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15 Greatest Da Vinci War Machines

2010-05-05T03:03:00.000-07:00

The Great Leonardo Da Vinci designed many weapons, including giant crossbows, machine guns, siege towers, cluster bombs and even a precursor to the modern-day tank.

1. Leonardo Da Vinci’s Terminator

Leonardo Da Vinci’s mechanical knight was not discovered until 1957, when Carlo Pedretti discovered it, hidden amongst Da Vinci’s countless designs. The mechanical knight, first sketched by DaVinci in 1495, was mentioned in 1974, in the Codex Madrid edited by Ladislao Reti, but there was no attempt to reconstruct it until 1996 when Mark Rosheim published an independent study of the robot, followed by a joint enterprise with the Florence Institute and Museum of the History of Science.

However, it was not until 2002 that Rosheim built a complete physical model of the robot for a BBC documentary. Since then, a soldier on wheels labelled, “Leonardo’s robot” has been included in countless exhibitions and museums.

In the 2007 Mario Taddei made a new research on Da Vinci’s original documents finding enough data to build a version of the soldier robot, more closely related to the original drawings. This robot was designed just for defensive purposes, not for war or theatre. Its movements are somewhat restricted since the arms only move right and left when pulled with a rope. This particular model is shown in various exhibitions around the world and the Tadei’s research results are published in the book, Leonardo Da Vinci’s Robots.

2. Machine Gun

The multi-barrelled machine gun was a weapon with remarkable firepower. Da Vinci sketched this rolling artillery battery around 1480 while in Florence, perhaps as a calling card to a warrior prince in need of a military architect. A hand crank adjusts elevation, and reloading is a major challenge – especially when under fire .

Though capable of rapid-fire which later model machine guns became noted for, this his housed an ingenious aiming and loading mechanism. By widening the field of fire, the fan-like shape of Da Vinci’s prototype made it a potentially effective weapon against a line of advancing troops. In addition Da Vinci’s design was easy to move around on the battlefield because it was lightweight and mounted on wheels.

3. Cluster Bomb

To make the bombard, or cannon, a weapon already known at the time, even more deadly, Da Vinci also designed large projectiles, comprised of round shells fitted around iron spacers and stitched inside a pliable casing. Once fired, this invention exploded into many fragments with that had greater range and impact than a single cannon-ball.

4. Scythed chariots

This is one of Leonardo’s most beautiful manuscripts. His sketches horse drawn reveal carriages covered with sharp, swirling blades that moved in the thick of battle slashing through everything in their wake. The rotating blades were specifically designed to sever the limbs from its victims. In one of his drawings, Da Vinci illustrated the carnage in such gruesome detail that his notation indicated that his contraption probably would wreak as much havoc on friends as on foes.

5. Barrage Cannon

This drawing is on of the first page of the Codex Atlanticus. The drawing itself is very complete and quite fascinating, illustrating the plan of a bombard with sixteen radial cannons. The most interesting aspect of the project is the centre of the bombard itself, housing a pair of mechanical paddles and gear wheels, providing only a partial glimpse of the possibilities of massive weapon.

6. Tank

This is perhaps one of the most famous of Da Vinci’s projects. His idea of reaping panic and destruction among enemy troops was envisioned in this tortoise-shaped vehicle, reinforced with metal plates, and ringed with cannons. In a job application to the Duke of Milan, Da Vinci boasted “I can make armoured cars, safe and unassailable, which will enter the close ranks of the enemy with their artillery, and no company of soldiers is so great that they will not break through them. And behind these the infantry will be able to follow quite unharmed and without any opposition.” Da Vinci’s precursor to the modern tank surely could have created “shock and awe” on the 15th-century battlefield, the design contained some serious flaws. Even with several modifications to the original plans he continued to be faced with a number of unresolved problems and eventually abandoned the project.

7. Wall Defense

Leonardo designed complex and ingenious methods of defence. Here, when the walls are under attack, the soldiers hidden behind the battlements could quickly and easily ward-off enemies and their single movement by using a system of levers. As the enemy used ladders in an attempt to breach the walls, the levers were engaged to move the rails built into the walls that the ladders were leaning on, causing them to become unstable and eventually fall.

