Incubator escapee wiki

Metric tensor

Article Discussion
Revision as of 21:58, 19 May 2025 by 37.160.51.111 (talk) (→‎Intrinsic definitions of a metric: Added wl)
(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)

Template:Short description Template:About

In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold Template:Mvar (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point Template:Mvar of Template:Mvar is a bilinear form defined on the tangent space at Template:Mvar (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric field on Template:Mvar consists of a metric tensor at each point Template:Mvar of Template:Mvar that varies smoothly with Template:Mvar.

A metric tensor Template:Mvar is positive-definite if Template:Math for every nonzero vector Template:Mvar. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying infinitesimal distance on the manifold. On a Riemannian manifold Template:Mvar, the length of a smooth curve between two points Template:Mvar and Template:Mvar can be defined by integration, and the distance between Template:Mvar and Template:Mvar can be defined as the infimum of the lengths of all such curves; this makes Template:Mvar a metric space. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner).Template:Fact

While the notion of a metric tensor was known in some sense to mathematicians such as 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. The metric tensor is an example of a tensor field.

The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. 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.

IntroductionEdit

Carl Friedrich Gauss in his 1827 Disquisitiones generales circa superficies curvas (General investigations of curved surfaces) considered a surface parametrically, with the Cartesian coordinates Template:Mvar, Template:Mvar, and Template:Mvar of points on the surface depending on two auxiliary variables Template:Mvar and Template:Mvar. Thus a parametric surface is (in today's terms) a vector-valued function

<math>\vec{r}(u,\,v) = \bigl( x(u,\,v),\, y(u,\,v),\, z(u,\,v) \bigr)</math>

depending on an ordered pair of real variables Template:Math, and defined in an open set Template:Mvar in the Template:Mvar-plane. 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.

One natural such invariant quantity is the length of a curve drawn along the surface. Another is the angle between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the area of a piece of the surface. The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor.

The metric tensor is <math display=inline> \begin{bmatrix} E & F \\ F & G \end{bmatrix} </math> in the description below; E, F, and G in the matrix can contain any number as long as the matrix is positive definite.

Arc lengthEdit

If the variables Template:Mvar and Template:Mvar are taken to depend on a third variable, Template:Mvar, taking values in an interval Template:Math, then Template:Math will trace out a parametric curve in parametric surface Template:Mvar. The arc length of that curve is given by the integral

<math> \begin{align}
 s &= \int_a^b\left\|\frac{d}{dt}\vec{r}(u(t),v(t))\right\|\,dt \\[5pt]
   &= \int_a^b \sqrt{u'(t)^2\,\vec{r}_u\cdot\vec{r}_u + 2u'(t)v'(t)\, \vec{r}_u\cdot\vec{r}_v + v'(t)^2\,\vec{r}_v\cdot\vec{r}_v}\, dt \,,

\end{align}</math>

where <math> \left\| \cdot \right\| </math> represents the Euclidean norm. Here the chain rule has been applied, and the subscripts denote partial derivatives:

<math>\vec{r}_u = \frac{\partial \vec{r}}{\partial u}\,, \quad \vec{r}_v = \frac{\partial \vec{r}}{\partial v}\,.</math>

The integrand is the restriction<ref>More precisely, the integrand is the pullback of this differential to the curve.</ref> to the curve of the square root of the (quadratic) differential

Template:NumBlk

where

Template:NumBlk

The quantity Template:Mvar in (Template:EquationNote) is called the line element, while Template:Math is called the first fundamental form of Template:Mvar. Intuitively, it represents the principal part of the square of the displacement undergone by Template:Math when Template:Mvar is increased by Template:Mvar units, and Template:Mvar is increased by Template:Mvar units.

