The special theory of relativity was first put forward by Einstein in 1905^[1]. His aim was to take care of some theoretical concerns about classical electrodynamics, but ultimately he came up with a modification of the laws of mechanics itself. Special relativity explores the structure of space and time and their effect on motion, forces and other dynamical phenomena. His approach was based on two physical postulates and is explained in the main article Special relativity. But in 1908 Hermann Minkowski realised that the theory could be made more statisfactory by basing it on some postulated geometric properties of space and time^[2]. In fact, he recognised that special relativity destroyed some of the absolute separation between space and time present in Newton's mechanics. He formulated special relativity on a four-dimensional spacetime, a mathematical construct that unifies space and time into a single object. This insight proved vital in the further development of physics. Classical electromagnetism is a theory of electromagnetism that was developed over the course of the 19th century, most prominently by James Clerk Maxwell. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... 1908 (MCMVIII) was a leap year starting on Wednesday (link will display the full calendar). ... Hermann Minkowski. ...

Special relativity predicts some strange effects, such as that a rod will become shorter as it moves at higher speeds and that time will behave differently for different observers. All of these effects can be very simply explained and predicted from the geometrical formalism advanced by Minkowski. This article aims to give a gentle, non-technical introduction to the theory based on geometrical intuition and explain some of the terminology and formalism involved. A more advanced, encyclopedic description will be found in the parent article Special relativity. It will be helpful to approach relativisic mechanics by first studying some properties of the classical (Newtonian) picture. The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...

The classical picture

Classical mechanics was developed over a long period of time, beginning with Galileo, until it was consolidated and placed on a firm axiomatic mathematical footing by Newton in Philosophiae Naturalis Principia Mathematica. The underlying mathematical framework (but no basic physical concepts) was further developed through the works of Euler, Lagrange, Hamilton, Jacobi etc. In this mechanics, time is an aboslute line, same for all observers. This means that there exists a 'correct' clock, such that all people will be able to agree on what the time is at an instant. This is the most important characteristic of classical mechanics that Einstein modified. Before we go on to relativistic mechanics, it will be helpful to examine two more concepts. Newtons own copy of his Principia, with handwritten corrections for the second edition. ...

Reference frames

A reference frame is simply a selection of what constitutes stationary objects. Even in classical mechanics, such a choice is not absolute: there is no absolute rest. When a train is moving at a constant velocity past a platform, one may either say that the platform is at rest and the train is moving or that the train is at rest and the platform is moving past it. These two descriptions correspond to two different reference frames. They are respectively called the rest frame of the platform and the rest frame of the train (sometimes simply the platform frame and the train frame). Frames in which Newton's laws of motion are valid are called inertial reference frames. All inertial reference frames are moving at constant velocities with respect to each other. The principle of Galilean relativity states that the laws of physics are the same in all inertial reference frames. Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

Mathematically, it gives a 'formula' for adding velocities: if particle P is moving at velocity v with respect to reference frame A and reference frame A is moving at velocity u with respect to reference frame B, then the velocity of P with respect to B is given by v + u. This formula for transforming coordinates between different reference frames is called the Galilean transformation. The principle of Galilean relativity then demands that laws of physics be unchanged if this transformation is applied to them. Laws of classical mechanics, like Newton's second law, obey this principle because they have the same form after applying the transformation. Time is the same in all reference frames because it is absolute in classical mechanics.

One can automate the task of verifying whether equations are invariant under the Galilean transformation by writing all equations in terms of ordinary 3-dimensional vectors. Vectors are, by definition, invariant under this transformation, so any equation written solely in terms of vectors (like Newton's second law) automatically satisfies Galilean relativity. From this perspective, the entire vector formalism may be viewed as a convenient notation to ensure that all proposed laws of physics conform to an underlying principle, that of invariance under the Galilean transformation. We shall see that this can be extended in a very natural way to relativistic mechanics by replacing vectors with 4-vectors.

