Einstein’s special theory of relativity tells us how the Universe looks to
an observer moving at a steady speed. Because the speed of light is the same
for all such observers, moving clocks run slow, moving rulers shrink, and
moving masses become even more massive
ALTHOUGH it was Albert Einstein who developed the special theory of
relativity at the beginning of the 20th century, the principle of relativity
on which that theory is based goes back a further two centuries, to the time
of Isaac Newton. That principle holds that the same laws of physics apply for
all observers moving in straight lines at constant speeds relative to one
another. Such individuals are known as inertial observers. Einstein’s insight,
which led to a new way of looking at the Universe, lay in applying the
principle of relativity to the behaviour of light.
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The speed of light had actually been measured for the first time in 1675 by
the Dane Ole Rømer, from observations of the timing of eclipses of the
moons of Jupiter. By the end of the 19th century, this speed, now usually
denoted by c, had been determined reasonably accurately. It is just under 300
thousand kilometres a second (3Ă—108 metres per second).
During the 1860s, the Scottish physicist James Clerk Maxwell developed his
mathematical representation of the way in which electromagnetic disturbances
are transmitted, or propagated. He found that such disturbances can be
described in terms of an electric component and a magnetic component, moving
together through space (see Inside Science, No 34). The varying electric part
of the wave creates a varying magnetic wave, which in turn creates the varying
electric wave. Maxwell’s equations include a constant which specifies the
speed with which electromagnetic waves move. That speed is precisely the speed
of light (now known to be 2.99792458Ă—108 metres per second).
Maxwell had proved that light is a form of electromagnetic wave.
This raises a puzzle about the nature of light. A modernised version of an
old Newtonian picture has an Observer A setting up some apparatus on the
platform of a railway station to carry out some simple mechanical experiments.
He or she will get exactly the same results (in line with Newton’s laws of
mechanics) as Observer B, who carries out the same experiments aboard a train
moving smoothly through the station at a steady speed. But if the train is
moving at 100 kilometres per hour, and observer A rolls a ball at a speed of 1
kilometre per hour along the platform, in the same direction as the train,
Observer B will measure the speed of the ball as 99 kilometres per hour. Here,
measured velocity is relative to the motion of the observer. But what is the
speed of light relative to when we measure it?
The light puzzle
An absolute constant
IN THE Newtonian picture, if Observer A shines a torch along the platform
in the direction that the train is moving, Observer B should measure the speed
of light from the torch as 100 kilometres per hour less than that measured by
Observer A. Yet according to Maxwell’s equations there is only one speed of
light, c. During the late 19th century, physicists thought that there must be
an “ether”, an invisible, undetectable fluid filling all space (even
penetrating materials such as glass), in which light waves move, just as at
sea waves move across the surface of the water. They reasoned that the unique
speed of light defined by Maxwell’s equations referred to the speed through
the ether, and they expected that it would, indeed, be possible to measure
different speeds of light for observers moving through the ether in different
ways.
But no experiment ever showed any such effect. In particular, the physicist
Albert Michelson and chemist Edward Morley, working in the US, found no
evidence of any influence on the speed of light caused by the Earth’s motion
through the ether (see Box). The Irish physicist George Fitzgerald and the
Dutch physicist Hendrik Lorentz, working independently of each other, were the
first to realise that the results of the Michelson-Morley experiment could be
accounted for if all bodies, large or small, moving through the ether were
shortened. The extent to which fast-moving objects are shortened in this way
is called the Fitzgerald-Lorentz contraction.
Space and time trouble
A relativistic stretch
THE STRANGENESS of the new picture of the Universe that resulted from
Einstein’s acceptance that the speed of light is an absolute constant
(published in 1905) can be seen by imagining another simple experiment
involving a train. If Observer B (who, because he is travelling on the train
can, by the principle of relativity, regard the train as at rest) sets up a
source of light in the middle of the carriage, and sends off two pulses of
light in opposite directions at the same time, they will reach the end walls
of the carriage simultaneously. But this is not what Observer A sees. From the
platform, the speed of each of the two light pulses is exactly the same as the
speed measured by Observer B. But while the light pulses are moving outward
from the centre of the carriage, the train itself moves forward. Observer A
sees the rearward moving pulse hit the back wall of the carriage before the
forward moving pulse hits the front wall. In other words, a pair of events
(the arrival of the light pulses at the carriage ends) that are considered to
be simultaneous by one observer are seen to occur at different times for an
observer who is moving relative to the first observer. Simultaneity is
relative.