Using matrix notation, the first fundamental form becomes

<math>ds^2 =
 \begin{bmatrix} du & dv \end{bmatrix}
 \begin{bmatrix} E & F \\ F & G \end{bmatrix}
 \begin{bmatrix} du \\ dv \end{bmatrix}

</math>

Coordinate transformationsEdit

Suppose now that a different parameterization is selected, by allowing Template:Mvar and Template:Mvar to depend on another pair of variables Template:Math and Template:Math. Then the analog of (Template:EquationNote) for the new variables is Template:NumBlk

The chain rule relates Template:Math, Template:Math, and Template:Math to Template:Mvar, Template:Mvar, and Template:Mvar via the matrix equation

Template:NumBlk

where the superscript T denotes the matrix transpose. The matrix with the coefficients Template:Mvar, Template:Mvar, and Template:Mvar arranged in this way therefore transforms by the Jacobian matrix of the coordinate change

<math>
 J = \begin{bmatrix}
   \frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'} \\
   \frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}

\end{bmatrix}\,.</math>

A matrix which transforms in this way is one kind of what is called a tensor. The matrix

<math>\begin{bmatrix} E & F \\ F & G \end{bmatrix}</math>

with the transformation law (Template:EquationNote) is known as the metric tensor of the surface.

Invariance of arclength under coordinate transformationsEdit

Template:Harvtxt first observed the significance of a system of coefficients Template:Mvar, Template:Mvar, and Template:Mvar, that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form (Template:EquationNote) is invariant under changes in the coordinate system, and that this follows exclusively from the transformation properties of Template:Mvar, Template:Mvar, and Template:Mvar. Indeed, by the chain rule,

<math>\begin{bmatrix} du \\ dv \end{bmatrix} =
 \begin{bmatrix}
   \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\
   \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'}
 \end{bmatrix}
 \begin{bmatrix} du' \\ dv' \end{bmatrix}

</math>

so that

<math>\begin{align}
 ds^2
 &=
   \begin{bmatrix} du & dv \end{bmatrix}
   \begin{bmatrix} E & F \\ F & G \end{bmatrix}
   \begin{bmatrix} du \\ dv \end{bmatrix} \\[6pt]
 &=
   \begin{bmatrix} du' & dv' \end{bmatrix}
   \begin{bmatrix}
     \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\[6pt]
     \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'}
   \end{bmatrix}^\mathsf{T}
   \begin{bmatrix} E & F \\ F & G \end{bmatrix}
   \begin{bmatrix}
     \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\[6pt]
     \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'}
   \end{bmatrix}
   \begin{bmatrix} du' \\ dv' \end{bmatrix} \\[6pt]
 &=
   \begin{bmatrix} du' & dv' \end{bmatrix}
   \begin{bmatrix} E' & F' \\ F' & G' \end{bmatrix}
   \begin{bmatrix} du' \\ dv' \end{bmatrix}\\[6pt]
 &= (ds')^2 \,.

\end{align}</math>

Length and angleEdit

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. In contemporary terms, the metric tensor allows one to compute the dot product of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface Template:Mvar can be written in the form

<math>\mathbf{p} = p_1\vec{r}_u + p_2\vec{r}_v</math>

for suitable real numbers Template:Math and Template:Math. If two tangent vectors are given:

<math>\begin{align}
 \mathbf{a} &= a_1\vec{r}_u + a_2\vec{r}_v \\
 \mathbf{b} &= b_1\vec{r}_u + b_2\vec{r}_v

\end{align}</math>

then using the bilinearity of the dot product,

<math>\begin{align}
 \mathbf{a} \cdot \mathbf{b}
   &= a_1 b_1 \vec{r}_u\cdot\vec{r}_u + a_1b_2 \vec{r}_u\cdot\vec{r}_v + a_2b_1 \vec{r}_v\cdot\vec{r}_u + a_2 b_2 \vec{r}_v\cdot\vec{r}_v \\[8pt]
   &= a_1 b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G. \\[8pt]
   &= \begin{bmatrix} a_1 & a_2 \end{bmatrix}
        \begin{bmatrix} E & F \\ F & G \end{bmatrix}
        \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \,.