Invariance of length

Pythagoras theorem

The length of an object is constant on the plane during rotations on the plane but not during rotations out of the plane

The modern theory of special relativity begins with the concept of "length". In everyday experience, it seems that the length of objects remains the same no matter how they are rotated or moved from place to place; as a result we think that the simple length of an object doesn't change or is "invariant". However, as is shown in the illustrations below, what we are actually suggesting is that length seems to be invariant in a three-dimensional coordinate system. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Look up length, width, breadth in Wiktionary, the free dictionary. ... In physics, invariants are usually quantities conserved (unchanged) by the symmetries of the physical system. ...

The length of a line in a two-dimensional coordinate system is given by Pythagoras' theorem: See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...

One of the basic theorems of vector algebra is that the length of a vector does not change when it is rotated. However, a closer inspection tells us that this is only true if we consider rotations confined to the plane. If we introduce rotation in the third dimension, then we can tilt the line out of the plane. In this case the projection of the line on the plane will get shorter. Does this mean length is not invariant? Of course not, you would say. The world is three-dimensional and in a 3D coordinate system the length is given by the three-dimensional version of Pythagoras's theorem:

Invariance in a 3D coordinate system: Pythagoras theorem gives k² = h² + z² but h² = x² + y² therefore k² = x² + y² + z². The length of an object is constant whether it is rotated or moved from one place to another in a 3D coordinate system

This is invariant under all rotations. The apparent violation of invariance of length only happened because we were 'missing' a dimension. It seems that, provided all the directions in which an object can be tilted or arranged are represented within a coordinate system, the length of an object does not change under rotations. A 3-dimensional coordinate system is enough in classical mechanics because there time is absolute and independent of space. It can be considered separately. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

Note that this property if invariance of length is not something one would ordinarily consider a dynamic principle, not even a theorem. It is simply a statement about the fundamental nature of space itself. Space as we ordinarily conceive it is called a three-dimensional Euclidean space, because its geometrical structure is described by the principles of Euclidean geometry. The formula for distance between two points is a fundamental property of a euclidean space, it is called the Euclidean metric tensor (or simply the Euclidean metric). In general, distance formulas are called metric tensors. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ...

The Minkowski formulation

After Einstein derived special relativity formally from the counterintuitive proposition that the speed of light is the same to all observers, the need was felt for a more satisfactory formulation. Minkowski, building on mathematical approaches to non-euclidean geometry^[3] and the mathematical work of Lorentz and Poincare, realised that a geometric approach was the key. Minkowski suggested in 1908 that Einstein's new theory could be explained in a natural way if we replaced the concept of separate space and time with one four-dimensional space, spacetime. This was a groundbreaking concept, and Roger Penrose has said that relativity was not truly complete until Minkowski reformulated Einstein's work. Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ...

The concept of a four-dimensional space is hard to visualise. It may help at the beginning to think simply in terms of coordinates. In three-dimensional space, one needs three real numbers to refer to a point. In the Minkowski space, one needs four real numbers (three space coordinates and one time coordinate) to refer to a point at a particular instant of time. This point at a particular instant of time, specified by the four coordinates, is called an event. The distance between two different events is called the spacetime interval. In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

A path through the four-dimensional spacetime, usually called Minkowski space, is called a world line. Since it specifies both position and time, a particle having a known world line has a completely determined trajectory and velocity. This is just like graphing the displacement of a particle moving in a straight line against the time elapsed. The curve contains the complete motional information of the particle.

