partial derivative of metric tensor

** DefTensor: Defining symmetric metric tensor metrich@-a,-bD. v {\displaystyle g} In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. Tensor shortcuts for easy entry of tensors. The inverse of Sg is a mapping T*M → TM which, analogously, gives an abstract formulation of "raising the index" on a covector field. Any covector field α has components in the basis of vector fields f. These are determined by, Denote the row vector of these components by, Under a change of f by a matrix A, α[f] changes by the rule. There is thus a natural one-to-one correspondence between symmetric bilinear forms on TpM and symmetric linear isomorphisms of TpM to the dual T∗pM. In a basis of vector fields f = (X1, ..., Xn), any smooth tangent vector field X can be written in the form. μ g 6 0 [itex]\mathcal{L}_M(g_{kn}) = -\frac{1}{4\mu{0}}g_{kj} g_{nl} F^{kn} F^{jl} \\ When {\displaystyle r} One of the core ideas of general relativity is that the metric (and the associated geometry of spacetime) is determined by the matter and energy content of spacetime. where Dy denotes the Jacobian matrix of the coordinate change. In order for the metric to be symmetric we must have. 1,b. The entries of the matrix G[f] are denoted by gij, where the indices i and j have been raised to indicate the transformation law (5). {\displaystyle v} Similarly, when t → [6] This isomorphism is obtained by setting, for each tangent vector Xp ∈ TpM. where where is a partial derivative, is the metric tensor, (4) where is the radius vector, and (5) Therefore, for an orthogonal curvilinear coordinate system, by this definition, (6) The symmetry of definition (6) means that (7) (Walton 1967). From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point. d In particular, the length of a tangent vector a is given by, and the angle θ between two vectors a and b is calculated by, The surface area is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. Connection coe cients are antisymmetric in their lower indices. The curvature of spacetime is then given by the Riemann curvature tensor which is defined in terms of the Levi-Civita connection ∇. , and defines For Lorentzian metric tensors satisfying the, This section assumes some familiarity with, Invariance of arclength under coordinate transformations, The energy, variational principles and geodesics, The notation of using square brackets to denote the basis in terms of which the components are calculated is not universal. Partial, covariant, total, absolute and Lie derivative routines for any dimension and any order. {\displaystyle ct} , the metric can be evaluated on The metric tensor with respect to arbitrary (possibly curvilinear) coordinates qi is given by, The unit sphere in ℝ3 comes equipped with a natural metric induced from the ambient Euclidean metric, through the process explained in the induced metric section. > Partial differentiation of a tensor is in general not a tensor. μ for some uniquely determined smooth functions v1, ..., vn. We note that the quantities V1, V., and Velas are the components of the same third-order tensor Vt with respect to different tenser bases, i.e. t Given two tangent vectors the linear functional on TpM which sends a tangent vector Yp at p to gp(Xp,Yp). d The flat space metric (or Minkowski metric) is often denoted by the symbol η and is the metric used in special relativity. Let, Under a change of basis f ↦ fA for a nonsingular matrix A, θ[f] transforms via, Any linear functional α on tangent vectors can be expanded in terms of the dual basis θ. where a[f] denotes the row vector [ a1[f] ... an[f] ]. The nondegeneracy of Let U be an open set in ℝn, and let φ be a continuously differentiable function from U into the Euclidean space ℝm, where m > n. The mapping φ is called an immersion if its differential is injective at every point of U. Associated to any metric tensor is the quadratic form defined in each tangent space by, If qm is positive for all non-zero Xm, then the metric is positive-definite at m. If the metric is positive-definite at every m ∈ M, then g is called a Riemannian metric. G The definition of the covariant derivative does not use the metric in space. More generally, for a tensor of arbitrary rank, the covariant derivative is the partial derivative plus a connection for each upper index, minus a connection for each lower index. {\displaystyle g} Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d(p, q) whose value at a pair of points p and q is the distance from p to q. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner). 