\end{align}</math>

This is plainly a function of the four variables Template:Math, Template:Math, Template:Math, and Template:Math. It is more profitably viewed, however, as a function that takes a pair of arguments Template:Math and Template:Math which are vectors in the Template:Mvar-plane. That is, put

<math>g(\mathbf{a}, \mathbf{b}) = a_1b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G \,.</math>

This is a symmetric function in Template:Math and Template:Math, meaning that

<math>g(\mathbf{a}, \mathbf{b}) = g(\mathbf{b}, \mathbf{a})\,.</math>

It is also bilinear, meaning that it is linear in each variable Template:Math and Template:Math separately. That is,

<math>\begin{align}
 g\left(\lambda\mathbf{a} + \mu\mathbf{a}', \mathbf{b}\right) &= \lambda g(\mathbf{a}, \mathbf{b}) + \mu g\left(\mathbf{a}', \mathbf{b}\right),\quad\text{and} \\
 g\left(\mathbf{a}, \lambda\mathbf{b} + \mu\mathbf{b}'\right) &= \lambda g(\mathbf{a}, \mathbf{b}) + \mu g\left(\mathbf{a}, \mathbf{b}'\right)

\end{align}</math>

for any vectors Template:Math, Template:Math, Template:Math, and Template:Math in the Template:Mvar plane, and any real numbers Template:Mvar and Template:Mvar.

In particular, the length of a tangent vector Template:Math is given by

<math> \left\| \mathbf{a} \right\| = \sqrt{g(\mathbf{a}, \mathbf{a})}</math>

and the angle Template:Mvar between two vectors Template:Math and Template:Math is calculated by

<math>\cos(\theta) = \frac{g(\mathbf{a}, \mathbf{b})}{ \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| } \,.</math>

AreaEdit

The surface area is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surface Template:Mvar is parameterized by the function Template:Math over the domain Template:Mvar in the Template:Mvar-plane, then the surface area of Template:Mvar is given by the integral

<math>\iint_D \left|\vec{r}_u \times \vec{r}_v\right|\,du\,dv</math>

where Template:Math denotes the cross product, and the absolute value denotes the length of a vector in Euclidean space. By Lagrange's identity for the cross product, the integral can be written

<math>\begin{align}
      &\iint_D \sqrt{\left(\vec{r}_u\cdot\vec{r}_u\right) \left(\vec{r}_v\cdot\vec{r}_v\right) - \left(\vec{r}_u\cdot\vec{r}_v\right)^2}\,du\,dv \\[5pt]
  ={} &\iint_D \sqrt{EG - F^2}\,du\,dv\\[5pt]
  ={} &\iint_D \sqrt{\det \begin{bmatrix} E & F \\ F & G \end{bmatrix}}\, du\, dv

\end{align}</math>

where Template:Math is the determinant.

DefinitionEdit

Let Template:Mvar be a smooth manifold of dimension Template:Mvar; for instance a surface (in the case Template:Math) or hypersurface in the Cartesian space <math>\R^{n+1}</math>. At each point Template:Math there is a vector space Template:Math, called the tangent space, consisting of all tangent vectors to the manifold at the point Template:Mvar. A metric tensor at Template:Mvar is a function Template:Math which takes as inputs a pair of tangent vectors Template:Math and Template:Math at Template:Mvar, and produces as an output a real number (scalar), so that the following conditions are satisfied: 

 g_p(aU_p + bV_p, Y_p) &= ag_p(U_p, Y_p) + bg_p(V_p, Y_p) \,, \quad \text{and} \\
 g_p(Y_p, aU_p + bV_p) &= ag_p(Y_p, U_p) + bg_p(Y_p, V_p) \,.

\end{align}</math>

A metric tensor field Template:Mvar on Template:Mvar assigns to each point Template:Mvar of Template:Mvar a metric tensor Template:Math in the tangent space at Template:Mvar in a way that varies smoothly with Template:Mvar. More precisely, given any open subset Template:Mvar of manifold Template:Mvar and any (smooth) vector fields Template:Mvar and Template:Mvar on Template:Mvar, the real function <math display="block">g(X, Y)(p) = g_p(X_p, Y_p)</math> is a smooth function of Template:Mvar.

