. Time dilation and length contraction, which cause

the age difference, are symmetrical phenomena, whereas the twin paradox is asymmetrical.

This is what makes the twin paradox paradoxical. So what, then, is responsible for breaking

the symmetry? It is simply the fact that one of the twins must change direction in order

to return to the other twin. Actually, the longer the spacetime path taken to get from A to B,

the shorter the proper time, that is the time measured by the traveler, will be. Perhaps this

doesn't make sense intuitively, but can be shown mathematically as follows. Let's compare

proper time for two

Fig.9

for two spacetime paths between A and B. The shortest one is Dt1, the longer one is Dt2.

can also be written as

The elapsed proper time along the first world line is then

This is clearly longer than Dt2 since dx2 is always positive.

It is often argued that the cause of the asymmetry is the fact that one of the twins is

accelerated. This, however, does not solve the triplet paradox, where no body is actually

accelerated. If triplet 2 recedes from triplet 1 and then encounters triplet 3, who is

heading for triplet 1, triplet 3 will be younger than triplet 1, assuming that triplet 3

synchronized his clock with triplet 2 during their encounter. Thus it seems that all that

needs to be invoked is a change in the direction of motion, irregardless of wether a body

has actually felt an acceleration. However, the triplet paradox can probably be solved

by invoking acceleration since triplet 3 in some point in time had to turn around in order

to head back to triplet 1.

So far the arguments have been fairly abstract, and the question is whether the phenomena

of time dilation, length contraction and the twin paradox are real in the sense that they

could actually be observed. The problem is that these phenomena become important only

at extremely high speeds, close to the speed of light. However, several experiments have

confirmed the reality of these effects. For example, without time dilation, cosmic ray

muons would never make it to the ground, since the half life is too short. In another

experiment, one atomic clock, having circled the earth in an airplane, has been found to

go slower than a second atomic clock that remained on ground, thereby experimentally

confiming the reality of the twin "paradox".

As far as length contraction goes, the question is what is meant by a real physical

phenomenon. Of course, objects are not really sqeezed together in the sense that stress

or strain forces could be measured. But in three dimensions, length contraction might

not even be apparent. According to Terrel, objects sometimes appear to be contracted, but

only because they are rotated. A sphere, for example, would appear to be rotated - not

distorted to an ellipsoid. Even after almost a century of STR, the subject of how moving

bodies would appear if c = 100m/s remains contraversial.

Appendix

References

1. Resnick, Introduction to the Special Theory of Relativity.

2. Arya, P. Atam. Introduction to Classical Mechanics. Allen and Bacon, 1990.

3. Ray, Christopher. Time, Space, and Philosophy. Routledge, 1991.