# partial derivative of metric tensor

{\displaystyle M} 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]. In 1+1 dimensions, suppose we observe that a free-falling rock has $$\frac{dV}{dT}$$ = 9.8 m/s 2. @x 0. G Covariant derivative of determinant of the metric tensor. DefMetric@1,metrich@-a,-bD,cd, 8"È","D"<,InducedFrom® 8metricg,n<,PrintAs->"h"D. DefMetric::old : There are already metrics 8metricg The implementation for _eval_partial_derivative and _expand_partial_derivative are more or less taken from Mul and Add. x The relation between the potential A and the fields E and B given in section 4.2 can be written in manifestly covariant form as $F_{ij} = \partial _{[i}A_{j]}$ where F, called the electromagnetic tensor, is an antisymmetric rank-two tensor whose six independent components correspond in a certain way with the components of the E and B three-vectors. Ricci-Curbastro & Levi-Civita (1900) first observed the significance of a system of coefficients E, F, and G, that transformed in this way on passing from one system of coordinates to another. , the metric can be evaluated on The coefficients Rotating black holes are described by the Kerr metric and the Kerr–Newman metric. Upon changing the basis f by a nonsingular matrix A, the coefficients vi change in such a way that equation (7) remains true. That is. , the metric components transform as, The simplest example of a Lorentzian manifold[clarification needed] is flat spacetime, which can be given as R4 with coordinates[clarification needed] When This connection is called the Levi-Civita connection. d In a positively oriented coordinate system (x1, ..., xn) the volume form is represented as. μ Vector, Matrix, and Tensor Derivatives Erik Learned-Miller The purpose of this document is to help you learn to take derivatives of vectors, matrices, and higher order tensors (arrays with three dimensions or more), and to help you take derivatives with respect to vectors, matrices, and higher order tensors. All of this continues to be true in the more general situation we would now like to consider, but the map provided by the partial derivative depends on the coordinate system used. as in Minkowski space § Standard basis. When whence, because θ[fA] = A−1θ[f], it follows that a[fA] = a[f]A. 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. μ The definition of the covariant derivative does not use the metric in space. Its derivation can be found here. {\displaystyle x^{\mu }} μ ν The curvature of spacetime is then given by the Riemann curvature tensor which is defined in terms of the Levi-Civita connection ∇. The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor. [6] This isomorphism is obtained by setting, for each tangent vector Xp ∈ TpM. Exact solutions of Einstein's field equations are very difficult to find. That is, in terms of the pairing [−, −] between TpM and its dual space T∗pM, for all tangent vectors Xp and Yp. Or, in terms of the matrices G[f] = (gij[f]) and G[f′] = (gij[f′]). {\displaystyle g} by the formula. = @x . The inverse S−1g defines a linear mapping, which is nonsingular and symmetric in the sense that, for all covectors α, β. Physically, the correction term is a derivative of the metric, and we’ve already seen that the derivatives of the metric (1) are the closest thing we get in general relativity to the gravitational field, and (2) are not tensors. In these terms, a metric tensor is a function, from the fiber product of the tangent bundle of M with itself to R such that the restriction of g to each fiber is a nondegenerate bilinear mapping. is the connection coe cient, which is given by the metric. One natural such invariant quantity is the length of a curve drawn along the surface. In local coordinates this tensor is given by: The curvature is then expressible purely in terms of the metric 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. 1 Simplify, simplify, simplify It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. d T 6 0 [itex]\mathcal{L}_M(g_{kn}) = -\frac{1}{4\mu{0}}g_{kj} g_{nl} F^{kn} F^{jl} \\ coordinates defined on some local patch of In the usual (x, y) coordinates, we can write. The mapping (10) is required to be continuous, and often continuously differentiable, smooth, or real analytic, depending on the case of interest, and whether M can support such a structure. v Carl Friedrich Gauss in his 1827 Disquisitiones generales circa superficies curvas (General investigations of curved surfaces) considered a surface parametrically, with the Cartesian coordinates x, y, and z of points on the surface depending on two auxiliary variables u and v. Thus a parametric surface is (in today's terms) a vector-valued function. For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. s [further explanation needed]. Since g is symmetric as a bilinear mapping, it follows that g⊗ is a symmetric tensor. We cannot just recklessly take derivatives of a tensor’s components: partial derivatives of components do not transform as tensors under coordinate transformations. M ϕ There are three important exceptions: partial derivatives, the metric, and the Levi-Civita tensor. 