Glossary |
Brief, simple definitions of terms
which may be used on this site. Closed Set -- In R^{n}, a set which contains all of its limit points -- in other words, every point which is not in a closed set is surrounded by an open ball which is also not in the set. In general, a closed set is one whose complement is open. Compact Set -- In R^{n}, a set which is closed and bounded. In general, a compact set is one for which every open cover has a finite subcover. Contravariant Tensor -- A tensor which follows the inverse transformation law from a covariant tensor. In particular, ordinary vectors are contravariant tensors. So, contravariant tensors include ordinary vectors and all tensor products of ordinary vectors, and covariant tensors include covectors and all tensor products of covectors. Any linear map of covectors to the real numbers is called contravariant, and any linear map of vectors to the real numbers is called covariant. CAVEAT -- Some authors reverse the meanings of the terms "contravariant tensor" and "covariant tensor". If you encounter either word out of context, be sure you know what the author's intention is. Covariant Tensor -- When we change from one set of coordinates to another in a vector space, a covariant tensor transforms according to the same rules that the basis vectors follow. If the change of basis matrix is P, such that the new coordinates of a vector X are found via X' = PX, then the new basis vectors, expressed in terms of the old basis, are B' = P^{-1}B. A covariant rank 1 tensor follows a similar-looking rule: F' = P^{-1}F. At any rate, this is the traditional justification for the name "covariant"; it's not a very strong argument, in my opinion. CAVEAT -- Some authors reverse the meanings of the terms "contravariant tensor" and "covariant tensor". If you encounter either word out of context, be sure you know what the author's intention is. Covector -- also called a dual vector -- A linear map from vectors into the real numbers. The set of covectors on a particular vector space also forms a vector space. In R^{n} with the usual Cartesian metric, the covectors are naturally isomorphic to the vectors. In that case, a vector may be viewed as a linear map from vectors to real numbers via dot product. In more general vector spaces there is no such distinguished, natural isomorphism between vectors and covectors. Dual Vector -- Another name for a covector. The name "dual vector" is used by Wald and many differential geometry books. It's more cumbersome than covector, however, which is why I've used the latter term on this website. Flat Space -- Space in which the metric is the Lorentz metric: the square of the proper distance between any two points is -Δt^{2}+Δx^{2}+Δy^{2}+Δz^{2} (assuming an appropriate coordinate system is used). When mass is present, and hence there is a "gravitational field", space is not flat: the metric is not Lorentz. Special relativity describes the behavior of things in flat space. In curved space, the metric varies from point to point, and the proper distance between points must be found by integrating rather than just taking the differences of the coordinates of the start and end points. General relativity is needed to obtain a complete description of things in curved space. Geodesic -- A (timelike) geodesic is the path followed by a particle in free fall. In locally flat coordinates, a geodesic represents motion in a straight line at constant velocity. A spaceship in orbit follows a geodesic; a book sitting on a table does not (it's not in free fall -- the table is pushing on it). More generally, a geodesic may be defined as a smooth path whose tangent vector's covariant derivative with respect to itself is zero. Equivalently, the the tangent vector of a geodesic is "parallel transported" by the geodesic (if you slide it along the geodesic, it remains a tangent vector to the geodesic -- it doesn't twist). Equivalently, a geodesic may be defined as a curve which traverses the longest proper distance between any two nearby points along the path. These definitions are most apparently equivalent if we view them in locally flat coordinates. Index of Refraction -- In vacuum, the speed of light (or any electromagnetic radiation) is c, or 299,792,458 meters per second. In any material other than vacuum, light travels more slowly than c. For a particular material, the ratio of c to the speed of electromagnetic radiation in that material is its refractive index. The refractive index of a particular material typically varies depending on the frequency of the radiation. Vacuum has a refractive index of 1. The refractive index of air is very slightly larger. Typical glass has a refractive index for visible light of roughly 1.5, which means light travels about 2/3 as fast in glass as it does in vacuum. The refractive index of material used in printed circuit boards is the determining factor in signal speed along the traces of the boards. The refractive index of printed circuit board material may be on the order of 4 or larger, which means signals in PC boards typically travel at no more than 1/4 c. Locally Flat Coordinates -- A coordinate system in which, at one particular point P, space appears to be "flat". The metric at P is the ordinary Lorentz metric used in special relativity, and the derivative of the metric, in all directions, is zero at P. In locally flat coordinates, objects at P behave as they do in Newtonian mechanics: if they're stationary and no force is applied, they remain stationary. If they're moving, they keep moving. The local coordinates of an object in free fall are locally flat. A coordinate system in a spaceship in orbit is locally flat. The coordinate system used at a fixed point on the surface of the Earth is not locally flat. Manifold -- A "smooth" object which, near each point, "looks like" ordinary Cartesian space. A 2 dimensional manifold is very much like a smooth rubber sheet, or a plane. A 3-dimensional manifold is like the world we live in. Relativity represents "space-time" as a 4-dimensional manifold. A "flat" N-dimensional manifold is exactly like ordinary N-dimensional Cartesian space; a "curved" manifold may be thought of as being a smoothly curved structure embedded in a higher dimensional Cartesian space. A 2-sphere is a 2-dimensional curved manifold, typically embedded in 3-dimensional space. A field of vectors on a manifold defines a vector at each point on the manifold. An electric field, for example, defines a vector field on the spacetime manifold. (For a more precise definition of a manifold, see any differential geometry text.) MCRF -- Momentarily Comoving Reference Frame -- An inertial frame of reference which happens to be moving in the same direction, at the same speed, as an object or an accelerated frame which we're examining. Many questions in relativity can only be addressed in an inertial frame. In many cases, when the frame under consideration isn't inertial, such questions can still be addressed by considering the MCRF of the accelerated frame. Open Ball -- In R^{n}, the set of all points on the interior of an n-dimensional sphere of nonzero radius. Open Cover -- A collection of "open" sets which completely "covers", or "contains", another set or a manifold. A "finite" open cover is one that contains a finite number of open sets. Open Set -- In R^{n}, a set which contains an open ball about every point. On a general manifold, an open set is one whose projection into R^{n} via any set of coordinate functions is open. In a general topological space, an open set is just, well, an open set (the concept is fundamental in that case, and the "topology" on the space is exactly the specification of which sets are considered "open"). Partition of Unity -- A collection of smooth functions, F_{i}, and an open cover, O_{i}, such that only a finite number of the F_{i} are nonzero at any point, and at every point, Σ_{i}F_{i} = 1. Partitions of unity are typically used to "stitch together" functions which are defined only locally into global functions which cover an entire manifold. Smooth Function -- One that's infinitely differentiable. Most interesting functions in physics are smooth. Tensor -- A multilinear map from covectors and vectors into the real numbers. A vector maps covectors to real numbers and is hence a "rank (1,0)" tensor. A covector maps vectors into real numbers and is hence a "rank (0,1)" tensor. In Cartesian space, R^{n}, the ordinary dot product maps pairs of vectors into real numbers, and is a "rank (0,2)" tensor. Page created in 2004. Reformated on 11/16/06. Most recent minor additions, 1/04/09 |