8. Catapult

The basic design of the catapult had been in use for hundreds of years before Da Vinci embarked upon improving it. He actually came up with several different models. This particular design uses a double leaf spring to produce an enormous amount of energy in order to propel stone projectiles or incendiary materials over great distances. Loading of the two large leaf springs was accomplished using a hand crank on the side of the catapult.

9. Fortress

Leonardo designed this fortress with the idea of rendering it safe from the attack. The elaborate shape is innovative and presumably could have been an effective defence against the impact of deadly artillery projectiles.

The Da Vinci fortress could be considered by many as very modern in its design with its circular towers and the slightly inclined exterior walls designed to absorb attacks from firearms. The lord of the castle lived in the centre of the complex, which, according to original drawings also features a secret underground passage. In addition, the fortress features two levels of concentric walls, the tops of which are rounded, in order to help deflect the impact of cannon fire. Small openings make it possible for those fighting from within to return fire with minimum risk of injury from the outside.

10. Dismountable cannon

Cannons were very heavy and the carriages used to transport them were often unwieldy. Leonardo deigned a structure that could be easily dismantled and transported, thus permitting the cannon to be easily moved about.

11. Springald

The Springald, a device that throws large bolts or stones resembles a contemporary crossbow with inward swinging arms. Examples of springalds were drawn by Leonardo da Vinci during a period when he was also drawing powder-propelled weapons. Though several reconstructed examples can be found, there are no known archaeological finds of these machines. It is quite probable that this is because materials used to make them were recycled when they were no longer useful.

12. Da Vinci’s Helicopter

Leonardo Da Vinci is credited with having first thought of a machine for vertical flight. His sketch of the airscrew dated 1493, was not discovered until the 19th century. It consisted of a platform mounted by a helical screw driven by a rudimentary system, not unlike that of rubber band-powered model aircraft. Da Vinci’s notes state “if this instrument in the form of a screw were well made of linen, the pores of which had been stopped with starch, it should, upon being turned sharply, rise into the air in a spiral”. His design, however, was never put to any use.

Da Vinci left his imprint on aeronautics through his work with ornithopters and helicopter models and is said to have begun the first sound experiments in search of a practical heavier-than-air flying machine. He was convinced that if man were to able to realize his long held dream of travelling in the sky above him, it would happen by a flying machine based on the principle of the helicopter. Slightly more than two hundred years later, his prediction proved to be true.

13. Armoured vessel

The drawing depicting Da Vinci’s armoured vessel shows a light vessel fitted with a prow protected by metal and used to ram enemy ships. A rotating covering shield, which opens during the boarding phase of the attack, is also featured.

The covering shield provided protection against enemy ships and allowed the vessel to approach the enemy without the cannon being observed. The shield would not be opened to reveal the cannon until after the armoured vessel rammed an enemy ship, or got too close to evade it. The shields are attached to a system of winches which open very quickly, enhancing the element of surprise. Once lowered into the water, the shields could also function as a brake to offset the recoil of the cannon. The shields were closed through a system of manually operated winches.

14. Giant crossbow

The crossbow is so big that the six wheels are set at a slight angle in order to increase its stability. This gigantic Da Vinci crossbow launches heavy balls, rather than arrows.

The bow is made with flexible wooden pieces bound together by cords and held in place by pivoting pins. It has a span of approximately thirteen meters and it is stretched by a complex screw mechanism. Da Vinci installed winches to regulate the traction at the rear sides of the bow which also set a second screw mechanism, designed to reduce the force required to tighten the bow in motion.

15. Da Vinci Siege Weapon

The model proposed by Leonardo represents a machine designed for attacking defensive walls, consisting of a mobile structure with an armoured bridge that rests on the walls of an enemy fortress, while the troops to penetrate the city or castle.

In addition to new machines, Da Vinci proposes classical systems for use in assaulting enemy city walls. The ladder is fixed to a special support, made up of partially toothed wheel grips into a worm screw. A crank turns the wheel back and forth that lifts and lowers the ladder.

Lorentz transformation

2010-05-05T02:57:00.001-07:00

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A visualisation of the Lorentz transformation (full animation). Only one space coordinate is considered. The thin solid lines crossing at right angles depict the time and distance coordinates of an observer at rest; the skewed solid straight lines depict the coordinate grid of a moving observer.

