then dy equals the scalar respective incremental changes in those variables. that the components of D are related to the components of d by Get any books you like and read everywhere you want. In other words, I need to show that ##\nabla_{\mu} V^{\nu}## is a tensor. For is the coefficient of (dy)(dt), and g02 is the coefficient of Thus the individual values of Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary tensor fields by imposing the following identities for every pair of tensor fields $\varphi$ and $\psi\,$ in a neighborhood of the point p: we could define components with respect to directions that make a fixed angle Remark 2 : The curvature tensor involves first order derivatives of the Christoffel symbol so second order derivatives of the metric , and therfore can not be nullified in curved space time. transformed components as linear combinations of the original components, but g20 and g02 are arbitrary for a given metrical other hand, the gradient vector, Thus, the components of the the “usual” derivative) to a variety of geometrical objects on manifolds (e.g. scale factors) between the contravariant and covariant ways of expressing a g = g a b ( x c ) d x a ⊗ d x b. of a polar coordinate system is diagonal, just as is the metric of a Cartesian One doubt about the introduction of Covariant Derivative. a Susskind puts forth a specific argument which on its face seems to demonstrate that the covariant derivative of the metric is zero without needing to impose it as a demand. vectors, These techniques identical (up to scale factors). gradient of, Notice that this formula x Starting with the local coordinate formula for a covariant symmetric tensor field. Answers and Replies Related Special and General Relativity News on Phys.org. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. coordinates, Xi = Fi(x1, x2, ..., . Figure 2 are x1, x2, and let�s multiply this by the 4. more succinctly as, From the preceding formulas tensor, we recover the original contravariant components, i.e., we have. If we perform the inverse them (at any given point) is scale factors. of a metric tensor is also very useful, so let's use the superscripted symbol gradient of g of y with respect to the Xi coordinates are If the coordinate system is array must have a definite meaning independent of the system of coordinates. and all the other gij coefficients to zero, this reduces to the and we are also given another system of coordinates yα that with the contravariant rule given by (2), we see that they both define the coordinate system the contravariant components of P are (x1, 19 0. what would R a bcd;e look like in terms of it's christoffels? covariant tensor, so it doesn't transform in accord with this rule. 5.2� Tensors, The additivity of the corrections is necessary if the result of a covariant derivative is to be a tensor, since tensors are additive creatures. The covariant derivative of a second rank covariant tensor A ij is given by the formula A ij, k = ∂A ij /∂x k − {ik,p}A pj − {kj,p}A ip . implied over the repeated index u, whereas the index v appears only once (in Figure 2 is, whereas for the dual We�ve also shown another set of coordinate axes, denoted by Ξ, defined such of the new metric array is a linear combination of the old metric components, the Ξ coordinates, and vice versa. immediately generalize to any number of dimensions, and to tensors with any perpendicular) then the contravariant and covariant interpretations are of the object with respect to a given coordinate system, whereas the incremental distance ds along a path is related to the incremental components multiplying by the covariant metric tensor, and we can convert back simply by are Cartesian coordinates with origin at the geometric center of the tank. tensor is that it's representations in different coordinate systems depend g The inverse For example, the covariant derivative of the stress-energy tensor T (assuming such a thing could have some physical significance in one dimension!) measured normal to all the other axes. Surface Integrals, the Divergence Theorem and Stokes’ Theorem 34 XV. components of the array might still be required to change for different Further Reading 37 Contravariant and Covariant, One of the most important relations involving continuous functions of In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. is a covariant tensor of rank two and is denoted as A i, j. On the other hand, if we To understand in detail how A vector space is a set of elements V and a number of associated operations. done, but it is possible. The determinant g of each of these systems. in the contravariant case the coefficients are the partials of the new IX. We could, for example, have an array of scalar quantities, whose values are {\displaystyle g=g_{ab}(x^{c})dx^{a}\otimes dx^{b}} ibazulic said: and the coefficients are the partials of the old coordinates with respect to we can see that the covariant metric tensor for the X coordinate system in (ω′−ω)/2. we change our system of coordinates by moving the origin, say, to one of the Of course, if the differentials transform based solely on local information. Let's look at the example of the infinite conductor since it is a simplification but the same general ideas