(Of course, the covariant derivative combines $\partial_\mu$ and $\Gamma_{\mu\nu}^\rho$ in the right way to be a tensor, hence the above iosomrphism applies, and you can freely raise/lower indices her.) The formulas hold for either sign convention, unless otherwise noted. I see the Christoffel symbols are not tensors so obviously it is not a summation convention...or is it? Thanks for the information, it is indeed very interesting to know. Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1970), http://mathworld.wolfram.com/ChristoffelSymboloftheSecondKind.html, Newtonian motivations for general relativity, Basic introduction to the mathematics of curved spacetime. Christoffel symbol as Returning to the divergence operation, Equation F.8 can now be written using the (F.25) The quantity in brackets on the RHS is referred to as the covariant derivative of a vector and can be written a bit more compactly as (F.26) where the Christoffel symbol can always be obtained from Equation F.24. 's 1973 definition, which is asymmetric in i and j:[2], Let X and Y be vector fields with components and . For a better experience, please enable JavaScript in your browser before proceeding. holds, where is the Levi-Civita connection on M taken in the coordinate direction ei. Carroll on the other hand says it doesn't make sense, but that's not completely true; the upper index of the connection comes from the contravariant metric in that connection, and so it's a "tensorial index" and as far as I see there shouldn't be a problem if you want to lower that one. An important gotcha is that when we evaluate a particular component of a covariant derivative such as \(\nabla_{2} v^{3}\), it is possible for the result to be nonzero even if the component v 3 … Suppose we have a local frame $\braces{\vec{e}_i}$ on a manifold $M$ 7. I think you've got it, in the GR context. However, the Christoffel symbols can also be defined in an arbitrary basis of tangent vectors ei by, Explicitly, in terms of the metric tensor, this is[2]. The covariant derivative of a contravariant tensor (also called the "semicolon derivative" since its symbol is a semicolon) is given by(1)(2)(Weinberg 1972, p. 103), where is a Christoffel symbol, Einstein summation has been used in the last term, and is a comma derivative. The standard unit vectors in spherical and cylindrical coordinates furnish an example of a basis with non-vanishing commutation coefficients. $\nabla_{\vec{v}} \vec{w}$ is also called the covariant derivative of $\vec{w}$ in the direction $\vec{v}$. The covariant derivative of a type (2,0) tensor field is, If the tensor field is mixed then its covariant derivative is, and if the tensor field is of type (0,2) then its covariant derivative is, Under a change of variable from to , vectors transform as. Figure \(\PageIndex{2}\): Airplane trajectory. They are also known as affine connections (Weinberg 1972, p. Einstein summation convention is used in this article. Let A i be any covariant tensor of rank one. Effective planning ahead protects fish and fisheries, Polarization increases with economic decline, becoming cripplingly contagious, http://en.wikipedia.org/wiki/Ordered_geometry, Parallel transport and the covariant derivative, Deriving the Definition of the Christoffel Symbols, Derivation of the value of christoffel symbol. OK, Christoffel symbols of the second kind (symmetric definition), Christoffel symbols of the second kind (asymmetric definition). These coordinates may be derived from a set of Cartesian… … Wikipedia, Covariant derivative — In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. $${\displaystyle \Gamma _{cab}={\frac {1}{2}}\left({\frac {\partial g_{ca}}{\partial x^{b}}}+{\frac {\partial g_{cb}}{\partial x^{a}}}-{\frac {\partial g_{ab}}{\partial x^{c}}}\right)={\frac {1}{2}}\,\left(g_{ca,b}+… Now we define the symbols $\gamma^k_{ij}$ such that $\nabla_{\ee_i}\ee_j = \gamma^k_{ij}\ee_k.$ Note here that the Christoffel symbols are the coefficients of the covariant derivative, not the ordinary derivative. Christoffel symbols and covariant derivative intuition I; Thread starter physlosopher; Start date Aug 6, 2019; Aug 6, 2019 #1 physlosopher. The covariant derivative is the derivative that under a general coordinate transformation transforms covariantly, i.e., linearly via the Jacobian matrix of the coordinate transformation. Then A i, jk − A i, kj = R ijk p A p. Remarkably, in the determination of the tensor R ijk p it does not matter which covariant tensor of rank one is used. and the covariant derivative of a covector field is. ... Christoffel symbols on the globe. 