Special relativity

physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein

Special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physics theory regarding the relationship between space and time. In Albert Einstein's original pedagogical treatment, it is based on two postulates:

  1. The laws of physics are invariant (i.e. identical) in all inertial systems (non-accelerating frames of reference).
  2. The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.
     Light cone
World line


  • It is regrettable that Voigt's original ideas were unnoticed and, hence, did not play a role in the vigorous development of special relativity, somewhat similar to Poincare's work in 1905. Lorentz, Poincare, Einstein and others did not refer to Voigt's paper in their works. It appears that young Pauli was an early physicist to mention Voigt's transformation in his book 'Theory of Relativity', published in 1921, but no further comment was made.
    • Chinese Journal of Physics (2001) Vol. 39, p. 212.
  • The great merit of Minkowski was to show that an absolute world could nevertheless be imagined, although it was a far different world from that of classical physics. In Minkowski's world the absolute which supersedes the absolute length and duration of classical physics is the Einsteinian interval. ...
    Thus suppose that, as measured in our Galilean frame of reference, two flashes occur at points A and B, situated at a distance l apart, and suppose the flashes are separated in time by an interval t. If we change our frame of reference, both l and t will change in value, becoming l' and t' respectively, exhibiting by their changes the relativity of length and duration. In Minkowski's words, "Henceforth space and time themselves are mere shadows." On the other hand, the mathematical construct   will remain invariant, and so we shall have   It is this invariant expression, which involves both length and duration, or both space and time, which constitutes the Einsteinian interval; and the objective world which it cannotes is the world of four-dimensional space-time. The Einsteinian interval... remains the same for all observers, just as distance alone or duration alone were mistakenly believed to remain the same for all observers in classical physics. ...the Einsteinian interval still remains an invariant as measured for all frames of reference, whether accelerated or not.
  • A four dimensional continuum described by the co ordinates x1, x2, x3, x4, was called "world" by Minkowski, who also termed a point-event a "world point." From a "happening" in three-dimensional space, physics becomes, as it were, an "existence" in the four-dimensional world. This four dimensional "world" bears a close similarity to the three-dimensional "space" of Euclidean analytical geometry. ...We can regard Minkowski's "world" in a formal manner as a four-dimensional Euclidean space (with imaginary time co-ordinate); the Lorentz transformation corresponds to a "rotation" of the co-ordinate system in the four-dimensional world.
  • The discovery of Minkowski... is to be found... in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude,  , proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry. ...These inadequate remarks can give the reader only a vague notion of the important idea contributed by Minkowski. Without it the general theory of relativity... would perhaps have got no farther than its long clothes.
  • For over 200 years the equations of motion enunciated by Newton were believed to describe nature correctly, and the first time that an error in these laws was discovered, the way to correct it was also discovered. Both the error and its correction were discovered by Einstein in 1905.

    Newton’s Second Law, which we have expressed by the equation   was stated with the tacit assumption that m is a constant, but we now know that this is not true, and that the mass of a body increases with velocity. In Einstein’s corrected formula m has the value   where the “rest mass” m0 represents the mass of a body that is not moving and c is the speed of light, which is about 3×105 km⋅sec−1 or about 186,000 mi⋅sec−1.

