Minkowski space

In theoretical physics, Minkowski space is often contrasted with Euclidean space. While a Euclidean space has only spacelike dimensions, a Minkowski space also has one timelike dimension. The isometry group of a Euclidean space is the Euclidean group and for a Minkowski space it is the Poincaré group. History[edit] In 1905 (published 1906) it was noted by Henri Poincaré that, by taking time to be the imaginary part of the fourth spacetime coordinate √−1 ct, a Lorentz transformation can be regarded as a rotation of coordinates in a four-dimensional Euclidean space with three real coordinates representing space, and one imaginary coordinate, representing time, as the fourth dimension. The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. For further historical information see references Galison (1979), Corry (1997), Walter (1999). Structure[edit] The Minkowski inner product[edit] Standard basis[edit] where

Gravitation Gravitation, or gravity, is a natural phenomenon by which all physical bodies attract each other. It is most commonly recognized and experienced as the agent that gives weight to physical objects, and causes physical objects to fall toward the ground when dropped from a height. During the grand unification epoch, gravity separated from the electronuclear force. History of gravitational theory Scientific revolution Modern work on gravitational theory began with the work of Galileo Galilei in the late 16th and early 17th centuries. Newton's theory of gravitation In 1687, English mathematician Sir Isaac Newton published Principia, which hypothesizes the inverse-square law of universal gravitation. Newton's theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not be accounted for by the actions of the other planets. Equivalence principle Formulations of the equivalence principle include: General relativity Specifics

Hayashi track Stellar evolution tracks (blue lines) for the pre-main-sequence. The nearly-vertical curves are Hayashi tracks. Low-mass stars have nearly vertical evolution tracks until they arrive on the main sequence. For more massive stars, the Hayashi track bends lefts into the Henyey track. Even more massive stars are born directly onto the Henyey track. The end (leftmost point) of every track is labeled with the star's mass in solar masses, and represents its position on the main sequence. years old lie along the curve labeled , and similarly for the other 3 isochrones. The Hayashi track is a luminosity–temperature relationship obeyed by infant stars of less than 3 solar masses in the pre-main-sequence phase of stellar evolution. At an end of a low- or intermediate-mass star's life, the star follows an analogue of the Hayashi track, but in reverse—it increases in luminosity, expands, and stays at roughly the same temperature, eventually becoming a red giant. History[edit] where . Derivation[edit] .

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