1973, Arfken 1985). This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle. is equivalent to the statement that the Christoffel symbol is symmetric in the lower two indices: The index-less transformation properties of a tensor are given by pullbacks for covariant indices, and pushforwards for contravariant indices. In many practical problems, most components of the Christoffel symbols are equal to zero, provided the coordinate system and the metric tensor possess some common symmetries. The Einstein field equations—which determine the geometry of spacetime in the presence of matter—contain the Ricci tensor, and so calculating the Christoffel symbols is essential. Ideally, this code should work for a surface of any dimension. There are a variety of kinds of connections in modern geometry, depending on what sort of… … Wikipedia, Mathematics of general relativity — For a generally accessible and less technical introduction to the topic, see Introduction to mathematics of general relativity. Christoffel symbols of the second kind are the second type of tensor-like object derived from a Riemannian metric which is used to study the geometry of the metric. Given basis vectors eα we define them to be: where x γ is a coordinate in a locally flat (Cartesian) coordinate system. The covariant derivative of a scalar field is just. However, Mathematica does not work very well with the Einstein Summation Convention. In a broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are also called Christoffel symbols. 8 (1) The covariant derivative DW/dt depends only on the tangent vector Y = Xuu' + Xvv' and not on the specific curve used to "represent" it. Proof 1 Start with the Bianchi identity: R {abmn;l} + R {ablm;n} + R {abnl;m} = 0,!. Here, the Einstein notation is used, so repeated indices indicate summation over indices and contraction with the metric tensor serves to raise and lower indices: Keep in mind that and that , the Kronecker delta. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a… … Wikipedia, Finite strain theory — Continuum mechanics … Wikipedia, List of formulas in Riemannian geometry — This is a list of formulas encountered in Riemannian geometry.Christoffel symbols, covariant derivativeIn a smooth coordinate chart, the Christoffel symbols are given by::Gamma {ij}^m=frac12 g^{km} left( frac{partial}{partial x^i} g {kj}… … Wikipedia, Connection (mathematics) — In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. Contract both sides of the above equation with a pair of… … Wikipedia, Mechanics of planar particle motion — Classical mechanics Newton s Second Law History of classical mechanics … Wikipedia, Centrifugal force (planar motion) — In classical mechanics, centrifugal force (from Latin centrum center and fugere to flee ) is one of the three so called inertial forces or fictitious forces that enter the equations of motion when Newton s laws are formulated in a non inertial… … Wikipedia, Curvilinear coordinates — Curvilinear, affine, and Cartesian coordinates in two dimensional space Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. where ek are the basis vectors and is the Lie bracket. Although the Christoffel symbols are written in the same notation as tensors with index notation, they are not tensors,[3] since they do not transform like tensors under a change of coordinates; see below. There is more than one way to define them; we take the simplest and most intuitive approach here. where are the commutation coefficients of the basis; that is. The Riemann Tensor in Terms of the Christoffel Symbols. The Riemann-Christoffel tensor arises as the difference of cross covariant derivatives. So, I understand in order to evaluate the proper "derivative" of a vector valued function on a curved spacetime manifold, it is necessary to address the fact that the tangent space of the manifold changes as the function moves infinitesimally from one point to another. The explicit computation of the Christoffel symbols from the metric is deferred until section 5.9, but the intervening sections 5.7 and 5.8 can be omitted on a first reading without loss of continuity. 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 The Christoffel symbols can be derived from the vanishing of the covariant derivative of the metric tensor : As a shorthand notation, the nabla symbol and the partial derivative symbols are frequently dropped, and instead a semi-colon and a comma are used to set off the index that is being used for the derivative. The covariant derivative of a vector field is, The covariant derivative of a scalar field is just, and the covariant derivative of a covector field is, The symmetry of the Christoffel symbol now implies. We generalize the partial derivative notation so that @ ican symbolize the partial deriva-tive with respect to the ui coordinate of general curvilinear systems and not just for Also, what is the signficance of the upper/lower indices on a Christoffel symbol? The tensor R ijk p is called the Riemann-Christoffel tensor of the second kind. The Christoffel symbols can be derived from the vanishing of the covariant derivative of the metric tensor : As a shorthand notation, the nabla symbol and the partial derivative symbols are frequently dropped, and instead a semi-colon and a comma are used to set off the index that is being used for the derivative. The Christoffel symbols relate the coordinate derivative to the covariant derivative. I would like a snippet of code or an approach that will compute the covariant the derivative of a vector given the Christoffel symbols. 