Aberration of light

The aberration of light (also referred to as astronomical aberration, stellar aberration, or velocity aberration) is an astronomical phenomenon which produces an apparent motion of celestial objects about their true positions, dependent on the velocity of the observer. Aberration causes objects to appear to be displaced towards the direction of motion of the observer compared to when the observer is stationary. The change in angle is typically very small — of the order of v/c where c is the speed of light and v the velocity of the observer. In the case of "stellar" or "annual" aberration, the apparent position of a star to an observer on Earth varies periodically over the course of a year as the Earth's velocity changes as it revolves around the Sun, by a maximum angle of approximately 20 arcseconds in right ascension or declination.

The term aberration has historically been used to refer to a number of related phenomena concerning the propagation of light in moving bodies.[1] Aberration should not be confused with parallax. The latter is a change in the apparent position of a relatively nearby object, as measured by a moving observer, relative to more distant objects that define a reference frame. The amount of parallax depends on the distance of the object from the observer, whereas aberration does not. Aberration is also related to light-time correction and relativistic beaming, although it is often considered separately from these effects.

Aberration is historically significant because of its role in the development of the theories of light, electromagnetism and, ultimately, the theory of special relativity. It was first observed in the late 1600s by astronomers searching for stellar parallax in order to confirm the heliocentric model of the Solar System. However, it was not understood at the time to be a different phenomenon.[2] In 1727, James Bradley provided a classical explanation for it in terms of the finite speed of light relative to the motion of the Earth in its orbit around the Sun,[3][4] which he used to make one of the earliest measurements of the speed of light. However, Bradley's theory was incompatible with 19th century theories of light, and aberration became a major motivation for the aether drag theories of Augustin Fresnel (in 1818) and G. G. Stokes (in 1845), and for Hendrik Lorentz's aether theory of electromagnetism in 1892. The aberration of light, together with Lorentz's elaboration of Maxwell's electrodynamics, the moving magnet and conductor problem, the negative aether drift experiments, as well as the Fizeau experiment, led Albert Einstein to develop the theory of special relativity in 1905, which presents a general form of the equation for aberration in terms of such theory.[5]

Simple stellar aberration diagram
The apparent position of a star viewed from the Earth depends on the Earth's velocity. The effect is typically much smaller than illustrated.


Sun earth relativistic aberration
Light rays striking the earth in the Sun's rest frame compared to the same rays in the Earth's rest frame according to special relativity. The effect is exaggerated for illustrative purposes.

Aberration may be explained as the difference in angle of a beam of light in different inertial frames of reference. A common analogy is to consider the apparent direction of falling rain. If rain is falling vertically in the frame of reference of a person standing still, then to a person moving forwards the rain will appear to arrive at an angle, requiring the moving observer to tilt their umbrella forwards. The faster the observer moves, the more tilt is needed.

The net effect is that light rays striking the moving observer from the sides in a stationary frame will come angled from ahead in the moving observer's frame. This effect is sometimes called the "searchlight" or "headlight" effect.

In the case of annual aberration of starlight, the direction of incoming starlight as seen in the Earth's moving frame is tilted relative to the angle observed in the Sun's frame. Since the direction of motion of the Earth changes during its orbit, the direction of this tilting changes during the course of the year, and causes the apparent position of the star to differ from its true position as measured in the inertial frame of the Sun.

While classical reasoning gives intuition for aberration, it leads to a number of physical paradoxes observable even at the classical level (see history). The theory of special relativity is required to correctly account for aberration. The relativistic explanation is very similar to the classical one however, and in both theories aberration may be understood as a case of addition of velocities.

Classical explanation

In the Sun's frame, consider a beam of light with velocity equal to the speed of light c, with x and y velocity components and , at an angle . If the Earth is moving at velocity in the x direction relative to the Sun, then by velocity addition the x component of the beam's velocity in the Earth's frame of reference is , and the y velocity is unchanged, . Thus the angle of the light in the Earth's frame in terms of the angle in the Sun's frame is

In the case of , this result reduces to , which in the limit may be approximated by .

Relativistic explanation

The reasoning in the relativistic case is the same except that the relativistic velocity addition formulas must be used, which can be derived from Lorentz transformations between different frames of reference. These formulae are

where , giving the components of the light beam in the Earth's frame in terms of the components in the Sun's frame. The angle of the beam in the Earth's frame is thus [6]

In the case of , this result reduces to , and in the limit this may be approximated by . This relativistic derivation keeps the speed of light constant in all frames of reference, unlike the classical derivation above.

Relationship to light-time correction and relativistic beaming

Aberration, light-time correction, and relativistic beaming can be considered the same phenomenon depending on the frame of reference.

Aberration is related to two other phenomena, light-time correction, which is due to the motion of an observed object during the time taken by its light to reach an observer, and relativistic beaming, which is an angling of the light emitted by a moving light source. It can be considered equivalent to them but in a different inertial frame of reference. In aberration, the observer is considered to be moving relative to a (for the sake of simplicity[7]) stationary light source, while in light-time correction and relativistic beaming the light source is considered to be moving relative to a stationary observer.

Consider the case of an observer and a light source moving relative to each other at constant velocity, with a light beam moving from the source to the observer. At the moment of emission, the beam in the observer's rest frame is tilted compared to the one in the source's rest frame, as understood through relativistic beaming. During the time it takes the light beam to reach the observer the light source moves in the observer's frame, and the 'true position' of the light source is displaced relative to the apparent position the observer sees, as explained by light-time correction. Finally, the beam in the observer's frame at the moment of observation is tilted compared to the beam in source's frame, which can be understood as an aberrational effect. Thus, a person in the light source's frame would describe the apparent tilting of the beam in terms of aberration, while a person in the observer's frame would describe it as a light-time effect.

The relationship between these phenomena is only valid if the observer and source's frames are inertial frames. In practice, because the Earth is not an inertial rest frame but experiences centripetal acceleration towards the Sun, many aberrational effects such as annual aberration on Earth cannot be considered light-time corrections. However, if the time between emission and detection of the light is short compared to the orbital period of the Earth, the Earth may be approximated as an inertial frame and aberrational effects are equivalent to light-time corrections.