If observers who move relative to one another cannot agree on the
simultaneity of events, they cannot agree on the outcome of measurements
involving time. This is seen most clearly in the phenomenon known as time
dilation. Imagine a clock carried by Observer B on the moving train. This is a
perfect, ideal clock which measures time by how long it takes light to bounce
back and forth between two mirrors, at right angles to the direction in which
the train is moving. A passage of a light pulse from one mirror to the other
and back again represents one tick of the clock. Because the observer knows
the distance between the mirrors and the speed of light, the time taken for
one tick is precisely determined.
But now look at things from the point of view of Observer A, on the
platform. While the light pulse is travelling up and down, the train with the
clock and its mirrors move forward. So the path travelled by the light pulse
is around two sides of a triangle. Simple geometry shows that this path is
longer than the up-and-down distance between the two mirrors at rest. But the
speed of light is a constant. So the time taken for one tick of the moving
clock is longer, according to Observer A, than for one tick of an identical
clock sitting on the platform. But by the principle of relativity, Observer B
can make the same calculation, regarding the train as at rest, and infers that
the clock on the platform is running slow.
It is important to realise that the situation is symmetrical. There is no
paradox here once we remember that simultaneity is relative. We cannot compare
the readings of the separated clocks “at the same instant” until we decide
what “at the same instant” means, and our two observers have different views
on this. Since the light pulse in the moving clock travels, in effect, along
the hypotenuses of two right-angled triangles, it is easy to calculate the
size of the time dilation. If ν is the speed of the moving clock, the time
stretches by exactly a factor 1/(1-ν2/c2)½,
an expression which appears in many relativistic calculations.
Although the time dilation, and its size, can be seen most clearly in terms
of this special kind of clock, this is a real effect that applies to all
moving clocks, including the “clocks” that regulate the activity and ageing of
the human body (biological clocks). Experiments involving fast-moving but
short-lived subatomic particles have shown that their lifetimes are indeed
extended, by just this relativistic factor (see Box 2).
The problem with defining simultaneity also affects the way different
inertial observers will measure the length of the same object. Suppose that
there is a long stick lying on a table in the train as it passes through the
station. If Observer A wants to measure the length of the stick, this involves
making a simultaneous observation of the position of each end of the stick,
and the observers will not agree that each other’s measurements were made
simultaneously. The other way to measure the length of the stick is for
Observer A to count how many ticks of the platform clock it takes for the
stick to pass a certain point on the platform. But, according to Observer B,
Observer A’s clock is running slow. Again, they will not agree on the length
of the stick. Either way, from the point of view of Observer A the moving
stick is shrunk in the direction of its motion by a factor
(1-ν2/c2)½.
This Fitzgerald-Lorentz contraction applies also to the train, and to
Observer B. Everything is shrunk, in the direction of its motion, by the usual
relativistic factor. For Concorde travelling at mach 2, the contraction is
only 2 parts in 1 million million-less than the width of an atom.
Trouble with mass
E=mc2
AS THE speed of a moving object increases relative to an observer in an
inertial frame (a situation in which the same laws of physics apply to both
object and observer), the object will be increasingly foreshortened by the
contraction effect, and its clocks will tick ever more slowly. At the speed of
light, the moving object’s length in the direction of motion is zero, and time
“stands still” for it. The speed of light is a limiting velocity for all
objects. This is expressed in the relativistic expression for the addition of
velocities, which takes account of the differences in length and time
intervals measured in different frames of reference. If Observer B, riding on
the train at velocity ν1, fires a projectile out ahead of the
train at a velocity ν2, as measured from the train, then
Observer A, on the platform of the station, will measure the velocity of the
projectile not as ν1+ν2, but as
((ν1+ν2)/(1+
(ν1ν2)/c2)).
If the velocities are small compared with that of light, this reduces to
the familiar addition rule we know from everyday life. But as long as
ν1 and ν2 are both less than c the projectile will
never be seen to move faster than c by any inertial observer. A few minutes
playing with a calculator should convince you – for example, if
ν1 and ν2 are both 0.75c, Observer A sees the
projectile moving at 0.96c, not at 1.5c.
But what has happened to the momentum of the projectile? When the gun on
the train fired the projectile forward, the projectile was given energy, and
its momentum increased. That energy cannot have disappeared. Because momentum
is the product of mass and velocity, if the velocity of the projectile has not
increased sufficiently to absorb the energy input, the mass must have
increased. In other words, moving objects have more mass than they do when at
rest. This led Einstein to conclude, in a paper published in 1906, that the
mass of a moving object is equal to its mass when stationary divided by the
familiar relativistic factor,
(1-ν2/c2)½.