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. In flat space in Cartesian coordinates, the partial derivative operator is a map from (k, l) tensor fields to (k, l + 1) tensor fields, which acts linearly on its arguments and obeys the Leibniz rule on tensor products. That is. x Let's look at the partial derivative first. ). In many cases, whenever a calculation calls for the length to be used, a similar calculation using the energy may be done as well. 0 where the dxi are the coordinate differentials and ∧ denotes the exterior product in the algebra of differential forms. r μ Metric tensor of spacetime in general relativity written as a matrix, Local coordinates and matrix representations, Friedmann–Lemaître–Robertson–Walker metric, fundamental theorem of Riemannian geometry, Basic introduction to the mathematics of curved spacetime, https://en.wikipedia.org/w/index.php?title=Metric_tensor_(general_relativity)&oldid=979589164, Articles which use infobox templates with no data rows, Wikipedia articles needing clarification from August 2017, Wikipedia articles needing clarification from May 2017, Creative Commons Attribution-ShareAlike License, This page was last edited on 21 September 2020, at 15:56. μ 1 Simplify, simplify, simplify ** DefTensor: Defining antisymmetric tensor epsilonmetrich@a,bD. Another is the angle between a pair of curves drawn along the surface and meeting at a common point. u In this case, the spacetime interval is written as, The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. g g d = @x . det The covariance of the components of a[f] is notationally designated by placing the indices of ai[f] in the lower position. The following notation is used: The metric tensor, gµν, has a signature of +2 and g = |det(gµν)|. , the interval is timelike and the square root of the absolute value of M {\displaystyle u} In other words, the components of a vector transform contravariantly (that is, inversely or in the opposite way) under a change of basis by the nonsingular matrix A. s d Some of them are without the event horizon or can be without the gravitational singularity. In the above coordinates, the matrix representation of η is, (An alternative convention replaces coordinate Certain metric signatures which arise frequently in applications are: Let f = (X1, ..., Xn) be a basis of vector fields, and as above let G[f] be the matrix of coefficients, One can consider the inverse matrix G[f]−1, which is identified with the inverse metric (or conjugate or dual metric). Whereas the metric itself provides a way to measure the length of (or angle between) vector fields, the inverse metric supplies a means of measuring the length of (or angle between) covector fields; that is, fields of linear functionals. Given local coordinates Furthermore, it is usually demanded that the field equations be at most of second.order in the derivatives of both sets of field functions. With coordinates. The Schwarzschild metric describes an uncharged, non-rotating black hole. means that this matrix is non-singular (i.e. , In components, (9) is. by the formula. ( 2 s Covariant derivative of determinant of the metric tensor. Throughout this article we work with a metric signature that is mostly positive (− + + +); see sign convention. When If the surface M is parameterized by the function r→(u, v) over the domain D in the uv-plane, then the surface area of M is given by the integral, where × denotes the cross product, and the absolute value denotes the length of a vector in Euclidean space. , the interval is spacelike and the square root of By the universal property of the tensor product, any bilinear mapping (10) gives rise naturally to a section g⊗ of the dual of the tensor product bundle of TM with itself, The section g⊗ is defined on simple elements of TM ⊗ TM by, and is defined on arbitrary elements of TM ⊗ TM by extending linearly to linear combinations of simple elements. In the usual (x, y) coordinates, we can write. Let M be a smooth manifold of dimension n; for instance a surface (in the case n = 2) or hypersurface in the Cartesian space ℝn + 1. and more generally that the components of a metric tensor in primed coordinate system could be expressed in non primed coordinates as: Each of the partial derivatives is a function of the primed coordinates so, for a region close to the event point P, we can expand these derivatives … ν To account for charge, the metric must satisfy the Einstein Field equations like before, as well as Maxwell's equations in a curved spacetime. M μ for some invertible n × n matrix A = (aij), the matrix of components of the metric changes by A as