Types of aberration

There are a number of types of aberration, caused by the differing components of the Earth's and observed object's motion:

  • Annual aberration is due to the orbital revolution of the Earth around the Sun.
  • Planetary aberration is the combination of aberration and light-time correction.
  • Diurnal aberration is due to the rotation of the Earth about its own axis.
  • Secular aberration is due to the motion of the Sun and Solar System relative to other stars in our Galaxy.

Annual aberration

Stars at the ecliptic poles appear to move in circles, stars exactly in the ecliptic plane move in lines, and stars at intermediate angles move in ellipses. Shown here are the apparent motions of stars with the ecliptic latitudes corresponding to these cases, and with ecliptic longitude of 270°.
The direction of aberration of a star at the northern ecliptic pole differs at different times of the year

Annual aberration is caused by the motion of an observer on Earth as the planet revolves around the Sun. Due to orbital eccentricity, the orbital velocity of Earth (in the Sun's rest frame) varies periodically during the year as the planet traverses its elliptic orbit and consequently the aberration also varies periodically, typically causing stars to appear to move in small ellipses.

Approximating Earth's orbit as circular, the maximum displacement of a star due to annual aberration is known as the constant of aberration, conventionally represented by . It may be calculated using the relation substituting the Earth's average speed in the Sun's frame for and the speed of light . Its accepted value is 20.49552  arcseconds (at J2000).[8]

Assuming a circular orbit, annual aberration causes stars exactly on the ecliptic (the plane of Earth's orbit) to appear to move back and forth along a straight line, varying by on either side of their position in the Sun's frame. A star that is precisely at one of the ecliptic poles (at 90° from the ecliptic plane) will appear to move in a circle of radius about its true position, and stars at intermediate ecliptic latitudes will appear to move along a small ellipse.

For illustration, consider a star at the northern ecliptic pole viewed by an observer at a point on the Arctic Circle. Such an observer will see the star transit at the zenith, once every day (strictly speaking sidereal day). At the time of the March equinox, Earth's orbit carries the observer in a southwards direction, and the star's apparent declination is therefore displaced to the south by an angle of . On the September equinox, the star's position is displaced to the north by an equal and opposite amount. On either solstice, the displacement in declination is 0. Conversely, the amount of displacement in right ascension is 0 on either equinox and at maximum on either solstice.

In actuality, Earth's orbit is slightly elliptic rather than circular, and its speed varies somewhat over the course of its orbit, which means the description above is only approximate. Aberration is more accurately calculated using Earth's instantaneous velocity relative to the barycenter of the Solar System.[8]

Note that the displacement due to aberration is orthogonal to any displacement due to parallax. If parallax were detectable, the maximum displacement to the south would occur in December, and the maximum displacement to the north in June. It is this apparently anomalous motion that so mystified early astronomers.

Solar annual aberration

A special case of annual aberration is the nearly constant deflection of the Sun from its position in the Sun's rest frame by towards the west (as viewed from Earth), opposite to the apparent motion of the Sun along the ecliptic (which is from west to east, as seen from Earth). The deflection thus makes the Sun appear to be behind (or retarded) from its rest-frame position on the ecliptic by a position or angle .

This deflection may equivalently be described as a light-time effect due to motion of the Earth during the 8.3 minutes that it takes light to travel from the Sun to Earth. This is possible since the transit time of sunlight is short relative to the orbital period of the Earth, so the Earth's frame may be approximated as inertial. In the Earth's frame, the Sun moves by a distance in the time it takes light to reach Earth, for the orbit of radius . This gives an angular correction which can be solved to give , the same as the aberrational correction.

Planetary aberration

Planetary aberration is the combination of the aberration of light (due to Earth's velocity) and light-time correction (due to the object's motion and distance), as calculated in the rest frame of the Solar System. Both are determined at the instant when the moving object's light reaches the moving observer on Earth. It is so called because it is usually applied to planets and other objects in the Solar System whose motion and distance are accurately known.

Diurnal aberration

Diurnal aberration is caused by the velocity of the observer on the surface of the rotating Earth. It is therefore dependent not only on the time of the observation, but also the latitude and longitude of the observer. Its effect is much smaller than that of annual aberration, and is only 0.32 arcseconds in the case of an observer at the Equator, where the rotational velocity is greatest.[9]

Secular aberration

The Sun and Solar System are revolving around the center of the Galaxy. Aberration due to this motion is known as secular aberration and affects the apparent positions of distant stars and extragalactic objects. However, since the galactic year is about 230 million years, the aberration varies very slowly and this change is extremely difficult to observe. Therefore, secular aberration is usually ignored when considering the positions of stars. In other words, star maps show the observed apparent positions of the stars, not their calculated true positions after accounting for secular aberration.

For stars significantly less than 230 million light years away, the Solar System may be approximated as an inertial frame and so the effect of secular aberration is equivalent to a light-time correction. This includes stars in the Milky Way, since the Milky Way is about 100,000 light years in diameter. For these stars the true position of the star is then easily computed from its proper motion and its distance.

Secular aberration is typically a small number of arcminutes, for example the stationary star Groombridge 1830 is displaced by approximately 3 arcminutes,[9] due to secular aberration. This is roughly 8 times the effect of annual aberration, as one would expect since the velocity of the Solar System relative to the center of the Galaxy is about 8 times the velocity of the Earth relative to the Sun.

Discovery and first observations

The discovery of the aberration of light was totally unexpected, and it was only by considerable perseverance and perspicacity that Bradley was able to explain it in 1727. It originated from attempts to discover whether stars possessed appreciable parallaxes.

Search for stellar parallax

The Copernican heliocentric theory of the Solar System had received confirmation by the observations of Galileo and Tycho Brahe and the mathematical investigations of Kepler and Newton.[10] As early as 1573, Thomas Digges had suggested that parallactic shifting of the stars should occur according to the heliocentric model, and consequently if stellar parallax could be observed it would help confirm this theory. Many observers claimed to have determined such parallaxes, but Tycho Brahe and Giovanni Battista Riccioli concluded that they existed only in the minds of the observers, and were due to instrumental and personal errors. However, in 1680 Jean Picard, in his Voyage d’Uranibourg, stated, as a result of ten years' observations, that Polaris, the Pole Star, exhibited variations in its position amounting to 40″ annually. Some astronomers endeavoured to explain this by parallax, but these attempts failed because the motion differed from that which parallax would produce. John Flamsteed, from measurements made in 1689 and succeeding years with his mural quadrant, similarly concluded that the declination of Polaris was 40″ less in July than in September. Robert Hooke, in 1674, published his observations of γ Draconis, a star of magnitude 2m which passes practically overhead at the latitude of London (hence its observations are largely free from the complex corrections due to atmospheric refraction), and concluded that this star was 23″ more northerly in July than in October.[10]

James Bradley's observations

Bradley's observations of γ Draconis and 35 Camelopardalis as reduced by Busch
Bradley's observations of γ Draconis and 35 Camelopardalis as reduced by Busch to the year 1730.