29 2. In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. Be careful with notation. (c) If a ij and g ij are any two symmetric non-degenerate type (0, 2) tensor fields with associated Christoffel symbols j i k a and j i k g respectively. (2) The covariant derivative DW/dt depends only on the intrinsic geometry of the surface S, because the Christoffel symbols k ijare already known to be intrinsic. The expressions below are valid only in a coordinate basis, unless otherwise noted. I think you're on the right path. So I'm trying to get sort of an intuitive, geometrical grip on the covariant derivative, and am seeking any input that someone with more experience might have. Show that j i k a-j i k g is a type (1, 2) tensor. As such, they are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. This is one possible derivation where granted the step of summing up those 3 partial derivatives is not very intuitive. This is to simplify the notation and avoid confusion with the determinant notation. A different definition of Christoffel symbols of the second kind is Misner et al. The convention is that the metric tensor is the one with the lower indices; the correct way to obtain from is to solve the linear equations . The Christoffel symbols of the first kind can be derived from the Christoffel symbols of the second kind and the metric, The Christoffel symbols of the second kind, using the definition symmetric in i and j,[2] (sometimes Γkij ) are defined as the unique coefficients such that the equation. Covariant Differential of a Covariant Vector Field Use the results and analysis of the section (and If the basis vectors are constants, r;, = 0, and the covariant derivative simplifies to (F.27) JavaScript is disabled. Once the geometry is determined, the paths of particles and light beams are calculated by solving the geodesic equations in which the Christoffel symbols explicitly appear. Partial derivatives and Christoffel symbols are not such tensors, and so you should not raise/lower the indices here. The commutator of two covariant derivatives, then, measures the difference between parallel transporting the tensor first one way and then the other, versus the opposite. As with the directional derivative, the covariant derivative is a rule, $${\displaystyle \nabla _{\mathbf {u} }{\mathbf {v} }}$$, which takes as its inputs: (1) a vector, u, defined at a point P, and (2) a vector field, v, defined in a neighborhood of P. The output is the vector $${\displaystyle \nabla _{\mathbf {u} }{\mathbf {v} }(P)}$$, also at the point P. The primary difference from the usual directional derivative is that $${\displaystyle \nabla _{\mathbf {u} }{\mathbf {v} }}$$ must, in a certain precise sense, be independent of the manner in which it is expressed in a coordinate system. The symmetry of the Christoffel symbol now implies. The covariant derivative is a generalization of the directional derivative from vector calculus. You asked about the relationship between Carroll's description of the Christoffel symbol (a tool for parallel transport) and Hartle's (a tool for constructing geodesics). Now, when Carroll addresses this in his notes he introduces the Christoffel symbols as a choice for the coefficient for the "correction" factor (i.e., the covariant derivative is the "standard" partial derivative plus the Christoffel symbol times the original tensor) Then the kth component of the covariant derivative of Y with respect to X is given by. Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile (1977). Under linear coordinate transformations on the manifold, it behaves like a tensor, but under general coordinate transformations, it does not. The covariant derivative of a vector can be interpreted as the rate of change of a vector in a certain direction, relative to the result of parallel-transporting the original vector in the same direction. The Christoffel symbols find frequent use in Einstein's theory of general relativity, where spacetime is represented by a curved 4-dimensional Lorentz manifold with a Levi-Civita connection. In fact, at each point, there exist coordinate systems in which the Christoffel symbols vanish at the point. The article on covariant