Components of the metricEdit

{{#invoke:Hatnote|hatnote}} The components of the metric in any basis of vector fields, or frame, Template:Math are given by<ref>The notation of using square brackets to denote the basis in terms of which the components are calculated is not universal. The notation employed here is modeled on that of Template:Harvtxt. Typically, such explicit dependence on the basis is entirely suppressed.</ref> Template:NumBlk The Template:Math functions Template:Math form the entries of an Template:Math symmetric matrix, Template:Math. If

<math>v = \sum_{i=1}^n v^iX_i \,, \quad w = \sum_{i=1}^n w^iX_i</math>

are two vectors at Template:Math, then the value of the metric applied to Template:Mvar and Template:Mvar is determined by the coefficients (Template:EquationNote) by bilinearity:

<math>g(v, w) = \sum_{i,j=1}^n v^iw^jg\left(X_i,X_j\right) = \sum_{i,j=1}^n v^iw^jg_{ij}[\mathbf{f}]</math>

Denoting the matrix Template:Math by Template:Math and arranging the components of the vectors Template:Mvar and Template:Mvar into column vectors Template:Math and Template:Math,

<math>g(v,w) = \mathbf{v}[\mathbf{f}]^\mathsf{T} G[\mathbf{f}] \mathbf{w}[\mathbf{f}] = \mathbf{w}[\mathbf{f}]^\mathsf{T} G[\mathbf{f}]\mathbf{v}[\mathbf{f}]</math>

where Template:MathT and Template:MathT denote the transpose of the vectors Template:Math and Template:Math, respectively. Under a change of basis of the form

<math>\mathbf{f}\mapsto \mathbf{f}' = \left(\sum_k X_ka_{k1},\dots,\sum_k X_ka_{kn}\right) = \mathbf{f}A</math>

for some invertible Template:Math matrix Template:Math, the matrix of components of the metric changes by Template:Mvar as well. That is,

<math>G[\mathbf{f}A] = A^\mathsf{T} G[\mathbf{f}]A</math>

or, in terms of the entries of this matrix,

<math>g_{ij}[\mathbf{f}A] = \sum_{k,l=1}^n a_{ki}g_{kl}[\mathbf{f}]a_{lj} \, .</math>

For this reason, the system of quantities Template:Math is said to transform covariantly with respect to changes in the frame Template:Math.

Metric in coordinatesEdit

A system of Template:Mvar real-valued functions Template:Math, giving a local coordinate system on an open set Template:Mvar in Template:Mvar, determines a basis of vector fields on Template:Mvar

<math>\mathbf{f} = \left(X_1 = \frac{\partial}{\partial x^1}, \dots, X_n = \frac{\partial}{\partial x^n}\right) \,.</math>

The metric Template:Mvar has components relative to this frame given by

<math>g_{ij}\left[\mathbf{f}\right] = g\left(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}\right) \,.</math>

Relative to a new system of local coordinates, say

<math>y^i = y^i(x^1, x^2, \dots, x^n),\quad i=1,2,\dots,n</math>

the metric tensor will determine a different matrix of coefficients,

<math>g_{ij}\left[\mathbf{f}'\right] = g\left(\frac{\partial}{\partial y^i}, \frac{\partial}{\partial y^j}\right).</math>

This new system of functions is related to the original Template:Math by means of the chain rule

<math>\frac{\partial}{\partial y^i} = \sum_{k=1}^n \frac{\partial x^k}{\partial y^i}\frac{\partial}{\partial x^k}</math>

so that

<math>g_{ij}\left[\mathbf{f}'\right] = \sum_{k,l=1}^n \frac{\partial x^k}{\partial y^i} g_{kl}\left[\mathbf{f}\right]\frac{\partial x^l}{\partial y^j}.</math>

Or, in terms of the matrices Template:Math and Template:Math,

<math>G\left[\mathbf{f}'\right] = \left((Dy)^{-1}\right)^\mathsf{T} G\left[\mathbf{f}\right] (Dy)^{-1}</math>

where Template:Mvar denotes the Jacobian matrix of the coordinate change.