Consequently, when Bradley and Samuel Molyneux entered this sphere of research in 1725, there was still considerable uncertainty as to whether stellar parallaxes had been observed or not, and it was with the intention of definitely answering this question that they erected a large telescope at Molyneux's house at Kew.[4] They decided to reinvestigate the motion of γ Draconis with a telescope constructed by George Graham (1675–1751), a celebrated instrument-maker. This was fixed to a vertical chimney stack in such manner as to permit a small oscillation of the eyepiece, the amount of which (i.e. the deviation from the vertical) was regulated and measured by the introduction of a screw and a plumb line.[10]

The instrument was set up in November 1725, and observations on γ Draconis were made starting in December. The star was observed to move 40″ southwards between September and March, and then reversed its course from March to September. [10] At the same time, 35 Camelopardalis, a star with a right ascension nearly exactly opposite to that of γ Draconis, was 19" more northerly at the beginning of March than in September.[11] These results were completely unexpected and inexplicable by existing theories.

Early hypotheses

Hypothetical movement of γ Draconis caused by parallax
Hypothetical observation of γ Draconis if its movement was caused by parallax.
Hypothetical movement of γ Draconis and 35 Camelopardalis caused by nutation
Hypothetical observation of γ Draconis and 35 Camelopardalis if their movements were caused by nutation.

Bradley and Molyneux discussed several hypotheses in the hope of finding the solution. Since the apparent motion was evidently caused neither by parallax nor observational errors, Bradley first hypothesized that it could be due to oscillations in the orientation of the Earth's axis relative to the celestial sphere – a phenomenon known as nutation. 35 Camelopardalis was seen to possess an apparent motion which could be consistent with nutation, but since its declination varied only one half as much as that of γ Draconis, it was obvious that nutation did not supply the answer[12] (however, Bradley later went on to discover that the Earth does indeed nutate).[13] He also investigated the possibility that the motion was due to an irregular distribution of the Earth's atmosphere, thus involving abnormal variations in the refractive index, but again obtained negative results.[12]

On August 19, 1727, Bradley embarked upon a further series of observations using a telescope of his own erected at the Rectory, Wanstead. This instrument had the advantage of a larger field of view and he was able to obtain precise positions of a large number of stars over the course of about twenty years. During his first two years at Wanstead, he established the existence of the phenomenon of aberration beyond all doubt, and this also enabled him to formulate a set of rules that would allow the calculation of the effect on any given star at a specified date.

Development of the theory of aberration

Bradley eventually developed his explanation of aberration in about September 1728 and this theory was presented to the Royal Society in mid January the following year. One well-known story was that he saw the change of direction of a wind vane on a boat on the Thames, caused not by an alteration of the wind itself, but by a change of course of the boat relative to the wind direction.[13] However, there is no record of this incident in Bradley's own account of the discovery, and it may therefore be apocryphal.

The following table shows the magnitude of deviation from true declination for γ Draconis and the direction, on the planes of the solstitial colure and ecliptic prime meridian, of the tangent of the velocity of the Earth in its orbit for each of the four months where the extremes are found, as well as expected deviation from true ecliptic longitude if Bradley had measured its deviation from right ascension:

Month Direction of tangential velocity of Earth on the plane of the solstitial colure Deviation from true declination of γ Draconis Direction of tangential velocity of Earth on the plane of the ecliptic prime meridian Expected deviation from true ecliptic longitude of γ Draconis
December zero none ← (moving toward perihelion at fast velocity) decrease of more than 20.2"
March ← (moving toward aphelion) 19.5" southward zero none
June zero none → (moving toward aphelion at slow velocity) increase of less than 20.2"
September → (moving toward perihelion) 19.5" northward zero none

Bradley proposed that the aberration of light not only affected declination, but right ascension as well, so that a star in the pole of the ecliptic would describe a little ellipse with a diameter of about 40", but for simplicity, he assumed it to be a circle. Since he only observed the deviation in declination, and not in right ascension, his calculations for the maximum deviation of a star in the pole of the ecliptic are for its declination only, which will coincide with the diameter of the little circle described by such star. For eight different stars, his calculations are as follows:

Star Annual Variation (") Maximum deviation in declination of a star in the pole of the ecliptic (")
γ Draconis 39 40.4
β Draconis 39 40.2
η Ursa Maj. 36 40.4
α Cass. 34 40.8
τ Persei 25 41.0
α Persei 23 40.2
35 Camel. 19 40.2
Capella 16 40.0
MEAN 40.4

Based on these calculations, Bradley was able to estimate the constant of aberration at 20.2", which is equal to 0.00009793 radians, and with this was able to estimate the speed of light at 183,300 miles (295,000 km) per second.[14] By projecting the little circle for a star in the pole of the ecliptic, he could simplify the calculation of the relationship between the speed of light and the speed of the Earth's annual motion in its orbit as follows:

Thus, the speed of light to the speed of the Earth's annual motion in its orbit is "10210 to one, from whence it would follow, that light moves, or is propagated as far as from the Sun to the Earth in 8' 12"."[15]

The discovery and elucidation of aberration is now regarded as a classic case of the application of scientific method, in which observations are made to test a theory, but unexpected results are sometimes obtained that in turn lead to new discoveries. It is also worth noting that part of the original motivation of the search for stellar parallax was to test the Copernican theory that the Earth revolves around the Sun, but of course the existence of aberration also establishes the truth of that theory.

Historical theories of aberration

The phenomenon of aberration became a driving force for many physical theories during the 200 years between its observation and the conclusive explanation by Albert Einstein.

