# special theory of relativity

Albert Einstein at the Swiss patent office during the time he was developing the theory of special relativity.

The special theory of relativity is a theory put forward by Albert Einstein in 1905 that is based on two postulates:

As in Newtonian mechanics, there is a set of mathematical transformations that relate the spacetime coordinates used by different observers in such a way that the laws of nature appear the same to all observers. In special relativity however, the spacetime transformations have the essential property of leaving the speed of light unchanged, irrespective of the velocity of the observer. These transformations result in a number of surprising relativistic effects.

Among the predictions of the special theory is that no material object can be accelerated to reach light-speed – a result which appears to rule out practical interstellar travel for all but the nearest stars. However, the special theory provides an escape clause in that for objects (such as a very fast spacecraft and its crew) moving at significant fractions of the speed of light, time slows down. In principle, this time dilation would allow astronauts to travel as far as they liked within as short a period as they liked, simply by moving at a sufficiently high speed. The problem is that the time dilation effect does not operate for those who stay at home, so that the star-hopping astronauts would return home to find their relatives and friends had aged more than them or were possibly even long-dead.

## How Einstein was led to the special theory

As a sixteen-year-old Einstein started pondering the kind of thought experiments that would become a hallmark of his career. What would you see, he wondered, if you rode astride a beam of light or held a mirror and looked into it while traveling at the speed of light? Over the next ten years, he continued to question the nature of space, time, and light, and to ask how physics would need to change if it were to address all circumstances in which objects are moving relative to one another. Amazingly he brought about a revolution in physics while working as a technical expert third class at the Berne patent office. Isolated from academe, unheard of by eminent professors at the venerable universities of Europe, he whiled away his spare hours rethinking the notion of time and mulling over the relationship between the speed of light and the speed of an observer. During this period of anonymity, his closest confidante was Michele Besso, a fellow patent clerk, whom he'd first met at a musical evening in 1896, both being keen violinists. Of Besso, Einstein said: "I could not have got a better sounding board in the whole of Europe."

For over a year, Einstein had been stymied by a problem he couldn't see a way around. It was simply this: if the speed of light is a universal constant, as he guessed it was, then the usual "Galilean" addition of velocities doesn't work. According to the physics of Galileo and Newton, if you walk down the aisle of a train that's moving along the track, then your speed relative to the ground equals the train's speed plus your walking speed. This seems like perfectly good commonsense and is what you'd actually find if you took measurements, unless you could measure with atomic precision, in which case you'd notice a puzzling discrepancy. In the same way, if the train has a lamp at the front then, according to Newtonian science, the speed of the rays leaving the lamp ought to equal the train's speed plus the speed of light. But if the speed of light doesn't depend on how the source is moving, as Einstein believed, this simple addition can't be right, otherwise you could easily break the light barrier. Then, "one momentous day in May," recalled Einstein, inspiration dawned. He visited Besso and explained what was on his mind: "Today I come here," he said, "to battle against that problem with you." Together they looked at the question from every angle, until suddenly Einstein saw the answer to his difficulty: if the speed of light is always the same, then space and time must be changeable. The notion that space and time formed a fixed backdrop against which the drama of the universe played out – Newton's viewpoint – was wrong. Space and time, length and duration, would be measured differently depending on the relative motion of the observer.

On Jun 30, 1905, Einstein's paper that formed the basis of what would eventually
be called the special theory of relativity was published. Modestly titled
(translated from German) "On the Electrodynamics of Moving Bodies," it was
very unusual as scientific papers go.^{1} No references appeared
in it to any other papers or to the work of any other researcher, merely
an acknowledgment of the "loyal assistance of my friend M. Besso." Of it,
C. P. Snow has said:^{2}

There is a good deal of verbal commentary. The conclusions, the bizarre conclusions, emerge as though with the greatest of ease: the reasoning is unbreakable. It looks as though he had reached the conclusions by pure thought, unaided, without listening to the opinions of others. To a surprisingly large extent, that is precisely what he had done.

There's a directness about the Jun '05 paper, a clarity, that comes across without the need for advanced math. The seemingly ad hoc transformations of Lorentz and Fitzgerald follow naturally and inevitably, in Einstein's hands, from simple geometry and the well-known theorem of Pythagoras. And all of this simplicity and clarity – in fact, the whole theory of special relativity – stems from just two basic postulates that Einstein lays down early on. The first is that the laws of physics are identical in all inertial frames: that is, to all observers who are at a rest or traveling with constant velocity relative to one another. The second is that the speed of light is the same in any inertial frame, whether the light is given off by a body at rest or in a state of uniform motion.

Right at the start of his paper, Einstein does some impressive spring-cleaning – dispensing with the ether along with any lingering notions of absolute space and an absolute state of rest:

[T]he introduction of a light-ether will prove to be superfluous since, according to the view to be developed here, neither will a space in absolute rest endowed with special properties be introduced nor will a velocity vector be associated with a point of empty space in which electromagnetic processes take place.