In physics, the Lorentz transformation, named after the Dutch physicist Hendrik Lorentz, describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It reflects the surprising fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events.

The Lorentz transformation was originally the result of attempts by Lorentz and others to explain observed properties of light propagating in what was presumed to be the luminiferous aether; Albert Einstein later reinterpreted the transformation to be a statement about the nature of both space and time, and he independently re-derived the transformation from his postulates of special relativity. The Lorentz transformation supersedes the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). According to special relativity, this is only a good approximation at relative speeds much smaller than the speed of light.

If space is homogeneous, then the Lorentz transformation must be a linear transformation. Also, since relativity postulates that the speed of light is the same for all observers, it must preserve the spacetime interval between any two events in Minkowski space. The Lorentz transformation describes only the transformations in which the spacetime event at the origin is left fixed, so they can be considered as a rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.

Galilean transformation

2010-05-05T02:55:00.000-07:00

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The Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. This is the passive transformation point of view. The equations below, although apparently obvious, break down at speeds that approach the speed of light due to physics described by Einstein's theory of relativity.

Galileo formulated these concepts in his description of uniform motion ^[1] The topic was motivated by Galileo's description of the motion of a ball rolling down a ramp, by which he measured the numerical value for the acceleration of gravity, at the surface of the Earth. The descriptions below are another mathematical notation for this concept.

[hide]

[edit] Translation

Standard configuration of coordinate systems for Galilean transformations.

In essence, the Galilean transformations embody the intuitive notion of addition and subtraction of velocities. The assumption that time can be treated as absolute is at the heart of the Galilean transformations.

This assumption is abandoned in the Lorentz transformations. These relativistic transformations are deemed applicable to all velocities, whilst the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.

The notation below describes the relationship under the Galilean transformation between the coordinates (x,y,z,t) and (x′,y′,z′,t′) of a single arbitrary event, as measured in two coordinate systems S and S', in uniform relative motion (velocity v) in their common x and x’ directions, with their spatial origins coinciding at time t=t'=0: ^[2] ^[3] ^[4] ^[5]

Note that the last equation expresses the assumption of a universal time independent of the relative motion of different observers.

[edit] Galilean transformations

Diagram 1. Views of spacetime along the world line of an accelerating observer.

Vertical direction indicates time. Horizontal indicates distance, the dashed line is the spacetime trajectory of the observer. The lower half of the diagram shows events in the past. Upper half shows future events. The small dots are arbitrary events in spacetime.

The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime shears when the observer accelerates.

Under the Erlangen program, the space-time (no longer spacetime) of nonrelativistic physics is described by the symmetry group generated by Galilean transformations, spatial and time translations and rotations.

The Galilean symmetries (interpreted as active transformations):

Spatial translations:

Time translations:

Shear mappings:

Rotations and Reflections:

where R is an orthogonal matrix.

[edit] Central extension of the Galilean group

The Galilean group: Here, we will only look at its Lie algebra. It's easy to extend the results to the Lie group. The Lie algebra of L is spanned by E, P_i, C_i and L_ij (antisymmetric tensor) subject to commutators (operators of the form [,]), where

H is generator of time translations (Hamiltonian), P_i is generator of translations (momentum operator), C_i is generator of Galileian boosts and L_ij stands for a generator of rotations (angular momentum operator).

We can now give it a central extension into the Lie algebra spanned by H', P'_i, C'_i, L'_ij (antisymmetric tensor), M such that M commutes with everything (i.e. lies in the center, that's why it's called a central extension) and