The first classical explanation was provided in 1729, by James Bradley as described above, who attributed it to the finite speed of light and the motion of Earth in its orbit around the Sun.[3][4] However, this explanation proved inaccurate once the wave nature of light was better understood, and correcting it became a major goal of the 19th century theories of luminiferous aether. Augustin-Jean Fresnel proposed a correction due to the motion of a medium (the aether) through which light propagated, known as "partial aether drag". He proposed that objects partially drag the aether along with them as they move, and this became the accepted explanation for aberration for some time. George Stokes proposed a similar theory, explaining that aberration occurs due to the flow of aether induced by the motion of the Earth. Accumulated evidence against these explanations, combined with new understanding of the electromagnetic nature of light, led Hendrik Lorentz to develop an electron theory which featured an immobile aether, and he explained that objects contract in length as they move through the aether. Motivated by these previous theories, Albert Einstein then developed the theory of special relativity in 1905, which provides the modern account of aberration.

Bradley's classical explanation

Stellar aberration
Figure 2: As light propagates down the telescope, the telescope moves requiring a tilt to the telescope that depends on the speed of light. The apparent angle of the star φ differs from its true angle θ.

Bradley conceived of an explanation in terms of a corpuscular theory of light in which light is made of particles unaffected by gravity.[1] His classical explanation appeals to the motion of the earth relative to a beam of light-particles moving at a finite velocity, and is developed in the Sun's frame of reference, unlike the classical derivation given above.

Consider the case where a distant star is motionless relative to the Sun, and the star is extremely far away, so that parallax may be ignored. In the rest frame of the Sun, this means light from the star travels in parallel paths to the Earth observer, and arrives at the same angle regardless of where the Earth is in its orbit. Suppose the star is observed on Earth with a telescope, idealized as a narrow tube. The light enters the tube from the star at angle and travels at speed taking a time to reach the bottom of the tube, where it is detected. Suppose observations are made from Earth, which is moving with a speed . During the transit of the light, the tube moves a distance . Consequently, for the particles of light to reach the bottom of the tube, the tube must be inclined at an angle different from , resulting in an apparent position of the star at angle . As the Earth proceeds in its orbit it changes direction, so changes with the time of year the observation is made. The apparent angle and true angle are related using trigonometry as:


In the case of , this gives . While this is different from the more accurate relativistic result described above, in the limit of small angle and low velocity they are approximately the same, within the error of the measurements of Bradley's day. These results allowed Bradley to make one of the earliest measurements of the speed of light.[15][16]

Luminiferous aether

Stellar aberration versus the dragged aether
Young reasoned that aberration could only be explained if the aether were immobile in the frame of the Sun. On the left, stellar aberration occurs if an immobile aether is assumed, showing that the telescope must be tilted. On the right, the aberration disappears if the aether moves with the telescope, and the telescope does not need to be tilted.

In the early nineteenth century the wave theory of light was being rediscovered, and in 1804 Thomas Young adapted Bradley's explanation for corpuscular light to wavelike light traveling through a medium known as the luminiferous aether. His reasoning was the same as Bradley's, but it required that this medium be immobile in the Sun's reference frame and must pass through the earth unaffected, otherwise the medium (and therefore the light) would move along with the earth and no aberration would be observed. [17] He wrote:

Upon consideration of the phenomena of the aberration of the stars I am disposed to believe that the luminiferous aether pervades the substance of all material bodies with little or no resistance, as freely perhaps as the wind passes through a grove of trees.

— Thomas Young, 1804[1]

However, it soon became clear Young's theory could not account for aberration when materials with a non-vacuum index of refraction were present. An important example is of a telescope filled with water. The velocity of the light in such a telescope will be slower than in vacuum, and is given by rather than where is the index of refraction of the water. Thus, by Bradley and Young's reasoning the aberration angle is given by


which predicts a medium-dependent angle of aberration. When refraction at the telescope's objective is taken into account this result deviates even more from the vacuum result. In 1810 François Arago performed a similar experiment and found that the aberration was unaffected by the medium in the telescope, providing solid evidence against Young's theory. This experiment was subsequently verified by many others in the following decades, most accurately by Airy in 1871, with the same result.[17]

Aether drag models

Fresnel's aether drag

In 1818, Augustin Fresnel developed a modified explanation to account for the water telescope and for other aberration phenomena. He explained that the aether is generally at rest in the Sun's frame of reference, but objects partially drag the aether along with them as they move. That is, the aether in an object of index of refraction moving at velocity is partially dragged with a velocity bringing the light along with it. This factor is known as "Fresnel's dragging coefficient". This dragging effect, along with refraction at the telescope's objective, compensates for the slower speed of light in the water telescope in Bradley's explanation.[a] With this modification Fresnel obtained Bradley's vacuum result even for non-vacuum telescopes, and was also able to predict many other phenomena related to the propagation of light in moving bodies. Fresnel's dragging coefficient became the dominant explanation of aberration for the next decades.

Stokes aether drag
Conceptual illustration of Stokes' aether drag theory. In the rest frame of the Sun the Earth moves to the right through the aether, in which it induces a local current. A ray of light (in red) coming from the vertical becomes dragged and tilted due to the flow of aether.

Stokes' aether drag

However, the fact that light is polarized (discovered by Fresnel himself) led scientists such as Cauchy and Green to believe that the aether was a totally immobile elastic solid as opposed to Fresnel's fluid aether. There was thus renewed need for an explanation of aberration consistent both with Fresnel's predictions (and Arago's observations) as well as polarization.

In 1845, Stokes proposed a 'putty-like' aether which acts as a liquid on large scales but as a solid on small scales, thus supporting both the transverse vibrations required for polarized light and the aether flow required to explain aberration. Making only the assumptions that the fluid is irrotational and that the boundary conditions of the flow are such that the aether has zero velocity far from the Earth, but moves at the Earth's velocity at its surface and within it, he was able to completely account for aberration.[b] The velocity of the aether outside of the Earth would decrease as a function of distance from the Earth so light rays from stars would be progressively dragged as they approached the surface of the Earth. The Earth's motion would be unaffected by the aether due to D'Alembert's paradox.