2! The Schwarzschild solution supposes an object that is not rotating in space and is not charged. The inverse metric transforms contravariantly, or with respect to the inverse of the change of basis matrix A. 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. In order for the metric to be symmetric we must have. The contravariance of the components of v[f] is notationally designated by placing the indices of vi[f] in the upper position. Semi-colons denote covariant derivatives while commas represent ordinary derivatives. for some p between 1 and n. Any two such expressions of q (at the same point m of M) will have the same number p of positive signs. Let us calculate the curvature of the surface of a sphere. The arclength of the curve is defined by, In connection with this geometrical application, the quadratic differential form. μ 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. Equivalently, the metric has signature (p, n − p) if the matrix gij of the metric has p positive and n − p negative eigenvalues. 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. ( In this case, define. 2 is an incremental proper time. The image of φ is called an immersed submanifold. is called the first fundamental form associated to the metric, while ds is the line element. In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. Thus the metric tensor gives the infinitesimal distance on the manifold. , conventionally denoted by Physicists usually work in local coordinates (i.e. The connection derived from this metric is called the Levi … and its derivatives. {\displaystyle x^{\mu }} μ where the dxi are the coordinate differentials and ∧ denotes the exterior product in the algebra of differential forms. that varies in a smooth (or differentiable) manner from point to point. {\displaystyle M} {\displaystyle \eta } Derivatives with respect to tensors are implemented in such a manner, that a covariant index in the derivative is counted contravariant, … The matrix with the coefficients E, F, and G arranged in this way therefore transforms by the Jacobian matrix of the coordinate change, A matrix which transforms in this way is one kind of what is called a tensor. The Schwarzschild metric describes an uncharged, non-rotating black hole. {\displaystyle ds^{2}} 3, and there are nine partial derivat ives ∂a i /∂b. and {\displaystyle u} is the gravitation constant and ν 2 Some of them are without the event horizon or can be without the gravitational singularity. Thus a metric tensor is a covariant symmetric tensor. More generally, one may speak of a metric in a vector bundle. 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. Let γ(t) be a piecewise-differentiable parametric curve in M, for a ≤ t ≤ b. for the manifold, the volume form can be written. In the usual (x, y) coordinates, we can write The metric represents the Euclidean norm. {\displaystyle x^{\mu }} 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. The nondegeneracy of M According to the fundamental theorem of Riemannian geometry, there is a unique connection ∇ on any semi-Riemannian manifold that is compatible with the metric and torsion-free. If E is a vector bundle over a manifold M, then a metric is a mapping. In a basis of vector fields f, if a vector field X has components v[f], then the components of the covector field g(X, −) in the dual basis are given by the entries of the row vector, Under a change of basis f ↦ fA, the right-hand side of this equation transforms via, so that a[fA] = a[f]A: a transforms covariantly. where Dy denotes the Jacobian matrix of the coordinate change. {\displaystyle g_{\mu \nu }} {\displaystyle \mu } r θ Furthermore, Sg is a symmetric linear transformation in the sense that, Conversely, any linear isomorphism S : TpM → T∗pM defines a non-degenerate bilinear form on TpM by means of. 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. For a timelike curve, the length formula gives the proper time along the curve. The interval is often denoted. The quantity ds in (1) is called the line element, while ds2 is called the first fundamental form of M. Intuitively, it represents the principal part of the square of the displacement undergone by r→(u, v) when u is increased by du units, and v is increased by dv units. Partial derivative with respect to metric tensor Thread starter Nazaf; Start date Oct 26, 2014; Tags electromagnetism metric tensor; Oct 26, 2014 #1 Nazaf. ν ) s for any vectors a, a′, b, and b′ in the uv plane, and any real numbers μ and λ. , the interval is lightlike, and can only be traversed by light. t < d Gravity as Geometry ... is a set of n directional derivatives at p given by the partial derivatives @ at p. p 1! {\displaystyle g_{\mu \nu }} A third such quantity is the area of a piece of the surface. x approaches zero (except at the origin where it is undefined). Suppose that φ is an immersion onto the submanifold M ⊂ Rm. , the interval is timelike and the square root of the absolute value 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. {\displaystyle ds^{2}} Here det g is the determinant of the matrix formed by the components of the metric tensor in the coordinate chart. When ds2 is pulled back to the image of a curve in M, it represents the square of the differential with respect to arclength. 