[edit] Notes

^ Galileo 1638 Discorsi e Dimostrazioni Matematiche, intorno á due nuoue scienze 191 - 196, published by Lowys Elzevir (Louis Elsevier), Leiden, or Two New Sciences, English translation by Henry Crew and Alfonso de Salvio 1914, reprinted on pages 515-520 of On the Shoulders of Giants: The Great Works of Physics and Astronomy. Stephen Hawking, ed. 2002 ISBN 0-7624-1348-4
^ Mould, Richard A. (2002), Basic relativity, Springer-Verlag, ISBN 0-387-95210-1, http://books.google.be/books?id=lfGE-wyJYIUC&pg=PA42 , Chapter 2 §2.6, p. 42
^ Lerner, Lawrence S. (1996), Physics for Scientists and Engineers, Volume 2, Jones and Bertlett Publishers, Inc, ISBN 0-7637-0460-1, http://books.google.be/books?id=B8K_ym9rS6UC&pg=PA1047 , Chapter 38 §38.2, p. 1046,1047
^ Serway, Raymond A.; Jewett, John W. (2006), Principles of Physics: A Calculus-based Text, Fourth Edition, Brooks/Cole - Thomson Learning, ISBN 0-534-49143-X, http://books.google.be/books?id=1DZz341Pp50C&pg=PA261 , Chapter 9 §9.1, p. 261
^ Hoffmann, Banesh (1983), Relativity and Its Roots, Scientific American Books, ISBN 0-486-40676-8, http://books.google.be/books?id=JokgnS1JtmMC&pg=PA83 , Chapter 5, p. 83

Albert Einstein

2010-05-05T02:53:00.000-07:00

Born on: March 14, 1879

Born in: Ulm, Württemberg, Germany

Died on: April 18, 1955

Nationality: German, Swiss, American

Career: Physicist

Albert Einstein was a German born physicist, who is known for his phenomenal contribution to theoretical physics. His best works include ‘Theory of Relativity and specifically mass-energy equivalence, ‘E = mc^2’. He even received a Nobel Prize in Physics, in the year 1921. Einstein published over 300 scientific works and over 150 non-scientific works. The legendary scientist is highly revered by the physics community. In the year 1999, ‘Time magazine’ named him the ‘Person of the Century’. As for the rest of the world, the name "Einstein" is synonymous with genius.

Childhood and Early Life

Albert Einstein was born in a Jewish family in Ulm, Württemberg, Germany, on March 14, 1879. His father, Hermann Einstein was a salesman and engineer, while his mother’s name was Pauline Einstein. In 1880, his family moved to Munich, where his father founded an electrical equipment manufacturing company, Elektrotechnische Fabrik J. Einstein & Cie, along with his uncle. As a child, Einstein never observed Jewish religious practices and attended a Catholic elementary school. Although he faced speech difficulties in elementary school, he remained a top student.

At the tender age of five, Einstein was greatly influenced by a pocket compass given to him by his father. The movement of the needle had an everlasting impression on his young mind. In effect, as he grew, he started building models and mechanical devices, showing deep interest in mathematics. By the age of twelve, he had learnt Euclidean geometry and studied calculus. After his father’s business failed, Einstein’s family moved to Pavia. During this time, he wrote his first scientific work, "The Investigation of the State of Aether in Magnetic Fields".

Einstein did not even complete his high school and directly applied to ETH, the Swiss Federal Institute of Technology, in Zürich, Switzerland. Though he could not get through the entrance exam, he did exceptionally well in mathematics and physics. At the age of 16, he performed his first famous thought experiment, visualizing traveling alongside a beam of light. Einstein completed his secondary school from Aarau, Switzerland and fell in love with Marie, the daughter of his professor. He graduated at the age of 17 and renounced his German citizenship.

Thereafter, he enrolled in the mathematics program at ETH, while Marie moved onto Olsberg, for a teaching post. Einstein met his future wife, Mileva Mariæ in 1896. She was the only woman studying mathematics at ETH. Within a few years, both of them grew fond of each other and fell in love. In 1900, Einstein graduated with a degree in physics. After graduation, he could not find a teaching job and after two years of struggle, was employed at the Federal Office for Intellectual Property, the patent office, as an assistant examiner.

Early Works

While working in the patent office, Einstein’s four papers were published in the Annalen der Physik, the leading German physics journal. These papers have come to be known as Annus Mirabilis Papers. His papers were based on photoelectric effect, Brownian movement, electrodynamics and mass-energy equivalence. At the age of 26, Einstein was awarded a PhD by the University of Zurich. In 1910, he wrote a paper on critical opalescence, thereby explaining why the sky is blue. During 1909, Einstein wrote another paper, this time explaining the photon concept.

In 1911, Einstein became an associate professor at the University of Zurich and soon, was a full professor at the Charles University of Prague. Here, Einstein published a paper about the effects of gravity on light, specifically the gravitational redshift and the gravitational deflection of light. In 1912, he accepted professorship at ETH and in 1915, published a paper on general theory of relativity. In 1917, Einstein published an article on stimulated emission as well as a paper on the cosmological constant.

