Space has dimensions. Vectors {basis vector} can lie along coordinate axes. Other vectors are basis-vector linear combinations.
contravariant
For point, coefficients a(i) times basis vectors e(i) result in point coordinates x(i): a(i) * e(i) = x(i), where i is number of dimensions. Points can move. New coefficients x(i) of same dimensions e(i) relate to old coefficients a(i): a(i) * e(i) = x(i).
covariant
For points, coordinate system can change. New-dimension coefficients a(i) relate to old dimensions e(i), because new dimensions are linear transformations x(i) of old dimensions: a(i) = x(i) * e(i).
relation
Covariant components relate to contravariant components. If basis vectors are orthogonal, covariant components and contravariant components are equal. If basis vectors are curved coordinates, a(i) = g(i,j) * a(j), where a(i) and a(j) are coefficients and g(i,j) is tensor relating basis vectors e(i) ... e(j). Some g(i,j) components are for covariance, some for contravariance, and some for both. g(i,j) elements are functions of curved-space positions. g(i,j) elements are 1 or 0 for flat space with orthogonal basis vectors.
Tensor can express coordinates {coordinate system}. For example, space points can be vectors: a*i + b*j + c*k. Coordinates can be perpendicular or not perpendicular. Physical quantity depends on coordinates and can use tensor form. For example, momentum can be vector: mass * (a*i + b*j + c*k).
Surface can deviate from flatness {curvature, tensor}.
local
Curvature is at point and is local property.
intrinsic
Curvature is intrinsic to surface and does not depend on outside reference points.
curve
For curves, curvature at point is curvature-radius reciprocal.
surface
For surface points, curvature R is product of reciprocals of maximum curvature radius R1 and minimum curvature radius R2: R = (1 / R1) * (1 / R2). Rotating plane around normal to surface can find both.
surface: triangle
Triangles are three geodesics. Curvature is triangle-angle sum minus pi radians, all divided by triangle area: r = (angle sum - pi) / area.
surface: sphere
Surface curvature is area by surface projection onto a sphere, divided by surface area: r = (projection area) / area.
surface: normals
Curvature is solid angle in radians by normals to surface over surface region, divided by surface area.
surface: volume
Volume can also find curvature.
surface: hexagon
To measure curvature around point, use regular hexagon around the point and measure angles.
bending
If surface bends without stretching, curvature stays constant, because one radius increases and other radius decreases proportionally.
space-time: curvature tensor
In space-time, geodesic deviations, measured along each dimension, are metric second derivative and are second-order differential forms. The result is fourth-order tensor whose matrix coefficients have one covariant and three contravariant indices. For four-dimensional space-time, tensor has 20 independent terms. Metric coefficients are potentials.
Tensor is skew-symmetric and cyclic symmetric and is commutative, with diagonal terms equal zero.
space-time: Weyl tensor
Tensors {Weyl tensor} can measure gravity-field tidal distortion.
space-time: Riemann curvature tensor
At points, total space curvature depends on Ricci tensors and Weyl tensors. Curved space can be Euclidean space {Riemannian manifold} locally. Curved space-time can be Minkowski space {Lorentzian manifold} {pseudo-Riemannian manifold} {semi-Riemannian manifold} locally.
space-time: distance curvature
Second-order tensor {distance curvature} can derive from vector curvature, by contracting matrix coefficients.
space-time: scalar curvature
Tensor trace is scalar invariant. First-order tensors {scalar curvature} can derive from distance curvatures, by contracting to put metric-coefficient second derivatives into linear forms. Scalar curvature is the only such invariant. Physical world has four dimensions, because only such manifold results in curvature invariance.
Functions have functions {form, mathematics}.
types
Functions are 0-form manifolds.
Exterior derivatives of functions with basis vectors are 1-forms {differential form} {covector} {covariant vector} {1-form, basis}. Tensors operate on 1-forms to give real numbers. Linear operations on vectors can give integral numbers of equally spaced phase surfaces, of de-Broglie particle waves, through which vector passes. Force, energy, momentum, and velocity directional derivatives are 1-form gradients. Anticommutative tensor products and wedge products of two 1-forms are 2-forms. Anticommutative wedge products of three 1-forms are 3-forms. Antisymmetric covariant tensors are k-forms.
algebra
Exterior-derivative wedge products form exterior algebra (Elie Cartan).
dual
p-forms and (n-p)-forms are duals on n-manifolds {Hodge star}. 1-form and vector field are duals and interact to make scalar products. Tangent vector has covector dual.
Two manifold points have shortest path {geodesic, tensor}| between them. Geodesic is straightest possible direction between two points.
metric
Quadratic differential linear metric forms can measure geodesic length: ds^2. Geodesic length is sum from points i = 1 to i = m, and from points j = 1 to j = m, of g(i, j) * du(i) * du(j). Coefficients g(i, j) = Du(i) / Du(j), where D are partial derivatives and u are coordinates.
geometry
Geodesic metric defines surface geometry at manifold points.
linear
Using only local operations allows geodesic to be linear.
operator
Geodesic metric operates on vectors to give squared lengths. Squared length can be greater than zero {space-like vector}, less than zero {time-like vector}, or equal to zero {light-like vector}.
space-time
In four-dimensional space-time, particles move along maximum spatial-length lines, which is the shortest possible time as measured in particle reference frame. In flat space-time, geodesics are straight lines. On spheres, geodesics are on great circles.
For surface, quadratic differential linear form can measure geodesic length {metric, length}|: ds^2.
Four-dimensional space has vector curvature with 4x4x4x4 terms, which are linear function-second-derivative combinations. However, terms are in pairs, so four-dimensional vector curvature can contract to 16 independent terms in 4x4 matrices {Ricci tensor}. The other four terms are vector components. Ricci tensor measures volume change, as gravity causes space to contract, and equals energy-momentum tensor. Ricci tensor equals mass-energy density, which is pressure.
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Date Modified: 2022.0225