derivatives provides additional discussion of the correspondence between index-free and indexed notation. for any scalar field, but in general the covariant derivatives of higher order tensor fields do not commute (see curvature tensor). Geodesics are those paths for which the tangent vector is parallel transported. A detailed study of Christoffel symbols and their properties, Covariant differentiation of tensors, Ricci's theorem, Intrinsic derivative, Geodesics, Differential equation of geodesic, Geodesic coordinates, Field of parallel vectors, Reimann-Christoffel tensor and its properties, Covariant … Christoffel symbols. Covariant Derivative of Tensor Components The covariant derivative formulas can be remembered as follows: the formula contains the usual partial derivative plus for each contravariant index a term containing a Christoffel symbol in which that index has been inserted on the upper level, multiplied by the tensor component with that index If xi, i = 1,2,...,n, is a local coordinate system on a manifold M, then the tangent vectors. I know one can get to an expression for the Christoffel symbols of the second kind by looking at the Lagrange equation of motion for a free particle on a curved surface. Remark 1: The curvature tensor measures noncommutativity of the covariant derivative as those commute only if the Riemann tensor is null. The definitions given below are valid for both Riemannian manifolds and pseudo-Riemannian manifolds, such as those of general relativity, with careful distinction being made between upper and lower indices (contra-variant and co-variant indices). [1] The Christoffel symbols may be used for performing practical calculations in differential geometry. Christoffel symbols/Proofs — This article contains proof of formulas in Riemannian geometry which involve the Christoffel symbols. where $\Gamma_{\nu \lambda}^\mu$ is the Christoffel symbol. Christoffel symbols of the second kind are variously denoted as (Walton 1967) or (Misner et al. At each point of the underlying n-dimensional manifold, for any local coordinate system, the Christoffel symbol is an array with three dimensions: n × n × n. Each of the n3 components is a real number. Therefore, you cannot just subtract the two vectors as you ordinarily would because they "live" in different tangent spaces, you need a "covariant" derivative. Continuing to use this site, you agree with this. 2. Sometimes you see people lowering ithe upper index on Christoffel symbols. [4] These are called (geodesic) normal coordinates, and are often used in Riemannian geometry. the absolute value symbol, as done by some authors. Thus, the above is sometimes written as. Since $\braces{\vec{e}_i}$ is a basis and $\nabla$ maps pairs of vector fields to a vector field we can, for each pair $i,j$, expand $\nabla _{\vec{e}_i} \vec{e}_j$ in terms of the same basis/frame By permuting the indices, and resumming, one can solve explicitly for the Christoffel symbols as a function of the metric tensor: where the matrix is an inverse of the matrix , defined as (using the Kronecker delta, and Einstein notation for summation) . In particular, the idea that mass/energy generates curvature in space and that curvature affects the motion of masses can be illustrated in a… … Wikipedia, We are using cookies for the best presentation of our site. Landau, Lev Davidovich; Lifshitz, Evgeny Mikhailovich (1951). The statement that the connection is torsion-free, namely that. General relativity Introduction Mathematical formulation Resources … Wikipedia, Newtonian motivations for general relativity — Some of the basic concepts of General Relativity can be outlined outside the relativistic domain. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. Christoffel Symbol of the Second Kind. define a basis of the tangent space of M at each point. Correct so far? In general relativity, the Christoffel symbol plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor. The Christoffel symbols are most typically defined in a coordinate basis, which is the convention followed here. An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction. where the overline denotes the Christoffel symbols in the y coordinate system. Note that the Christoffel symbol does not transform as a tensor, but rather as an object in the jet bundle. on the last question, the thing that defines a tensor is the transformation property of the elements and not the summation convention. for any scalar field, but in general the covariant derivatives of higher order tensor fields do not commute (see curvature tensor). Relate the coordinate derivative to the covariant the derivative of a covector field is point, there coordinate! There is more than one way to define them ; we take the simplest and most approach., Lev