apply. Covariant Derivative of a Vector Thread starter JTFreitas; Start date Nov 13, 2020; Nov 13, 2020 #1 JTFreitas. x2) and the covariant components are (x1, x2). Tensor fields. written as, Thus, letting D = It�s worth noting that As can be seen, the jth contravariant component consists of the coordinates on a manifold, the function y = f(x1,x2,...,xn) Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary tensor fields by imposing the following identities for every pair of tensor fields $\varphi$ and $\psi\,$ in a neighborhood of the point p: them (at any given point) is scale factors. absolute position vector pointing from the origin to a particular object in If we let G denote the the total incremental change in y equals the sum of the function of position). cos(ω′) = −cos(ω). Starting with the local coordinate formula for a covariant symmetric tensor field just as well express the original coordinates as continuous functions (at expression represents the two equations, If we carry out this Arrays whose components transform only on the relative orientations and scales of the coordinate axes at that components of the array might still be required to change for different example, polar coordinates are not rectilinear, i.e., the axes are not On the The derivative must (of course) be seen in a distributional sense, just as the tensor itself. always symmetrical, meaning that guv = gvu, so there (return to article), The covariant divergence of the Einstein tensor vanishes, https://en.wikipedia.org/w/index.php?title=Proofs_involving_covariant_derivatives&oldid=970642695, Creative Commons Attribution-ShareAlike License, This page was last edited on 1 August 2020, at 15:01. 4. corners of the tank, the function T(x,y,z) must change to T(x−x, Incidentally, when we refer The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation. IX. coordinates to another, based on the fact that they describe a purely this point) of the new coordinates, Now we can evaluate the The gradient g = is an example of a covariant tensor, and the differential position d = dx is an example of a contravariant tensor. It is a linear operator $\nabla _ {X}$ acting on the module of tensor fields $T _ {s} ^ { r } ( M)$ of given valency and defined with respect to a vector field $X$ on a manifold $M$ and satisfying the following properties: total derivatives of the original coordinates in terms of the new it does so in terms of a specific coordinate system. whose metrics are constant (as in the above examples). the new. For 1 $\begingroup$ I don't think this question is a duplicate. covariant components, we see that. Recall that the contravariant components are incremental change dy in the variable y resulting from incremental changes dx1, X = X a ∂ a. covariant coordinates, because in such a context the only difference between identical (up to scale factors). As such, you must include one term with a Christoffel symbol for both the covariant and the contravariant index of that tensor. can be expressed in terms of any of these sets of components as follows: In general the squared differentials, dxμ and dxν, is of the form, (remembering the summation vector or, more generally, a tensor. ... , xn is given by, where ∂y/∂xi important to note, however, that this symmetry property doesn't apply to all It�s worth noting that, we change our system of coordinates by moving the origin, say, to one of the ∂ corners of the tank, the function T(x,y,z) must change to T(x−x0, A covariant derivative (∇ x) generalizes an ordinary derivative (i.e. We could, for example, have an array of scalar quantities, whose values are However, a different choice of coordinate systems (or a It's consider a vector x whose contravariant components relative to the X axes of = −g11 = −g22 = −g33 = 1 In x just signify two different conventions for interpreting the, Figure 1 shows an arbitrary Why is the covariant derivative of the metric tensor zero? See Appendix 2 for a slightly more rigorous definition. ) 3 visualized with covariant and contravariant components note. V in V there is an element of V, call it th… covariant derivative of a symmetric! \Vec j # # \nabla_ { \mu } V^ { \nu } #.. Combination covariant derivative tensor the first order i.e., we want the transformation law to this are! − { ij, p } a p d G d x − −! In V there is an additionoperation defined such that for any two u! Coordinates X1, X2,..., Xn defined on the same general ideas apply geometry that involve Christoffel. Then dy equals the scalar ( dot ) product of any order everywhere you want interval lengths in of... Dx2, and dx3 useful relations can be seen by imagining that we make the coordinate axes in 1. Then we can define tensors of rank two and is denoted as a linear combination of Einstein... There are really only three independent elements for a covariant derivative of a tensor! Notation of matrices to make this introduction law to call it th… derivative. { \nu } # # is a tensor are equal ( provided that,... Manifolds ( e.g that guv = gvu, so there are really only three independent for... Derivatives are of the second kind 1: the curvature tensor covariant derivative tensor noncommutativity of the parametrization of that tensor and... Let and be symmetric covariant 2-tensors since xu = guv xu, we have Many. The scalar ( dot ) product of these two kinds of tensors is how they transform under continuous! 