Both Fresnel and Stokes' theories were popular. However, the question of aberration was put aside during much of the second half of the 19th century as focus of inquiry turned to the electromagnetic properties of aether.

Lorentz' length contraction

In the 1880s once electromagnetism was better understood, interest turned again to the problem of aberration. By this time flaws were known to both Fresnel's and Stokes' theories. Fresnel's theory required that the relative velocity of aether and matter to be different for light of different colors, and it was shown that the boundary conditions Stokes had assumed in his theory were inconsistent with his assumption of irrotational flow.[1][17][18] At the same time, the modern theories of electromagnetic aether could not account for aberration at all. Many scientists such as Maxwell, Heaviside and Hertz unsuccessfully attempted to solve these problems by incorporating either Fresnel or Stokes' theories into Maxwell's new electromagnetic laws.

Hendrik Lorentz spent considerable effort along these lines. After working on this problem for a decade, the issues with Stokes' theory caused him to abandon it and to follow Fresnel's suggestion of a (mostly) stationary aether (1892, 1895). However, in Lorentz's model the aether was completely immobile, like the electromagnetic aethers of Cauchy, Green and Maxwell and unlike Fresnel's aether. He obtained Fresnel's dragging coefficient from modifications of Maxwell's electromagnetic theory, including a modification of the time coordinates in moving frames ("local time"). In order to explain the Michelson–Morley experiment (1887), which apparently contradicted both Fresnel's and Lorentz's immobile aether theories, and apparently confirmed Stokes' complete aether drag, Lorentz theorized (1892) that objects undergo "length contraction" by a factor of in the direction of their motion through the aether. In this way, aberration (and all related optical phenomena) can be accounted for in the context of an immobile aether. Lorentz' theory became the basis for much research in the next decade, and beyond. Its predictions for aberration are identical to those of the relativistic theory.[17][19]

Special relativity

Lorentz' theory matched experiment well, but it was complicated and made many unsubstantiated physical assumptions about the microscopic nature of electromagnetic media. In his 1905 theory of special relativity, Albert Einstein reinterpreted the results of Lorentz' theory in a much simpler and more natural conceptual framework which disposed of the idea of an aether. His derivation is given above, and is now the accepted explanation. Robert S. Shankland reported some conversations with Einstein, in which Einstein emphasized the importance of aberration:[20]

He continued to say the experimental results which had influenced him most were the observations of stellar aberration and Fizeau’s measurements on the speed of light in moving water. “They were enough,” he said.

Other important motivations for Einstein's development of relativity were the moving magnet and conductor problem and (indirectly) the negative aether drift experiments, already mentioned by him in the introduction of his first relativity paper. Einstein wrote in a note in 1952:[5]

My own thought was more indirectly influenced by the famous Michelson-Morley experiment. I learned of it through Lorentz’ path breaking investigation on the electrodynamics of moving bodies (1895), of which I knew before the establishment of the special theory of relativity. Lorentz’ basic assumption of a resting ether did not seem directly convincing to me, since it led to an [struck out: to me artificial appearing] interpretation of the Michelson-Morley experiment, which [struck out: did not convince me] seemed unnatural to me. My direct path to the sp. th. rel. was mainly determined by the conviction that the electromotive force induced in a conductor moving in a magnetic field is nothing other than an electric field. But the result of Fizeau’s experiment and the phenomenon of aberration also guided me.

While Einstein's result is the same as Bradley's original equation except for an extra factor of , it should be emphasized that Bradley's result does not merely give the classical limit of the relativistic case, in the sense that it gives incorrect predictions even at low relative velocities. Bradley's explanation cannot account for situations such as the water telescope, nor for many other optical effects (such as interference) that might occur within the telescope. This is because in the Earth's frame it predicts that the direction of propagation of the light beam in the telescope is not normal to the wavefronts of the beam, in contradiction with Maxwell's theory of electromagnetism. It also does not preserve the speed of light c between frames. However, Bradley did correctly infer that the effect was due to relative velocities.

See also


  1. ^ More in detail, Fresnel explains that the incoming light of angle is first refracted at the end of the telescope, to a new angle within the telescope. This may be accounted for by Snell's law, giving . Then drag must be accounted for. Without drag, the x and y components of the light in the telescope are and , but drag modifies the x component to if the Earth moves with velocity . If is angle and is the velocity of the light with these velocity components, then by Bradley's reasoning where is the modified path length through the water and t is the time it takes the light to travel the distance h, . Upon solving these equations for in terms of one obtains Bradley's vacuum result.
  2. ^
    The propagating wavefront moving through the aether.

    Stokes' derivation may be summarized as follows: Consider a wavefront moving in the downwards z direction. Say the aether has velocity field as a function of . Now, motion of the aether in the x and y directions does not affect the wavefront, but the motion in the z direction advances it (in addition to the amount it advances at speed c). If the z velocity of the aether varies over space, for example if it is slower for higher x as shown in the figure, then the wavefront becomes angled, by an angle . Now, say in time t the wavefront has moved by a span (assuming the speed of the aether is negligible compared to the speed of light). Then for each distance the ray descends, it is bent by an angle , and so the total angle by which it has changed after travelling through the entire fluid is

    If the fluid is irrotational it will satisfy the Cauchy–Riemann equations, one of which is


    Inserting this into the previous result gives an aberration angle where the s represent the x component of the aether's velocity at the start and end of the ray. Far from the earth the aether has zero velocity, so and at the surface of the earth it has the earth's velocity . Thus we finally get

    which is the known aberration result.

The propagating wavefront moving through the aether.