Fame

In 1919, Einstein’s gravitational deflection theory was confirmed by a team led by British astronomer. This won worldwide acclaim for Einstein and he became extremely famous. With this, he got an entry of in the scientific community, which was resented by many. In 1921, Einstein was awarded the Nobel Prize in Physics, for his contribution to Theoretical Physics and especially, for his discovery of the law of the photoelectric effect.

However, as per his settlement with his wife, he gave the Nobel Prize money to Melvic on their divorce. Einstein wanted to generalize his theory of gravitation in order to unify and simplify the fundamental laws of physics, particularly gravitation and electromagnetism. Though he was highly appreciated for his works in theoretical physics, with time, he started becoming isolated and unsuccessful in his research.

Death

Einstein died on April 18, 1955, due to internal bleeding caused by the rupture of an aortic aneurysm, which had previously been diagnosed and reinforced. The legendary scientist died in Princeton Hospital, at the age of 76, but worked till the very end. His remains were cremated and his ashes were scattered. Just before the cremation, Princeton Hospital pathologist Thomas Stoltz Harvey removed Einstein's brain for preservation, without the permission of his family. He hoped that the neuroscience of the future might someday be able to find out what made Einstein so intelligent.

Personal Life

Einstein married Mileva Mariæ on January 6, 1903 and had a daughter, Lieserl Einstein, born in early 1902. It is said that their relationship was more like a personal and intellectual partnership. Many people question whether she contributed to Einstein’s work, while historians believe that she did not make any major contributions. Albert and Mileva’s first son, Hans Albert Einstein was born on May 14, 1904, in Berne, Switzerland.

Their second son, Eduard was born in Munich on July 28, 1910. However, the couple got divorced in February 1919. In June 1919, Einstein married Elsa Löwenthal, who had nursed him through an illness. Elsa was maternally Albert's first cousin and paternally, his second cousin. Together, both of them raised Margot and Ilse, Elsa's daughters from her first marriage. Conversely, their union produced no children.

General Relativity

2010-05-05T02:49:00.002-07:00

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.

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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!

Special Relativity

2010-05-05T02:49:00.001-07:00

Newton's laws of motion give us a complete description of the behavior moving objects at low speeds. The laws are different at speeds reached by the particles at SLAC.

Einstein's Special Theory of Relativity describes the motion of particles moving at close to the speed of light. In fact, it gives the correct laws of motion for any particle. This doesn't mean Newton was wrong, his equations are contained within the relativistic equations. Newton's "laws" provide a very good approximate form, valid when v is much less than c. For particles moving at slow speeds (very much less than the speed of light), the differences between Einstein's laws of motion and those derived by Newton are tiny. That's why relativity doesn't play a large role in everyday life. Einstein's theory supersedes Newton's, but Newton's theory provides a very good approximation for objects moving at everyday speeds.

Einstein's theory is now very well established as the correct description of motion of relativistic objects, that is those traveling at a significant fraction of the speed of light.

Because most of us have little experience with objects moving at speeds near the speed of light, Einstein's predictions may seem strange. However, many years of high energy physics experiments have thoroughly tested Einstein's theory and shown that it fits all results to date.

Theoretical Basis for Special Relativity

Einstein's theory of special relativity results from two statements -- the two basic postulates of special relativity:

The speed of light is the same for all observers, no matter what their relative speeds.
The laws of physics are the same in any inertial (that is, non-accelerated) frame of reference. This means that the laws of physics observed by a hypothetical observer traveling with a relativistic particle must be the same as those observed by an observer who is stationary in the laboratory.

Given these two statements, Einstein showed how definitions of momentum and energy must be refined and how quantities such as length and time must change from one observer to another in order to get consistent results for physical quantities such as particle half-life. To decide whether his postulates are a correct theory of nature, physicists test whether the predictions of Einstein's theory match observations. Indeed many such tests have been made -- and the answers Einstein gave are right every time!

The Speed of Light is the same for all observers.

The first postulate -- the speed of light will be seen to be the same relative to any observer, independent of the motion of the observer -- is the crucial idea that led Einstein to formulate his theory. It means we can define a quantity c, the speed of light, which is a fundamental constant of nature.

