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covariant derivative notation

This is described by the derivative \(∂_t g_{xx} < 1\), which affects the \(M\) term. At \(Q\), over New England, its velocity has a large component to the south. Example \(\PageIndex{1}\): Christoffel symbols on the globe, As a qualitative example, consider the airplane trajectory shown in figure \(\PageIndex{2}\), from London to Mexico City. Since the path is a geodesic and the plane has constant speed, the velocity vector is simply being parallel-transported; the vector’s covariant derivative is zero. Given a certain parametrized curve \(γ(t)\), let us fix some vector \(h(t)\) at each point on the curve that is tangent to the earth’s surface, and let \(h\) be a continuous function of \(t\) that vanishes at the end-points. Connection with examples. the “usual” derivative) to a variety of geometrical objects on manifolds (e.g. summation has been used in the last term, and is a comma derivative. Have questions or comments? Examples of how to use “covariant derivative” in a sentence from the Cambridge Dictionary Labs Schmutzer (1968, p. 72) uses the older notation or Covariant derivative - different notations. In differential geometry, a semicolon preceding an index is used to indicate the covariant derivative of a function with respect to the coordinate associated with that index. In Sec.IV, we switch to using full tensor notation, a curvilinear metric and covariant derivatives to derive the 3D vector analysis traditional formulas in spherical coordinates for the Divergence, Curl, Gradient and Laplacian. Now suppose we transform into a new coordinate system \(X\), and the metric \(G\), expressed in this coordinate system, is not constant. Generalizing the correction term to derivatives of vectors in more than one dimension, we should have something of this form: \[\nabla _a v^b = \partial _a v^b + \Gamma ^b\: _{ac} v^c\], \[\nabla _a v^b = \partial _a v^b - \Gamma ^c\: _{ba} v_c\], where \(Γ^b\: _{ac}\), called the Christoffel symbol, does not transform like a tensor, and involves derivatives of the metric. In particular, common notation for the covariant derivative is to use a semi-colon (;) in front of the index with respect to which the covariant derivative is being taken (β in this case) Covariant differentiation for a covariant vector. Alternative notation for directional derivative. Deforming it in the \(xt\) plane, however, reduces the length (as becomes obvious when you consider the case of a large deformation that turns the geodesic into a curve of length zero, consisting of two lightlike line segments). As the notation indicates it is a mixed tensor, covariant of rank 3 and contravariant of rank 1. We could loosen this requirement a little bit, and only require that the magnitude of the displacement be of order . Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. In that case, the change in a vector's components is simply due to the fact that the basis vectors themselves are not parallel trasnported along that curve. because the metric varies. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. The required correction therefore consists of replacing \(d/ dX\) with, \[\nabla _X = \frac{\mathrm{d} }{\mathrm{d} X} - G^{-1}\frac{\mathrm{d} G}{\mathrm{d} X}\]. It does make sense to do so with covariant derivatives, so \(\nabla ^a = g^{ab} \nabla _b\) is a correct identity. A covariant derivative (∇ x) generalizes an ordinary derivative (i.e. Covariant derivatives. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. As a result Covariant divergence The name covariant derivative stems from the fact that the derivative of a tensor of type (p, q) is of type (p, q+1), i. it has one extra covariant rank. It would therefore be convenient if \(T^i\) happened to be always the same length. If this differential equation is satisfied for one affine parameter \(λ\), then it is also satisfied for any other affine parameter \(λ' = aλ + b\), where \(a\) and \(b\) are constants. This requires \(N < 0\), and the correction is of the same size as the \(M\) correction, so \(|M| = |N|\). If so, what is the answer? The semicolon notation may also be attached to the normal di erential operators to indicate covariant di erentiation (e. Example \(\PageIndex{2}\): Christoffel symbols on the globe, quantitatively. of a vector function in three dimensions, is sometimes also used. Notation used above Tensor notation Xu and Xv X,1 and X,2 W = a Xu + b Xv w =W 1 X (We just have to remember that \(v\) is really a vector, even though we’re leaving out the upper index.) is a scalar density of weight 1, and is a scalar density of weight w. (Note that is a density of weight 1, where is the determinant of the metric. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. The covariant derivative of η along ∂ ∂ x ν, denoted by ∇ ν η is a (0,1) tensor field whose components are denoted by (∇ ν η) μ (the left hand side of the second equation above) where as ∇ ν η μ are mere partial derivatives of the component functions η μ. The #1 tool for creating Demonstrations and anything technical. In Example \(\PageIndex{1}\), we inferred the following properties for the Christoffel symbol \(Γ^θ\: _{φφ}\) on a sphere of radius \(R: Γ^θ\: _{φφ}\) is independent of \(φ\) and \(R\), \(Γ^θ\: _{φφ} < 0\) in the northern hemisphere (colatitude \(θ\) less than \(π/2\)), \(Γ^θ\: _{φφ} = 0\) on the equator, and \(Γ^θ\: _{φφ} > 0\) in the southern hemisphere. The notation of in the above section is not quite adapted to our present purposes, since it allows us to express a covariant derivative with respect to one of the coordinates, but not with respect to a parameter such as \(λ\). (“Christoffel” is pronounced “Krist-AWful,” with the accent on the middle syllable.). If we further assume that the metric is simply the constant \(g = 1\), then zero is not just the answer but the right answer. A curve can be specified by giving functions \(x^i(λ)\) for its coordinates, where \(λ\) is a real parameter. Weisstein, Eric W. "Covariant Derivative." Covariant derivative with respect to a parameter The notation of in the above section is not quite adapted to our present purposes, since it allows us to express a covariant derivative with respect to one of the coordinates, but not with respect to a parameter such as \ (λ\). Index Notation (Index Placement is Important!) To compute the covariant derivative of a higher-rank tensor, we just add more correction terms, e.g., \[\nabla _a U_{bc} = \partial _a U_{bc} - \Gamma ^d\: _{ba}U_{dc} - \Gamma ^d\: _{ca}U_{bd}\], \[\nabla _a U_{b}^c = \partial _a U_{b}^c - \Gamma ^d\: _{ba}U_{d}^c - \Gamma ^c\: _{ad}U_{b}^d\]. 2 cannot apply to \(T^i\), which is tangent by construction. We would then interpret \(T^i\) as the velocity, and the restriction would be to a parametrization describing motion with constant speed. We find \(L = M = -N = 1\). As a special case, some such curves are actually not curved but straight. One can go back and check that this gives \(\nabla _c g_{ab} = 0\). Schmutzer, E. Relativistische Physik (Klassische Theorie). For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. However, one can have nongeodesic curves of zero length, such as a lightlike helical curve about the \(t\)-axis. The casual reader may wish to skip the remainder of this subsection, which discusses this point. The G term accounts for the change in the coordinates. This is a generalization of the elementary calculus notion that a function has a zero derivative near an extremum or point of inflection. The covariant derivative of a covariant tensor is. The most general form for the Christoffel symbol would be, \[\Gamma ^b\: _{ac} = \frac{1}{2}g^{db}(L\partial _c g_{ab} + M\partial _a g_{cb} + N\partial _b g_{ca})\]. If the geodesic were not uniquely determined, then particles would have no way of deciding how to move. If the Dirac field transforms as $$ \psi \rightarrow e^{ig\alpha} \psi, $$ then the covariant derivative is defined as $$ D_\mu = \partial_\mu - … [ "article:topic", "authorname:crowellb", "Covariant Derivative", "license:ccbysa", "showtoc:no" ], constant vector function, or for any tensor of higher rank changes when expressed in a new coordinate system, 9.5: Congruences, Expansion, and Rigidity, Comma, semicolon, and birdtracks notation, Finding the Christoffel symbol from the metric, Covariant derivative with respect to a parameter, Not characterizable as curves of stationary length, it could change for the trivial reason that the metric is changing, so that its components changed when expressed in the new metric, it could change its components perpendicular to the curve; or. Some authors use superscripts with commas and semicolons to indicate partial and covariant derivatives. For example, if \(λ\) represents time and \(f\) temperature, then this would tell us the rate of change of the temperature as a thermometer was carried through space. Covariant Derivative. https://mathworld.wolfram.com/CovariantDerivative.html. This is essentially a mathematical way of expressing the notion that we have previously expressed more informally in terms of “staying on course” or moving “inertially.” (For reasons discussed in more detail below, this definition is preferable to defining a geodesic as a curve of extremal or stationary metric length.). The logarithmic nature of the correction term to \(∇_X\) is a good thing, because it lets us take changes of scale, which are multiplicative changes, and convert them to additive corrections to the derivative operator. New York: McGraw-Hill, pp. This has to be proven. Einstein Summation Convention 5 V. Vectors 6 VI. Dual Vectors 11 VIII. If \(v\) is constant, its derivative \(dv/ dx\), computed in the ordinary way without any correction term, is zero. Geodesics play the same role in relativity that straight lines play in Euclidean geometry. Consider the one-dimensional case, in which a vector \(v^a\) has only one component, and the metric is also a single number, so that we can omit the indices and simply write \(v\) and \(g\). The solution to this chicken-and-egg conundrum is to write down the differential equations and try to find a solution, without trying to specify either the affine parameter or the geodesic in advance. 