Tuesday, September 25, 2007

From the Ashes of the First Stars

Above, an artist's impression shows a primordial quasar as it might have been, surrounded by sheets of gas, dust, stars and early star clusters. Exacting observations of three distant quasars now indicate emission of very specific colors of the element iron. These Hubble Space Telescope observations, which bolster recent results from the WMAP mission, indicate that a whole complete cycle of stars was born, created this iron, and died within the first few hundred million years of the universe.

Young Stars Hatching in Orion

The latest image released from the Spitzer Space Telescope shows infant stars “hatching” in the head of Orion. Astronomers think that a supernova 3 million years ago sent shockwaves through the region, collapsing clouds of gas and dust, and beginning a new generation of star formation.

The region imaged by Spitzer is called Barnard 30, located about 1,300 light-years from Earth in the constellation of Orion. More specifically, it’s located right beside the star considered to be Orion’s head, Lambda Orionis.

Since the region is shrouded in dark clouds of gas and dust that obscure visible light images, this was an ideal target for Spitzer, which can peer right through them in the infrared spectrum. The tints of orange-red glow are dust particles warmed by the newly forming stars. The reddish-pink dots are the young stars themselves, embedded in the clouds of gas and dust.

Crater for moon settlement

Although ESA’s SMART-1 was smashed into the Moon in 2006, it had the opportunity to gather a tremendous amount of science. Its view of this crater in particular has given ESA scientists the feeling that they might be looking at the perfect spot for a future permanent base on the Moon.

Crater Plaskett sits very close to the Moon’s north pole. This means it’s bathed in eternal sunlight. This would provide plenty of solar energy for future explorers, and creates a predictable temperature - it’s only hot, not hot and cold. Nearby craters bathed in eternal darkness might contain large stores of water ice that could be used for air, fuel and drinking water.



Crater Plaskett might provide a good first step for exploration of the Solar System. It’s close enough that astronauts would still be able to see the Earth. Help could arrive within days, if necessary, and communications would be almost instantaneous. But it’s remote enough to help mission planners understand what would be involved for future, longer duration missions on the Moon, and eventually to Mars.

SMART-1 ended its mission on September 3, 2006, when it ran out of fuel and crashed into the lunar surface. Scientists will be studying its data and images for years.

The Brightest Supernova Ever

The brightest stellar explosion ever recorded may be a long-sought new type of supernova, according to observations by NASA's Chandra X-ray Observatory and ground-based optical telescopes. This discovery indicates that violent explosions of extremely massive stars were relatively common in the early universe, and that a similar explosion may be ready to go off in our own galaxy.

"This was a truly monstrous explosion, a hundred times more energetic than a typical supernova," said Nathan Smith of the University of California at Berkeley, who led a team of astronomers from California and the University of Texas in Austin. "That means the star that exploded might have been as massive as a star can get, about 150 times that of our sun. We've never seen that before."
Astronomers think many of the first stars in the Universe were this massive, and this new supernova may thus provide a rare glimpse of how those first generation stars died. It is unprecedented, however, to find such a massive star and witness its death. The discovery of the supernova, known as SN 2006gy, provides evidence that the death of such massive stars is fundamentally different from theoretical predictions.

"Of all exploding stars ever observed, this was the king," said Alex Filippenko, leader of the ground-based observations at the Lick Observatory at Mt. Hamilton, Calif., and the Keck Observatory in Mauna Kea, Hawaii. "We were astonished to see how bright it got, and how long it lasted."

The Chandra observation allowed the team to rule out the most likely alternative explanation for the supernova: that a white dwarf star with a mass only slightly higher than the sun exploded into a dense, hydrogen-rich environment. In that event, SN 2006gy should have been 1,000 times brighter in X-rays than what Chandra detected.

Monday, September 17, 2007

Easy explanation of relativity theory

In the late 19th century scientists attempted to measure the absolute velocity of the earth using the equations of Maxwell and Galileo. Maxwell's equations gave the velocity of the speed of light, and Galileo's gave the way to measure differences between moving and stationery systems. The following is an example of how one would use these to find a velocity:

Imagine a spaceship that could move at a very high speed. Light from behind was shining past this spaceship. If a measurement of the speed of the light gave a result of 2x108m/s, then one can expect the speed of the ship to be 1x108m/s - the difference between the known speed of light (3x108m/s) and the speed measured from the spaceship.

This model could not be used until 1887, because the measuring apparatus was too imprecise. However, when this idea was finally used to try and determine the velocity of the earth, the experiment produced a remarkable result - the earth had zero velocity. Naturally scientists at the time were completely baffled by this and tried to discover where and how the laws of physics had failed.

Several scientists attempted to explain this anomaly before Einstein solved the problem with the Special Theory of Relativity - among them were Henri Poincare who suggested that it was impossible to determine an absolute velocity; H.A. Lorentz who came up with the transformation for motion; Michelson and Morley whose experiment it was that failed and started everyone thinking.

It was left to Einstein to solve the problem with the publication of the Special Theory of Relativity in 1905. He gave up the idea of an absolute velocity, and abolished the idea of the "ether", the mysterious substance through which scientists though light travelled.
Albert Einstein is the most well known physicist of the 20th century. Most famous for his Theory of Relativity, Einstein is also ranked high amongst the scientists responsible for the emergence of quantum mechanics for his proof of light traveling as a particle.

Albert Einstein was born on March 14 1879 to a middle class Jewish family in Ulm, Germany. In 1886 he began his school career in Munich. He disliked school because of the mindless drilling that was involved and he preferred to study at home where he gained an interest in mathematics and science. He began studying Calculus at age twelve at the Luitpold gymnasium. It was at about this time that his studies came into conflict with his deep religious feelings. His realisation that the Bible could not be literally true created his lifelong distrust of authority. He was granted Swiss citizenship a year after graduating from the Polytechnic Institute in Zurich.

He avoided compulsory military service thanks to his flat feet and varicose veins, but he was denied university assistantship. He then began moving around from post to post as a temporary teacher. Through a university contact he eventually gained a permanent job at the Swiss Patent Office as a technical expert, third class. In 1906, four years later, he was promoted to technical expert second class. During this time he wrote a fair amount of theoretical physics literature on a wide range of subjects. Many of these papers, written during his free time, were published and one thesis on a new determination of molecular dimensions earned him a doctorate from the University of Zurich.

During that same year of 1905 Einstein wrote two papers that turned the science world upside down. The first, on the photoelectric effect, contradicted previous perceptions of electromagnetic energy based on Maxwell's equations, and helped establish the nascent science of quantum mechanics. The second linked important parts of mechanics and Maxwell's electrodynamics to form The Special Theory of Relativity. The most important and famous part of this theory was his equation of energy and mass, E=mc2 , which was an undisputed display of pure genius.

He continued to work at the patent office until 1909 in which time he had extended the Special Theory of Relativity to include phenomena involving acceleration. He made significant contributions to the Quantum Theory and in 1908 became a lecturer at the University of Bern after submitting a further thesis for the constitution of radiation. In 1909 he left the patent office and his lectureship at Bern for the University of Zurich, where he was a professor for 2 years before being appointed a full professor at the Karl-Ferdinand University in Prague. By this time Einstein, at age 32, was recognised internationally as a leading scientist and physicist. A year later he began his work on the General Theory of Relativity. He moved to Zurich that same year to take up a chair at the Eidgenssische Technische Hochschule.