Einstein restores the principle of relativity to physics because he insists that physical laws have to look the same to all observers traveling at constant velocity – exactly as Galileo and Newton had done. But the only way he could do this and keep the speed of light constant in his theory, no matter what the motion of the source, was to allow space and time to be flexible. From Einstein's postulates the Lorentz transformation flows as a matter of course: time and length really do shrink along the direction of movement, although these "relativistic" effects only kick in at velocities that are a significant fraction of that of light itself.

Oddly enough, Einstein never talked about the Michelson-Morley experiment even though his seminal paper gave a complete explanation of its null results. Despite being the main reason that other scientists, such as Poincaré and Lorentz, were questioning the established notions about space and time, the experiment was never acknowledged by Einstein as having any influence on his thinking. Even more strangely, Poincaré and Einstein seemed to go out of their way to avoid crediting each other. Poincaré was closing in on relativity theory himself at the time Einstein achieved his breakthrough and, perhaps with the help of others, would have come to similar conclusions, sooner rather than later. Yet Einstein mentions him only once in all of his papers, while Poincaré never wrote a word on relativity in which he refers to Einstein. Lorentz, on the other hand, was praised by both and often cited by them, even though Lorentz never fully accepted relativity. Here is the man after whom the transformation equations at the heart of special relativity are named speaking in 1913:

As far as this lecturer is concerned he finds a certain satisfaction in the older interpretation according to which the ether possesses at least some substantiality, space and time can be sharply separated, and simultaneity without further specification can be spoken of. Finally it should be noted that the daring assertion that one can never observe velocities larger than the velocity of light contains a hypothetical restriction of what is accessible to us, a restriction which cannot be accepted without some reservation.

Long before he uttered these words, however, Lorentz had found himself in a minority as acceptance of the special theory of relativity grew, especially following Max Planck's endorsement of it.

On September 27, 1905, Einstein sent out a sequel to his June paper in which
he proved what's become the most famous formula in all of science.^{3} Previously, it had been thought that energy couldn't be created or destroyed, a claim known as the law of conservation
of energy. A similar conservation law was believed to hold true for mass.
But what Einstein showed, again starting from his two basic postulates,
is that energy and mass are interrelated – the one can turn into the
other. How much energy you get for a given investment of mass is decided
by the equation *E = mc*^{2}, where *c* is the speed
of light. Because *c*^{2} (*c* times *c*) is,
when measured in conventional units, is a very large number, a tiny amount
of mass yields a vast outpouring of energy. In time this relationship would
prove to be the secret behind the energy production of stars and the key
to the most destructive weapons ever conceived by humankind. And it would
play a part too in Einstein's formulation of a new theory of gravity.

Einstein melded energy and mass with his *E* = *mc*^{2}.
But it was one of his old university professors, the Lithuanian-born mathematician
Hermann Minkowski, who showed
that space and time were also inextricably connected. When Einstein was
an undergraduate at the Swiss Federal Institute of Technology in Zurich
(also known as the Eidgenössische Technische Hochschule, or ETH) in the
late 1890s, Minkowski was less than impressed by him. "A lazy dog," he called
him, because of his apathy toward work. So, no one was more surprised than
Minkowski when his miscreant of a student made good: "Oh, that Einstein,
always cutting lectures – I really would not have thought him capable
of it." But it didn't take Minkowski long to grasp the importance of relativity
theory and to plunge into it with whole-hearted enthusiasm.

According to the Lorentz transformation, length contracts as time slows
down or "dilates," suggesting a complementary relationship between the two.
Minkowski took this idea to its natural conclusion: time *(t*) is
just another coordinate like the three coordinates of space (*x*, *y*, *z*), so that every happening or "event" can be uniquely
pinpointed in space and time by a quartet of values – its coordinates
(*x*, *y*, *z*, *t*) in the continuum of spacetime.
Minkowski summed up these ideas in a lecture he gave in Sep 1908:

Henceforth space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

In this strange but elegant new world of Minkowski spacetime, we're all travelers in a four dimensional realm. At each moment you exist at a specific point in spacetime, and your life, from birth to death, can be charted as a unique trajectory through spacetime – a complicated twisting path that Minkowski called a world-line. If you move relative to others, space and time change for you as determined by the Lorentz transformation. Move very quickly indeed, at speeds that are a sizable fraction of the speed of light, and these changes become marked: space is effectively traded for time – length contracting as time dilates, while energy of motion turns into increased mass. The mathematician sees these effects in terms of symmetry operations: the Lorentz transformation is equivalent to rotations and translations in the multi-dimensionality of spacetime.

### References

1. Einstein, A. Zur Elektrodynamik bewegter Körper (On the Electrodynamics
of Moving Bodies). *Annalen der Physik* 17 (1905): 891-921.

2. Snow, C. P. *Variety of Men*, 85-86. Harmondsworth, U.K.: Penguin
Books, 1969.

3. Einstein, A. Ist die Trägheit eines Körpers von seinem Energieinhalt
Abhängig? (Does the Inertia of a Body Depend Upon Its Energy Content?). *Annalen der Physik* 18 (1905): 639-641.