Note that this is quite different from the motion of ordinary, massive objects. If I am driving down the freeway at 50 miles per hour relative to the road, a car traveling in the same direction at 55 mph has a speed of only 5 mph relative to me, while a car coming in the opposite direction at 55 mph approaches me at a rate of 105 mph. Their speed relative to me depends on my motion as well as on theirs.

Physics is the same for all inertial observers.

This second postulate is really a basic though unspoken assumption in all of science -- the idea that we can formulate rules of nature which do not depend on our particular observing situation. This does not mean that things behave in the same way on the earth and in space, e.g. an observer at the surface of the earth is affected by the earth's gravity, but it does mean that the effect of a force on an object is the same independent of what causes the force and also of where the object is or what its speed is.

Einstein developed a theory of motion that could consistently contain both the same speed of light for any observer and the familiar addition of velocities described above for slow-moving objects. This is called the special theory of relativity, since it deals with the relative motions of objects.

Note that Einstein's General Theory of Relativity is a separate theory about a very different topic -- the effects of gravity.

Relativistic Definitions

Physicists call particles with v/c comparable to 1 "relativistic" particles. Particles with v/c <<>

Gamma()

The measurable effects of relativity are based on gamma. Gamma depends only on the speed of a particle and is always larger than 1. By definition:

c is the speed of light
v is the speed of the object in question

For example, when an electron has traveled ten feet along the accelerator it has a speed of 0.99c, and the value of gamma at that speed is 7.09. When the electron reaches the end of the linac, its speed is 0.99999999995c where gamma equals 100,000.

What do these gamma values tell us about the relativistic effects detected at SLAC? Notice that when the speed of the object is very much less than the speed of light (v << c), gamma is approximately equal to 1. This is a non-relativistic situation (Newtonian).

Momentum

For non-relativistic objects Newton defined momentum, given the symbol p, as the product of mass and velocity -- p = m v. When speed becomes relativistic, we have to modify this definition -- p = gamma (mv)

Notice that this equation tells you that for any particle with a non-zero mass, the momentum gets larger and larger as the speed gets closer to the speed of light. Such a particle would have infinite momentum if it could reach the speed of light. Since it would take an infinite amount of force (or a finite force acting over an infinite amount of time) to accelerate a particle to infinite momentum, we are forced to conclude that a massive particle always travels at speeds less than the speed of light.

Some text books will introduce the definition m₀ for the mass of an object at rest, calling this the "rest mass" and define the quantity (M = gamma m₀) as the mass of the moving object. This makes Newton's definition of momentum still true provided you choose the correct mass. In particle physics, when we talk about mass we always mean mass of an object at rest and we write it as m and keep the factor of gamma explicit in the equations.

Energy

Probably the most famous scientific equation of all time, first derived by Einstein is the relationship E = mc².

This tells us the energy corresponding to a mass m at rest. What this means is that when mass disappears, for example in a nuclear fission process, this amount of energy must appear in some other form. It also tells us the total energy of a particle of mass m sitting at rest.

Einstein also showed that the correct relativistic expression for the energy of a particle of mass m with momentum p is E² = m²c⁴ + p²c². This is a key equation for any real particle, giving the relationship between its energy (E), momentum ( p), and its rest mass (m).

If we substitute the equation for p into the equation for E above, with a little algebra, we get E = gamma mc², so energy is gamma times rest energy. (Notice again that if we call the quantity M =gamma m the mass of the particle then E = Mc² applies for any particle, but remember, particle physicists don't do that.)

Let's do a calculation. The rest energy of an electron is 0.511 MeV. As we saw earlier, when an electron has gone about 10 feet along the SLAC linac, it has a speed of 0.99c and a gamma of 7.09. Therefore, using the equation E = gamma x the rest energy, we can see that the electron's energy after ten feet of travel is 7.09 x 0.511 MeV = 3.62 MeV. At the end of the linac, where gamma = 100,000, the energy of the electron is 100,000 x 0.511 MeV = 51.1 GeV.

The energy E is the total energy of a freely moving particle. We can define it to be the rest energy plus kinetic energy (E = KE + mc²) which then defines a relativistic form for kinetic energy. Just as the equation for momentum has to be altered, so does the low-speed equation for kinetic energy (KE = (1/2)mv²). Let's make a guess based on what we saw for momentum and energy and say that relativistically KE = gamma(1/2)mv². A good guess, perhaps, but it's wrong.

