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Stress–energy Tensor
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity. Definition The stress–energy tensor involves the use of superscripted variables (''not'' exponents; see tensor index notation and Einstein summation notation). If Cartesian coordinates in SI units are used, then the components of the position four-vector are given by: , , , and , where ''t'' is time in seconds, and ''x'', ''y'', and ''z'' are distances in meters. The stress–energy tensor is defined as the tensor '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Index Notation
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century. A component of a tensor is a real number that is used as a coefficient of a basis element for the tensor space. The tensor is the sum of its components multiplied by their corresponding basis elements. Tensors and tensor fields can be expressed in terms of their components, and operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comoving And Proper Distances
In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects. ''Proper distance'' roughly corresponds to where a distant object would be at a specific moment of cosmological time, which can change over time due to the expansion of the universe. ''Comoving distance'' factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space (though this may change due to other, local factors, such as the motion of a galaxy within a cluster). Comoving distance and proper distance are defined to be equal at the present time. At other times, the Universe's expansion results in the proper distance changing, while the comoving distance remains constant. Comoving coordinates Although general relativity allows one to formulate the laws of physics using arbitrary coordinates, some coordinate choices are more natural or easier to work with. Com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Energy Density
In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly called specific energy or . Often only the ''useful'' or extractable energy is measured, which is to say that inaccessible energy (such as rest mass energy) is ignored. In cosmological and other general relativistic contexts, however, the energy densities considered are those that correspond to the elements of the stress–energy tensor and therefore do include mass energy as well as energy densities associated with pressure. Energy per unit volume has the same physical units as pressure and in many situations is synonymous. For example, the energy density of a magnetic field may be expressed as and behaves like a physical pressure. Likewise, the energy required to compress a gas to a certain volume may be determined by multiplying the difference between the gas pressure and the external pressure by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion Tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves (or rather the rotation of the Frenet–Serret frame about the tangent vector). In the geometry of surfaces, the ''geodesic torsion'' describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames "roll" along a curve "without twisting". More generally, on a differentiable manifold equipped with an affine connection (that is, a connection in the tangent bundle), torsion and curvature form the two fundamental invariants of the connection. In this context, torsion gives an intrinsic characterization of how tangent spaces twist about a curve when they are parallel transported; whereas curvature describes how the tangent spaces roll al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Tensor
In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The tensor has application in general relativity and special relativity, as well as quantum mechanics, relativistic quantum mechanics, and quantum field theory. The special Euclidean group SE(''d'') of direct isometries is generated by translations and rotations. Its Lie algebra is written \mathfrak(d). This article uses Cartesian coordinates and tensor index notation. Background on Noether currents The Noether current for translations in space is momentum, while the current for increments in time is energy. These two statements combine into one in spacetime: translations in spacetime, i.e. a displacement between two events, is generated by the four-momentum ''P''. Conservation of four-momentum is given by the continuity equation: :\partial_\nu T^ = 0\,, where T^ \, is the stress–energy tensor, and ∂ are partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein–Cartan Theory
In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation similar to general relativity. The theory was first proposed by Élie Cartan in 1922. Einstein–Cartan theory is the simplest Poincaré gauge theory. Overview Einstein–Cartan theory differs from general relativity in two ways: (1) it is formulated within the framework of Riemann–Cartan geometry, which possesses a locally gauged Lorentz symmetry, while general relativity is formulated within the framework of Riemannian geometry, which does not; (2) an additional set of equations are posed that relate torsion to spin. This difference can be factored into by first reformulating general relativity onto a Riemann–Cartan geometry, replacing the Einstein–Hilbert action over Riemannian geometry by the Palatini action over Riemann–Cartan geometry; and second, removing the zero torsion constraint from the Palatini action, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitation (book)
''Gravitation'' is a widely adopted textbook on Albert Einstein's general theory of relativity, written by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. It was originally published by W. H. Freeman and Company in 1973 and reprinted by Princeton University Press in 2017. It is frequently abbreviated ''MTW'' (for its authors' last names). The cover illustration, drawn by Kenneth Gwin, is a line drawing of an apple with cuts in the skin to show the geodesics on its surface. The book contains 10 parts and 44 chapters, each beginning with a quotation. The bibliography has a long list of original sources and other notable books in the field. While this may not be considered the best introductory text because its coverage may overwhelm a newcomer, and even though parts of it are now out of date, it remains a highly valued reference for advanced graduate students and researchers. Content Subject matter After a brief review of special relativity and flat spacetime, ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-momentum
In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy and three-momentum , where is the particle's three-velocity and the Lorentz factor, is p = \left(p^0 , p^1 , p^2 , p^3\right) = \left(\frac E c , p_x , p_y , p_z\right). The quantity of above is ordinary non-relativistic momentum of the particle and its rest mass. The four-momentum is useful in relativistic calculations because it is a Lorentz covariant vector. This means that it is easy to keep track of how it transforms under Lorentz transformations. The above definition applies under the coordinate convention that . Some authors use the convention , which yields a modified definition with . It is also possible to define covaria ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the ''x''-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of analytic geometry. Common coordinate systems Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the ''number line''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate of a po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector (geometric)
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a '' directed line segment'', or graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by \overrightarrow . A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word ''vector'' means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from ''A'' to ''B''. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-vector
In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (,) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). Four-vectors describe, for instance, position in spacetime modeled as Minkowski space, a particle's four-momentum , the amplitude of the electromagnetic four-potential at a point in spacetime, and the elements of the subspace spanned by the gamma matrices inside the Dirac algebra. The Lorentz group may be represented by 4×4 matrices . The action of a Lorentz transformation o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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