Late in 1915 he published the definitive version of the General Theory of Relativity. In 1919 British eclipse expeditions confirmed predictions derived from the General Theory of Relativity and Einstein was idolised by the press the world over.

In 1921 Einstein visited the U.S.A for the first time to raise funds for the planned Hebrew University of Jerusalem. He did lecture a few times on relativity and he received the Barnard Medal. That same year he was awarded the Nobel Prize for his work on the photoelectric effect in 1905, although he was not present for the award.For the next 6 years Einstein travelled around the world, receiving the Copley Medal of the Royal Society in 1925 and the Gold Medal of the Royal Astronomical Society in 1926. His schedule proved too hectic, for in 1928 Einstein experienced a physical collapse due to overwork. Although he did recover, he had to take things easy for the next two years.

He resumed his international visits in 1930 and in December of 1932, while he was in the U.S.A. the Nazis came to power, seizing his property after he had revoked his citizenship. He was granted permanent residence in America in 1935. At Princeton he resumed his quest to unify electromagnetic and gravitational phenomena in a theory he called the Unified Field Theory. He failed despite devoting the last 25 years of his life to this theory. In 1940 he was granted American citizenship and he made many contributions to world peace, he himself being a pacifist. By 1949 he was unwell and he began preparing for death by drawing up a will. He was offered the Presidency of Israel following the death of its first president in 1952 and, although it was difficult for him to do so, he declined the offer.

A week before his death Einstein signed his last letter. It was a letter to Bertrand Russell in which he agreed that his name should be placed on a manifesto urging all nations to give up nuclear weapons. It is fitting that one of his last acts was to argue, as he had done all his life, for world peace. He died peacefully on April 18 1955 at the age of 76.
Albert Einstein is the most well known physicist of the 20th century. Most famous for his Theory of Relativity, Einstein is also ranked high amongst the scientists responsible for the emergence of quantum mechanics for his proof of light traveling as a particle.

Albert Einstein was born on March 14 1879 to a middle class Jewish family in Ulm, Germany. In 1886 he began his school career in Munich. He disliked school because of the mindless drilling that was involved and he preferred to study at home where he gained an interest in mathematics and science. He began studying Calculus at age twelve at the Luitpold gymnasium. It was at about this time that his studies came into conflict with his deep religious feelings. His realisation that the Bible could not be literally true created his lifelong distrust of authority. He was granted Swiss citizenship a year after graduating from the Polytechnic Institute in Zurich.

He avoided compulsory military service thanks to his flat feet and varicose veins, but he was denied university assistantship. He then began moving around from post to post as a temporary teacher. Through a university contact he eventually gained a permanent job at the Swiss Patent Office as a technical expert, third class. In 1906, four years later, he was promoted to technical expert second class. During this time he wrote a fair amount of theoretical physics literature on a wide range of subjects. Many of these papers, written during his free time, were published and one thesis on a new determination of molecular dimensions earned him a doctorate from the University of Zurich.

During that same year of 1905 Einstein wrote two papers that turned the science world upside down. The first, on the photoelectric effect, contradicted previous perceptions of electromagnetic energy based on Maxwell's equations, and helped establish the nascent science of quantum mechanics. The second linked important parts of mechanics and Maxwell's electrodynamics to form The Special Theory of Relativity. The most important and famous part of this theory was his equation of energy and mass, E=mc2 , which was an undisputed display of pure genius.

He continued to work at the patent office until 1909 in which time he had extended the Special Theory of Relativity to include phenomena involving acceleration. He made significant contributions to the Quantum Theory and in 1908 became a lecturer at the University of Bern after submitting a further thesis for the constitution of radiation. In 1909 he left the patent office and his lectureship at Bern for the University of Zurich, where he was a professor for 2 years before being appointed a full professor at the Karl-Ferdinand University in Prague. By this time Einstein, at age 32, was recognised internationally as a leading scientist and physicist. A year later he began his work on the General Theory of Relativity. He moved to Zurich that same year to take up a chair at the Eidgenssische Technische Hochschule.

Late in 1915 he published the definitive version of the General Theory of Relativity. In 1919 British eclipse expeditions confirmed predictions derived from the General Theory of Relativity and Einstein was idolised by the press the world over.

In 1921 Einstein visited the U.S.A for the first time to raise funds for the planned Hebrew University of Jerusalem. He did lecture a few times on relativity and he received the Barnard Medal. That same year he was awarded the Nobel Prize for his work on the photoelectric effect in 1905, although he was not present for the award.For the next 6 years Einstein travelled around the world, receiving the Copley Medal of the Royal Society in 1925 and the Gold Medal of the Royal Astronomical Society in 1926. His schedule proved too hectic, for in 1928 Einstein experienced a physical collapse due to overwork. Although he did recover, he had to take things easy for the next two years.

He resumed his international visits in 1930 and in December of 1932, while he was in the U.S.A. the Nazis came to power, seizing his property after he had revoked his citizenship. He was granted permanent residence in America in 1935. At Princeton he resumed his quest to unify electromagnetic and gravitational phenomena in a theory he called the Unified Field Theory. He failed despite devoting the last 25 years of his life to this theory. In 1940 he was granted American citizenship and he made many contributions to world peace, he himself being a pacifist. By 1949 he was unwell and he began preparing for death by drawing up a will. He was offered the Presidency of Israel following the death of its first president in 1952 and, although it was difficult for him to do so, he declined the offer.

A week before his death Einstein signed his last letter. It was a letter to Bertrand Russell in which he agreed that his name should be placed on a manifesto urging all nations to give up nuclear weapons. It is fitting that one of his last acts was to argue, as he had done all his life, for world peace. He died peacefully on April 18 1955 at the age of 76.Symmetry in physics is generally defined as the ability of something to remain the same after undergoing a certain operation. A sphere for example has total symmetry because it looks exactly the same after being turned in any way. A cylinder on the other hand only has left right symmetry because it only remains the same if the rotation is around the vertical axis.

Not only does the object itself have to be considered, but any outside influences as well. For example if we have a machine that relies on gravity or oxygen then moving it to another place does not necessarily mean that symmetry will hold, so the operation, in this case displacement, would have to be performed on all the components relied upon by that machine to function. Physical phenomena remain unchanged (therefore their laws remain unchanged) after undergoing operations such as:

Displacement in space and time
Rotation around a fixed axis
Constant velocity in a straight line
Reflection in space
Reversal of time
Displacement in space is a seemingly obvious case as is displacement in time. Velocity in a straight line means that if we have an apparatus in a moving vehicle it would work in exactly the same way as it would if it wasn't moving provided the velocity and direction do not change. Reflection in space means that if we had two objects one looking exactly like the others mirror image, they would work exactly the same. It cannot as yet be proven, but it is believed that the physical laws hold true under the reversal of time.

Two operations that seem to conform to symmetry but do not are a change of scale and constant rotation at a fixed angular velocity. Symmetry does not hold under a change of scale because larger things deteriorate faster than smaller things. For example if we had a small bridge over a small space it would last longer than a larger bridge over a larger space and if gravity was increased according to the scale as well then the larger bridge would deteriorate even faster. Therefore increasing something in scale does not mean it will remain the same. An object rotating around a fixed point at a constant angular velocity will experience centrifugal forces. These forces are not around when that object is still, thus the object would have different forces on it and symmetry would not hold.