    • Richard Feynman, The Feynman Lectures on Physics, Vol. I, Ch. 15. The Special Theory of Relativity
  • The impressions received by the two observers A0 and A would be alike in all respects. It would be impossible to decide which of them moves or stands still with respect to the ether, and there would be no reason for preferring the times and lengths measured by the one to those determined by the other, nor for saying that either of them is in possession of the "true" times or the "true" lengths. This is a point which Einstein has laid particular stress on, in a theory in which he starts from what he calls the principle of relativity, i.e., the principle that the equations by means of which physical phenomena may be described are not altered in form when we change the axes of coordinates for others having a uniform motion of translation relatively to the original system.
    I cannot speak here of the many highly interesting applications which Einstein has made of this principle. His results concerning electromagnetic and optical phenomena ...agree in the main with those which we have obtained... the chief difference being that Einstein simply postulates what we have deduced, with some difficulty and not altogether satisfactorily, from the fundamental equations of the electromagnetic field. By doing so, he may certainly take credit for making us see in the negative result of experiments like those of Michelson, Rayleigh and Brace, not a fortuitous compensation of opposing effects, but the manifestation of a general and fundamental principle.
    Yet, I think, something may also be claimed in favour of the form in which I have presented the theory. I cannot but regard the ether, which can be the seat of an electromagnetic field with its energy and vibrations, as endowed with a certain degree of substantiality, however different it may be from all ordinary matter. ...it seems natural not to assume at starting that it can never make any difference whether a body moves through the ether or not, and to measure distances and lengths of time by means of rods and clocks having a fixed position relatively to the ether.
    It would be unjust not to add that, besides the fascinating boldness of its starting point, Einstein's theory has another marked advantage over mine. Whereas I have not been able to obtain for the equations referred to moving axes exactly the same form as for those which apply to a stationary system, Einstein has accomplished this by means of a system of new variables slightly different from those which I have introduced.
    • Hendrik Lorentz, The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat (1916) Ch. V Optical Phenomena in Moving Bodies.
  • Between 1968 and 2005 I’ve learned a lot about explaining special relativity. One pedagogical discovery has been especially valuable. Anybody wishing to understand the subject must be able to visualize how certain events taking place, say, in a railroad station, are described from the point of view of a passenger passing through that station on a uniformly moving train and, conversely, how events taking place on such a train appear to a person standing in the station. Without the ability to translate from one such description to another, one cannot begin to understand relativity. But all introductions to relativity that I know of, including my own 1968 book, take the ability to do this for granted. They immediately require the reader to apply this unused, undeveloped, often nonexistent skill to some highly counterintuitive phenomena.
    In explaining relativity this process often leads to descriptions from two different perspectives, which appear, at first glance, to contradict each other. Faced with an apparent paradox, people who have never before thought about transforming station descriptions to train descriptions and vice versa quite reasonably assume that they must have done something wrong in the transcription. Rather than seeking an understanding of why the contradiction is only apparent, they lose confidence in the analytical technique that gave rise to it.
    In this respect the pedagogy of the standard approach to relativity is terrible. One introduces a crucial and unfamiliar conceptual technique— changing descriptions from one “frame of reference” to another—by immediately applying it to some unusual and highly counterintuitive cases. The most important thing I learned in teaching relativity to many generations of Cornell undergraduates, none of them science majors, is that one must begin teaching them the technique of changing frames of reference by applying that technique to some entirely commonplace, highly intuitive examples. There are many such ways to develop these skills, and they enable one to learn much that is not at all obvious, though never paradoxical.
    • N. David Mermin, It’s About Time : Understanding Einstein’s Relativity (2005)
  • Another thing I have learned since 1968 is that one should emphasize as early as possible that although objects moving at the speed of light famously behave in some very strange ways, the behavior of objects moving at speeds comparable to the speed of light can be just as peculiar. The peculiarity of motion at the speed of light is just a special case of a more general peculiarity of all motion, which becomes prominent only at extremely high speeds. That more general peculiarity can be expressed by an elementary but precise rule that it is possible and useful to formulate at a very early stage of the subject.
    • N. David Mermin, It’s About Time : Understanding Einstein’s Relativity (2005)
  • I. Redefine the foot.
    II. View nonrelativistic collisions from different frames.
    III. Immediately introduce relativistic velocity addition law.
    IV. Immediately introduce relativity of simultaneity.
    V. Give numerical illustrations.
    VI. Minkowski diagrams, direct from Einstein’s postulates.
    VII. Don’t bother with the spacetime Lorentz transformation.
  • The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. 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.
    • Hermann Minkowski, Address to the 80th Assembly of German Natural Scientists and Physicians, (Sep 21, 1908)
  • The whole world appears resolved into such world-lines. And I should like to say beforehand that, according to my opinion, it would be possible for the physical laws to find their fullest expression as correlations of these world-lines.
    • Hermann Minkowski, Space and Time (1909), Tr. Ganesh Prasad in: Bulletin of the Calcutta
  • The word postulate of relativity... appears to me very stale... I should rather like to give this statement the name Postulate of the absolute world (or briefly, world-postulate).
  • In a Newtonian view, space and time are separate and different. Symmetries of the laws of physics are combinations of rigid motions of space and an independent shift in time. But... these transformations do not leave Maxwell's equations invariant. Pondering this, the mathematicians Henri Poincaré and Hermann Minkowski were led to a new view of the symmetries of space and time, on a purely mathematical level. If they had described these symmetries in physical terms, they would have beaten Einstein to relativity, but they avoided physical speculations. They did understand that symmetries in the laws of electromagnetism do not affect space and time independently but mix them up. The mathematical scheme describing these intertwined changes is known as the Lorentz group, after the physicist, Hendrik Lorentz.
    • Ian Stewart, Why Beauty Is Truth: The History of Symmetry (2008)
  • Poincaré came very close to inventing special relativity in the years 1900-1904, showing in particular that Lorentz transformations form a group; hence in the case of the Poincaré group, the name is accurate.
by Martin Gardner
  • The strangest explanation [for the Michelson–Morley experiment] was put forth by an Irish physicist, George Francis Fitzgerald. Perhaps, he said, the ether wind puts pressure on a moving object, causing it to shrink a bit in the direction of motion. To determine the length of a moving object, its length at rest must be multiplied by the following simple formula, in which   is the velocity of the object multiplied by itself,   is the velocity of light multiplied by itself:  .
  • The speed of light in an unobtainable limit; when this is reached the formula becomes   which reduces to 0. ...In other words, if an object could obtain the speed of light, it would have no length at all in the direction of its motion!
  • Lorentz made an important addition to his original theory. He introduced changes in time. Clocks, he said, would be slowed down by the ether wind, and in just such a way as to make the velocity of light always measure 299,800 meters per second.
  • Einstein, following the steps of Ernst Mach... said... There is no ether wind. He did not say that there was no ether; only... [that the ether] is of no value in measuring uniform motion.
  • The special theory of relativity carries the classical relativity of Newton forward another step. It says that in addition to being unable to detect the train's motion by a mechanical experiment, it is also impossible to detect its motion by an optical experiment.
  • It is not possible to measure uniform motion in any absolute way.
  • In the special theory of relativity, the speed of light becomes... a new absolute. ...Regardless of the motion of its source, light always moves through space with the same constant speed.
  • There is no absolute time throughout the universe by which absolute simultaneity can be measured. Absolute simultaneity of distant events is a meaningless concept.
  • If an astronaut traveled as fast as light his clock would stop completely.
  • If two spaceships are in relative motion, an observer on each ship will measure the other ship as contracted slightly in the direction of its motion. ...The theory does not say that each ship is shorter than the other; it says that astronauts on each ship measure the other ship as shorter.
  • Two ships are passing each other with uniform speed close to that of light. As they pass, a beam of light on the other ship is sent from the ceiling to the floor. There it strikes a mirror and is reflected back to the ceiling again. You will see the path of this light as a V [shape]. If you had sufficiently accurate instruments (of course no such instrument exists), you could clock the time it takes this light beam to traverse the V-shaped path. By dividing the length of the path by the time, you obtain the speed of light. ...an astronaut inside the other ship is doing the same thing [measuring his light beam's speed]. From his point of view... the light simply goes down and up along the same line, obviously a shorter distance than along the V that you observed. ...he also obtains the speed of light. ...But his light path is shorter. ...There is only one possible explanation: his clock is slower. Of course, the situation is perfectly symmetrical. If you send a beam down and up inside your ship, he will see its path as V-shaped. He will deduce that your clock is slower.
  • All three variables—length, time, mass—are covered by the same Lorentz contraction [   ]... Length and the rate of clocks vary in the same direct proportion, so the formula is the same for each. Mass... varies in the inverse proportion...  .
  • The speed of light can never be reached. If it were reached, the outside observer would find that the ship had shrunk to zero length, had acquired an infinite mass, and was exerting an infinite force with its rocket motors. Astronauts inside the ship would observe no changes in themselves, but they would find the cosmos hurtling backward with the speed of light, cosmic time at a standstill, every star flattened to a disk and infinitely massive.