Now here is an exercise for the interested reader. Calculate the quantity KE = E - mc² for the case of v very much smaller than c, and show that it is the usual expression for kinetic energy (1/2 mv²) plus corrections that are proportional to (v/c)² and higher powers of (v/c). The complicated result of this exercise points out why it is not useful to separate the energy of a relativistic particle into a sum of two terms, so when particle physicists say "the energy of a moving particle" they mean the total energy, not the kinetic energy.

Another interesting fact about the expression that relates E and p above (E² = m²c⁴ + p²c²), is that it is also true for the case where a particle has no mass (m=0). In this case, the particle always travels at a speed c, the speed of light. You can regard this equation as a definition of momentum for such a mass-less particle. Photons have kinetic energy and momentum, but no mass!

In fact Einstein's relationship tells us more, it says Energy and mass are interchangeable. Or, better said, rest mass is just one form of energy. For a compound object, the mass of the composite is not just the sum of the masses of the constituents but the sum of their energies, including kinetic, potential, and mass energy. The equation E=mc² shows how to convert between energy units and mass units. Even a small mass corresponds to a significant amount of energy.

In the case of an atomic explosion, mass energy is released as kinetic energy of the resulting material, which has slightly less mass than the original material.
In any particle decay process, some of the initial mass energy becomes kinetic energy of the products.

Even in chemical processes there are tiny changes in mass which correspond to the energy released or absorbed in a process. When chemists talk about conservation of mass, they mean that the sum of the masses of the atoms involved does not change. However, the masses of molecules are slightly smaller than the sum of the masses of the atoms they contain (which is why molecules do not just fall apart into atoms). If we look at the actual molecular masses, we find tiny mass changes do occur in any chemical reaction.

At SLAC, and in any particle physics facility, we also see the reverse effect -- energy producing new matter. In the presence of charged particles a photon (which only has kinetic energy) can change into a massive particle and its matching massive antiparticle. The extra charged particle has to be there to absorb a little energy and more momentum, otherwise such a process could not conserve both energy and momentum. This process is one more confirmation of Einstein's special theory of relativity. It also is the process by which antimatter (for example the positrons accelerated at SLAC) is produced.

Units of Mass, Energy, and Momentum

Instead of using kilograms to measure mass, physicists use a unit of energy -- the electron volt. It is the energy gained by one electron when it moves through a potential difference of one volt. By definition, one electron volt (eV) is equivalent to 1.6 x 10^-19 joules.

Lets look at an example of how this energy unit works. The rest mass of an electron is 9.11 x 10^-31kg. Using E = mc² and a calculator we get:

E = 9.11 x 10^-31kg x (3 x 10⁸m/s)² = 8.199 x 10^-14joules

This gives us the energy equivalent of one electron. So, whether we say we have 9.11 x 10^-31 kg or 8.199 x 10^-14joules, we really talking about the same thing -- an electron. Physicists go one stage further and convert the joules to electron volts. This gives the mass of an electron as 0.511 MeV (about half a million eV).

So if you ask a high energy physicist what the mass of an electron is, you'll be told the answer in units of energy. You can blame Einstein for that!

Eagle-eyed readers will notice that if you solve E=mc² for m, you get m=E/c², so the unit of energy should be eV/c². What happened to the c²? It's very simple, particle physicists choose units of length so that the speed of light = 1! How can we do that? Quite easily, as long as everyone understands the system. All we have to do is use a conversion factor to get back the "real" (i.e. everyday) units, if we want them.

Not only are mass and energy measured in eV, so is momentum. It makes life so much easier than dividing by c² or c all the time.

There is more information available on units in relativistic physics.

Peculiar Relativistic Effects

Length Contraction and Time Dilation

One of the strangest parts of special relativity is the conclusion that two observers who are moving relative to one another, will get different measurements of the length of a particular object or the time that passes between two events.

Consider two observers, each in a space-ship laboratory containing clocks and meter sticks. The space ships are moving relative to each other at a speed close to the speed of light. Using Einstein's theory:

Each observer will see the meter stick of the other as shorter than their own, by the same factor gamma (- defined above). This is called length contraction.
Each observer will see the clocks in the other laboratory as ticking more slowly than the clocks in his/her own, by a factor gamma. This is called time dilation.