Constant velocity in a straight line is what Special Relativity is all about. However, the idea that constant velocity in a straight line is symmetrical did not come from Einstein, but was stated by Newton in one of his corollaries to the laws of motion. He stated, "the motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line". This means that if a spaceship was moving uniformly forward in a straight line, any experiments performed and any phenomena measured will give the same results as if the spaceship were not moving at all. This is why the experiment to determine the velocity of the earth gave a result of zero - zero is the result it would have given if the earth were not moving at all, so zero is the result it must give when moving uniformly in a straight line. It is when this principle is applied together with the principle that the speed of light also remains the same under all conditions that the strange consequences of relativity become apparent.

The principles of Symmetry are very nice, but we need some way to make them work. We do this by using mathematical devices known as transformations.

What is a transformation? A transformation is a formula which takes co-ordinates in one system, and gives us their corresponding co-ordinates in another. For example, there is a transformation which will give us the co-ordinates of a system in a system whose origin has been rotated relative to ours. Transformations are a vital part of physics, especially as they help us ensure that results are consistent - if we apply the standard transformations, our laws should come out the same before and after.

It was this seemingly simple problem which caused so many headaches at the end of the last century. It appeared as if James Clerk Maxwell's equations governing the speed of light did not obey these transformations - thus violating the principle of relativity. It was Lorentz who first suggested that Maxwell's laws were correct, and Newton's needed changing, and he did so by introducing his Lorentz transformations, which are at the heart of many Relativistic phenomena (these formulas are quite complex, and are given in the Advanced section). Einstein took this idea, and so first derived the Lorentz transformations and introduce the ground shaking ideas of Relativity.

Theory of every thing

Flucidity is a new way of thinking. It is what scientists call a "theory of everything" except that unlike any grand theory it can actually be applied to everything, not just physics. It is also easy enough for anyone to understand and can even be used for the very simple as well as the very complex.

Flucidity is a completely ridiculous idea until you actually begin to use it. Its best feature is that it can be used by anyone, in minutes.

You can think of Flucidity as a language. We use language all the time to do everything from solving problems to developing relationships to making our lives better. We also have the language of life, made possible by just four letters of DNA. Flucidity is a language of languages, expressed in four simple "letters" called elements. Knowing how it works will enable you to achieve results limited only by your imagination

History topics

History topic: Special relativity

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The classical laws of physics were formulated by Newton in the Principia in 1687. According to this theory the motion of a particle has to be described relative to an inertial frame in which the particle, not subjected to external forces, will move at a constant velocity in a straight line. Two inertial frames are related in that they move in a fixed direction at a constant speed with respect to each other. Time in the frames differs by a constant and all times can be described relative to an absolute time. This 17th Century theory was not challenged until the 19th Century when electric and magnetic phenomena were studied theoretically.

It had long been known that sound required a medium to travel through and it was quite natural to postulate a medium for the transmission of light. Such a medium was called the ether and many 19th Century scientists postulated an ether with various properties. Cauchy, Stokes, Thomson and Planck all postulated ethers with differing properties and by the end of the 19th Century light, heat, electricity and magnetism all had their respective ethers.

A knowledge that the electromagnetic field was spread with a velocity essentially the same as the speed of light caused Maxwell to postulate that light itself was an electromagnetic phenomenon. Maxwell wrote an article on Ether for the 1878 edition of Encyclopaedia Britannica. He proposed the existence of a single ether and the article tells of a failed attempt by Maxwell to measure the effect of the ether drag on the earth's motion. He also proposed an astronomical determination of the ether drag by measuring the velocity of light using Jupiter's moons at different positions relative to the earth.

Prompted by Maxwell's ideas, Michelson began his own terrestrial experiments and in 1881 he reported

The result of the hypothesis of a stationary ether is shown to be incorrect, and the necessary conclusion follows that the hypothesis is erroneous.


Lorentz wrote a paper in 1886 where he criticised Michelson's experiment and really was not worried by the experimental result which he dismissed being doubtful of its accuracy. Michelson was persuaded by Thomson and others to repeat the experiment and he did so with Morley, again reporting that no effect had been found in 1887. It appeared that the velocity of light was independent of the velocity of the observer. [Michelson and Morley were to refine their experiment and repeat it many times up to 1929.]

Also in 1887 Voigt first wrote down the transformations

x' = x - vt, y' = y/g, z' = z/g, t' = t - vx/c2

and showed that certain equations were invariant under these transformations. These transformations, with a different scale factor, are now known as the Lorentz equations and the group of Lorentz transformations gives the geometry of special relativity. All this was unknown to Voigt who was writing on the Doppler shift when he wrote down the transformations.

Voigt corresponded with Lorentz about the Michelson-Morley experiment in 1887 and 1888 but Lorentz does not seem to have learnt of the transformations at that stage. Lorentz however was now greatly worried by the new Michelson-Morley experiment of 1887.

In 1889 a short paper was published by the Irish physicist George FitzGerald in Science. The paper The ether and the earth's atmosphere takes up less than half a page and is non-technical. FitzGerald pointed out that the results of the Michelson-Morley experiment could be explained only if

... the length of material bodies changes, according as they are moving through the ether or across it, by an amount depending on the square of the ratio of their velocities to that of light.

Lorentz was unaware of FitzGerald's paper and in 1892 he proposed an almost identical contraction in a paper which now took the Michelson-Morley experiment very seriously. When it was pointed out to Lorentz in 1894 that FitzGerald had published a similar theory he wrote to FitzGerald who replied that he had sent an article to Science but I do not know if they ever published it . He was glad to know that Lorentz agreed with him for I have been rather laughed at for my view over here . Lorentz took every opportunity after this to acknowledge that FitzGerald had proposed the idea first. Only FitzGerald, who did not know if his paper had been published, believed that Lorentz had published first!

Larmor wrote an article in 1898 Ether and matter in which he wrote down the Lorentz transformations (still not written down by Lorentz) and showed that the FitzGerald-Lorentz contraction was a consequence.

Lorentz wrote down the transformations, now named after him, in a paper of 1899, being the third person to write them down. He, like Larmor, showed that the FitzGerald-Lorentz contraction was a consequence of the Lorentz transformations.

The most amazing article relating to special relativity to be published before 1900 was a paper of Poincaré La mesure du temps which appeared in 1898. In this paper Poincaré says

... we have no direct intuition about the equality of two time intervals.
The simultaneity of two events or the order of their succession, as well as the equality of two time intervals, must be defined in such a way that the statements of the natural laws be as simple as possible.


By 1900 the concept of the ether as a material substance was being questioned. Paul Drude wrote

The conception of an ether absolutely at rest is the most simple and the most natural - at least if the ether is conceived to be not a substance but merely space endowed with certain physical properties.

Poincaré, in his opening address to the Paris Congress in 1900, asked Does the ether really exist? In 1904 Poincaré came very close to the theory of special relativity in an address to the International Congress of Arts and Science in St Louis. He pointed out that observers in different frames will have clocks which will

... mark what on may call the local time. ... as demanded by the relativity principle the observer cannot know whether he is at rest or in absolute motion.

The year that special relativity finally came into existence was 1905. June of 1905 was a good month for papers on relativity, on the 5th June Poincaré communicated an important work Sur la dynamique de l'electron while Einstein's first paper on relativity was received on 30th June. Poincaré stated that It seems that this impossibility of demonstrating absolute motion is a general law of nature. After naming the Lorentz transformations after Lorentz, Poincaré shows that these transformations, together with the rotations, form a group.