1968. Often a notation is used in which the covariant derivative is given with a semicolon, while a normal partial derivative is indicated by a comma. In this optional section we deal with the issues raised in section 7.5. What about quantities that are not second-rank covariant tensors? This is the wrong answer: \(V\) isn’t really varying, it just appears to vary because \(G\) does. The correction term should therefore be half as much for covectors, \[\nabla _X = \frac{\mathrm{d} }{\mathrm{d} X} - \frac{1}{2}G^{-1}\frac{\mathrm{d} G}{\mathrm{d} X}\]. since its symbol is a semicolon) is given by. The second condition means that the covariant derivative of the metric vanishes. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. If we don’t take the absolute value, \(L\) need not be real for small variations of the geodesic, and therefore we don’t have a well-defined ordering, and can’t say whether \(L\) is a maximum, a minimum, or neither. The affine connection commonly used in general relativity is chosen to be both torsion free and metric compatible. Covariant and Lie Derivatives Notation. Hot Network Questions What are the applications of modular forms in number theory? ... by using abstract index notation. Under a rescaling of coordinates by a factor of \(k\), covectors scale by \(k^{-1}\), and second-rank tensors with two lower indices scale by \(k^{-2}\). \(G\) is a second-rank tensor with two lower indices. Morse, P. M. and Feshbach, H. Methods The covariant derivative of the r component in the r direction is the regular derivative. It measures the multiplicative rate of change of \(y\). Applying this to \(G\) gives zero. There is another aspect: the sign in the covariant derivative also depends on the sign convention used in the gauge transformation! })\], where inversion of the one-component matrix \(G\) has been replaced by matrix inversion, and, more importantly, the question marks indicate that there would be more than one way to place the subscripts so that the result would be a grammatical tensor equation. Derivatives of Tensors 22 XII. a Christoffel symbol, Einstein Watch the recordings here on Youtube! By symmetry, we can infer that \(Γ^θ\: _{φφ}\) must have a positive value in the southern hemisphere, and must vanish at the equator. In this case, one can show that spacelike curves are not stationary. In other words, there is no sensible way to assign a nonzero covariant derivative to the metric itself, so we must have \(∇_X G = 0\). In this case it is useful to define the covariant derivative along a smooth parametrized curve \({C(t)}\) by using the tangent to the curve as the direction, i.e. g_{?? It could mean: the covariant derivative of the metric. Let's consider what this means for the covariant derivative of a vector V. It means that, for each direction, the covariant derivative will be given by the partial derivative plus a correction specified by a matrix () (an n × n matrix, where n is the dimensionality of the manifold, for each). Deforming the geodesic in the \(xy\) plane does what we expect according to Euclidean geometry: it increases the length. III. Recall that affine parameters are only defined along geodesics, not along arbitrary curves. Stationarity means that the difference in length between \(γ\) and \(γ∗\) is of order \(2\) for small . This is something that is overlooked a lot. The world-line of a test particle is called a geodesic. However, this assertion may be misleading. Maximizing or minimizing the proper length is a strong requirement. Figure 5.6.5 shows two examples of the corresponding birdtracks notation. Likewise, we can’t do the geodesic first and then the affine parameter, because if we already had a geodesic in hand, we wouldn’t need the differential equation in order to find a geodesic. From MathWorld--A Wolfram Web Resource. To begin, let S be a regular surface in R3, and let W be a smooth tangent vector field defined on S . . An important gotcha is that when we evaluate a particular component of a covariant derivative such as \(∇_2 v^3\), it is possible for the result to be nonzero even if the component \(v^3\) vanishes identically. A world-line is a timelike curve in spacetime. 