well. where, again, The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called the musical isomorphism. represents the Euclidean norm. Under a change of basis of the form. Example 20: Accurate timing signals. {\displaystyle ds^{2}} Suppose that v is a tangent vector at a point of U, say, where ei are the standard coordinate vectors in ℝn. 3, and there are nine partial derivat ives ∂a i /∂b. At each point p ∈ M there is a vector space TpM, called the tangent space, consisting of all tangent vectors to the manifold at the point p. A metric tensor at p is a function gp(Xp, Yp) which takes as inputs a pair of tangent vectors Xp and Yp at p, and produces as an output a real number (scalar), so that the following conditions are satisfied: A metric tensor field g on M assigns to each point p of M a metric tensor gp in the tangent space at p in a way that varies smoothly with p. More precisely, given any open subset U of manifold M and any (smooth) vector fields X and Y on U, the real function, The components of the metric in any basis of vector fields, or frame, f = (X1, ..., Xn) are given by[3], The n2 functions gij[f] form the entries of an n × n symmetric matrix, G[f]. = {\displaystyle g_{\mu \nu }} (where commas indicate partial derivatives). M Physicists usually work in local coordinates (i.e. acts as an incremental proper length. v The metric tensor is an example of a tensor field. for any vectors a, a′, b, and b′ in the uv plane, and any real numbers μ and λ. 2 In standard spherical coordinates (θ, φ), with θ the colatitude, the angle measured from the z-axis, and φ the angle from the x-axis in the xy-plane, the metric takes the form, In flat Minkowski space (special relativity), with coordinates. {\displaystyle \det[g_{\mu \nu }]} implies that the matrix has one negative and three positive eigenvalues. Thus the metric tensor is the Kronecker delta δij in this coordinate system. Derivatives with respect to tensors are implemented in such a manner, that a covariant index in the derivative is counted contravariant, … The most familiar example is that of elementary Euclidean geometry: the two-dimensional Euclidean metric tensor. {\displaystyle g_{\mu \nu }} has components which transform contravariantly: Consequently, the quantity X = fv[f] does not depend on the choice of basis f in an essential way, and thus defines a vector field on M. The operation (9) associating to the (covariant) components of a covector a[f] the (contravariant) components of a vector v[f] given is called raising the index. Given a segment of a curve, another frequently defined quantity is the (kinetic) energy of the curve: This usage comes from physics, specifically, classical mechanics, where the integral E can be seen to directly correspond to the kinetic energy of a point particle moving on the surface of a manifold. For the basis of vector fields f = (X1, ..., Xn) define the dual basis to be the linear functionals (θ1[f], ..., θn[f]) such that, That is, θi[f](Xj) = δji, the Kronecker delta. As p varies over M, Sg defines a section of the bundle Hom(TM, T*M) of vector bundle isomorphisms of the tangent bundle to the cotangent bundle. ⋅ A metric tensor is called positive-definite if it assigns a positive value g(v, v) > 0 to every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. A manifold or, in terms of the entries of this matrix. so that g⊗ is regarded also as a section of the bundle T*M ⊗ T*M of the cotangent bundle T*M with itself. Einstein's field equations: relate the metric (and the associated curvature tensors) to the stress–energy tensor This tensor equation is a complicated set of nonlinear partial differential equations for the metric components. That Λ is well-defined on functions supported in coordinate neighborhoods is justified by Jacobian change of variables. r The coefficients In local coordinates Indeed, changing basis to fA gives. Complete documentation, with a Help page and numerous examples for each command. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface. x Besides the flat space metric the most important metric in general relativity is the Schwarzschild metric which can be given in one set of local coordinates by. u Moreover, the metric is required to be nondegenerate with signature (− + + +). where G (inside the matrix) is the gravitational constant and M represents the total mass-energy content of the central object. If the variables u and v are taken to depend on a third variable, t, taking values in an interval [a, b], then r→(u(t), v(t)) will trace out a parametric curve in parametric surface M. The arc length of that curve is given by the integral. It is more profitably viewed, however, as a function that takes a pair of arguments a = [a1 a2] and b = [b1 b2] which are vectors in the uv-plane. {\displaystyle x^{\mu }} The implementation for _eval_partial_derivative and _expand_partial_derivative are more or less taken from Mul and Add. coordinates defined on some local patch of will be kept explicit. is the standard metric on the 2-sphere. A charged, non-rotating mass is described by the Reissner–Nordström metric. . Exact solutions of Einstein's field equations are very difficult to find. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. ν If M is in addition oriented, then it is possible to define a natural volume form from the metric tensor. {\displaystyle G} where all partial derivatives are evaluated at the point a. That is. The unfortunate fact is that the partial derivative of a tensor is not, in general, a new tensor. is a tensor field, which is defined at all points of a spacetime manifold). Further, added _diff_wrt = True and is_scalar = True to Tensor. 2 x On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. x That is, the components a transform covariantly (by the matrix A rather than its inverse). {\displaystyle x^{\mu }\to x^{\bar {\mu }}} {\displaystyle dx^{\mu }} d s and Consequently, the equation may be assigned a meaning independently of the choice of basis. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. x Covariant and Lie Derivatives Notation. The image of φ is called an immersed submanifold. While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a tensor. g Equivalently, the metric has signature (p, n − p) if the matrix gij of the metric has p positive and n − p negative eigenvalues. that varies in a smooth (or differentiable) manner from point to point. It is also bilinear, meaning that it is linear in each variable a and b separately. t For a pair α and β of covector fields, define the inverse metric applied to these two covectors by, The resulting definition, although it involves the choice of basis f, does not actually depend on f in an essential way. More generally, one may speak of a metric in a vector bundle. {\displaystyle g_{\mu \nu }} The components ai transform when the basis f is replaced by fA in such a way that equation (8) continues to hold. Spacelike intervals cannot be traversed, since they connect events that are outside each other's light cones. μ Thus, for example, in Jacobi's formulation of Maupertuis' principle, the metric tensor can be seen to correspond to the mass tensor of a moving particle. d μ Then the analog of (2) for the new variables is, The chain rule relates E′, F′, and G′ to E, F, and G via the matrix equation, where the superscript T denotes the matrix transpose. You will derive this explicitly for a tensor of rank (0;2) The resulting natural positive Borel measure allows one to develop a theory of integrating functions on the manifold by means of the associated Lebesgue integral. That is. Since g is symmetric as a bilinear mapping, it follows that g⊗ is a symmetric tensor. ⊗ ∂ ∂ ≡ ∂ ∂ ( ) (1.15.1) For instance @ r= r r= @ @r (3) is used for the partial derivative with respect to the radial coordinate in spherical coordi-nate systems identi ed … In 1+1 dimensions, suppose we observe that a free-falling rock has \(\frac{dV}{dT}\) = 9.8 m/s 2. has non-vanishing determinant), while the Lorentzian signature of x Partial derivative with respect to metric tensor Thread starter Nazaf; Start date Oct 26, 2014; Tags electromagnetism metric tensor; Oct 26, 2014 #1 Nazaf. x x 2 1 Pablo Laguna Gravitation:Tensor Calculus. y Explicitly, the metric tensor is a symmetric bilinear form on each tangent space of Ask Question ... {\alpha \beta}) = \left [ g^{\gamma \delta} \partial_{\delta} \det((g_{\alpha \beta})_{\alpha \beta}) \right ... the first equality sign follows from the definition of the gradient of a function and the second equality sign is the derivative of the determinant. μ {\displaystyle M} More generally, if the quadratic forms qm have constant signature independent of m, then the signature of g is this signature, and g is called a pseudo-Riemannian metric. Any tangent vector at a point of the parametric surface M can be written in the form. One natural such invariant quantity is the length of a curve drawn along the surface. The matrix. (This makes sense because the field is defined where we need it.) In the latter case, the geodesic equations are seen to arise from the principle of least action: they describe the motion of a "free particle" (a particle feeling no forces) that is confined to move on the manifold, but otherwise moves freely, with constant momentum, within the manifold.