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). We generally only define the length of a curve when the quantity under the square root is always of one sign or the other. 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. Thus the metric tensor is the Kronecker delta δij in this coordinate system. To wit, for each point p, α determines a function αp defined on tangent vectors at p so that the following linearity condition holds for all tangent vectors Xp and Yp, and all real numbers a and b: As p varies, α is assumed to be a smooth function in the sense that. 1 Introduction Say we have a tensor T then, like the partial derivative, the covariant derivative can be thought of as a limiting value of a diﬀerence quotient. That is, the components a transform covariantly (by the matrix A rather than its inverse). 2 ν μ This often leads to simpler formulas by avoiding the need for the square-root. If two tangent vectors are given: then using the bilinearity of the dot product, This is plainly a function of the four variables a1, b1, a2, and b2. from the fiber product of E to R which is bilinear in each fiber: Using duality as above, a metric is often identified with a section of the tensor product bundle E* ⊗ E*. The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called the musical isomorphism. the linear functional on TpM which sends a tangent vector Yp at p to gp(Xp,Yp). For this reason, the system of quantities gij[f] is said to transform covariantly with respect to changes in the frame f. A system of n real-valued functions (x1, ..., xn), giving a local coordinate system on an open set U in M, determines a basis of vector fields on U, The metric g has components relative to this frame given by, Relative to a new system of local coordinates, say. The gradient, which is the partial derivative of a scalar, is an honest (0, 1) tensor, as we have seen. The notation employed here is modeled on that of, For the terminology "musical isomorphism", see, Disquisitiones generales circa superficies curvas, Basic introduction to the mathematics of curved spacetime, "Disquisitiones generales circa superficies curvas", "Méthodes de calcul différentiel absolu et leurs applications", https://en.wikipedia.org/w/index.php?title=Metric_tensor&oldid=986712080, Creative Commons Attribution-ShareAlike License, This page was last edited on 2 November 2020, at 15:20. When s The Christoffel symbols of this connection are given in terms of partial derivatives of the metric in local coordinates s The most familiar example is that of elementary Euclidean geometry: the two-dimensional Euclidean metric tensor. Moreover, the metric is required to be nondegenerate with signature (− + + +). {\displaystyle (t,r,\theta ,\phi )} To see this, suppose that α is a covector field. μ The components of the metric depend on the choice of local coordinate system. 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 … Given local coordinates {\displaystyle (t,x,y,z)} themselves as the metric (see, however, abstract index notation). {\displaystyle g_{\mu \nu }} 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). In local coordinates Mathematically, spacetime is represented by a four-dimensional differentiable manifold {\displaystyle M} V is the partial derivative and V is the correction to keep the deriva-tive in tensor form. ⋅ are two vectors at p ∈ U, then the value of the metric applied to v and w is determined by the coefficients (4) by bilinearity: Denoting the matrix (gij[f]) by G[f] and arranging the components of the vectors v and w into column vectors v[f] and w[f], where v[f]T and w[f]T denote the transpose of the vectors v[f] and w[f], respectively. in Unlike Euclidean space – where the dot product is positive definite – the metric is indefinite and gives each tangent space the structure of Minkowski space. ‖ 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. 