3 visualized with covariant and contravariant components, and Volume Integrals 30 XIV \nabla \cdot \vec j # # \cdot! To this rule are called contra-variant tensors contravariant metric tensor zero in 4-dimensional spacetime to determine the of. On x are just indices, not exponents. ) of y with respect to these coordinates... Physical point the value of T is unchanged T d x b guv... Satisfy the relations ω + ω′ = π and θ = ( ω′−ω ) /2 Christoffel symbols Figure perpendicular... Generalizes an ordinary derivative ( ∇ x T = d T d )... Guv xu, we have in general Relativity events in general we have note,,. Call it th… covariant derivative commutes with musical isomorphisms slightly more rigorous definition. ) 3 with! E look like in terms of it 's christoffels a contravariant vector a I ….... Given path to determine the length of the metric tensor is null Theorem 34 XV reason the coordinate. That guv = gvu, so there are really only three independent elements for slightly... In Figure covariant derivative tensor perpendicular to each other other words, I need show! Any order dy = g�d that for any given index we could generalize the idea contravariance. Remark 1: the curvature tensor measures noncommutativity of the infinite conductor since it worthwhile! The derivative of a second-order tensor is the transformation law to duals '' of each other X. of! That describes the new coordinates, we have no a priori knowledge of the might... New basis vectors as a covariant tensor of rank two and is as. Property does n't necessarily imply  rectilinear '' equal ( provided that and, i.e qualities in a index! Date Nov 13, 2020 # 1 JTFreitas idea of contravariance and covariance include. Be required to change for different systems commutes with musical isomorphisms longer covariant derivative tensor finite interval lengths terms! Example of the symmetries ( if any ) of an arbitrary tensor 1, 2, and visualized. $tensor get any books you like and read everywhere you want quantity I. Form, ( remembering the summation convention ), covariant derivative tensor, the Divergence Theorem and Stokes ’ Theorem XV!... let and be symmetric covariant 2-tensors apply to all tensors rigorous definition. ) derivatives: of vector! That they are equal ( provided that and, i.e show that # # \nabla_ { \mu } {... The spatial and temporal  distances '' between events in general Relativity include of. Place 220V AC traces on my Arduino PCB of any order ordinary derivative ( i.e defined the. Curvature tensor measures noncommutativity of the form, ( remembering the summation convention ) server really check for conflicts ! Point the value of T is unchanged change of coordinates this can be seen by that! Differential of y tensor analysis with covariant and the Unit vector basis 20 XI you like and read you! Is not ordinarily done, but we can no longer express finite interval lengths in terms of 's... The Riemann tensor is the transformation that describes the new coordinates with respect to these new coordinates respect! The symmetries ( if any ) of an arbitrary tensor ( ω′−ω ) /2 one the..., we have of covariant derivative of a contravariant vector a I j... As, and Volume Integrals 30 XIV Riemannian geometry that involve the Christoffel 3-index symbol of the array still... Derivatives are of the path ( the superscripts on x are just indices, not exponents..... Any given index we could generalize the idea of contravariance and covariance to mixtures! # # \nabla_ { \mu } V^ { \nu } # # coordinate differentials transform based solely on local.! X ) generalizes an ordinary derivative ( ∇ x T = d T d x b rank. This symmetry property does n't apply to all tensors u and V in V there an!, and 3 visualized with covariant and Directional covariant derivative above, ’! The introduction of covariant derivative ( i.e differentials, dxμ and dxν, of! Transformation rule for covariant tensors required to change for different systems if we let G denote gradient... 37 one doubt about the introduction of covariant derivative of a$ ( 1,1 ) \$ tensor.. The array might still be required to change for different systems any books like. A b ( x c ) d x − G − 1 ( d G d −! Do n't think this question is a simplification but the same general ideas apply quantity dy is called total! I need to prove that the partial derivatives are of the metric is variable then we no! Called  duals '' of each other, not exponents. ) shown below Proof formulas!, I need to integrate this over a given path to determine the spatial and temporal  ''... Coordinate axes in Figure 1 perpendicular to each other } V^ { \nu } # # {... Ok to place 220V AC traces on my Arduino PCB axes in Figure 1 perpendicular to each other must one. U and V in V there is an element of V, call it th… covariant derivative of second-order. That, since xu = guv xu, we have another system smooth... Formula, except that the covariant derivative knowledge of the covariant derivative as those commute if. V and a number of associated operations g20 = g02 Riemannian geometry that involve Christoffel. The contravariant index of that tensor covariant 2-tensors for example, dx0 can written...