Einstein's paper is remarkable for the different approach it takes. It is not presented as an attempt to explain experimental results, it is presented because of its beauty and simplicity. In the introduction Einstein says

... the 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.

Inertial frames are introduced which, by definition, are in uniform motion with respect to each other. The whole theory is based on two postulates:-

1. The laws of physics take the same form in all inertial frames.
2. In any inertial frame, the velocity of light c is the same whether the light is emitted by a body at rest or by a body in uniform motion.

Einstein now deduced the Lorentz transformations from his two postulates and, like Poincaré proves the group property. Then the FitzGerald-Lorentz contraction is deduced. Also in the paper Einstein mentions the clock paradox. Einstein called it a theorem that if two synchronous clocks C1 and C2 start at a point A and C2 leaves A moving along a closed curve to return to A then C2 will run slow compared with C1. He notes that no paradox results since C2 experiences acceleration while C1 does not.

In September 1905 Einstein published a short but important paper in which he proved the famous formula

E = mc2.

The first paper on special relativity, other than by Einstein, was written in 1908 by Planck. It was largely due to the fact that relativity was taken up by someone as important as Planck that it became so rapidly accepted. At the time Einstein wrote the 1905 paper he was still a technical expert third class at the Bern patent office. Also in 1908 Minkowski published an important paper on relativity, presenting the Maxwell-Lorentz equations in tensor form. He also showed that the Newtonian theory of gravitation was not consistent with relativity.

The main contributors to special relativity were undoubtedly Lorentz, Poincaré and, of course, the founder of the theory Einstein. It is therefore interesting to see their respective reactions to the final formulation of the theory. Einstein, although he spent many years thinking about how to formulate the theory, once he had found the two postulates they were immediately natural to him. Einstein was always reluctant to acknowledge that the steps which others were taking due to the Michelson-Morley experiment had any influence on his thinking.

Poincaré's reaction to Einstein's 1905 paper was rather strange. When Poincaré lectured in Göttingen in 1909 on relativity he did not mention Einstein at all. He presented relativity with three postulates, the third being the FitzGerald-Lorentz contraction. It is impossible to believe that someone as brilliant as Poincaré had failed to understand Einstein's paper. In fact Poincaré never wrote a paper on relativity in which he mentioned Einstein. Einstein himself behaved in a similar fashion and Poincaré is only mentioned once in Einstein's papers. Lorentz, however, was praised by both Einstein and Poincaré and often cited in their work.

Lorentz himself poses a puzzle. Although he clearly understood Einstein's papers, he did not ever seem to accept their conclusions. He gave a lecture in 1913 when he remarked how rapidly relativity had been accepted. He for one was less sure.

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.

Despite Lorentz's caution the special theory of relativity was quickly accepted. In 1912 Lorentz and Einstein were jointly proposed for a Nobel prize for their work on special relativity. The recommendation is by Wien, the 1911 winner, and states

... While Lorentz must be considered as the first to have found the mathematical content of the relativity principle, Einstein succeeded in reducing it to a simple principle. One should therefore assess the merits of both investigators as being comparable...


Einstein never received a Nobel prize for relativity. The committee was at first cautious and waited for experimental confirmation. By the time such confirmation was available Einstein had moved on to further momentous work.


Article by: J J O'Connor and E F Robertson


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February 1996
MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Special_relativity.html]

general relativity

Einstein's 1916 paper
on General Relativity




In 1916 Einstein expanded his Special Theory to include the effect of gravitation on the shape of space and the flow of time.

This theory, referred to as the General Theory of Relativity, proposed that matter causes space to curve.
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Embedding Diagrams
Picture a bowling ball on a stretched rubber sheet.


GIF Image (62K)
The large ball will cause a deformation in the sheet's surface. A baseball dropped onto the sheet will roll toward the bowling ball. Einstein theorized that smaller masses travel toward larger masses not because they are "attracted" by a mysterious force, but because the smaller objects travel through space that is warped by the larger object. Physicists illustrate this idea using embedding diagrams.

Contrary to appearances, an embedding diagram does not depict the three-dimensional "space" of our everyday experience. Rather it shows how a 2D slice through familiar 3D space is curved downwards when embedded in flattened hyperspace. We cannot fully envision this hyperspace; it contains seven dimensions, including one for time! Flattening it to 3D allows us to represent the curvature. Embedding diagrams can help us visualize the implications of Einstein's General Theory of Relativity.


The Flow of Spacetime
Another way of thinking of the curvature of spacetime was elegantly described by Hans von Baeyer. In a prize-winning essay he conceives of spacetime as an invisible stream flowing ever onward, bending in response to objects in it s path, carrying everything in the universe along its twists and turns.


This is a basic postulate of the Theory of General Relativity. It states that a uniform gravitational field (like that near the Earth) is equivalent to a uniform acceleration.

What this means, in effect, is that a person cannot tell the difference between (a) standing on the Earth, feeling the effects of gravity as a downward pull and (b) standing in a very smooth elevator that is accelerating upwards at just the right rate of exactly 32 feet per second squared.
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In both cases, a person would feel the same downward pull of gravity. Einstein asserted that these effects were actually the same. A far cry from Newton's view of gravity as a force acting at a distance!

Gravitational Time Dilation
Einstein's Special Theory of Relativity predicted that time does not flow at a fixed rate: moving clocks appear to tick more slowly relative to their stationary counterparts. But this effect only becomes really significant at very high velocities that app roach the speed of light.
When "generalized" to include gravitation, the equations of relativity predict that gravity, or the curvature of spacetime by matter, not only stretches or shrinks distances (depending on their direction with respect to the gravitational field) but also w ill appear to slow down or "dilate" the flow of time.

In most circumstances in the universe, such time dilation is miniscule, but it can become very significant when spacetime is curved by a massive object such as a black hole. For example, an observer far from a black hole would observe time passing extremely slowly for an astronaut falling through the hole's boundary. In fact, the distant observer would never see the hapless victim actually fall in. His or her time, as measured by the observer, would appear to stand still. The slowing of time near a very simple black hole has been simulated on supercomputers at NCSA and visualized in a computer-generated animation.


Grappling With Relativity
In the decade after its publication in 1916, Einstein's Theory of General Relativity led to a burst of experimental activity in which many of its predictions were vindicated. These predictions were encapsulated in a series of field equations that laid the foundation for all subsequent research into relativity and partly for modern cosmology as well.

The Math Behind Einstein's Vision
The mathematics behind the Einstein Field Equations not only presented a formidable challenge to solve, but also led to seemingly bizarre consequences, particularly those of black holes and gravitatio nal waves. At the time they were postulated, both were dismissed by many experts as mathematical aberrations. It remains to be seen whether either truly exist.
Rest assured that the next section will further illuminate your grasp of relativity -- without math overload!



Forward to The Einstein Field Equations
Return to What's So Special About Relativity?
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Einstein papers onrelativity

ON THE ELECTRODYNAMICS
OF MOVING BODIES
By A. Einstein
June 30, 1905
It is known that Maxwell's electrodynamics--as usually understood at the present time--when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise--assuming equality of relative motion in the two cases discussed--to electric currents of the same path and intensity as those produced by the electric forces in the former case.

Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the ``light medium,'' suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.1 We will raise this conjecture (the purport of which will hereafter be called the ``Principle of Relativity'') to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies. The introduction of a ``luminiferous ether'' will prove to be superfluous inasmuch as the view here to be developed will not require an ``absolutely stationary space'' provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.