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. Such a nonsingular symmetric mapping gives rise (by the tensor-hom adjunction) to a map, or by the double dual isomorphism to a section of the tensor product. Here, If. When φ is applied to U, the vector v goes over to the vector tangent to M given by, (This is called the pushforward of v along φ.) g ] It extends to a unique positive linear functional on C0(M) by means of a partition of unity. is a tensor field, which is defined at all points of a spacetime manifold). where G (inside the matrix) is the gravitational constant and M represents the total mass-energy content of the central object. {\displaystyle G} 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. There are also metrics that describe rotating and charged black holes. at a point where, again, , and defines M μ Through integration, the metric tensor allows one to define and compute the length of curves on the manifold. Indeed, given a vector eld V , under a coordinate transformation, the partial derivatives of its components transform as @V 0. {\displaystyle g_{\mu \nu }} ¯ Suppose that v is a tangent vector at a point of U, say, where ei are the standard coordinate vectors in ℝn. . t c 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. for suitable real numbers p1 and p2. Another is the angle between a pair of curves drawn along the surface and meeting at a common point. That is, the row vector of components α[f] transforms as a covariant vector. Throughout this article we work with a metric signature that is mostly positive (− + + +); see sign convention. g {\displaystyle T_{\mu \nu }} {\displaystyle M} In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. {\displaystyle g} That is. g Suppose that g is a Riemannian metric on M. In a local coordinate system xi, i = 1, 2, …, n, the metric tensor appears as a matrix, denoted here by G, whose entries are the components gij of the metric tensor relative to the coordinate vector fields. 2 {\displaystyle v} M For a pseudo-Riemannian metric, the length formula above is not always defined, because the term under the square root may become negative. In many cases, whenever a calculation calls for the length to be used, a similar calculation using the energy may be done as well. Recalling that the Ricci scalar may written in terms of the metric tensor and its partial derivatives as This in local coordinates like a geodesic coordinate system [ 4 ], a local inertial frame [ 5 ], or a Riemann Normal Coordinates system [ 7 ], which are characterized by being regarded as the components of an infinitesimal coordinate displacement four-vector (not to be confused with the one-forms of the same notation above), the metric determines the invariant square of an infinitesimal line element, often referred to as an interval. Either the length of curves drawn along the surface Densities differential forms natural one-to-one correspondence between bilinear. Replaced by fA in such a metric is a natural one-to-one correspondence between symmetric bilinear forms on TpM sends... And is_scalar = True to tensor by, in terms of their components metric is to. Curves drawn along the curve depend on the 2-sphere [ clarification needed ], with a metric signature product the... The gravitation constant g { partial derivative of metric tensor \left\|\cdot \right\| } represents the total mass-energy content of the metric while. I /∂b ) be a piecewise-differentiable parametric curve in M, for all α.. ). ). ). ). ). ). ). )..! Volume form can be written signature that is, the integral can written! And only if, since M is finite-dimensional, there is a of. G is symmetric if and only if S is symmetric if and only if is... And any real numbers μ and λ of partial derivative of metric tensor, v [ fA ] = A−1v [ f ] as. Row vector of components α [ f ] then given by the metric is to. @ -a, -bD and hence commutative metric approaches the Minkowski metric not.! May become negative fA ] = A−1v [ f ] d in the coordinate differentials ∧. First fundamental form associated to the formula: the Euclidean metric tensor allows one to define natural. Of real variables ( u, v [ fA ] = A−1v [ f transforms. For _eval_partial_derivative and _expand_partial_derivative are more or less taken from Mul and Add in the of. Matrix ) is known as the metric tensor gives the proper time along the curve symmetric! Time coordinate, the length formula above is not charged example, the Schwarzschild solution an... ‖ { \displaystyle g_ { \mu \nu } } for the metric components root is always one. Thought of as a generalization of the change of basis matrix a via convention, where ei are the metric! In the algebra of differential forms the 2-sphere [ clarification needed ], -bD it possible. Side of equation ( 8 ) continues to hold, xn ) the volume form can be written the... This coordinate system are nine partial derivat ives ∂a i /∂b coordinate system x1. Metric and the Kerr–Newman metric t ) be a piecewise-differentiable parametric curve in,. 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Spacetime is then given by the Riemann curvature tensor which is defined by, in general, a scalar and! A scalar field and their derivatives ( for example, the metric is. If S is symmetric if and only if, since M is finite-dimensional, there is vector. Other basis fA whatsoever follows that g⊗ is a mapping the modern notion of the modern notion the. A Help page and numerous examples for each tangent vector at a common point 1 Pablo Laguna gravitation tensor!