Signature of a metricEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Associated to any metric tensor is the quadratic form defined in each tangent space by

<math>q_m(X_m) = g_m(X_m,X_m) \,, \quad X_m\in T_mM.</math>

If Template:Math is positive for all non-zero Template:Math, then the metric is positive-definite at Template:Mvar. If the metric is positive-definite at every Template:Math, then Template:Mvar is called a Riemannian metric. More generally, if the quadratic forms Template:Math have constant signature independent of Template:Mvar, then the signature of Template:Mvar is this signature, and Template:Mvar is called a pseudo-Riemannian metric.<ref>Template:Harvnb</ref> If Template:Mvar is connected, then the signature of Template:Mvar does not depend on Template:Mvar.<ref>Template:Harvnb</ref>

By Sylvester's law of inertia, a basis of tangent vectors Template:Math can be chosen locally so that the quadratic form diagonalizes in the following manner

<math>q_m\left(\sum_i\xi^iX_i\right) = \left(\xi^1\right)^2+\left(\xi^2\right)^2+\cdots+\left(\xi^p\right)^2 - \left(\xi^{p+1}\right)^2-\cdots-\left(\xi^n\right)^2</math>

for some Template:Mvar between 1 and Template:Mvar. Any two such expressions of Template:Mvar (at the same point Template:Mvar of Template:Mvar) will have the same number Template:Mvar of positive signs. The signature of Template:Mvar is the pair of integers Template:Math, signifying that there are Template:Mvar positive signs and Template:Math negative signs in any such expression. Equivalently, the metric has signature Template:Math if the matrix Template:Math of the metric has Template:Mvar positive and Template:Math negative eigenvalues.

Certain metric signatures which arise frequently in applications are:

Inverse metricEdit

Let Template:Math be a basis of vector fields, and as above let Template:Math be the matrix of coefficients

<math>g_{ij}[\mathbf{f}] = g\left(X_i,X_j\right) \,.</math>

One can consider the inverse matrix Template:Math, which is identified with the inverse metric (or conjugate or dual metric). The inverse metric satisfies a transformation law when the frame Template:Math is changed by a matrix Template:Mvar via

Template:NumBlk

The inverse metric transforms contravariantly, or with respect to the inverse of the change of basis matrix Template:Mvar. 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.

To see this, suppose that Template:Mvar is a covector field. To wit, for each point Template:Mvar, Template:Mvar determines a function Template:Math defined on tangent vectors at Template:Mvar so that the following linearity condition holds for all tangent vectors Template:Math and Template:Math, and all real numbers Template:Mvar and Template:Mvar:

<math>\alpha_p \left(aX_p + bY_p\right) = a\alpha_p \left(X_p\right) + b\alpha_p \left(Y_p\right)\,.</math>

As Template:Mvar varies, Template:Mvar is assumed to be a smooth function in the sense that

<math>p \mapsto \alpha_p \left(X_p\right)</math>

is a smooth function of Template:Mvar for any smooth vector field Template:Mvar.

Any covector field Template:Mvar has components in the basis of vector fields Template:Math. These are determined by

<math>\alpha_i = \alpha \left(X_i\right)\,,\quad i = 1, 2, \dots, n\,.</math>

Denote the row vector of these components by

<math>\alpha[\mathbf{f}] = \big\lbrack\begin{array}{cccc} \alpha_1 & \alpha_2 & \dots & \alpha_n \end{array}\big\rbrack \,.</math>

Under a change of Template:Math by a matrix Template:Mvar, Template:Math changes by the rule

<math>\alpha[\mathbf{f}A] = \alpha[\mathbf{f}]A \,.</math>

That is, the row vector of components Template:Math transforms as a covariant vector.

