Minkowski diagram
Minkowski diagram with resting frame (x,t), moving frame (x′,t′), light cone, and hyperbolas marking out time and space with respect to the origin. The Minkowski diagram, also known as a spacetime diagram, was developed in 1908 by Hermann Minkowski and provides an illustration of the properties of space and time in the special theory of relativity. It allows a quantitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical equations. The term Minkowski diagram is used in both a generic and particular sense. In general, a Minkowski diagram is a graphic depiction of a portion of Minkowski space, often where space has been curtailed to a single dimension. These two-dimensional diagrams portray worldlines as curves in a plane that correspond to motion along the spatial axis. Basics[edit] A photon moving right at the origin corresponds to the yellow track of events, a straight line with a slope of 45°. Different scales on the axes. History
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
Metric (mathematics)
In differential geometry, the word "metric" may refer to a bilinear form that may be defined from the tangent vectors of a differentiable manifold onto a scalar, allowing distances along curves to be determined through integration. It is more properly termed a metric tensor. d : X × X → R (where R is the set of real numbers). For all x, y, z in X, this function is required to satisfy the following conditions: d(x, y) ≥ 0 (non-negativity, or separation axiom)d(x, y) = 0 if and only if x = y (identity of indiscernibles, or coincidence axiom)d(x, y) = d(y, x) (symmetry)d(x, z) ≤ d(x, y) + d(y, z) (subadditivity / triangle inequality). Conditions 1 and 2 together produce positive definiteness. A metric is called an ultrametric if it satisfies the following stronger version of the triangle inequality where points can never fall 'between' other points: For all x, y, z in X, d(x, z) ≤ max(d(x, y), d(y, z)) d(x, y) = d(x + a, y + a) for all x, y and a in X. If a modification of the triangle inequality
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] .
Lorentz transformation
In physics, the Lorentz transformation (or transformations) is named after the Dutch physicist Hendrik Lorentz. It was the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. The Lorentz transformation is in accordance with special relativity, but was derived before special relativity. The transformations describe how measurements of space and time by two observers are related. The Lorentz transformation is a linear transformation. In Minkowski space, the Lorentz transformations preserve the spacetime interval between any two events. History[edit] Many physicists, including Woldemar Voigt, George FitzGerald, Joseph Larmor, and Hendrik Lorentz himself had been discussing the physics implied by these equations since 1887.[1] Lorentz transformation for frames in standard configuration[edit] Boost in the x-direction[edit] These are the simplest forms.
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Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold [1][2] (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite. Instead a weaker condition of nondegeneracy is imposed. Introduction[edit] Manifolds[edit] In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. An -dimensional differentiable manifold is a generalisation of -dimensional Euclidean space. See Manifold, differentiable manifold, coordinate patch for more details. Tangent spaces and metric tensors[edit] Associated with each point in an -dimensional differentiable manifold is a tangent space (denoted ). we can express this as The map is symmetric and bilinear so if are tangent vectors at a point to the manifold That
Length contraction
In physics, length contraction is the phenomenon of a decrease in length measured by an observer of objects which are traveling at any non-zero velocity relative to the observer. This contraction (more formally called Lorentz contraction or Lorentz–FitzGerald contraction after Hendrik Lorentz and George FitzGerald) is usually only noticeable at a substantial fraction of the speed of light. Length contraction is only in the direction parallel to the direction in which the observed body is travelling. where L0 is the proper length (the length of the object in its rest frame), L is the length observed by an observer in relative motion with respect to the object, v is the relative velocity between the observer and the moving object, c is the speed of light, and the Lorentz factor, γ(v), is defined as In this equation it is assumed that the object is parallel with its line of movement. History[edit] Basis in relativity[edit] Another method is to use a clock indicating its proper time and . . Since