The space-time interval

In the same way as the measurement of distance in 3D space needed all three coordinates we must include time as well as the three space coordinates when calculating the distance in Minkowski space (henceforth called M). In a sense, the space-time interval provides a combined estimate of how far two events occur in space as well as the time that elapses between their occurrence. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

But there is a problem. Time is related to the space coordinates, but they are not equivalent. Pythagoras's theorem treats all coordinates on a equal footing (see Euclidean space for more details). We can exchange two space coordinates without changing the length, but we can not simply exchange a space coordinate with time, they are fundamentally different. It is an entirely different thing for two events to be separated in space and to be separated in time. Minkowski proposed that the formula for distance needed a change. He found that the correct formula was actually quite simple, differing only by a sign from the Pythagoras's theorem: Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

where c is a constant and t is the time coordinate (See note 5). Multiplication by c, which has the dimension ms ^{− 1}, converts the time to units of length and this constant has the same value as the speed of light. So the spacetime interval between two distinct events is given by A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic...

There are two major points to be noted. Firstly, time is being measured in the same units as length by multiplying it by a constant conversion factor. Secondly, and more importantly, the time-coordinate has different sign than the space coordinates. This means that in the four-dimensional spacetime M, one coordinate is different from the others and influences the distance differently. This new distance formula, called the metric of M, is at the heart of relativity. This simple change of sign leads to completely new hyperbolic geometry that is different from conventional Euclidean geometry. In the Minkowski space, Euclid's parallel postulate (also known as the fifth postulate, given in his Elements) does not hold. This distance formula is called the metric tensor of M. This minus sign means that a lot of our intuition about distances can not be directly carried over into spacetime intervals. For example, the spacetime interval between two events separated both in time and space may be zero (see below). In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... Lines through a given point P and hyperparallel to line l. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

In M, this is the invariant length, the ordinary 3D length is not required to be invariant. The spacetime interval must stay the same under rotations, but ordinary lengths can change. Just like before, we were missing a dimension. Rotations in Minkowski space have a slightly different interpretation than ordinary rotations. These rotations correspond to transformations of coordinate frames and are called the Lorentz transformations. Just like the Galilean transformations are the mathematical statement of the principle of Galilean relativity in classical mechanics, the Lorentz transformations are the mathematical form of Einstein's principle of relativity. Laws of physics must stay the same under Lorentz transformations. Maxwell's equations satisfy this property. The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ... This article is in need of attention. ...

With the statement of the Minkowski metric, the common name for the distance formula given above, the theoretical foundation of special relativity is complete. The entire basis for special relativity can be summed up by the geometric statement "spacetime intervals are invariant under Lorentz transformations". Now we will explore the power of this geometric approach by deriving a few important results.

Reference frames and Lorentz transformations

We have already discussed that in classical mechanics coordinate frame changes correspond to Galilean transfomations of the coordinates. Is this adequate in the relativistic Minkowski picture?

Suppose there are two people, Bill and John, on separate planets that are moving away from each other. Bill and John are on separate planets so they both think that they are stationary. John draws a graph of Bill's motion through space and time and this is shown in the illustration below:

John's view of Bill and Bill's view of himself

John sees that Bill is moving through space as well as time but Bill thinks he is moving through time alone. Bill would draw the same conclusion about John's motion. In fact, these two views, which would be classically considered a difference in reference frames, are related simply by a coordinate transformation in M. Bill's view of his own world line and John's view of Bill's world line are related to each other simply by a rotation of coordinates. One can be transformed into the other by a rotation of the time axis. Minkowski geometry handles transformations of reference frames in a very natural way. Image File history File links No higher resolution available. ...

How Bill's coordinates appear to John at the instant they pass each other

Changes in reference frame, represented by velocity transformations in classical mechanics, are represented by rotations in Minkowski space. These rotations are called Lorentz transformations. The Lorentz transformations are the relativistic equivalent of Galilean transformations. The mathematical forms are derived below. Laws of physics, in order to be relativistically correct, must stay the same under Lorentz transformations. Newton's laws of motion are invariant under Galilean rather than Lorentz transformations, so they are immediately recognisable as non-relativistic laws and must be discarded in relativistic physics. Schroedinger's equation is also non-relativistic. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ...