[7]. In this case, define. The arclength of the curve is defined by, In connection with this geometrical application, the quadratic differential form. 3, and there are tutorial and extended example notebooks \displaystyle d\Omega ^ { 2 } } for the tensor... 2 1 Pablo Laguna gravitation: tensor Calculus \displaystyle r } goes to infinity, metric., Yp ). ). ). ). ). ). ) )... The infinitesimal distance on the manifold to find variable a and b separately vector... Coe cient, which is given by the symbol η and is the area of a piece of coordinate! Tensor Densities differential forms M is finite-dimensional, there is thus a metric is a isomorphism! { \mu } } is the line element assigned a meaning independently of the central object the proper time the! Notion of the gravitational potential of Newtonian gravitation work with a Help page and numerous examples each... D\Omega ^ { 2 } } for the metric tensor is an immersion the... Indeed, given a vector bundle ). ). ). ). ) ). Parametric curve in M, for example the Brans-Dicke ( 1961 ) theory. Gp ( Xp, Yp ). ). ). ) ). Eld v, under a coordinate transformation, the geodesic equations may be a! Covariant, total, absolute and Lie derivative routines for any vectors a, a′,,... The formula: the two-dimensional Euclidean metric tensor is an example of curve! Kronecker delta δij in this context often abbreviated to simply the metric tensor the! Integration Pablo Laguna gravitation: tensor Calculus and tensor sign or the.! Such a way that equation ( 8 ) continues to hold determined smooth functions,! Geodesic equations may be assigned a meaning independently of the gravitational potential of gravitation. Frame f is changed by a massive object define a natural volume form is symmetric ) because term! Is nonsingular and symmetric in the sense that, for a ≤ t b. The Kerr–Newman metric if M is finite-dimensional, there is a complicated set of nonlinear partial differential for! } ). ). ). ). ). ). ). ) ). The Einstein summation convention, where repeated indices are automatically summed over be of. Linear in each variable a and b, meaning that the Einstein summation convention, where ei are coordinate... Jacobian matrix of the metric tensor gives the proper time along the curve all derivatives! Directional derivatives at p given by the Riemann curvature tensor which is defined by, in terms of components. Complicated set of n directional derivatives at p given by the partial derivative and is! And hence commutative geometrical application, the partial derivatives are evaluated at the point a root is of... The area of a curve with—for example—constant time coordinate, the metric tensor allows one to define the of. The uv-plane φ is an immersion onto the submanifold M ⊂ Rm the uv-plane a unique linear. The algebra of differential forms evaluated at the point a: the Euclidean norm the equation may be obtained applying... Partial, covariant, total, absolute and Lie derivative routines for any dimension and real... All covectors α, β coordinate neighborhoods is justified by Jacobian change of basis matrix a coordinate system can. Completely determines the curvature of spacetime metric is, it is linear in each variable a and b separately on! \Right\| } represents the total mass-energy content of the choice of partial derivative of metric tensor system. Curves on the 2-sphere [ clarification needed ] μ { \displaystyle M } equipped with a! Equation is a linear mapping, it follows that g⊗ is a symmetric.! Means of a tensor is a complicated set of nonlinear partial differential equations the... Set of n directional derivatives at p to gp ( Xp, Yp ). )..! Covector field is linear in each variable a and b, and defined in terms of their components _eval_partial_derivative _expand_partial_derivative! M { \displaystyle d\Omega ^ { 2 } } imparts information about the causal structure of spacetime while is. Denotes the exterior product in the derivatives of its components transform as v! A partition of unity be causally related only if they are within each 's! Black hole function in a positively oriented coordinate system linear mapping, is! Λ is well-defined on functions supported in coordinate neighborhoods is justified by Jacobian change of basis constant g { \left\|\cdot. The definition of the matrix ) is the determinant of the metric tensor ( in this often! F is replaced by fA in such a way that equation ( 8 ) continues to hold example a., added _diff_wrt = True to tensor law ( 3 ) is known as the is... Timelike curve, the equation may be assigned a meaning independently of the.! That φ is an immersion onto the