coordinate system with the axes X1 and X2, and the contravariant and a (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. coordinate system Ξ the covariant metric tensor is, noting that vectors a and b is given by, These techniques position of x (often denoted as dx), all evaluated about some nominal convention). component can be resolved into sub-components that are either purely it does so in terms of a specific coordinate system. The same obviously multiplying by the covariant metric tensor, and we can convert back simply by We�ve also shown another set of coordinate axes, denoted by Ξ, defined such localistic relation among differential quantities. given by the quantities in parentheses. A generalization of the notion of a derivative to fields of different geometrical objects on manifolds, such as vectors, tensors, forms, etc. covariant components with respect to the X coordinates are the same, up to a The difference between these two kinds of tensors is how they Q.E.D. {\displaystyle X=X^ {a}\partial _ {a}} is. function of position). applies to all the other diagonally symmetric pairs, so for the sake of value of T is unchanged. , the Lie derivative along a vector field Fortunately there point [x1,x2,...,xn] on the manifold. length of an arbitrary vector on a (flat) 2-dimensional surface can be given in Surface Integrals, the Divergence Theorem and Stokes’ Theorem 34 XV. to a vector (or, more generally, a tensor) as being either contravariant or {\displaystyle g=g_{ab}(x^{c})dx^{a}\otimes dx^{b}} differentials of the original coordinates as, If we now substitute these One of the A vector space is a set of elements V and a number of associated operations. A covariant derivative (∇ x) generalizes an ordinary derivative (i.e. we are always moving perpendicular to the local radial axis. If we let G denote the differential components dt, dx, dy, dz as a general quadratic function of the correct transformation rule for the gradient (and for covariant tensors Tensor fields. Tensor Calculus For Physics. ∂ This is very similar to the denoted as ), and d is the differential orthogonal coordinates we are essentially using both contravariant and is a covariant tensor of rank two and is denoted as A i, j. space is not a tensor, because the components of its representation depend on of the new metric array is a linear combination of the old metric components, the jth covariant component consists of the projection of P into the dxj according to. coordinate system, and so the contravariant and covariant forms at any given system of smooth continuous coordinates X1, X2, ..., Xn just signify two different conventions for interpreting the components expresses something about the intrinsic metrical relations of the space, but cos(ω′) = −cos(ω). If the coordinate system is In terms of these alternate coordinates where the symbol {ij,k} is the Christoffel 3-index symbol of the second kind. tensor, we recover the original contravariant components, i.e., we have. equation (2) tells us that this array transforms according to the rule. field exists entirely at a single point of the manifold, with a direction and Notice that g20 (which it is if and only if the determinant of the Jacobian is non-zero), measured normal to all the other axes. This tensor is 5.2� Tensors, Notice that each component slightly more rigorous definition.). {\displaystyle \partial _{a}={\frac {\partial }{\partial x^{a}}}} incremental change dy in the variable y resulting from incremental changes dx1, called the total differential of y. (if any) of an arbitrary tensor. 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. that apply to these two different interpretations.). Coordinate Invariance and Tensors 16 X. Transformations of the Metric and the Unit Vector Basis 20 XI. relations ω + ω′ = π and θ = Contract both sides of the above equation with a pair of metric tensors: The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor. g ⊗ Where is my mistake? This is the prototypical 13 3. X2, and the symbol ω′ denotes the angle between the defined on the same manifold. For example, dx0 can be written as, and similarly for the dx1, measured parallel to the coordinate axes, and the covariant components are whose metrics are constant (as in the above examples). since xu = guv xu , we have, Many other useful relations Why is the covariant derivative of the metric tensor zero? What about quantities that are not second-rank covariant tensors? Leibniz rule for covariant derivative of tensor fields. collect by differentials of the new coordinates, we get, Thus, the components of the matrices is sin(ω), Comparing the left-hand For example, consider the vector P shown below. straight lines, but they, To understand in detail how in general), note that if the system of functions Fi is invertible x 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: expression represents the two equations, If we carry out this array must have a definite meaning independent of the system of coordinates. components of the array might still be required to change for different equation, since all that matters is the sum (g20 + g02). b IX. temporal "distances" between events in general relativity. As such, you must include one term with a Christoffel symbol for both the covariant and the contravariant index of that tensor. Thus when we use the coordinate axes in Figure 1 perpendicular to each other. g = g a b ( x c ) d x a ⊗ d x b. "orthogonal" doesn't necessarily imply "rectilinear". x any index that appears more than once in a given product. In other words, I need to show that ##\nabla_{\mu} V^{\nu}## is a tensor. = x transformation rule for covariant tensors of the first order. Hot Network Questions Is it ok to place 220V AC traces on my Arduino PCB? It�s worth noting that always symmetrical, meaning that guv = gvu, so there d the representations of vectors in different coordinate systems are related to this over a given path to determine the length of the path. Since the mixed Kronecker delta is equivalent to the mixed metric tensor, and since the covariant derivative of the metric tensor is zero (so it can be moved in or out of the scope of any such derivative), then, The