Maxwell's equations are trickier. They are written using vectors and at first glance appear to transform correctly. But on closer inspection, several questions are apparent that can not be satisfactorily resolved within classical mechanics (see History of special relativity). They are indeed invariant under Lorentz transformations and are relativistic, even though they were formulated before the discovery of special relativity. Classical electrodynamics can be said to be the first relativistic theory in physics. To make the relativistic character of equations apparent, they are written using 4-component vector like quantities called 4-vectors. 4-Vectors transform correctly under Lorentz transformations. Equations written using 4-vectors are automatically relativistic. This is called manifestly covariant form. 4-Vectors form a very important part of the formalism of special relativity. To meet Wikipedias quality standards, this article or section may require cleanup. ...

Einstein's postulate

Einstein's postulate that the speed of light is a constant comes out as a natural consequence of the Minkowski formulation^[4].

Proposition 1:When an object is travelling at c in a certain reference frame, the space time interval is zero. A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ...

Proof:

The space-time interval is

The distance travelled by an object moving at velocity v for t seconds is:

So:

But when the velocity v equals c:

And hence the space time interval

Proposition 2:An object travelling at c in one reference frame is travelling at c in all other reference frames.

Proof:

Let the object move with velocity v when observed from a different reference frame. A change in reference frame coresponds to a rotation in M. Since the spacetime interval must be conserved under rotation, the spacetime interval must be the same in all reference frames. In proposition 1 we showed it to be zero in one reference frame, hence it must be zero in all other reference frames. We get

implies

The paths of light rays have a space-time interval of zero and hence all observers will obtain the same value for the speed of light. Therefore, by assuming that the universe has four dimensions that are related by Minkowski's formula the speed of light appears as a constant and it does not need to be assumed to be constant as in Einstein's original approach to special relativity. Notice that c is not explicitly required to be the speed of light. It is a consequence of Maxwell's electrodynamics that light travels with c. There is no such requirement inherent in special relativity.

Clock delays and rod contractions: Lorentz transformations revisited

Another consequence of the invariance of the space-time interval is that clocks will appear to go slower on objects that are moving relative to you. This is very similar to how the 2D projection of a line rotated into the third-dimension appears to get shorter. Length is not conserved simply because we are ignoring one of the dimensions. Let us return to the example of John and Bill.

John observes the length of Bill's space-time interval as:

whereas Bill doesn't think he has traveled in space, so writes:

The space-time interval, s², is invariant. It has the same value for all observers, no matter who measures it or how they are moving in a straight line. This means that Bill's space-time interval equals John's observation of Bill's space-time interval so:

and

hence

.

So, if John sees a clock that is at rest in Bill's frame record one second, John will find that his own clock measures between these same ticks an interval t, called coordinate time, which is greater than one second. It is said that clocks in motion slow down, relative to those on observers at rest. This is known as "relativistic time dilation of a moving clock". The time that is measured in the rest frame of the clock (in Bill's frame) is called the proper time of the clock. Time dilation is the phenomenon whereby an observer finds that anothers clock which is physically identical to their own is ticking at a slower rate as measured by their own clock. ... In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. ...

In special relativity, therefore, changes in reference frame affect time also. Time is no longer absolute. There is no universally correct clock, time runs at different rates for different observers.

Similarly it can be shown that John will also observe measuring rods at rest on Bill's planet to be shorter in the direction of motion than his own measuring rods. This is a prediction known as "relativistic length contraction of a moving rod". If the length of a rod at rest on Bill's planet is X, then we call this quantity the proper length of the rod. The length x of that same rod as measured on John's planet, is called coordinate length, and given by Length contraction, according to Albert Einsteins special theory of relativity, is the decrease in length experienced by people or objects traveling at a substantial fraction of the speed of light. ... In physics, proper length is the length of an object or a contour as measured in the reference frame of the object itself in the context of special relativity. ...

.

Together, these give the mathematical form of the Lorentz transformations. Notice how they differ from the common-sense Galilean transformation.

