 # 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 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 The Minkowski inner product Standard basis where Related:  MathematicsPhysicsPhysics

Hilbert space The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer)—and ergodic theory, which forms the mathematical underpinning of thermodynamics. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. Definition and illustration Motivating example: Euclidean space Definition

Lorentz group The mathematical form of Basic properties The Lorentz group is a subgroup of the Poincaré group, the group of all isometries of Minkowski spacetime. The Lorentz transformations are precisely the isometries which leave the origin fixed. Thus, the Lorentz group is an isotropy subgroup of the isometry group of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the homogeneous Lorentz group while the Poincaré group is sometimes called the inhomogeneous Lorentz group. Mathematically, the Lorentz group may be described as the generalized orthogonal group O(1,3), the matrix Lie group which preserves the quadratic form on R4. The restricted Lorentz group arises in other ways in pure mathematics. Connected components Each of the four connected components can be categorized by which of these two properties its elements have: Lorentz transformations which preserve the direction of time are called orthochronous. P = diag(1, −1, −1, −1) T = diag(−1, 1, 1, 1). where

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 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

Spacetime symmetries Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems, spacetime symmetries finding ample application in the study of exact solutions of Einstein's field equations of general relativity. Physical motivation Physical problems are often investigated and solved by noticing features which have some form of symmetry. preserving geodesics of the spacetimepreserving the metric tensorpreserving the curvature tensor These and other symmetries will be discussed in more detail later. Mathematical definition A rigorous definition of symmetries in general relativity has been given by Hall (2004). on M. the term on the right usually being written, with an abuse of notation, as Killing symmetry A Killing vector field is one of the most important types of symmetries and is defined to be a smooth vector field that preserves the metric tensor:

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

Pascal's law Pascal's law or the principle of transmission of fluid-pressure is a principle in fluid mechanics that states that pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid such that the pressure variations (initial differences) remain the same. The law was established by French mathematician Blaise Pascal. Definition Pressure in water and air. Pascal's law applies only for fluids. Pascal's principle is defined as A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid This principle is stated mathematically as: ρ is the fluid density (in kilograms per cubic meter in the SI system); g is acceleration due to gravity (normally using the sea level acceleration due to Earth's gravity in metres per second squared); Explanation Pascal's principle applies to all fluids, whether gases or liquids. Applications See also References

Causal dynamical triangulation Causal dynamical triangulation (abbreviated as CDT) invented by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, and popularized by Fotini Markopoulou and Lee Smolin, is an approach to quantum gravity that like loop quantum gravity is background independent. This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. The Loops '05 conference, hosted by many loop quantum gravity theorists, included several presentations which discussed CDT in great depth, and revealed it to be a pivotal insight for theorists. It has sparked considerable interest as it appears to have a good semi-classical description. At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-d near the Planck scale, and reveals a fractal structure on slices of constant time. Introduction Derivation Advantages and Disadvantages Related theories See also References

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

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