So, when the velocity of a particle or rocket approaches the speed of
light, their masses become inversely proportional to their size. For a
particle, the smaller the volume that it crowds itself into, the greater its
mass. Due to the Fitzgerald-Lorentz contraction, a rocket travelling at
1.5Ă—105 kilometres per second (0.5c) would contract 15 per
cent, and under the same conditions the mass of a particle would increase 15
per cent. At 2.6Ă—105 kilometres per second (0.875c) the
rocket’s contraction would be 50 per cent and the mass of the particle 100 per
cent greater. At the speed of light (1c) the rocket would contract 100 per
cent and the mass of the particle would be infinitely great.
One way of interpreting this increase in mass as velocity increases is in
terms of kinetic energy. The mass of the moving object exceeds its rest mass
by an amount equal to the kinetic energy divided by c2. The
increase in mass is equal to the increase in energy divided by c2,
so Einstein inferred that the rest mass itself is equivalent to an energy
E/c2. In other words, E=mc2. This equivalence
between energy and mass is true for all forms of energy so, for example, we
would expect a battery to lose mass (not atoms) as it discharges. However, for
a car battery this loss is roughly one millionth of one millionth of a
kilogram – impossible to measure.
Einstein’s equation relating energy and mass provided a fundamental answer
to the puzzle of radioactivity: why radium continuously emits heat. Marie and
Pierre Curie isolated radium in 1898, two years after Antoine Becquerel
discovered radioactivity. Pierre Curie had found in 1903 that 1 gram of radium
emits 100 calories per hour, day after day, year after year. Einstein argued
that when a radioactive element disintegrated some of its mass was turned into
energy in line with the equation E=mc2. This energy is
carried off as energy of motion (heat) by the fragments of the atom. The
energy in 1 gram of radium would produce (3Ă—1010)2
ergs, that is 9Ă—1014 joules. Thus, 1 gram of radium would
contain enough energy to keep a one-kilowatt electric light bulb burning for
2850 years.
Proof of the pudding
Relativity rules!
STRANGE though the predictions of the special theory of relativity are,
they have all been born out by experiment – not least in the conversion of
rest mass into energy in the explosion of an atomic bomb.
For example, when electrically charged particles are accelerated by
magnetic fields in machines like those at the European Nuclear Research
Centre, CERN, the force required to keep those particles whirling around a
circular tube increases, by exactly the amount predicted by special
relativity, as the mass of the particles increases. Special relativity is
established beyond doubt as a description of how the Universe looks to
inertial observers (it does not deal with accelerations, or gravity; they are
covered by the general theory of relativity – see Inside Science No. 31). But
the most important evidence that relativity rules comes not from direct
measurements of time dilation, conversion of mass into energy, and so on, but
at a more subtle level, within the atom.
It was only by incorporating special relativity into the new quantum
description of particles that the British physicist Paul Dirac, in 1928,
developed a satisfactory mathematical description of the behaviour of the
electron (and other subatomic particles). The relativistic version of quantum
mechanics provides the understanding of the behaviour of electrons in atoms,
and the way in which they occupy stable “shells” around the nucleus – the
basis for modern chemistry. A little later, by merging special relativity and
quantum mechanics with electromagnetism, physicists developed a theory known
as quantum electrodynamics, or QED, which gives a complete description of the
way charged particles interact (see also Inside Science Nos. 15 and 17). The
success of QED has led to the construction of a similar theory to describe the
behaviour of quarks (quantum chromodynamics, or QCD) and is the archetype for
attempts to develop a grand unified theory incorporating all the forces of
nature except gravity. The whole edifice of modern particle physics rests upon
Dirac’s incorporation of special relativity into quantum mechanics in the
1920s.
Spacetime geometry
A new perspective
BOTH space and time lose their status as absolute properties of nature in
Einstein’s theory. But shortly after Einstein published his special theory,
the Russian mathematician Hermann Minkowski showed that a combination of space
and time to produce four-dimensional spacetime assumes a fundamental
significance.
This can best be understood by thinking of the way in which an extended
object like a broom handle will appear to have a different length depending on
its orientation to the observer. Viewed head on, it appears to have no length
at all; viewed sideways on, its full length is seen; and at intermediate
angles it is appropriately foreshortened. We can measure the real length of
the broom handle by setting up a system of three axes at right angles to each
other with any orientation in three-dimensional space, and measuring the
apparent length of the handle along each axis. The actual length is then given
by the three-dimensional version of Pythagoras’s famous formula:
s2 = x2 + y2 + z2
Minkowski pointed out that all the strange results concerning space and
time that stem from special relativity could be understood in terms of the
foreshortened view of an object with different orientations in four-
dimensional spacetime. Giving an object such as a broom handle a four-
dimensional “length” means taking into account the instants of time at which
we observe the ends of the handle. If we make the observations at different
times, the handle has an extension in time as well as space – and since
different inertial observers do not agree on simultaneity, they will not agree
on the size of this time extension.