For a pair Template:Mvar and Template:Mvar of covector fields, define the inverse metric applied to these two covectors by

Template:NumBlk

The resulting definition, although it involves the choice of basis Template:Math, does not actually depend on Template:Math in an essential way. Indeed, changing basis to Template:Math gives

<math>\begin{align}
     &\alpha[\mathbf{f}A] G[\mathbf{f}A]^{-1} \beta[\mathbf{f}A]^\mathsf{T} \\
 ={} &\left(\alpha[\mathbf{f}]A\right) \left(A^{-1}G[\mathbf{f}]^{-1} \left(A^{-1}\right)^\mathsf{T}\right) \left(A^\mathsf{T}\beta[\mathbf{f}]^\mathsf{T}\right) \\
 ={} &\alpha[\mathbf{f}] G[\mathbf{f}]^{-1} \beta[\mathbf{f}]^\mathsf{T}.

\end{align} </math>

So that the right-hand side of equation (Template:EquationNote) is unaffected by changing the basis Template:Math to any other basis Template:Math whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrix Template:Math are denoted by Template:Math, where the indices Template:Mvar and Template:Mvar have been raised to indicate the transformation law (Template:EquationNote).

Raising and lowering indicesEdit

Template:See also In a basis of vector fields Template:Math, any smooth tangent vector field Template:Mvar can be written in the form

Template:NumBlk

for some uniquely determined smooth functions Template:Math. Upon changing the basis Template:Math by a nonsingular matrix Template:Mvar, the coefficients Template:Math change in such a way that equation (Template:EquationNote) remains true. That is,

<math>X = \mathbf{fA}v[\mathbf{fA}] = \mathbf{f}v[\mathbf{f}]\,.</math>

Consequently, Template:Math. 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 Template:Mvar. The contravariance of the components of Template:Math is notationally designated by placing the indices of Template:Math in the upper position.

A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields Template:Math define the dual basis to be the linear functionals Template:Math such that

<math>\theta^i[\mathbf{f}](X_j) = \begin{cases} 1 & \mathrm{if}\ i=j\\ 0&\mathrm{if}\ i\not=j.\end{cases}</math>

That is, Template:Math, the Kronecker delta. Let

<math>\theta[\mathbf{f}] = \begin{bmatrix}\theta^1[\mathbf{f}] \\ \theta^2[\mathbf{f}] \\ \vdots \\ \theta^n[\mathbf{f}]\end{bmatrix}.</math>

Under a change of basis Template:Math for a nonsingular matrix Template:Math, Template:Math transforms via

<math>\theta[\mathbf{f}A] = A^{-1}\theta[\mathbf{f}].</math>

Any linear functional Template:Mvar on tangent vectors can be expanded in terms of the dual basis Template:Mvar

Template:NumBlk

where Template:Math denotes the row vector Template:Math. The components Template:Math transform when the basis Template:Math is replaced by Template:Math in such a way that equation (Template:EquationNote) continues to hold. That is,

<math>\alpha = a[\mathbf{f}A]\theta[\mathbf{f}A] = a[\mathbf{f}]\theta[\mathbf{f}]</math>

whence, because Template:Math, it follows that Template:Math. That is, the components Template:Mvar transform covariantly (by the matrix Template:Mvar rather than its inverse). The covariance of the components of Template:Math is notationally designated by placing the indices of Template:Math in the lower position.

Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding Template:Math fixed, the function

<math>g_p(X_p, -) : Y_p \mapsto g_p(X_p, Y_p)</math>

of tangent vector Template:Math defines a linear functional on the tangent space at Template:Mvar. This operation takes a vector Template:Math at a point Template:Mvar and produces a covector Template:Math. In a basis of vector fields Template:Math, if a vector field Template:Mvar has components Template:Math, then the components of the covector field Template:Math in the dual basis are given by the entries of the row vector

<math>a[\mathbf{f}] = v[\mathbf{f}]^\mathsf{T} G[\mathbf{f}].</math>

Under a change of basis Template:Math, the right-hand side of this equation transforms via

<math>
 v[\mathbf{f}A]^\mathsf{T} G[\mathbf{f}A] =
   v[\mathbf{f}]^\mathsf{T} \left(A^{-1}\right)^\mathsf{T} A^\mathsf{T} G[\mathbf{f}]A =
   v[\mathbf{f}]^\mathsf{T} G[\mathbf{f}]A

</math>

so that Template:Math: Template:Mvar transforms covariantly. The operation of associating to the (contravariant) components of a vector field Template:MathT the (covariant) components of the covector field Template:Math, where

<math>a_i[\mathbf{f}] = \sum_{k=1}^n v^k[\mathbf{f}]g_{ki}[\mathbf{f}]</math>

is called lowering the index.