The Riddle of Gravitation (1968)

: From Newton to Einstein to Today's Exciting Theories, by Peter G. Bergmann.
  • All [of Newton's] fundamental laws of mechanics involved statements concerning accelerations, changes in the velocities... rather than the velocities themselves. These accelerations were tied to the distances between the bodies... [F]or collecting data relevant to an experimental confirmation of Newton's laws... one may consider equivalent all observers who, relative to one another, are engaged in straight-line and unaccelerated motion. ...Such an observer will be called an inertial observer; relative to him, the motion of a forcefree body will be unaccelerated. If an inertial observer is considered the hub of a scaffolding... one calls the whole framework an inertial frame of reference, or for short, an inertial frame. ...The equal validity of all inertial frames... and the non-existence of one frame representing absolute rest, is known as the principle of relativity. [It] remained unquestioned for about two hundred years. ...[T]here was no such thing as absolute rest, or absolute motion, for that matter, but only absolute acceleration... governed by the forces resulting from the proximity of other bodies.
  • Based on Faraday's earlier work, Maxwell stressed the notion of fields, in contrast to Newton's emphasis on the direct action of bodies on each other across empty space (action at a distance). Faraday and Maxwell regarded the effect of an electrically charged body as giving rise to stresses in its immediate surroundings... [and] in ever widening circles, gradually diminishing... These stresses... [i.e.,] the fields are intermediaries between the material particles and assume the burden of Newton's action at a distance. ...[O]ne set of Maxwell's equations is to the effect that, in the presence of a magnetic field which changes in the course of time, an electric field arises which is not caused by the presence of any electric charge. This [is] the law of electromagnetic induction... From his theory, Maxwell... predicted that magnetic fields propogate at... the speed of light. ...The laws of mechanics involve only accelerations, not velocities: the laws of electromagnetism involve a universal velocity [c]...
  • If there is such a thing as a universal speed... Newtonian physics... must be reviewed. As long as the laws of physics were concerned only with accelerations... no conceivable experiment... would lead to the selection of one particular frame of reference as fundamental. But if in empty space light propogates at the universal speed... then a careful determination of the apparent speed of light relative to laboratory apparatus should reveal the [absolute] velocity of that apparatus.... There should exist one frame of reference with respect to which light does travel everywhere at the speed c. Call this... the frame of absolute rest. ...[W]ith respect to any other frame...the apparent speed of light should be less than c in the direction in which the frame is traveling relative to the frame of absolute rest; it should be greater than c in the opposite direction.

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