The way to vary orientation with respect to time is by motion. Because
light travels at 300 000 kilometres per second, one second of time is
equivalent to 300 000 kilometres of space. But the distinction between space
and time arises because instead of adding the square of the time extension
into the four-dimensional version of Pythagoras’s equation, it has to be
subtracted. The true four-dimensional extension of an object is:
s2 = x2 + y2 +
z2 – c2t2
This spacetime interval is always measured to be the same by all inertial
observers. Although different observers will describe the length of a
particular object in different ways, and will measure the speed of a clock
attached to it in different ways, the correct combination of space and time
properties belonging to the object gives an unambiguous measure of its
extension in spacetime.
Einstein was initially reluctant to take this idea seriously, and regarded
it as an unnecessary piece of mathematical tinkering with his theory. But he
later realised that Minkowski’s geometrisation of the special theory provided
the key to a general theory of relativity, incorporating gravity. The special
theory of relativity is described by the geometry of flat four-dimensional
spacetime; curved four-dimensional spacetime describes the general theory.
Minkowski provided the link between the two theories – but that, as they say,
is another story.
The Michelson-Morley experiment
IF THE Earth is moving through the ether, and light travels at a constant
speed through the ether, then there should be a detectable influence of the
Earth’s motion on the measured speed of light. At the end of the 19th century,
Polish-born Albert Michelson and the American Edward Morley set out to measure
this influence at the Case Institute of Technology, in Cleveland.
Their experiment used the fact that light behaves as a wave. Michelson and
Morley devised an apparatus in which a beam of light from a single source was
split by a half-silvered mirror into two beams which travelled along paths at
right angles to each other for the same distance before being reflected back
to run together into a detector known as an interferometer. If the two beams
of light were out of step with one another, they would interfere, producing a
characteristic pattern of light and dark bands.
In fact, Michelson and Morley found that there was no interference – the
two beams of light remained precisely in step with one another, which meant
that they had each taken precisely the same time on their journey. In other
words, the two beams at right angles to each other moved at the same speed
relative to the Earth. Even when the experiment was repeated at different
times of year (at different positions in the Earth’s orbit around the Sun) no
effect caused by the Earth’s motion could be found. Light travels at the same
speed as measured by all inertial observers, whatever their own velocity.
How time dilation delays decay
IN 1941 the Italian physicist Bruno Rossi found that Einstein’s time
dilation formula answered a puzzle that confronted him in his research on
high-speed muons. These particles are created when cosmic rays strike the
ionosphere; they decay into other subatomic particles with a half-life of
about 2 microseconds. This means that if we start with a collection of muons
the number surviving halves every 2 microseconds. Even travelling nearly at
the speed of light, 300 000 kilometres per second, these muons can cover only
600 metres in 2 microseconds, so we should expect them to thin out rapidly as
they travel down towards the Earth. Rossi’s measurements told a different
story.
The figure shows the muons reaching two laboratories, A and B, set one
above the other on the side of a mountain. As it takes roughly 6 microseconds
(three half lives) for the muons to travel down the mountain, we might expect
A to receive only ⅛(=½×½×½) of
the flux that B measures. In fact, A’s count rate is about 80 per cent of B’s
– which is what we would expect if it took the muons only ⅓ of a half
life to travel between the two laboratories. In other words, time for the
muons seems to be going slow by a factor of 9 (â…“ of a half-life instead
of 3). This is just what the time dilation formula predicts for particles
travelling at 99.4 per cent of the speed of light.
Thirty years later, in 1971, two Amercian physicists, J. C. Hafele and R. E.
Keating, were able to confirm the existence of time dilation by transporting
very precise caesium-beam clocks around the world on commercial jets (their
clocks were accurate to 10-9 seconds). After 45 hours of flying,
the researchers found that their clocks ran slower than those on Earth. You
age a fraction of a second less while in flight than if you stay at home.
Further reading
Among the many textbooks which discuss the special theory of relativity,
Invitation to Physics, by Jay Pasachoff and Marc Kutner (Norton, 1981), is
particularly good. Clifford Will’s superb book about the general theory of
relativity, Was Einstein Right? (Basic Books, 1986), includes an appendix
spelling out the importance of special relativity to 20th century science.
Relativity Visualized, by Lewis Epstein, is a tour de force that explains all
the strange features of special relativity pictorially, with no equations
(Insight Press, San Francisco, 1987). And The Matter Myth, by Paul Davies and
John Gribbin (Viking, 1991) looks at the broader implications of the special
theory for our understanding of time. QED – the Strange Theory of Light and
Matter, by Richard Feynman (Penguin, 1990), is a wonderful description of the
results of combining special relativity with quantum mechanics.