In particle accelerators, particles are moving very close to the speed of light where the length and time effects are large. This has allowed us to clearly verify that length contraction and time dilation do occur.

Time Dilation for Particles

Particle processes have an intrinsic clock that determines the half-life of a decay process. However, the rate at which the clock ticks in a moving frame, as observed by a static observer, is slower than the rate of a static clock. Therefore, the half-life of a moving particles appears, to the static observer, to be increased by the factor gamma.

For example, let's look at a particle sometimes created at SLAC known as a tau. In the frame of reference where the tau particle is at rest, its lifetime is known to be approximately 3.05 x 10^-13 s. To calculate how far it travels before decaying, we could try to use the familiar equation distance equals speed times time. It travels so close to the speed of light that we can use c = 3x10⁸ m/sec for the speed of the particle. (As we will see below, the speed of light in a vacuum is the highest speed attainable.) If you do the calculation you find the distance traveled should be 9.15 x 10^-5 meters.

d = v t

d = (3 x 10⁸ m/sec)( 3.05 x 10^-13 s) = 9.15 x 10^-5 m

Here comes the weird part - we measure the tau particle to travel further than this!

Pause to think about that for a moment. This result is totally contradictory to everyday experience. If you are not puzzled by it, either you already know all about relativity or you have not been reading carefully.

What is the resolution of this apparent paradox? The answer lies in time dilation. In our laboratory, the tau particle is moving. The decay time of the tau can be seen as a moving clock. According to relativity, moving clocks tick more slowly than static clocks.

We use this fact to multiply the time of travel in the taus moving frame by gamma, this gives the time that we will measure. Then this time times c, the approximate speed of the tau, will give us the distance we expect a high energy tau to travel.

What is gamma in this case? It depends on the tau's energy. A typical SLAC tau particle has a gamma = 20. Therefore, we detect the tau to decay in an average distance of 20 x (9.15 x 10^-5 m) = 1.8 x 10^-3 m or approximately 1.8 millimeters. This is 20 times further than we expect it to go if we use classical rather than relativistic physics. (Of course, we actually observe a spread of decay times according to the exponential decay law and a corresponding spread of distances. In fact, we use the measured distribution of distances to find the tau half-life.)

Observations particles with a variety of velocities have shown that time dilation is a real effect. In fact the only reason cosmic ray muons ever reach the surface of the earth before decaying is the time dilation effect.

Length Contraction

Instead of analyzing the motion of the tau from our frame of reference, we could ask what the tau would see in its reference frame. Its half-life in its reference frame is 3.05 x 10^-13 s. This does not change. The tau goes nowhere in this frame.

How far would an observer, sitting in the tau rest frame, see an observer in our laboratory frame move while the tau lives?

We just calculated that the tau would travel 1.8 mm in our frame of reference. Surely we would expect the observer in the tau frame to see us move the same distance relative to the tau particle. Not so says the tau-frame observer -- you only moved 1.8 mm/gamma = 0.09 mm relative to me. This is length contraction.

How long did the tau particle live according to the observer in the tau frame? We can rearrange d = v x t to read t = d/v. Here we use the same speed, Because the speed of the observer in the lab relative to the tau is just equal to (but in the opposite direction) of the speed of the tau relative to the observer in the lab, so we can use the same speed. So time = 0.09 x 10^-3m/(3 x 10⁸)m/sec = 3.0 x 10^-13 sec. This is the half-life of the tau as seen in its rest frame, just as it should be!

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Lorentz transformation

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Galilean transformation

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Contents

[edit] Translation

[edit] Galilean transformations

[edit] Central extension of the Galilean group

[edit] Notes

Albert Einstein

General Relativity

Embedding Diagrams

The Flow of Spacetime

Gravitational Time Dilation

Grappling With Relativity

The Math Behind Einstein's Vision

Special Relativity

Theoretical Basis for Special Relativity

The Speed of Light is the same for all observers.

Physics is the same for all inertial observers.

Relativistic Definitions

Gamma()

Momentum

Energy

Units of Mass, Energy, and Momentum

Peculiar Relativistic Effects

Length Contraction and Time Dilation

Time Dilation for Particles

Length Contraction