0 g u http://dx.doi.org/10.3968%2Fj.ans.1715787020120504.2023, https://en.wikiversity.org/w/index.php?title=Gravitational_tensor&oldid=2090780, Creative Commons Attribution-ShareAlike License. This article is about metric tensors on real Riemannian manifolds. g {\displaystyle \rho _{0}}   is the square root of the determinant ρ x This section has the same smoothness as g: it is continuous, differentiable, smooth, or real-analytic according as g. The mapping Sg, which associates to every vector field on M a covector field on M gives an abstract formulation of "lowering the index" on a vector field. produsul vectorial în trei dimensiuni E.g. 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.     are the constants of acceleration field and pressure field, respectively, One natural such invariant quantity is the length of a curve drawn along the surface. 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. a matrix). {\displaystyle ~A_{\mu }=\left({\frac {\varphi }{c}},-\mathbf {A} \right)} Um tensor de ordem n em um espaço com três dimensões possui 3 n componentes. 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. ν Since g is symmetric as a bilinear mapping, it follows that g⊗ is a symmetric tensor. 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. k π  , which does not depend on the coordinates and time. σ μ u x 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. μ ε ( ν {\displaystyle ~R_{\mu \alpha }\Phi ^{\mu \alpha }=0}   The tensor product of commutative algebras is of constant use in algebraic geometry.For affine schemes X, Y, Z with morphisms from X and Z to Y, so X = Spec(A), Y = Spec(B), and Z = Spec(C) for some commutative rings A, B, C, the fiber product scheme is the affine scheme corresponding to the tensor product of algebras: × = ⁡ (⊗). {\displaystyle ~{\sqrt {-g}}d\Sigma ={\sqrt {-g}}cdtdx^{1}dx^{2}dx^{3}} 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. Using matrix notation, the first fundamental form becomes, Suppose now that a different parameterization is selected, by allowing u and v to depend on another pair of variables u′ and v′. μ ν In May 2016, Google announced its Tensor processing unit (TPU), an application-specific integrated circuit (ASIC, a hardware chip) built specifically for machine learning and tailored for TensorFlow. μ {\displaystyle ~f_{\mu \nu }} In the expanded form the equation for the field strengths with field sources are as follows: where The inverse metric transforms contravariantly, or with respect to the inverse of the change of basis matrix A. = 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. , ν ν ρ μ The contravariance of the components of v[f] is notationally designated by placing the indices of vi[f] in the upper position. for any vectors a, a′, b, and b′ in the uv plane, and any real numbers μ and λ. J −   is the electromagnetic tensor, {\displaystyle ~\eta } g The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor. For instance, if Eij is a tensor field, then M i jk = ∇ iE jk Bj = ∇ iE ij (8) also are tensor fields. g The gravitational tensor or gravitational field tensor, (sometimes called the gravitational field strength tensor) is an antisymmetric tensor, combining two components of gravitational field – the gravitational field strength and the gravitational torsion field – into one. where [7]. L Φ μ ν A tensor of order two (second-order tensor) is a linear map that maps every vector into a vector (e.g. V α   A figura 1 mostra um tensor de ordem 2 e seus nove componentes.   is the vector potential of the gravitational field, G A vector is a tensor of order one. According to the first of these equations, the gravitational field strength is generated by the matter density, and according to the second equation the circular torsion field is always accompanied by the mass current, or emerges due to the change in time of the gravitational field strength vector. In particular for some uniquely determined smooth functions v1, ..., vn. ‖ f Before 1968, it was the only known conformally invariant tensor that is algebraically independent of the Weyl tensor. {\displaystyle ~{\sqrt {-g}}} In many cases, whenever a calculation calls for the length to be used, a similar calculation using the energy may be done as well. When ds2 is pulled back to the image of a curve in M, it represents the square of the differential with respect to arclength. 0 μ More specifically, for m = 3, which means that the ambient Euclidean space is ℝ3, the induced metric tensor is called the first fundamental form. ε 2 , The length of a curve reduces to the formula: The Euclidean metric in some other common coordinate systems can be written as follows. 3     That is, the row vector of components α[f] transforms as a covariant vector. F J 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].   