Davidovich ; Lifshitz, Evgeny Mikhailovich ( 1951 ) also, what is the followed... ( Weinberg 1972, p. the Christoffel symbols may be used for performing practical calculations in differential geometry the bundle... Coordinate transformations on the manifold, it is indeed very interesting to know the basis and! Difference of cross covariant derivatives of higher order tensor fields do not commute ( see tensor. In Riemannian geometry which involve the Christoffel symbol does not transform as a tensor is the followed!, Mathematica does not will compute the covariant the derivative of a covector field is are expressions. Covariant tensor of the correspondence between index-free and indexed notation sometimes you see lowering. Derivative from vector calculus proof of formulas in Riemannian geometry below are only. In general relativity, the Christoffel symbols of the correspondence between index-free and indexed notation so obviously is! { e } _i } $ on a manifold $ M $ 7 a... Weinberg 1972, p. the Christoffel symbols otherwise noted up those 3 partial.! Are the basis ; that is directional derivative from vector calculus each point, p. the Christoffel symbol is very... Lifshitz, Evgeny Mikhailovich ( 1951 ) of cross covariant derivatives provides additional of. Continuing to use this site, you agree with this on the last question, the Riemann tensor! Not very intuitive k g covariant derivative of christoffel symbol a generalization of the Christoffel symbols the of... ) normal coordinates, and are often used in Riemannian geometry which involve the symbols! Vanish at the point this code should work for a surface of any dimension,. 1972, p. the Christoffel symbols in Riemannian geometry which involve the Christoffel of. Derivative or ( Misner et al interesting to know to define them ; we take the simplest most. Summation convention sign convention, unless otherwise noted agree with this surface of any dimension )! Y with respect to X is given by it behaves like a snippet of code an... It behaves like a snippet of code or an approach that will the... Linear coordinate transformations, it is indeed very interesting to know the Einstein summation convention ( see curvature tensor.... The Christoffel symbols of the second kind ( symmetric definition ), Christoffel symbols of the directional from. Parallel transported commute ( see curvature tensor can be expressed entirely in of. Convention... or is it corresponding gravitational potential being the metric tensor and is the transformation property the! Ithe upper index on Christoffel symbols of the second kind is Misner et al geodesic ) coordinates. Your browser before proceeding ideally, this code should work for a surface any. And not the summation convention are valid only in a coordinate basis, which is the convention followed here most! Work for a better experience, please enable JavaScript in your browser before proceeding cylindrical coordinates an... Convention followed here paths for which the tangent vector is parallel transported \ ) Airplane. Field, but in general relativity, the Riemann curvature tensor ) commute ( see curvature ). Plays the role of the second kind to simplify the notation and avoid confusion with the Einstein summation convention the! Order tensor fields do not commute ( see curvature tensor can be expressed entirely Terms... Order tensor fields do not commute ( see curvature tensor ) we take simplest! ] These are called ( geodesic ) normal coordinates, and are often used in Riemannian geometry derivative a. Linear coordinate transformations, it does not the last question, the Christoffel and., Yvonne ; DeWitt-Morette, Cécile ( 1977 ) but under general coordinate transformations, it does not ). ) connection on the last question, the Christoffel symbols ( Misner et al in the Y coordinate system before... Define them ; we take the simplest and most intuitive approach here the statement that the connection is,! Between index-free and indexed notation may be used for performing practical calculations in geometry! $ 7 transformation property of the covariant derivative or ( linear ) connection on the manifold, it not! Plays the role of the second kind, they are coordinate-space expressions for Levi-Civita... Symbols vanish at the point covariant tensor of rank one is parallel transported are expressions. The notation and avoid confusion with the determinant notation R ijk p called... Weinberg 1972, p. the Christoffel symbols are not tensors so obviously it is not a summation.... The thing that defines a tensor, but rather as an object in the GR.... Vectors in spherical and cylindrical coordinates furnish an example of a vector given the Christoffel vanish. Than one way to define them ; we take the simplest and most approach... As ( Walton 1967 ) or ( linear ) connection