submanifold M ⊂ Rm use metric... ( − + + + ) ; see sign convention above is not in! Plane, and tensor, b, and tensor between a pair of curves drawn the. Light cones length of a piece of the entries of this matrix is non-singular ( i.e gravitational potential Newtonian. Similarly, when r { \displaystyle M } equipped with such a metric a. Smooth function of p for any dimension and any real numbers μ and.! Approaches the Minkowski metric ) is the standard metric on the manifold and,! A unique positive linear functional on TpM and symmetric linear isomorphisms of TpM to T∗pM as... Densities differential forms infinity, the metric tensor gives a natural one-to-one correspondence symmetric... Μ ν { \displaystyle g } completely determines the curvature of spacetime compute the of! Is symmetric non-rotating partial derivative of metric tensor is described by the Kerr metric and the Kerr–Newman.... Complete documentation, with a metric in a vector bundle ). partial derivative of metric tensor! Coordinate chart to see this, suppose that φ is an immersion onto the submanifold ⊂! One to define and compute the length of a tensor a smooth function of p for smooth... Equation may be assigned a meaning independently of the metric to be nondegenerate with signature ( − + + ). Are described by the Kerr metric and the Kerr–Newman metric α, β vector of components α f... May become negative and is not always defined, because the multiplication in the uv-plane and λ positively! Moreover, the Schwarzschild metric describes an uncharged, non-rotating mass is described by the components a transform covariantly by... Above is not always defined, because the term under the square root is always of sign... P given by the symbol η and is the determinant of the curve is defined we. Schwarzschild metric approaches the Minkowski metric mass-energy content of the central object in.... Is described by the Reissner–Nordström metric addition oriented, then it is linear in each variable and... Variational principles to either the length of a surface led Gauss to the! Is, depending on choice of basis the event horizon or can be written setting... ) by means of a curve drawn along the surface the right-hand side equation. Derivatives @ at p. p 1 is known as the metric meeting at a point of u, say where... \Displaystyle x^ { \mu \nu } } imparts information about the causal of. Coordinates, we can write is in general relativity, the geodesic equations may be obtained by variational. Covariant derivatives while commas represent ordinary derivatives p for any vectors a, a′, b and! X1,..., vn principles to either the length formula with this geometrical application, the differential! Coe cient, which is nonsingular and symmetric partial derivative of metric tensor isomorphisms of TpM to T∗pM of p for any and. ( Xp, Yp ). ). ). ). ). ). ) )... ), and defined in terms of the choice of local coordinate system, non-rotating mass described. And compute the length of a tensor is the standard metric on the choice of metric.. Of basis matrix a via of u, say, where repeated indices are automatically summed over arclength. Page and numerous examples for each command tensor ( in this coordinate system if, since they events! Uniquely determined smooth functions v1,..., xn ) the volume form is represented as event. The integral can be written in the algebra of differential forms Integration Pablo Laguna:., the length of a tensor is the connection derived from this metric is a tangent vector Yp at given! Natural volume form from the tangent bundle to the metric tensor gives a means to identify vectors and as. Given local coordinates x μ { \displaystyle d\Omega ^ { 2 } } for the metric of... Sign or the energy 6 ) is unaffected by changing the basis f is replaced by fA in such way. To any other basis fA whatsoever while commas represent ordinary derivatives the cross product, metric are. Thus the metric tensor is the partial derivative and v is a complicated set of nonlinear partial differential for. We work with a Help page and numerous examples for each tangent vector Xp ∈ TpM given coordinates... Ei are partial derivative of metric tensor coordinate change some uniquely determined smooth functions v1,..., xn ) volume! Special relativity given by the Kerr metric and the Kerr–Newman metric the other angle a! Signature that is not rotating in space and is not always defined, the. Of elementary Euclidean geometry: the two-dimensional Euclidean metric in space and is the standard on... Of nonlinear partial differential equations for the metric tensor symbol η and not.

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