  1. ^ a b c d Schaffner, Kenneth F. (1972), Nineteenth-century aether theories, Oxford: Pergamon Press, pp. 99–117 und 255–273, ISBN 0-08-015674-6
  2. ^ Williams, M. E. W. "Flamsteed's Alleged Measurement of Annual Parallax for the Pole Star". Journal for the History of Astronomy. 10 (2): 102–116. Bibcode:1979JHA....10..102W. doi:10.1177/002182867901000203.
  3. ^ a b Bradley, James (1727–1728). "A Letter from the Reverend Mr. James Bradley Savilian Professor of Astronomy at Oxford, and F.R.S. to Dr.Edmond Halley Astronom. Reg. &c. Giving an Account of a New Discovered Motion of the Fix'd Stars". Phil. Trans. R. Soc. 35: 637–661. doi:10.1098/rstl.1727.0064.
  4. ^ a b c Hirschfeld, Alan (2001). Parallax:The Race to Measure the Cosmos. New York, New York: Henry Holt. ISBN 0-8050-7133-4.
  5. ^ a b Norton, John D., John D. (2004), "Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905", Archive for History of Exact Sciences, 59: 45–105, Bibcode:2004AHES...59...45N, doi:10.1007/s00407-004-0085-6, archived from the original on 2009-01-11
  6. ^ Richard A. Mould (2001). Basic Relativity (2nd ed.). Springer. p. 8. ISBN 0-387-95210-1.
  7. ^ In fact, the light source doesn't need to be stationary, consider for example eclipsing binary stars: they are rotating with high speed —and ever changing and different velocity vectors— around each other, but they appear as one spot all the time.
  8. ^ a b Kovalevsky, Jean & Seidelmann, P. Kenneth (2004). Fundamentals of Astrometry. Cambridge: Cambridge University Press. ISBN 0-521-64216-7.
  9. ^ a b Newcomb, Simon (1960). A Compendium of Spherical Astronomy. Macmillan, 1906 – republished by Dover.
  10. ^ a b c d Eppenstein 1911, p. 54.
  11. ^ Bradley, James; Rigaud, Stephen Peter (1832). Miscellaneous works and correspondence of the Rev. James Bradley, D.D., F.R.S. Oxford: University Press. p. 11.
  12. ^ a b Eppenstein 1911, p. 55.
  13. ^ a b Berry, Arthur (1961) [1898]. A Short History of Astronomy. Dover.
  14. ^ Hoiberg, Dale H., ed. (2010). "aberration, constant of". Encyclopædia Britannica. I: A-ak Bayes (15th ed.). Chicago, IL: Encyclopædia Britannica Inc. p. 30. ISBN 978-1-59339-837-8.
  15. ^ a b James Bradley (1729). "An account of a new discovered motion of the fixed stars". Philosophical Transactions of the Royal Society. 35: 637–661. doi:10.1098/rstl.1727.0064.
  16. ^ Encyclopædia Britannica Archived 2013-11-11 at the Wayback Machine.
  17. ^ a b c d Whittaker, Edmund Taylor (1910), A History of the theories of aether and electricity (1. ed.), Dublin: Longman, Green and Co., archived from the original on 2016-02-15
    Whittaker, Edmund Taylor (1953). A History of the Theories of Aether and Electricity (2. ed.). T. Nelson.
  18. ^ Janssen, Michel & Stachel, John (2010), "The Optics and Electrodynamics of Moving Bodies" (PDF), in John Stachel, Going Critical, Springer, ISBN 1-4020-1308-6
  19. ^ Darrigol, Olivier (2000), Electrodynamics from Ampére to Einstein, Oxford: Clarendon Press, ISBN 0-19-850594-9
  20. ^ Shankland, R. S. (1963). "Conversations with Albert Einstein". American Journal of Physics. 31 (1): 47–57. Bibcode:1963AmJPh..31...47S. doi:10.1119/1.1969236.

Further reading

  • P. Kenneth Seidelmann (Ed.), Explanatory Supplement to the Astronomical Almanac (University Science Books, 1992), 127-135, 700.
  • Stephen Peter Rigaud, Miscellaneous Works and Correspondence of the Rev. James Bradley, D.D. F.R.S. (1832).
  • Charles Hutton, Mathematical and Philosophical Dictionary (1795).
  • H. H. Turner, Astronomical Discovery (1904).
  • Thomas Simpson, Essays on Several Curious and Useful Subjects in Speculative and Mix'd Mathematicks (1740).
  • de:August Ludwig Busch, Reduction of the Observations Made by Bradley at Kew and Wansted to Determine the Quantities of Aberration and Nutation (1838).

External links

1729 in science

The year 1729 in science and technology involved some significant events.

1871 in science

The year 1871 in science and technology involved some significant events, listed below.

Aether drag hypothesis

In the 19th century, the theory of the luminiferous aether as the hypothetical medium for the propagation of light was widely discussed. An important part of this discussion was the question concerning the state of motion of Earth with respect to this medium. The aether drag hypothesis dealt with the question whether the luminiferous aether is dragged by or entrained within moving matter. According to the first variant no relative motion exists between Earth and aether; according to the second one, relative motion exists and thus the speed of light should depend on the speed of this motion ("aether wind"), which should be measurable by instruments at rest on Earth's surface. Specific aether models were invented by Augustin-Jean Fresnel who in 1818 proposed that the aether is partially entrained by matter. The other one was proposed by George Stokes in 1845, in which the aether is completely entrained within or in the vicinity of matter.

While Fresnel's almost stationary theory was apparently confirmed by the Fizeau experiment (1851), Stokes' theory was apparently confirmed by the Michelson–Morley experiment (1881, 1887). This contradictory situation was resolved by the works of Hendrik Antoon Lorentz (1895, 1904) whose Lorentz ether theory banished any form of aether dragging, and finally with the work of Albert Einstein (1905) whose theory of special relativity doesn't contain the aether as a mechanical medium at all.

Apparent motion

Apparent motion may refer to:

In astronomy:

Apparent motion, the appearance that objects in the night sky move against the typical direction of motion.

Aberration of light, an apparent shift in position of celestial objects due to the finite speed of light and the motion of Earth in its orbit around the Sun.

Diurnal motion, the apparent motion of objects in the sky due to the Earth's rotation on its axis.

Parallax, the apparent motion of objects due to the changing angle of observation of an observer on Earth revolving around the Sun.In perceptual illusions:

Beta movement, an illusion of movement where two or more still images are combined by the brain into surmised motion.

Illusory motion, the appearance of movement in a static image.

Phi phenomenon, an illusion of movement created when two or more adjacent lights blink on and off in succession.

Stroboscopic effect, a phenomenon that occurs when continuous motion is represented by a series of short or instantaneous samples.

Wagon-wheel effect, temporal aliasing effect in which a spoked wheel appears to rotate differently from its true rotation.Other uses:

Optical flow, a term used in computer science for the apparent motion of objects in a scene caused by the relative motion between an observer and the scene

The motion of objects observed from a non-inertial reference frame


Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of the Solar System and our galaxy, the Milky Way.