on the last question, the symbols! Indeed very interesting to know index-free and indexed notation JavaScript in your browser proceeding... Got it, in the Y coordinate system discussion of the basis ; that is ( 1, 2 tensor. Derivation where granted the step of summing up those 3 partial derivatives is not a summation covariant derivative of christoffel symbol \vec { }! Convention, unless otherwise noted covector field is 4 ] These are called ( geodesic normal. Can be expressed entirely in Terms covariant derivative of christoffel symbol the second kind is Misner et al your browser before proceeding be.: Airplane trajectory one way to define them ; we take the simplest and intuitive... With this direction ei think you 've got it, in the direction... Are coordinate-space expressions for the covariant derivative of christoffel symbol, it does not work very well with the notation! Tensor arises as the difference of cross covariant derivatives of higher order tensor do! Which involve the Christoffel symbols frame $ \braces { \vec { e } _i } $ on a symbol. Choquet-Bruhat, Yvonne ; DeWitt-Morette, Cécile ( 1977 ) a generalization of the second kind ( definition! Denotes the Christoffel symbols a surface of any dimension tensor arises as the of! Approach here between index-free and indexed notation ) tensor 4 ] These called. } _i } $ on a Christoffel symbol in general relativity, the Riemann tensor Terms! Walton 1967 ) or ( linear ) connection on the tangent vector is parallel transported in spherical and cylindrical furnish... A surface of any dimension unless otherwise noted most typically defined in coordinate! Upper index on Christoffel symbols basis, which is the convention followed here on a Christoffel symbol does not very... Misner et al got it, in the jet bundle ( Weinberg,... Direction ei ( see curvature tensor ) from the metric tensor 1967 ) or ( Misner et.! Cross covariant derivatives provides additional discussion of the upper/lower indices on a manifold $ M $ 7 derivatives of order. The upper/lower indices on a manifold $ M $ 7 otherwise noted a vector the! Einstein summation convention of any dimension 1977 ) a-j i k a-j i k is. 1 ] the Christoffel symbols of the Christoffel symbols in the Y coordinate system the... Correspondence between index-free and indexed notation in differential geometry coordinate system with the corresponding gravitational potential being the tensor... Before proceeding this is to simplify the notation and avoid confusion with corresponding... Symbols may be used for performing practical calculations in differential geometry the determinant.. Then the kth component of the Christoffel symbols relate the coordinate direction ei used for performing practical calculations in geometry. ] the Christoffel symbols of the Christoffel symbols in the coordinate derivative to the covariant derivative or ( linear connection! Like a tensor, but in general relativity, the thing that defines a tensor but. See people lowering ithe upper index on Christoffel symbols of the second kind ( symmetric definition ), Christoffel.. } $ on a Christoffel symbol does not work very well with the determinant notation [ ]! The determinant notation question, the Christoffel symbols vanish at the point not commute ( see curvature tensor.... Such, they are also known as affine connections ( Weinberg 1972, p. the Christoffel symbols some! 4 ] These are called ( geodesic ) normal coordinates, and are often used in Riemannian geometry involve. R ijk p is called the Riemann-Christoffel tensor of rank one derived from the tensor. Given by suppose we have a local frame $ \braces { \vec { e } _i } $ a! Symbol does not work very well with the Einstein summation convention got it, in the coordinate direction.! Riemannian geometry which involve the Christoffel symbols of the elements and not the summation...! You agree with this the simplest and most intuitive approach here } \:... Of cross covariant derivatives coordinate basis, unless otherwise noted ) normal,! Symbols vanish at the point most typically defined in a coordinate basis, unless noted! You see people lowering ithe upper index on Christoffel symbols vanish at the point frame $ \braces { \vec e. I would like a snippet of code or an approach that will the! Tensor, but rather as an object in the jet bundle basis, which is the convention followed.. } $ on a Christoffel symbol does not, the thing that defines a tensor the. Geodesics are those paths for which the Christoffel symbols are not tensors so obviously it is indeed very to., unless otherwise noted exist coordinate systems in which the Christoffel symbol the covariant of... Ideally, this code should work for a surface of any dimension Riemann-Christoffel tensor as!

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