The theory to be developed is based--like all electrodynamics--on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters.

I. KINEMATICAL PART
§ 1. Definition of Simultaneity
Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good.2 In order to render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the ``stationary system.''

If a material point is at rest relatively to this system of co-ordinates, its position can be defined relatively thereto by the employment of rigid standards of measurement and the methods of Euclidean geometry, and can be expressed in Cartesian co-ordinates.

If we wish to describe the motion of a material point, we give the values of its co-ordinates as functions of the time. Now we must bear carefully in mind that a mathematical description of this kind has no physical meaning unless we are quite clear as to what we understand by ``time.'' We have to take into account that all our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, ``That train arrives here at 7 o'clock,'' I mean something like this: ``The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.''3

It might appear possible to overcome all the difficulties attending the definition of ``time'' by substituting ``the position of the small hand of my watch'' for ``time.'' And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or--what comes to the same thing--to evaluate the times of events occurring at places remote from the watch.

We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this co-ordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought.

If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an ``A time'' and a ``B time.'' We have not defined a common ``time'' for A and B, for the latter cannot be defined at all unless we establish by definition that the ``time'' required by light to travel from A to B equals the ``time'' it requires to travel from B to A. Let a ray of light start at the ``A time'' from A towards B, let it at the ``B time'' be reflected at B in the direction of A, and arrive again at A at the ``A time'' .

In accordance with definition the two clocks synchronize if


We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:--

If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.
Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary clocks located at different places, and have evidently obtained a definition of ``simultaneous,'' or ``synchronous,'' and of ``time.'' The ``time'' of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event, this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock.

In agreement with experience we further assume the quantity


to be a universal constant--the velocity of light in empty space.

It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it ``the time of the stationary system.''

§ 2. On the Relativity of Lengths and Times
The following reflexions are based on the principle of relativity and on the principle of the constancy of the velocity of light. These two principles we define as follows:--

The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion.
Any ray of light moves in the ``stationary'' system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Hence

where time interval is to be taken in the sense of the definition in § 1.

Let there be given a stationary rigid rod; and let its length be l as measured by a measuring-rod which is also stationary. We now imagine the axis of the rod lying along the axis of x of the stationary system of co-ordinates, and that a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod. We now inquire as to the length of the moving rod, and imagine its length to be ascertained by the following two operations:--

(a)
The observer moves together with the given measuring-rod and the rod to be measured, and measures the length of the rod directly by superposing the measuring-rod, in just the same way as if all three were at rest.

(b)
By means of stationary clocks set up in the stationary system and synchronizing in accordance with § 1, the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring-rod already employed, which in this case is at rest, is also a length which may be designated ``the length of the rod.''
In accordance with the principle of relativity the length to be discovered by the operation (a)--we will call it ``the length of the rod in the moving system''--must be equal to the length l of the stationary rod.

The length to be discovered by the operation (b) we will call ``the length of the (moving) rod in the stationary system.'' This we shall determine on the basis of our two principles, and we shall find that it differs from l.

Current kinematics tacitly assumes that the lengths determined by these two operations are precisely equal, or in other words, that a moving rigid body at the epoch t may in geometrical respects be perfectly represented by the same body at rest in a definite position.

We imagine further that at the two ends A and B of the rod, clocks are placed which synchronize with the clocks of the stationary system, that is to say that their indications correspond at any instant to the ``time of the stationary system'' at the places where they happen to be. These clocks are therefore ``synchronous in the stationary system.''

We imagine further that with each clock there is a moving observer, and that these observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks. Let a ray of light depart from A at the time4 , let it be reflected at B at the time , and reach A again at the time . Taking into consideration the principle of the constancy of the velocity of light we find that


where denotes the length of the moving rod--measured in the stationary system. Observers moving with the moving rod would thus find that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous.

So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.

§ 3. Theory of the Transformation of Co-ordinates and Times from a Stationary System to another System in Uniform Motion of Translation Relatively to the Former
Let us in ``stationary'' space take two systems of co-ordinates, i.e. two systems, each of three rigid material lines, perpendicular to one another, and issuing from a point. Let the axes of X of the two systems coincide, and their axes of Y and Z respectively be parallel. Let each system be provided with a rigid measuring-rod and a number of clocks, and let the two measuring-rods, and likewise all the clocks of the two systems, be in all respects alike.

Now to the origin of one of the two systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other stationary system (K), and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring-rod, and the clocks. To any time of the stationary system K there then will correspond a definite position of the axes of the moving system, and from reasons of symmetry we are entitled to assume that the motion of k may be such that the axes of the moving system are at the time t (this ``t'' always denotes a time of the stationary system) parallel to the axes of the stationary system.

We now imagine space to be measured from the stationary system K by means of the stationary measuring-rod, and also from the moving system k by means of the measuring-rod moving with it; and that we thus obtain the co-ordinates x, y, z, and , , respectively. Further, let the time t of the stationary system be determined for all points thereof at which there are clocks by means of light signals in the manner indicated in § 1; similarly let the time of the moving system be determined for all points of the moving system at which there are clocks at rest relatively to that system by applying the method, given in § 1, of light signals between the points at which the latter clocks are located.

To any system of values x, y, z, t, which completely defines the place and time of an event in the stationary system, there belongs a system of values , , , , determining that event relatively to the system k, and our task is now to find the system of equations connecting these quantities.

In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time.

If we place x'=x-vt, it is clear that a point at rest in the system k must have a system of values x', y, z, independent of time. We first define as a function of x', y, z, and t. To do this we have to express in equations that is nothing else than the summary of the data of clocks at rest in system k, which have been synchronized according to the rule given in § 1.

From the origin of system k let a ray be emitted at the time along the X-axis to x', and at the time be reflected thence to the origin of the co-ordinates, arriving there at the time ; we then must have , or, by inserting the arguments of the function and applying the principle of the constancy of the velocity of light in the stationary system:--


Hence, if x' be chosen infinitesimally small,


or


It is to be noted that instead of the origin of the co-ordinates we might have chosen any other point for the point of origin of the ray, and the equation just obtained is therefore valid for all values of x', y, z.

An analogous consideration--applied to the axes of Y and Z--it being borne in mind that light is always propagated along these axes, when viewed from the stationary system, with the velocity gives us


Since is a linear function, it follows from these equations that


where a is a function at present unknown, and where for brevity it is assumed that at the origin of k, , when t=0.

With the help of this result we easily determine the quantities , , by expressing in equations that light (as required by the principle of the constancy of the velocity of light, in combination with the principle of relativity) is also propagated with velocity c when measured in the moving system. For a ray of light emitted at the time in the direction of the increasing



But the ray moves relatively to the initial point of k, when measured in the stationary system, with the velocity c-v, so that


If we insert this value of t in the equation for , we obtain


In an analogous manner we find, by considering rays moving along the two other axes, that


when


Thus


Substituting for x' its value, we obtain


where


and is an as yet unknown function of v. If no assumption whatever be made as to the initial position of the moving system and as to the zero point of , an additive constant is to be placed on the right side of each of these equations.

We now have to prove that any ray of light, measured in the moving system, is propagated with the velocity c, if, as we have assumed, this is the case in the stationary system; for we have not as yet furnished the proof that the principle of the constancy of the velocity of light is compatible with the principle of relativity.