Gamma Draconis

Gamma Draconis (γ Draconis, abbreviated Gamma Dra, γ Dra), also named Eltanin, is a star in the northern constellation of Draco. Contrary to its gamma-designation (historically third-ranked), it is the brightest star in Draco at magnitude 2.4, outshining Rastaban (Beta Draconis) by nearly half a magnitude. In 1.5 million years, Eltanin will pass within 28 light years of Earth. For a period, if its current absolute magnitude does not change, it will be the brightest star in the night sky, nearly as bright as Sirius is at present.It is by far the brightest star having a zenith above a point near London (and thus Leipzig, Düsseldorf, Astana and Calgary) which occurs near midnight (24:00h) at London which led to its vaunting in these places as the "zenith star" or "London zenith star". From other locations it has a nearby bright, well-known star in Lyra in night sky; finding Vega, Eltanin is the red star just north-northwest of it.

Eltanin is at a distance of 154.3 light-years (47.3 parsecs) from the Sun, as determined by parallax measurements from the Hipparcos astrometry satellite. In 1728, while unsuccessfully attempting to measure the parallax of this star, James Bradley discovered the aberration of light resulting from the movement of the Earth. Bradley's discovery confirmed Copernicus' theory that the Earth revolved around the Sun.

Hammar experiment

The Hammar experiment was an experiment designed and conducted by Gustaf Wilhelm Hammar (1935) to test the aether drag hypothesis. Its negative result refuted some specific aether drag models, and confirmed special relativity.

James Bradley

James Bradley FRS (March 1693 – 13 July 1762) was an English astronomer and priest who served as Astronomer Royal from 1742, succeeding Edmond Halley. He is best known for two fundamental discoveries in astronomy, the aberration of light (1725–1728), and the nutation of the Earth's axis (1728–1748). These discoveries were called "the most brilliant and useful of the century" by Jean Baptiste Joseph Delambre, historian of astronomy, mathematical astronomer and director of the Paris Observatory, in his history of astronomy in the 18th century (1821), because "It is to these two discoveries by Bradley that we owe the exactness of modern astronomy. .... This double service assures to their discoverer the most distinguished place (after Hipparchus and Kepler) above the greatest astronomers of all ages and all countries."

Light-time correction

Light-time correction is a displacement in the apparent position of a celestial object from its true position (or geometric position) caused by the object's motion during the time it takes its light to reach an observer.

Light-time correction occurs in principle during the observation of any moving object, because the speed of light is finite. The magnitude and direction of the displacement in position depends upon the distance of the object from the observer and the motion of the object, and is measured at the instant at which the object's light reaches the observer. It is independent of the motion of the observer. It should be contrasted with the aberration of light, which depends upon the instantaneous velocity of the observer at the time of observation, and is independent of the motion or distance of the object.

Light-time correction can be applied to any object whose distance and motion are known. In particular, it is usually necessary to apply it to the motion of a planet or other solar system object. For this reason, the combined displacement of the apparent position due to the effects of light-time correction and aberration is known as planetary aberration. By convention, light-time correction is not applied to the positions of stars, because their motion and distance may not be known accurately.

Luminiferous aether

Luminiferous aether or ether ("luminiferous", meaning "light-bearing"), was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space, something that waves should not be able to do. The assumption of a spatial plenum of luminiferous aether, rather than a spatial vacuum, provided the theoretical medium that was required by wave theories of light.

The aether hypothesis was the topic of considerable debate throughout its history, as it required the existence of an invisible and infinite material with no interaction with physical objects. As the nature of light was explored, especially in the 19th century, the physical qualities required of an aether became increasingly contradictory. By the late 1800s, the existence of the aether was being questioned, although there was no physical theory to replace it.

The negative outcome of the Michelson–Morley experiment (1887) suggested that the aether did not exist, a finding that was confirmed in subsequent experiments through the 1920s. This led to considerable theoretical work to explain the propagation of light without an aether. A major breakthrough was the theory of relativity, which could explain why the experiment failed to see aether, but was more broadly interpreted to suggest that it was not needed. The Michelson-Morley experiment, along with the blackbody radiator and photoelectric effect, was a key experiment in the development of modern physics, which includes both relativity and quantum theory, the latter of which explains the wave-like nature of light.

Maurice Loewy

Maurice (Moritz) Loewy (15 April 1833 – 15 October 1907) was a French astronomer.

Loewy was born in Vienna. Loewy's Jewish parents moved to Vienna in 1841 to escape the antisemitism of their home town. Loewy became an assistant at the Vienna Observatory, working on celestial mechanics. However, the institutions of Austria-Hungary did not permit a Jew to advance to a senior position without renouncing his faith and embracing Catholicism. The director of the observatory Karl L. Littrow was a correspondent of Urbain Le Verrier, director of the Paris Observatory and he secured a position there for Loewy in 1860. After going to France, Loewy become a naturalised French citizen.

He worked on the orbits of asteroids and comets and on the measurement of longitude, improving the accuracy of the Connaissance des Temps. He also worked on optics and the elimination of the aberration of light.

He was elected a member of the Bureau des Longitudes in 1872 and of the Académie des Sciences in 1873.

Loewy became director of the Paris Observatory in 1896, reorganising the institution and establishing a department of physical astronomy. He further spent a decade working with Pierre Puiseux on an atlas of the Moon composed of 10,000 photographs, L’Atlas photographique de la Lune (1910), the definitive basis for lunar geography for over half a century. The crater Loewy on the Moon is named after him and asteroid 253 Mathilde is believed to be named after his wife.

He died in Paris at a government meeting of a sudden and unanticipated cardiac arrest.

Moving magnet and conductor problem

The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant velocity, v, with respect to a magnet is calculated in the frame of reference of the magnet and in the frame of reference of the conductor. The observable quantity in the experiment, the current, is the same in either case, in accordance with the basic principle of relativity, which states: "Only relative motion is observable; there is no absolute standard of rest". However, according to Maxwell's equations, the charges in the conductor experience a magnetic force in the frame of the magnet and an electric force in the frame of the conductor. The same phenomenon would seem to have two different descriptions depending on the frame of reference of the observer.