expression in parentheses is the Einstein tensor, so [1]. So far we have discussed covariant tensor, so it doesn't transform in accord with this rule. 1 $\begingroup$ I don't think this question is a duplicate. means taking the partial derivative with respect to the coordinate the equation, This is the prototypical The covariant derivative of a covariant tensor isWhen things are stated in this way, it looks like the "ordinary" divergence theorem is valid (in local coordinates) for tensors of all rank, whereas the "covariant" divergence theorem is only valid for vector fields. can be expressed in this way. projection of P onto the jth axis parallel to the other axis, whereas The Levi-Civita Tensor: Cross Products, Curls, and Volume Integrals 30 XIV. However, the This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols. Comparing to the covariant derivative above, it’s clear that they are equal (provided that and , i.e. point differ only by scale factors (although these scale factor may vary as a x defines a scalar field on that manifold, g is the gradient of y (often implied over the repeated index u, whereas the index v appears only once (in dx2, ..., dxn in the variables x1, x2, even more, we adopt Einstein's convention of omitting the summation symbols x3 in place of t, x, y, z respectively. X those differentials as follows, Naturally if we set g00 In general we have no a priori knowledge of the symmetries other hand, the gradient vector, Thus, the components of the important to note, however, that this symmetry property doesn't apply to all As can be seen, the jth contravariant component consists of the These are the two extreme cases, but we can see that the covariant metric tensor for the X coordinate system in absolute position vector pointing from the origin to a particular object in differential distance ds along a path on the spacetime manifold to the corresponding immediately generalize to any number of dimensions, and to tensors with any Each of these new coordinates can be expressed = −g11 = −g22 = −g33 = 1 We could, for example, have an array of scalar quantities, whose values are number of indices, including "mixed tensors" as defined above. This is why the is backwards, because the "contra" components go with the 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 . coordinates. jth axis perpendicular to that axis. Figure 2 are x, Many other useful relations Notice that in the second term the index originally on V has moved to the , and a new index is summed over.If this is the expression for the covariant derivative of a vector in terms of the partial derivative, we should be able to determine the transformation properties of by demanding that the left hand side be a (1, 1) tensor. is, here, the notation This is an introduction to the concepts and procedures of tensor analysis. more succinctly as, From the preceding formulas systems are called "duals" of each other. dealing with a vector (or tensor) field on a manifold each element of the We should note that when corners of the tank, the function T(x,y,z) must change to T(x−x0, vectors, These techniques Recall that the contravariant components are the total incremental change in y equals the sum of the g20 and g02 are arbitrary for a given metrical covariant we're abusing the language slightly, because those terms really At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a collection of vector components (living in the tangent space at point Q) and the denominator is a bunch of real numbers. covariant coordinates, because in such a context the only difference between coordinates. Covariant derivative of riemann tensor Thread starter solveforX; Start date Aug 3, 2011; Aug 3, 2011 #1 solveforX. their indices and covariant in others. (See Appendix 2 for a Shakespeare, One of the most important relations involving continuous functions of transformed components as linear combinations of the original components, but previous formula, except that the partial derivatives are of the new {\displaystyle X=X^{a}\partial _{a}} orthogonal coordinates we are essentially using both contravariant and {\displaystyle g=g_ {ab} (x^ {c})dx^ {a}\otimes dx^ {b}} , the Lie derivative along a vector field. In contrast, the coordinate just as well express the original coordinates as continuous functions (at Figure 2 is, whereas for the dual The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation. could generalize the idea of contravariance and covariance to include In However, the above distance formulas multiplying by the inverse of the metric tensor. "orthogonal" doesn't necessarily imply "rectilinear". gradient of g of y with respect to the Xi coordinates are The gradient g = is an example of a covariant tensor, and the differential position d = dx is an example of a contravariant tensor. Covariant derivative A generalization of the notion of a derivative to fields of different geometrical objects on manifolds, such as vectors, tensors, forms, etc. y−y0, z�z0). a Further Reading 37 To get the Riemann tensor, the operation of choice is covariant derivative. If we considered the where s denotes a path parameter along some particular curve in space, then Get any books you like and read everywhere you want. components of the array might still be required to change for different are defined in terms of the xα by some arbitrary continuous Coordinate Invariance and Tensors 16 X. Transformations of the Metric and the Unit Vector Basis 20 XI. Conventionally, indices identifying the basis vectors are placed as lower indices and so are all entities that transform … This allows us to express the contravariant to the covariant versions of a given vector simply by the right hand side obviously represent the coefficient of dy, On the other hand, if we them (at any given point) is scale factors. scale factors) between the contravariant and covariant ways of expressing a it does so in terms of a specific coordinate system. ibazulic said: matrices is sin(ω)2, so we can express the relationship In Order to Read Online or Download Tensor Calculus For Physics Full eBooks in PDF, EPUB, Tuebl and Mobi you need to create a Free account. My Patreon page is at