At the time , when the origin of the co-ordinates is common to the two systems, let a spherical wave be emitted therefrom, and be propagated with the velocity c in system K. If (x, y, z) be a point just attained by this wave, then

x2+y2+z2=c2t2.
Transforming this equation with the aid of our equations of transformation we obtain after a simple calculation


The wave under consideration is therefore no less a spherical wave with velocity of propagation c when viewed in the moving system. This shows that our two fundamental principles are compatible.5

In the equations of transformation which have been developed there enters an unknown function of v, which we will now determine.

For this purpose we introduce a third system of co-ordinates , which relatively to the system k is in a state of parallel translatory motion parallel to the axis of ,*1 such that the origin of co-ordinates of system , moves with velocity -v on the axis of . At the time t=0 let all three origins coincide, and when t=x=y=z=0 let the time t' of the system be zero. We call the co-ordinates, measured in the system , x', y', z', and by a twofold application of our equations of transformation we obtain


Since the relations between x', y', z' and x, y, z do not contain the time t, the systems K and are at rest with respect to one another, and it is clear that the transformation from K to must be the identical transformation. Thus


We now inquire into the signification of . We give our attention to that part of the axis of Y of system k which lies between and . This part of the axis of Y is a rod moving perpendicularly to its axis with velocity v relatively to system K. Its ends possess in K the co-ordinates



and


The length of the rod measured in K is therefore ; and this gives us the meaning of the function . From reasons of symmetry it is now evident that the length of a given rod moving perpendicularly to its axis, measured in the stationary system, must depend only on the velocity and not on the direction and the sense of the motion. The length of the moving rod measured in the stationary system does not change, therefore, if v and -v are interchanged. Hence follows that , or


It follows from this relation and the one previously found that , so that the transformation equations which have been found become


where


§ 4. Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Moving Clocks
We envisage a rigid sphere6 of radius R, at rest relatively to the moving system k, and with its centre at the origin of co-ordinates of k. The equation of the surface of this sphere moving relatively to the system K with velocity v is


The equation of this surface expressed in x, y, z at the time t=0 is


A rigid body which, measured in a state of rest, has the form of a sphere, therefore has in a state of motion--viewed from the stationary system--the form of an ellipsoid of revolution with the axes


Thus, whereas the Y and Z dimensions of the sphere (and therefore of every rigid body of no matter what form) do not appear modified by the motion, the X dimension appears shortened in the ratio , i.e. the greater the value of v, the greater the shortening. For v=c all moving objects--viewed from the ``stationary'' system--shrivel up into plane figures.*2 For velocities greater than that of light our deliberations become meaningless; we shall, however, find in what follows, that the velocity of light in our theory plays the part, physically, of an infinitely great velocity.

It is clear that the same results hold good of bodies at rest in the ``stationary'' system, viewed from a system in uniform motion.

Further, we imagine one of the clocks which are qualified to mark the time t when at rest relatively to the stationary system, and the time when at rest relatively to the moving system, to be located at the origin of the co-ordinates of k, and so adjusted that it marks the time . What is the rate of this clock, when viewed from the stationary system?

Between the quantities x, t, and , which refer to the position of the clock, we have, evidently, x=vt and


Therefore,


whence it follows that the time marked by the clock (viewed in the stationary system) is slow by seconds per second, or--neglecting magnitudes of fourth and higher order--by .

From this there ensues the following peculiar consequence. If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by (up to magnitudes of fourth and higher order), t being the time occupied in the journey from A to B.

It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide.

If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the travelled clock on its arrival at A will be second slow. Thence we conclude that a balance-clock7 at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions.

§ 5. The Composition of Velocities
In the system k moving along the axis of X of the system K with velocity v, let a point move in accordance with the equations


where and denote constants.

Required: the motion of the point relatively to the system K. If with the help of the equations of transformation developed in § 3 we introduce the quantities x, y, z, t into the equations of motion of the point, we obtain


Thus the law of the parallelogram of velocities is valid according to our theory only to a first approximation. We set

*3
a is then to be looked upon as the angle between the velocities v and w. After a simple calculation we obtain*4


It is worthy of remark that v and w enter into the expression for the resultant velocity in a symmetrical manner. If w also has the direction of the axis of X, we get


It follows from this equation that from a composition of two velocities which are less than c, there always results a velocity less than c. For if we set , and being positive and less than c, then


It follows, further, that the velocity of light c cannot be altered by composition with a velocity less than that of light. For this case we obtain


We might also have obtained the formula for V, for the case when v and w have the same direction, by compounding two transformations in accordance with § 3. If in addition to the systems K and k figuring in § 3 we introduce still another system of co-ordinates k' moving parallel to k, its initial point moving on the axis of *5 with the velocity w, we obtain equations between the quantities x, y, z, t and the corresponding quantities of k', which differ from the equations found in § 3 only in that the place of ``v'' is taken by the quantity


from which we see that such parallel transformations--necessarily--form a group.

We have now deduced the requisite laws of the theory of kinematics corresponding to our two principles, and we proceed to show their application to electrodynamics.

II. ELECTRODYNAMICAL PART
§ 6. Transformation of the Maxwell-Hertz Equations for Empty Space. On the Nature of the Electromotive Forces Occurring in a Magnetic Field During Motion
Let the Maxwell-Hertz equations for empty space hold good for the stationary system K, so that we have


where (X, Y, Z) denotes the vector of the electric force, and (L, M, N) that of the magnetic force.

If we apply to these equations the transformation developed in § 3, by referring the electromagnetic processes to the system of co-ordinates there introduced, moving with the velocity v, we obtain the equations


where


Now the principle of relativity requires that if the Maxwell-Hertz equations for empty space hold good in system K, they also hold good in system k; that is to say that the vectors of the electric and the magnetic force--(, , ) and (, , )--of the moving system k, which are defined by their ponderomotive effects on electric or magnetic masses respectively, satisfy the following equations:--


Evidently the two systems of equations found for system k must express exactly the same thing, since both systems of equations are equivalent to the Maxwell-Hertz equations for system K. Since, further, the equations of the two systems agree, with the exception of the symbols for the vectors, it follows that the functions occurring in the systems of equations at corresponding places must agree, with the exception of a factor , which is common for all functions of the one system of equations, and is independent of and but depends upon v. Thus we have the relations


If we now form the reciprocal of this system of equations, firstly by solving the equations just obtained, and secondly by applying the equations to the inverse transformation (from k to K), which is characterized by the velocity -v, it follows, when we consider that the two systems of equations thus obtained must be identical, that . Further, from reasons of symmetry8 and therefore


and our equations assume the form


As to the interpretation of these equations we make the following remarks: Let a point charge of electricity have the magnitude ``one'' when measured in the stationary system K, i.e. let it when at rest in the stationary system exert a force of one dyne upon an equal quantity of electricity at a distance of one cm. By the principle of relativity this electric charge is also of the magnitude ``one'' when measured in the moving system. If this quantity of electricity is at rest relatively to the stationary system, then by definition the vector (X, Y, Z) is equal to the force acting upon it. If the quantity of electricity is at rest relatively to the moving system (at least at the relevant instant), then the force acting upon it, measured in the moving system, is equal to the vector (, , ). Consequently the first three equations above allow themselves to be clothed in words in the two following ways:--

If a unit electric point charge is in motion in an electromagnetic field, there acts upon it, in addition to the electric force, an ``electromotive force'' which, if we neglect the terms multiplied by the second and higher powers of v/c, is equal to the vector-product of the velocity of the charge and the magnetic force, divided by the velocity of light. (Old manner of expression.)
If a unit electric point charge is in motion in an electromagnetic field, the force acting upon it is equal to the electric force which is present at the locality of the charge, and which we ascertain by transformation of the field to a system of co-ordinates at rest relatively to the electrical charge. (New manner of expression.)
The analogy holds with ``magnetomotive forces.'' We see that electromotive force plays in the developed theory merely the part of an auxiliary concept, which owes its introduction to the circumstance that electric and magnetic forces do not exist independently of the state of motion of the system of co-ordinates.

