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Fermi Coordinates
In the mathematical theory of Riemannian geometry, there are two uses of the term Fermi coordinates. In one use they are local coordinates that are adapted to a geodesic. In a second, more general one, they are local coordinates that are adapted to any world line, even not geodesical. Take a future-directed timelike curve \gamma=\gamma(\tau), \tau being the proper time along \gamma in the spacetime M. Assume that p=\gamma(0) is the initial point of \gamma. Fermi coordinates adapted to \gamma are constructed this way. Consider an orthonormal basis of TM with e_0 parallel to \dot\gamma. Transport the basis \_along \gamma(\tau) making use of Fermi-Walker's transport. The basis \_ at each point \gamma(\tau) is still orthonormal with e_0(\tau) parallel to \dot\gamma and is non-rotated (in a precise sense related to the decomposition of Lorentz transformations into pure transformations and rotations) with respect to the initial basis, this is the physical meaning of Fermi-Walker's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line". The noun '' geodesic'' and the adjective ''geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
World Line
The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from concepts such as an "orbit" or a "trajectory" (e.g., a planet's ''orbit in space'' or the ''trajectory'' of a car on a road) by the ''time'' dimension, and typically encompasses a large area of spacetime wherein perceptually straight paths are recalculated to show their ( relatively) more absolute position states—to reveal the nature of special relativity or gravitational interactions. The idea of world lines originates in physics and was pioneered by Hermann Minkowski. The term is now most often used in relativity theories (i.e., special relativity and general relativity). Usage in physics In physics, a world line of an object (approximated as a point in space, e.g., a particle or observer) is the sequence of spacetime events correspon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermi–Walker Transport
Fermi–Walker transport is a process in general relativity used to define a coordinate system or reference frame such that all curvature in the frame is due to the presence of mass/energy density and not to arbitrary spin or rotation of the frame. Fermi–Walker differentiation In the theory of Lorentzian manifolds, Fermi–Walker differentiation is a generalization of covariant differentiation. In general relativity, Fermi–Walker derivatives of the spacelike vector fields in a frame field, taken with respect to the timelike unit vector field in the frame field, are used to define non-inertial and non-rotating frames, by stipulating that the Fermi–Walker derivatives should vanish. In the special case of inertial frames, the Fermi–Walker derivatives reduce to covariant derivatives. With a (-+++) sign convention, this is defined for a vector field ''X'' along a curve \gamma(s): :\frac=\frac - \left(X,\frac\right) V + (X,V)\frac, where is four-velocity, is the covariant der ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christoffel Symbol
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and the "architect of the atomic bomb". He was one of very few physicists to excel in both theoretical physics and experimental physics. Fermi was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity by neutron bombardment and for the discovery of transuranium elements. With his colleagues, Fermi filed several patents related to the use of nuclear power, all of which were taken over by the US government. He made significant contributions to the development of statistical mechanics, Quantum mechanics, quantum theory, and nuclear physics, nuclear and particle physics. Fermi's first major contribution involved the field of statistical mechanics. After Wolfgang Pauli formulated his Pauli exclusion principle, exclusion pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature ''at a point'' of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Reference Frame (flat Spacetime)
A proper reference frame in the theory of relativity is a particular form of accelerated reference frame, that is, a reference frame in which an accelerated observer can be considered as being at rest. It can describe phenomena in curved spacetime, as well as in "flat" Minkowski spacetime in which the spacetime curvature caused by the energy–momentum tensor can be disregarded. Since this article considers only flat spacetime—and uses the definition that special relativity is the theory of flat spacetime while general relativity is a theory of gravitation in terms of curved spacetime—it is consequently concerned with accelerated frames in special relativity. (For the representation of accelerations in inertial frames, see the article Acceleration (special relativity), where concepts such as three-acceleration, four-acceleration, proper acceleration, hyperbolic motion etc. are defined and related to each other.) A fundamental property of such a frame is the employment of the pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic Normal Coordinates
In differential geometry, normal coordinates at a point ''p'' in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of ''p'' obtained by applying the exponential map to the tangent space at ''p''. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point ''p'', thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point ''p'', and that the first partial derivatives of the metric at ''p'' vanish. A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at ''p'' only), and the geodesics through ''p'' are locally linear functions of ''t'' (the affine para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christoffel Symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group . As a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