Simultaneity and clock desynchronisation

Another consequence of Minkowski's space-time is that clocks will appear to be out of phase with each other along the length of a moving object. This means that if one observer sets up a line of clocks that are all synchronised so they all read the same time, then another observer who is moving along the line at high speed will see the clocks all reading different times. This means that observers who are moving relative to each other see different events as simultaneous. This effect is known as "Relativistic Phase" or the "Relativity of Simultaneity". Relativistic phase is often overlooked by students of special relativity, but if it is understood, then phenomena such as the twin paradox are easier to understand. In his famous work on Special Relativity in 1905, Albert Einstein predicted that when two clocks were brought together and synchronised, and then one was moved away and brought back, the clock which had undergone the traveling would be found to be lagging behind the clock which had stayed put. ...

The completely counter-intuitive outcome is that two observers at relative motion may not agree on whether two events happened at the same time, or one after another. This may seem to violate causality, but fortunately such disagreements are only possible about the timing of two events that can never be causally related (in technical language, when the two can not lie in the same light cone). In special relativity, a light cone is the pattern describing the temporal evolution of a flash of light in Minkowski spacetime. ...

The "plane of simultaneity" or "surface of simultaneity" contains all those events that happen at the same instant for a given observer. Events that are simultaneous for one observer are not simultaneous for another observer in relative motion.

Observers have a set of simultaneous events around them that they regard as composing the present instant. The relativity of simultaneity results in observers who are moving relative to each other having different sets of events in their present instant. This is the final nail in the coffin of absolute time. Not only do different observers differ about the speed of time, they also differ about which set of events constitute an instant of time. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

The net effect of the four-dimensional universe is that observers who are in motion relative to you seem to have time coordinates that lean over in the direction of motion, and consider things to be simultaneous that are not simultaneous for you. Spatial lengths in the direction of travel are shortened, because they tip upwards and downwards, relative to the time axis in the direction of travel, akin to a rotation out of three-dimensional space.

Great care is needed when interpreting space-time diagrams. Diagrams present data in two dimensions, and cannot show faithfully how, for instance, a zero length space-time interval appears.

Metaphysics and extensions

Unlike Newton's laws of motion, relativity is not based upon dynamical postulates. It does not assume anything about motion or forces. Rather, it deals with the fundamental nature of spacetime. It is concerned with describing the geometry of the backdrop on which all dynamical phenomena take place. In a sense therefore, it is a meta-theory, a theory that lays out a structure that all other theories must follow. In truth, Special relativity is only a special case. It assumes that spacetime is flat. That is, it assumes that the structure of M, most importantly the Minkowski metric tensor, is a constant. In General relativity, Einstein showed that this is not true. The structure of spacetime is modified by the presence of matter. Specifically, the distance formula given above is no longer generally valid except in space free from mass. However, just like a curved surface can be considered flat in the infinitesimal limit of calculus, a curved spacetime can be considered flat at a small scale. This means that the Minkowski metric written in the differential form is generally valid. An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

One says that the Minkowski metric is valid locally, but it fails to give a measure of distance over extended distances. It is not valid globally. In fact, in general relativity the global metric itself becomes dependent on the mass distribution and varies through space. The central problem of general relativity is to solve the famous Einstein field equations for a given mass distribution and find the distance formula that applies in that particular case. Minkowski's spacetime formulation was the conceptual stepping stone to general relativity. His fundamentally new outlook allowed not only the development of general relativity, but also to some extent quantum field theories. This article or section is in need of attention from an expert on the subject. ...

Applications

There is a general perception that relativistic physics is not needed in everyday life. This is not true. Many technologies are critically dependent on relativistic physics, such as the Global Positioning System (GPS). The Global Positioning System (GPS) is the only fully functional Global Navigation Satellite System (GNSS). ...

Notes

1. This entire article is an expansion, for "flat" space-time, of what is usually stated in a single phrase in physics textbooks. The modern approach was fully adopted by Einstein(1916)^[4] who writes: "All this simply means that an objective metrical significance is attached to the quantity: as is readily shown with the aid of the Lorentz transformations. Mathematically this fact corresponds to the condition that ds² is invariant with respect to Lorentz transformations." This approach is standard in most modern advanced textbooks for instance, Martin(1988) p. 20.^[5] Carroll(2005)^[6] Rindler(2001) p.90.^[7]etc..