Furthermore it is clear that the asymmetry mentioned in the introduction as arising when we consider the currents produced by the relative motion of a magnet and a conductor, now disappears. Moreover, questions as to the ``seat'' of electrodynamic electromotive forces (unipolar machines) now have no point.

§ 7. Theory of Doppler's Principle and of Aberration
In the system K, very far from the origin of co-ordinates, let there be a source of electrodynamic waves, which in a part of space containing the origin of co-ordinates may be represented to a sufficient degree of approximation by the equations


where


Here (, , ) and (, , ) are the vectors defining the amplitude of the wave-train, and l, m, n the direction-cosines of the wave-normals. We wish to know the constitution of these waves, when they are examined by an observer at rest in the moving system k.

Applying the equations of transformation found in § 6 for electric and magnetic forces, and those found in § 3 for the co-ordinates and the time, we obtain directly


where


From the equation for it follows that if an observer is moving with velocity v relatively to an infinitely distant source of light of frequency , in such a way that the connecting line ``source-observer'' makes the angle with the velocity of the observer referred to a system of co-ordinates which is at rest relatively to the source of light, the frequency of the light perceived by the observer is given by the equation


This is Doppler's principle for any velocities whatever. When the equation assumes the perspicuous form


We see that, in contrast with the customary view, when .

If we call the angle between the wave-normal (direction of the ray) in the moving system and the connecting line ``source-observer'' , the equation for *6 assumes the form


This equation expresses the law of aberration in its most general form. If , the equation becomes simply


We still have to find the amplitude of the waves, as it appears in the moving system. If we call the amplitude of the electric or magnetic force A or respectively, accordingly as it is measured in the stationary system or in the moving system, we obtain


which equation, if , simplifies into


It follows from these results that to an observer approaching a source of light with the velocity c, this source of light must appear of infinite intensity.

§ 8. Transformation of the Energy of Light Rays. Theory of the Pressure of Radiation Exerted on Perfect Reflectors
Since equals the energy of light per unit of volume, we have to regard , by the principle of relativity, as the energy of light in the moving system. Thus would be the ratio of the ``measured in motion'' to the ``measured at rest'' energy of a given light complex, if the volume of a light complex were the same, whether measured in K or in k. But this is not the case. If l, m, n are the direction-cosines of the wave-normals of the light in the stationary system, no energy passes through the surface elements of a spherical surface moving with the velocity of light:--


We may therefore say that this surface permanently encloses the same light complex. We inquire as to the quantity of energy enclosed by this surface, viewed in system k, that is, as to the energy of the light complex relatively to the system k.

The spherical surface--viewed in the moving system--is an ellipsoidal surface, the equation for which, at the time , is


If S is the volume of the sphere, and that of this ellipsoid, then by a simple calculation


Thus, if we call the light energy enclosed by this surface E when it is measured in the stationary system, and when measured in the moving system, we obtain


and this formula, when , simplifies into


It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law.

Now let the co-ordinate plane be a perfectly reflecting surface, at which the plane waves considered in § 7 are reflected. We seek for the pressure of light exerted on the reflecting surface, and for the direction, frequency, and intensity of the light after reflexion.

Let the incidental light be defined by the quantities A, , (referred to system K). Viewed from k the corresponding quantities are


For the reflected light, referring the process to system k, we obtain


Finally, by transforming back to the stationary system K, we obtain for the reflected light


The energy (measured in the stationary system) which is incident upon unit area of the mirror in unit time is evidently . The energy leaving the unit of surface of the mirror in the unit of time is . The difference of these two expressions is, by the principle of energy, the work done by the pressure of light in the unit of time. If we set down this work as equal to the product Pv, where P is the pressure of light, we obtain


In agreement with experiment and with other theories, we obtain to a first approximation


All problems in the optics of moving bodies can be solved by the method here employed. What is essential is, that the electric and magnetic force of the light which is influenced by a moving body, be transformed into a system of co-ordinates at rest relatively to the body. By this means all problems in the optics of moving bodies will be reduced to a series of problems in the optics of stationary bodies.

§ 9. Transformation of the Maxwell-Hertz Equations when Convection-Currents are Taken into Account
We start from the equations


where


denotes times the density of electricity, and (ux,uy,uz) the velocity-vector of the charge. If we imagine the electric charges to be invariably coupled to small rigid bodies (ions, electrons), these equations are the electromagnetic basis of the Lorentzian electrodynamics and optics of moving bodies.

Let these equations be valid in the system K, and transform them, with the assistance of the equations of transformation given in §§ 3 and 6, to the system k. We then obtain the equations


where


and


Since--as follows from the theorem of addition of velocities (§ 5)--the vector is nothing else than the velocity of the electric charge, measured in the system k, we have the proof that, on the basis of our kinematical principles, the electrodynamic foundation of Lorentz's theory of the electrodynamics of moving bodies is in agreement with the principle of relativity.

In addition I may briefly remark that the following important law may easily be deduced from the developed equations: If an electrically charged body is in motion anywhere in space without altering its charge when regarded from a system of co-ordinates moving with the body, its charge also remains--when regarded from the ``stationary'' system K--constant.

§ 10. Dynamics of the Slowly Accelerated Electron
Let there be in motion in an electromagnetic field an electrically charged particle (in the sequel called an ``electron''), for the law of motion of which we assume as follows:--

If the electron is at rest at a given epoch, the motion of the electron ensues in the next instant of time according to the equations


where x, y, z denote the co-ordinates of the electron, and m the mass of the electron, as long as its motion is slow.

Now, secondly, let the velocity of the electron at a given epoch be v. We seek the law of motion of the electron in the immediately ensuing instants of time.

Without affecting the general character of our considerations, we may and will assume that the electron, at the moment when we give it our attention, is at the origin of the co-ordinates, and moves with the velocity v along the axis of X of the system K. It is then clear that at the given moment (t=0) the electron is at rest relatively to a system of co-ordinates which is in parallel motion with velocity v along the axis of X.

From the above assumption, in combination with the principle of relativity, it is clear that in the immediately ensuing time (for small values of t) the electron, viewed from the system k, moves in accordance with the equations


in which the symbols , , , , , refer to the system k. If, further, we decide that when t=x=y=z=0 then , the transformation equations of §§ 3 and 6 hold good, so that we have


With the help of these equations we transform the above equations of motion from system k to system K, and obtain

· · · (A)

Taking the ordinary point of view we now inquire as to the ``longitudinal'' and the ``transverse'' mass of the moving electron. We write the equations (A) in the form


and remark firstly that , , are the components of the ponderomotive force acting upon the electron, and are so indeed as viewed in a system moving at the moment with the electron, with the same velocity as the electron. (This force might be measured, for example, by a spring balance at rest in the last-mentioned system.) Now if we call this force simply ``the force acting upon the electron,''9 and maintain the equation--mass × acceleration = force--and if we also decide that the accelerations are to be measured in the stationary system K, we derive from the above equations


With a different definition of force and acceleration we should naturally obtain other values for the masses. This shows us that in comparing different theories of the motion of the electron we must proceed very cautiously.