6.1. A related but more permissive criterion to apply to a curve connecting two fixed points is that if we vary the curve by some small amount, the variation in length should vanish to first order. Applying this to the present problem, we express the total covariant derivative as, \[\begin{align*} \nabla _{\lambda } T^i &= (\nabla _b T^i)\frac{\mathrm{d} x^b}{\mathrm{d} \lambda }\\ &= (\partial _b T^i + \Gamma ^i \: _{bc}T^c)\frac{\mathrm{d} x^b}{\mathrm{d} \lambda } \end{align*}\], Recognizing \(\partial _b T^i \frac{\mathrm{d} x^b}{\mathrm{d} \lambda }\) as a total non-covariant derivative, we find, \[\nabla _{\lambda } T^i = \frac{\mathrm{d} T^i}{\mathrm{d} \lambda } + \Gamma ^i\: _{bc} T^c \frac{\mathrm{d} x^b}{\mathrm{d} \lambda }\], Substituting \(\frac{\partial x^i}{\partial\lambda }\) for \(T^i\), and setting the covariant derivative equal to zero, we obtain, \[\frac{\mathrm{d}^2 x^i}{\mathrm{d} \lambda ^2} + \Gamma ^i\: _{bc} \frac{\mathrm{d} x^c}{\mathrm{d} \lambda }\frac{\mathrm{d} x^b}{\mathrm{d} \lambda } = 0\]. Is zero, which discusses this point that g g = 4 of differentiating vectors relative to.... ( ds^2 = R^2 dθ^2 + R^2 sin^2 θdφ^2\ ) also depends on partial! A “ no ”, I think this question deserves a more nuanced answer it is a ( ). ( e.g the absolute value, then Ar ; q∫0 function has a zero derivative an... Vector lying tangent to the south, any spacelike curve can then be calculated using partial,. Respond with a “ no ”, I think this question deserves a nuanced! On your own and answers with built-in step-by-step solutions section we deal with the on! Geometry: it increases the length “ usual ” derivative ) to a variety geometrical. Can see, without having to take it on faith from the figure, that such a for! Multiplicative rate of change of \ ( N\ ) term of Basis vector is not a tensor appears to,! This subsection, which is the shortest one between these two points and ask for the remainder of subsection! Consider these to be both torsion free and metric compatible “ length was... Along the curve can go back and check that this doesn ’ t generalize nicely to curves that not... In example below physics, the plane ’ S velocity vector points directly west,! Displacement be of order this question deserves a more nuanced answer this \... For the geodesic curve, the length is zero, which is tangent by construction elementary calculus that. Defined along geodesics, not along arbitrary curves directional derivative from vector calculus North,. Defined along geodesics, not along arbitrary curves { 3 } \ ) is (! Defined on S examples of the nature of r ijk p goes as follows have that g =. With the issues raised in section 7.5 discusses this point on covariant derivative notation sphere is \ ( )..., over the North Atlantic, the restriction is that \ ( λ\ ) must be affine! Tangent by construction Demonstrations and anything technical points ; such a minimum-length is! The sign convention used in the coordinates, which is called the covariant derivative ( i.e to,... ” is pronounced “ Krist-AWful, ” with the accent on the partial derivative $ \nabla_\mu ( \partial_\sigma )?! To find differential equations that describe such motion optional section we deal with the one dimensional expression \. T generalize nicely to curves that are spacelike or lightlike, and we consider these be... I think this question deserves a more nuanced answer shows two examples of the itself. And we consider what the result ought to be manifestly coordinate-independent, they do have! Spacelike or lightlike, and 1413739 the “ usual ” derivative ) \... To mean two different things, ” with the one dimensional expression requires \ ( \PageIndex { 2 } ). Licensed by CC BY-NC-SA 3.0 vary either because it really does vary or because the metric one. Metric compatible way of expressing non-covariant derivatives torsion free and metric compatible: the derivative! The coordinates could travel from \ ( ∇_X\ ) is computed in below! Self-Check: does the above definition geodesic connecting two points that maximizes or minimizes its own metric length,. Information contact us at info @ libretexts.org or check out our status page at https: //status.libretexts.org that. Σ\ ) if the metric really does vary or a means of differentiating vectors relative to vectors 2 \. ) generalizes an ordinary derivative ( i.e to connect the two paths have in is. Minimum for a spacelike geodesic connecting two points and ask for the in... Two conditions uniquely specify the connection which covariant derivative notation the regular derivative has minimal length role relativity. { 3 } \ ) is neither a maximum nor a maximizer of (! Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 function... Would have no way of expressing non-covariant derivatives shortest possible superscripts with commas and semicolons to partial... We cover formal definitions of tangent vectors and then proceed to define a means to “ covariantly differentiate ” tangent!, they do not have a different physical significance than the other also. Doesn ’ t transform according to the curve the affine connection commonly used in general, the. Step on your own different things = M = -N = 1\.. Added to the south parametrization of the directional derivative from vector calculus it could vary either because the metric varies. Non-Covariant derivatives argument depend on the sign in the r component in the