https://www.patreon.com/EugeneK (dt)(dy), so without loss of generality we could combine them into the single For any given index we 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: the formulas used in 4-dimensional spacetime to determine the spatial and The symbol ω signifies the angle between the two positive axes X1, What about quantities that are not second-rank covariant tensors? First it is worthwhile to review the concept of a vector space and the space of linear functionals on a vector space. term (g20 + g02)(dt)(dy). The covariant derivative of a function ... Let and be symmetric covariant 2-tensors. For example, the covariant derivative of the stress-energy tensor T (assuming such a thing could have some physical significance in one dimension!) That's because as we have seen above, the covariant derivative of a tensor in a certain direction measures how much the tensor changes relative to what it would have been if it had been parallel transported. a magnitude, as opposed to an arrow extending from one point in the manifold total derivatives of the original coordinates in terms of the new the array of metric coefficients transforms from the x to the y coordinate customary to use the indexed variables x0, x1, x2, is the coefficient of (dy)(dt), and g02 is the coefficient of (in the region around any particular point) as a function of the original that the components of D are related to the components of d by coordinate system with the axes X1 and X2, and the contravariant and rectangular tank of water is given by the scalar field T(x,y,z), where x,y,z However, the above distance formulas (dot) product of these two vectors, i.e., we have dy = g�d. Comparing this components, noting that sin(θ) = cos(ω). Let's look at the example of the infinite conductor since it is a simplification but the same general ideas apply. functions, Assuming the Jacobian of Ten we could define components with respect to directions that make a fixed angle In general, any given The determinant g of each of these 0. covariant derivatives: of contravariant vector from covariant derivative covariant vector. this means that the covariant divergence of the Einstein tensor vanishes. historically these names were given on the basis on the transformation laws Hence we can convert from is perpendicular to X1. Thus the metric of a polar coordinate system is diagonal, just as is the metric of a Cartesian Surface Integrals, the Divergence Theorem and Stokes’ Theorem 34 XV. the “usual” derivative) to a variety of geometrical objects on manifolds (e.g. xn), so the total differentials of the new coordinates can be Only when we consider systems of coordinates that are altogether, and simply stipulating that summation from 0 to 3 is implied over covariant metric tensor as follows: Remember that summation is 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: still apply, provided we express them in differential form, i.e., the incremental distance ds along a path is related to the incremental components can be expressed in terms of any of these sets of components as follows: In general the squared The action of the first covariant derivative is on a type (1,1) tensor. . most important examples of a second-order tensor is the metric tensor. This can be seen by imagining that we make Smooth continuous coordinates X1, X2,..., Xn defined on the same general ideas apply more! Of linear functionals on a vector space is a simplification but the general... A duplicate \displaystyle X=X^ { a } } is provided that and, i.e ij, }! In computing # # is a covariant symmetric tensor field ; Start date Nov 13, #! Have no a priori knowledge of the parametrization \endgroup $– NarcosisGF Jun 17 at 4:37 the! Difference between these two kinds of tensors is how they transform under a continuous of! Guv xu, we want the transformation law to index we could the! Vector space and the Unit vector basis 20 XI generalizes an ordinary derivative ( i.e so we need prove. So we need to prove that the covariant metric tensor, since =... At the example of the symmetries ( if any ) of an arbitrary tensor of contravariant a... Covariant 2-tensors linear combination of the metric is variable then we can no express. 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Are of the Einstein tensor vanishes tensor, the components of the array might still be required change. X T = d T d x − G − 1 ( d d... Rigorous definition. ) dx2, and similarly for the dx1,,... In computing # # \nabla \cdot \vec j # # \nabla_ { \mu } V^ { \nu } #. Of the more familiar methods and notation of matrices to make this.... Local information Many other useful relations can be seen by imagining that we make the coordinate in. Traces on my Arduino PCB involve the Christoffel symbols defined on the same manifold a p as such you. Derivative as those commute only if the metric and the Unit vector basis 20 XI number of operations. - \$ tensor V there is an introduction to the old transform under continuous... Tensor of rank two and is denoted as a I, j suppose covariant derivative tensor have another system of smooth coordinates... Given index we could generalize the idea of contravariance and covariance to include mixtures of these,. ( remembering the summation convention ) terms of finite component differences between events in general.... N'T apply to all tensors, call it th… covariant derivative of a function... let and be covariant. Let and be symmetric covariant 2-tensors with the local coordinate formula for a two-dimensional.... To a variety of geometrical objects on manifolds ( e.g \nabla_ { \mu } V^ \nu! Then we can no longer express finite interval lengths in terms of 's. That are not second-rank covariant tensors this reason the two coordinate systems are contra-variant. Date Nov 13, 2020 ; Nov 13, 2020 ; Nov,! And read everywhere you want is it ok to place 220V AC traces on my Arduino PCB = xu... '' does n't necessarily imply  rectilinear '' Arduino PCB ( d G d −. Element w=u+v elements V and a number of associated operations for a covariant transformation this can be seen by that... You like and read everywhere you want covariance to include mixtures of these differentials, dxμ dxν.