We remark that these results as to the mass are also valid for ponderable material points, because a ponderable material point can be made into an electron (in our sense of the word) by the addition of an electric charge, no matter how small.

We will now determine the kinetic energy of the electron. If an electron moves from rest at the origin of co-ordinates of the system K along the axis of X under the action of an electrostatic force X, it is clear that the energy withdrawn from the electrostatic field has the value . As the electron is to be slowly accelerated, and consequently may not give off any energy in the form of radiation, the energy withdrawn from the electrostatic field must be put down as equal to the energy of motion W of the electron. Bearing in mind that during the whole process of motion which we are considering, the first of the equations (A) applies, we therefore obtain


Thus, when v=c, W becomes infinite. Velocities greater than that of light have--as in our previous results--no possibility of existence.

This expression for the kinetic energy must also, by virtue of the argument stated above, apply to ponderable masses as well.

We will now enumerate the properties of the motion of the electron which result from the system of equations (A), and are accessible to experiment.

From the second equation of the system (A) it follows that an electric force Y and a magnetic force N have an equally strong deflective action on an electron moving with the velocity v, when . Thus we see that it is possible by our theory to determine the velocity of the electron from the ratio of the magnetic power of deflexion to the electric power of deflexion , for any velocity, by applying the law

This relationship may be tested experimentally, since the velocity of the electron can be directly measured, e.g. by means of rapidly oscillating electric and magnetic fields.

From the deduction for the kinetic energy of the electron it follows that between the potential difference, P, traversed and the acquired velocity v of the electron there must be the relationship

We calculate the radius of curvature of the path of the electron when a magnetic force N is present (as the only deflective force), acting perpendicularly to the velocity of the electron. From the second of the equations (A) we obtain

or


These three relationships are a complete expression for the laws according to which, by the theory here advanced, the electron must move.

In conclusion I wish to say that in working at the problem here dealt with I have had the loyal assistance of my friend and colleague M. Besso, and that I am indebted to him for several valuable suggestions.


--------------------------------------------------------------------------------

Footnotes
1.
The preceding memoir by Lorentz was not at this time known to the author.

2.
i.e. to the first approximation.

3.
We shall not here discuss the inexactitude which lurks in the concept of simultaneity of two events at approximately the same place, which can only be removed by an abstraction.

4.
``Time'' here denotes ``time of the stationary system'' and also ``position of hands of the moving clock situated at the place under discussion.''

5.
The equations of the Lorentz transformation may be more simply deduced directly from the condition that in virtue of those equations the relation x2+y2+z2=c2t2 shall have as its consequence the second relation .

6.
That is, a body possessing spherical form when examined at rest.

7.
Not a pendulum-clock, which is physically a system to which the Earth belongs. This case had to be excluded.

8.
If, for example, X=Y=Z=L=M=0, and N 0, then from reasons of symmetry it is clear that when v changes sign without changing its numerical value, must also change sign without changing its numerical value.

9.
The definition of force here given is not advantageous, as was first shown by M. Planck. It is more to the point to define force in such a way that the laws of momentum and energy assume the simplest form.

Editor's Notes
*1
In Einstein's original paper, the symbols (, H, Z) for the co-ordinates of the moving system k were introduced without explicitly defining them. In the 1923 English translation, (X, Y, Z) were used, creating an ambiguity between X co-ordinates in the fixed system K and the parallel axis in moving system k. Here and in subsequent references we use when referring to the axis of system k along which the system is translating with respect to K. In addition, the reference to system , later in this sentence was incorrectly given as ``k'' in the 1923 English translation.

*2
In the original 1923 English edition, this phrase was erroneously translated as ``plain figures''. I have used the correct ``plane figures'' in this document.

*3
This equation was incorrectly given in Einstein's original paper and the 1923 English translation as a=tan-1 wy/wx.

*4
The exponent of c in the denominator of the sine term of this equation was erroneously given as 2 in the 1923 edition of this paper. It has been corrected to unity here.

*5
``X'' in the 1923 English translation.

*6
Erroneously given as l' in the 1923 English translation, propagating an error, despite a change in symbols, from the original 1905 paper.

About this Edition
This edition of Einstein's On the Electrodynamics of Moving Bodies is based on the English translation of his original 1905 German-language paper (published as Zur Elektrodynamik bewegter Körper, in Annalen der Physik. 17:891, 1905) which appeared in the book The Principle of Relativity, published in 1923 by Methuen and Company, Ltd. of London. Most of the papers in that collection are English translations by W. Perrett and G.B. Jeffery from the German Das Relativatsprinzip, 4th ed., published by in 1922 by Tuebner. All of these sources are now in the public domain; this document, derived from them, remains in the public domain and may be reproduced in any manner or medium without permission, restriction, attribution, or compensation.

Numbered footnotes are as they appeared in the 1923 edition; editor's notes are preceded by asterisks (*) and appear in sans serif type. The 1923 English translation modified the notation used in Einstein's 1905 paper to conform to that in use by the 1920's; for example, c denotes the speed of light, as opposed the V used by Einstein in 1905.

This electronic edition was prepared by John Walker in November 1999. You can download a ready-to-print PostScript file of this document or the LaTeX source code used to create it from this site; both are supplied as Zipped archives. In addition, a PDF document is available which can be read on-line or printed. This HTML document was initially converted from the LaTeX edition with the LaTeX2HTML utility and the text and images subsequently hand-edited to produce this text.

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gravity

What Newton came up with.
Over three centuries ago, Isaac Newton invented a new kind of mathematics called calculus so that he could model motion in the natural world using mathematics. Calculus is about measuring change and so calculus became a vital tool in describing the motions of simple objects.
Newton was able to make a mathematical model that encompassed both objects falling because of gravity on Earth, and the motion of heavenly bodies in the skies.
Newton decided that the force of gravity on Earth was the same force that organized the motions of the moon around the Earth and the Earth and all the planets around the sun. He invented a formalism and developed mathematical formulas for calculating the size of the gravitational force both on Earth and in outer space.
One of the important formulas in Newton's model is his law for calculating the force of gravity between two objects 1 and 2 with mass m1 and m2, which are separated by a distance R:
F12 = F21 = G m1 m2 / R2
The constant G is a number that occurs in Nature, like the speed of light c. The constant G is called Newton's gravitational constant.
Newton's law of gravity winds up describing the observed motions of the planets extremely well. Another thing it models quite well is the way the gravitational force felt on the surface of a planet depends on the size and mass of the planet. For example, comparing the gravitational force at the surface of the Earth vs. the moon, we get
Fmoon/Fearth = (Mmoon/Mearth) (Rearth2/Rmoon2)
which is about 1/6, and the astronauts who walked on the moon felt it, too. You can see how much lighter lunar gravity is if you watch films of astronauts moonwalking.
This was an enormous thing Newton did - to invent a new kind of math to build a model that described in the same formula the observed motion of both falling objects on Earth and the planets in the heavens.
BUT unfortunately, Newtonian gravity falls apart when we try to combine it with what we've just learned about Special Relativity.