REMARK:The notation for each section carries on to the next. {\displaystyle \Phi _{\alpha }^{\mu }=g^{\mu \nu }\Phi _{\nu \alpha }} {\displaystyle ~G} In abstract indices the Bach tensor is given by ijk, G ijk and H i j are tensors, then J ijk = D ijk +G ijk K ijk‘ m = D ijk H ‘ m L ik‘ = D ijk H ‘ j (7) also are tensors. This is called the induced metric. η   and As shown earlier, in Euclidean 3-space, ( g i j ) {\displaystyle \left(g_{ij}\right)} is simply the Kronecker delta matrix. μ d {\displaystyle ~\mu \nu \sigma } the metric tensor will determine a different matrix of coefficients, This new system of functions is related to the original gij(f) by means of the chain rule. Likes jedishrfu. {\displaystyle ~c=c_{g}} μ The inverse of Sg is a mapping T*M → TM which, analogously, gives an abstract formulation of "raising the index" on a covector field. J 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. So that the right-hand side of equation (6) is unaffected by changing the basis f to any other basis fA whatsoever. {\displaystyle \varepsilon ^{0123}=1.}. ρ If E is a vector bundle over a manifold M, then a metric is a mapping.   {\displaystyle ~\rho _{0q}}   and the charge s R 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.  . The arclength of the curve is defined by, In connection with this geometrical application, the quadratic differential form. Upon changing the basis f by a nonsingular matrix A, the coefficients vi change in such a way that equation (7) remains true. In a basis of vector fields f = (X1, ..., Xn), any smooth tangent vector field X can be written in the form. 16 {\displaystyle \varepsilon ^{\mu \nu \sigma \rho }} {\displaystyle \left\|\cdot \right\|} {\displaystyle ~R}   is the product of differentials of the spatial coordinates. α     − 0 [1] N. Bourbaki, "Elements of mathematics. μ Algebra: Algebraic structures.   is the torsion field. {\displaystyle ~\eta _{\mu \nu }} In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. d 1 G where G (inside the matrix) is the gravitational constant and M represents the total mass-energy content of the central object. This might be a bit confusing, but it is the one dimensional version of what we call e.g. {\displaystyle ~D_{\mu }}   is the electromagnetic vector potential, ∫ = ν 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. 0 g where Dy denotes the Jacobian matrix of the coordinate change. {\displaystyle ~F_{\mu \nu }} ν Let A {\displaystyle A} and B {\displaystyle B} be symmetric covariant 2-tensors. (   is the gravitational constant. In the weak-field approximation Hamiltonian as the relativistic energy of a body with the mass If in (2) we use nonrecurring combinations 012, 013, 023 and 123 as the indices {\displaystyle ~J^{\mu }} c In Minkowski space the Ricci tensor The TPU was developed by … 1. = so that g⊗ is regarded also as a section of the bundle T*M ⊗ T*M of the cotangent bundle T*M with itself. α 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. {\displaystyle ~j^{\mu }} tensorul de curbură Riemann: 2 Tensorul metric ⁠(d) invers, bivectorii ⁠(d), de exemplu structura Poisson ⁠(d) … render at 1080p, then resize it … 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. 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. x + More generally, one may speak of a metric in a vector bundle. α 0 Γ d μ 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. ρ Tensor of gravitational field is defined by the gravitational four-potential of gravitational field   is the electric scalar potential, and   is the mass current density. [E.g. The law of transformation of these vectors in the transition from the fixed reference frame K into the reference frame K', moving at the velocity V along the axis X, has the following form: In the more general case where the velocity 16 {\displaystyle ~\sigma } x D ρ {\displaystyle ~J^{0}} {\displaystyle ~H=\int {(s_{0}J^{0}-{\frac {c^{2}}{16piG}}\Phi _{\mu \nu }\Phi ^{\mu \nu }+{\frac {c^{2}\varepsilon _{0}}{4}}F_{\mu \nu }F^{\mu \nu }+{\frac {c^{2}}{16\pi \eta }}u_{\mu \nu }u^{\mu \nu }+{\frac {c^{2}}{16\pi \sigma }}f_{\mu \nu }f^{\mu \nu }){\sqrt {-g}}dx^{1}dx^{2}dx^{3}},}. Any tangent vector at a point of the parametric surface M can be written in the form. V R   is the 4-potential of acceleration field, μ x ν {\displaystyle ~s_{0}} x A 0123   is the invariant 4-volume, = μ 2   η μ η Thus, for example, the geodesic equations may be obtained by applying variational principles to either the length or the energy. The tensor product is the category-theoretic product in the category of graphs and graph homomorphisms. D σ − 0   μ The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called the musical isomorphism. , {\displaystyle ~c=c_{g}}   d A basic knowledge of vectors, matrices, and physics is assumed.   α Another is the angle between a pair of curves drawn along the surface and meeting at a common point. J A third such quantity is the area of a piece of the surface. 4 The upshot is that the first fundamental form (1) is invariant under changes in the coordinate system, and that this follows exclusively from the transformation properties of E, F, and G. Indeed, by the chain rule, Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of tangent vectors to the surface, as well as the angle between two tangent vectors.