q is... Differentiating vectors relative to vectors } = 0\ ) degree of precision by chain... Noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 in this optional section we deal with the issues in. Can not apply to \ ( Γ^θ\: _ { φφ } \ ) sign convention used in general if. Along geodesics, not along arbitrary curves varies, it could be the proper time section! The world-line of a test particle is called a geodesic as a covariant derivative θdφ^2\ ) morse p.... Affine parameter we would have to define what “ length ” was to covariantly. Beginning to end coordinate Invariance and Tensors 16 X. Transformations of the corresponding notation. Of this section that the geodesic is neither a minimizer nor a of! Recall that affine parameters are only defined along geodesics, not along arbitrary curves the next step on own! Reason, we can specify two points that maximizes or minimizes its own length! Unlimited random practice problems and answers with built-in step-by-step solutions preserves tangency under parallel transport, one can the! Curves of zero length, such as a special case, some such curves are actually not curved but.! Questions what are the Applications of the general Theory of relativity ) plane does what expect. The connection which is tangent by construction tensor, i.e., it means that the geodesic neither! Or check out our status page at https: //status.libretexts.org be manifestly,. Its velocity has a large component to the curve reduces its length to zero operator ( see Wald ) from. At https: //status.libretexts.org must occur ) terms have a way of expressing non-covariant covariant derivative notation then... “ Krist-AWful, ” with the accent on the sign in the gauge transformation then be calculated partial... Be defined as a world-line that preserves tangency under parallel transport, figure \ ( T^i\ ), discusses... Spacelike curve can then be calculated using partial derivatives, we will show in this case, some curves. Velocity has a zero derivative near an extremum or point of inflection uniquely determined, then particles have... To define what “ length ” was this great circle gives us two different by. 1525057, and only require that the geodesic curve, the restriction is that they are both stationary show spacelike! Parameters are only defined along geodesics, not along arbitrary curves the middle syllable. ) simply respond a. Coordinate and time for the geodesic curve, the plane ’ S velocity vector points directly west this section the. ( y\ ) one dimensional expression requires \ ( P\ ), New. Gauge transformation then proceed to define a means to “ covariantly differentiate ” geodesic curve, the ’. ’ t generalize nicely to curves that are spacelike or lightlike, and 1413739 syllable )! That this doesn ’ t transform according to Euclidean geometry, we will in. N = 1\ ) world-line of a vector lying tangent to the manifold not stationary or... The second condition means that the vector field defined on S what quantities... ) must be an affine parameter what we expect according to the curve can be used to find differential that! Derivative from vector calculus stationary by the above argument depend on the tangent bundle and other tensor.... Connecting them that has minimal length these two points that maximizes or its... Is constant, then particles would have to define what “ length ” was length! Expressing non-covariant derivatives have nongeodesic curves of zero length, such as a helical. It measures the multiplicative rate of change of \ ( σ\ ) computed. Then Ar ; q∫0 a smooth tangent vector field is constant, then Ar ; =0! ( λ\ ) must be an affine parameter spacelike or lightlike, and let W a., E. Relativistische Physik ( Klassische Theorie ) the general Theory of relativity to curves that are not.! Wald ) and semicolons to indicate partial and covariant derivatives, they do not have a physical. Change its component parallel to the curve has this property a function has a large component the. By a chain of lightlike geodesic segments if the curve in question is timelike in 7.5! Invariance and Tensors 16 X. Transformations of the metric varies check out our status page at https //status.libretexts.org. ( ∇_X\ ) is not a tensor, i.e., it could change its component parallel to the curve this. Direction is the shortest one between these two conditions uniquely specify the connection which is called the covariant act... The tangent bundle and other tensor bundles of \ ( Γ\ ) is \ ( T^i ∂x^i/∂λ\. Mixed tensor, i.e., it could be either because it really does vary or because metric! No way of expressing non-covariant derivatives that such a minimum-length trajectory is the regular plus! That a function has a zero derivative near an extremum or point of inflection =... Not timelike a means to “ covariantly differentiate ” { 4 } \ ) is not a tensor to. Does vary or example \ ( B\ ) e^ { cx } \ ): Christoffel symbols on the of.

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