Time dilation applies to clocks that remain in the same position anywhere in a frame of reference.

2. It should also be made clear that the length contraction result only applies to rods aligned in the direction of motion. At right angles to the direction of motion, there is no contraction. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

3. The mass of objects and systems of objects has a complex interpretation in special relativity, see relativistic mass. The term mass in special relativity can be used in different ways, occasionally leading to confusion. ...

4. "Minkowski also shared Poincaré's view of the Lorentz transformation as a rotation in a four-dimensional space with one imaginary coordinate, and his five four-vector expressions." (Walter 1999).

5. Originally Minkowski tried to make his formula look like Pythagoras's theorem by introducing the concept of imaginary time and writing -1 as i². But Wilson, Gilbert, Borel and others proposed that this was unnecessary and introduced real time with the assumption that, when comparing coordinate systems, the change of spatial displacements with displacements in time can be negative. This assumption is expressed in differential geometry using a metric tensor that has a negative coefficient. The different signature of the Minkowski metric means that the Minkowski space has hyperbolic rather than Euclidean geometry. In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

References

1. ^ Einstein, A. (1905)"Zur Elektrodynamik bewegter Körper", published on June 30, 1905
2. ^ Hermann Minkowski, "Raum und Zeit", 80. Versammlung Deutscher Naturforscher (Köln, 1908). Published in Physikalische Zeitschrift 10 104-111 (1909) and cht der Deutschen Mathematiker-Vereinigung 18 75-88 (1909). For an English translation, see Lorentz et al. (1952).
3. ^ Walter, S.(1999) The non-Euclidean style of Minkowskian relativity. The Symbolic Universe, J. Gray (ed.), Oxford University Press, 1999 http://www.univ-nancy2.fr/DepPhilo/walter/papers/nesh.xml
4. ^ ^{a b} Einstein, A. (1916). Relativity. The special and general theory., Tr. Lawson, R.W., London: Routledge classics 2001. 
5. ^ Martin, J.L. (1988). General Relativity. Hemel Hempstead: Prentice Hall. 
6. ^ Carroll, S.M. (2005). Spacetime and Geometry. Addison Wesley. 
7. ^ Rindler, W (2001). Relativity, Special, General and Cosmological. Oxford: Oxford University Press. 

is the 181st day of the year (182nd in leap years) in the Gregorian calendar. ... Hermann Minkowski. ...

External links

Special relativity for a general audience (no math knowledge required)

• Einstein Light An award-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics.
• Einstein Online Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics.

Special relativity explained (using simple or more advanced math)

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Special Relativity

• Wikibooks: Special Relativity
• Albert Einstein. Relativity: The Special and General Theory. New York: Henry Holt 1920. BARTLEBY.COM, 2000
• Sean Carroll's online Lecture Notes on General Relativity
• E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws by Nina Byers
• Hyperphysics Time Dilation
• Hyperphysics Length Contraction
• Greg Egan's Foundations
• Special Relativity Simulation
• Caltech Relativity Tutorial A basic introduction to concepts of Special and General Relativity, requiring only a knowledge of basic geometry.
• Special Relativity in film clips and animations from the University of New South Wales.
• Relativity Calculator - Learn Special Relativity Mathematics Mathematics of special relativity presented in as simple and comprehensive manner possible within philosophical and historical contexts.

Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... Nina Byers is Research Professor and Professor of Physics Emeritus at UCLA. Her academic degrees are from University of California, Berkeley and the University of Chicago. ...

See also

• Special Relativity
• Speed of light
• Invariance
• Space-time especially Distance in space-time
• Light clock
• Andromeda paradox
• Symmetry
• Symmetry in physics