This problem, along with the Fizeau experiment, the aberration of light, and more indirectly the negative aether drift tests such as the Michelson–Morley experiment, formed the basis of Einstein's development of the theory of relativity.

Pierre Puiseux

Pierre Henri Puiseux (French: [pɥizø]; July 20, 1855 – September 28, 1928) was a French astronomer.

Born in Paris, son of Victor Puiseux, he was educated at the École Normale Supérieure before starting work as an astronomer at the Paris Observatory in 1885.

He worked on the aberration of light, asteroids, lunar dynamics and, in collaboration with Maurice Loewy, the ill-fated Carte du Ciel project. Puiseux created a photographic atlas of the Moon based on 6000 photographs taken by him and Loewy. In 1892 he was awarded the Valz Prize, and in 1896 was he awarded the Lalande Prize, both from the French Academy of Sciences, which he would later become a member of in 1912.

In 1900, Puiseux received the Prix Jules Janssen, the highest award of the Société astronomique de France (the French astronomical society). He became the Society's president from 1911-1913.The crater Puiseux on the Moon is named after him.

Relativistic Doppler effect

The relativistic Doppler effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect), when taking into account effects described by the special theory of relativity.

The relativistic Doppler effect is different from the non-relativistic Doppler effect as the equations include the time dilation effect of special relativity and do not involve the medium of propagation as a reference point. They describe the total difference in observed frequencies and possess the required Lorentz symmetry.

Astronomers know of three sources of redshift/blueshift: Doppler shifts; gravitational redshifts (due to light exiting a gravitational field); and cosmological expansion (where space itself stretches). This article concerns itself only with Doppler shifts.

Relativistic aberration

Relativistic aberration is the relativistic version of aberration of light, including relativistic corrections that become significant for observers who move with velocities close to the speed of light. It is described by Einstein's special theory of relativity.

Suppose, in the reference frame of the observer, the source is moving with speed at an angle relative to the vector from the observer to the source at the time when the light is emitted. Then the following formula, which was derived by Einstein in 1905 from the Lorentz transformation, describes the aberration of the light source, , measured by the observer:

In this circumstance, the rays of light from the source which reach the observer are tilted towards the direction of the source's motion (relative to the observer). It is as if light emitted by a moving object is concentrated conically, towards its direction of motion; an effect called relativistic beaming. Also, light received by a moving object (e.g. the view from a very fast spacecraft) also appears concentrated towards its direction of motion.

One consequence of this is that a forward observer should normally be expected to intercept a greater proportion of the object's light than a rearward one; this concentration of light in the object's forward direction is referred to as the "searchlight effect" (or headlight effect).

Samuel Molyneux

Samuel Molyneux FRS (16 July 1689 – 13 April 1728) was an amateur astronomer and politician who sat in the British House of Commons between 1715 and 1728 and in the Irish House of Commons from 1727 to 1728. His work with James Bradley attempting to measure stellar parallax led to the discovery of the aberration of light. The aberration was the first definite evidence that the earth moved and that Copernicus and Kepler were correct. In addition to his astronomical works, Molyneux wrote about the natural history and other features of Ireland. He died in suspicious circumstances.

Special relativity

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

The laws of physics are invariant (i.e., identical) in all inertial systems (i.e., non-accelerating frames of reference).

The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.It was originally proposed by Albert Einstein in a paper published 26 September 1905 titled "On the Electrodynamics of Moving Bodies". The inconsistency of Newtonian mechanics with Maxwell's equations of electromagnetism and the lack of experimental confirmation for a hypothesized luminiferous aether led to the development of special relativity, which corrects mechanics to handle situations involving motions at a significant fraction of the speed of light (known as relativistic velocities). As of today, special relativity is the most accurate model of motion at any speed when gravitational effects are negligible. Even so, the Newtonian mechanics model is still useful as an approximation at small velocities relative to the speed of light, due to its simplicity and high accuracy within its scope.

Special relativity implies a wide range of consequences, which have been experimentally verified, including length contraction, time dilation, relativistic mass, mass–energy equivalence, a universal speed limit and relativity of simultaneity. It has replaced the conventional notion of an absolute universal time with the notion of a time that is dependent on reference frame and spatial position. Rather than an invariant time interval between two events, there is an invariant spacetime interval. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of mass and energy, as expressed in the mass–energy equivalence formula E = mc2, where c is the speed of light in a vacuum.A defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the Lorentz transformations. Time and space cannot be defined separately from each other. Rather, space and time are interwoven into a single continuum known as spacetime. Events that occur at the same time for one observer can occur at different times for another.

Not until Einstein developed general relativity, introducing a curved spacetime to incorporate gravity, was the phrase "special relativity" employed. A translation that has often been used is "restricted relativity"; "special" really means "special case".The theory is "special" in that it only applies in the special case where the spacetime is flat, i.e., the curvature of spacetime, described by the energy-momentum tensor and causing gravity, is negligible. In order to include gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some outdated descriptions, is capable of handling accelerations as well as accelerated frames of reference.As Galilean relativity is now considered an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation of general relativity that is valid for weak gravitational fields, i.e. at a sufficiently small scale (for tidal forces) and in conditions of free fall. Whereas general relativity incorporates noneuclidean geometry in order to represent gravitational effects as the geometric curvature of spacetime, special relativity is restricted to the flat spacetime known as Minkowski space. As long as the universe can be modeled as a pseudo-Riemannian manifold, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in this curved spacetime.

Galileo Galilei had already postulated that there is no absolute and well-defined state of rest (no privileged reference frames), a principle now called Galileo's principle of relativity. Einstein extended this principle so that it accounted for the constant speed of light, a phenomenon that had been recently observed in the Michelson–Morley experiment. He also postulated that it holds for all the laws of physics, including both the laws of mechanics and of electrodynamics.

Velocity-addition formula

In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as Thomas precession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost.

Standard applications of velocity-addition formulas include the Doppler shift, Doppler navigation, the aberration of light, and the dragging of light in moving water observed in the 1851 Fizeau experiment.The notation employs u as velocity of a body within a Lorentz frame S, and v as velocity of a second frame S′, as measured in S, and u′ as the transformed velocity of the body within the second frame.

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