To raise the index, one applies the same construction but with the inverse metric instead of the metric. If Template:Math are the components of a covector in the dual basis Template:Math, then the column vector Template:NumBlk has components which transform contravariantly:

<math>v[\mathbf{f}A] = A^{-1}v[\mathbf{f}].</math>

Consequently, the quantity Template:Math does not depend on the choice of basis Template:Math in an essential way, and thus defines a vector field on Template:Mvar. The operation (Template:EquationNote) associating to the (covariant) components of a covector Template:Math the (contravariant) components of a vector Template:Math given is called raising the index. In components, (Template:EquationNote) is

<math>v^i[\mathbf{f}] = \sum_{k=1}^n g^{ik}[\mathbf{f}] a_k[\mathbf{f}].</math>

Induced metricEdit

Let Template:Mvar be an open set in Template:Math, and let Template:Mvar be a continuously differentiable function from Template:Mvar into the Euclidean space Template:Math, where Template:Math. The mapping Template:Mvar is called an immersion if its differential is injective at every point of Template:Mvar. The image of Template:Mvar is called an immersed submanifold. More specifically, for Template:Math, which means that the ambient Euclidean space is Template:Math, the induced metric tensor is called the first fundamental form.

Suppose that Template:Mvar is an immersion onto the submanifold Template:Math. The usual Euclidean dot product in Template:Math is a metric which, when restricted to vectors tangent to Template:Mvar, gives a means for taking the dot product of these tangent vectors. This is called the induced metric.

Suppose that Template:Mvar is a tangent vector at a point of Template:Mvar, say

<math>v = v^1\mathbf{e}_1 + \dots + v^n\mathbf{e}_n</math>

where Template:Math are the standard coordinate vectors in Template:Math. When Template:Mvar is applied to Template:Mvar, the vector Template:Mvar goes over to the vector tangent to Template:Mvar given by

<math>\varphi_*(v) = \sum_{i=1}^n \sum_{a=1}^m v^i\frac{\partial \varphi^a}{\partial x^i}\mathbf{e}_a\,.</math>

(This is called the pushforward of Template:Mvar along Template:Mvar.) Given two such vectors, Template:Mvar and Template:Mvar, the induced metric is defined by

<math>g(v,w) = \varphi_*(v)\cdot \varphi_*(w).</math>

It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields Template:Math is given by

<math>G(\mathbf{e}) = (D\varphi)^\mathsf{T}(D\varphi)</math>

where Template:Mvar is the Jacobian matrix:

<math>D\varphi = \begin{bmatrix}
 \frac{\partial\varphi^1}{\partial x^1} & \frac{\partial\varphi^1}{\partial x^2} &
   \dots  & \frac{\partial\varphi^1}{\partial x^n} \\[1ex]
 \frac{\partial\varphi^2}{\partial x^1} & \frac{\partial\varphi^2}{\partial x^2} &
   \dots  & \frac{\partial\varphi^2}{\partial x^n} \\
 \vdots                                 & \vdots                                 &
   \ddots & \vdots \\
 \frac{\partial\varphi^m}{\partial x^1} & \frac{\partial\varphi^m}{\partial x^2} &
   \dots  & \frac{\partial\varphi^m}{\partial x^n}

\end{bmatrix}.</math>

Intrinsic definitions of a metricEdit

The notion of a metric can be defined intrinsically using the language of fiber bundles and vector bundles. In these terms, a metric tensor is a function

Template:NumBlk

from the fiber product of the tangent bundle of Template:Mvar with itself to Template:Math such that the restriction of Template:Mvar to each fiber is a nondegenerate bilinear mapping

<math>g_p : \mathrm{T}_pM\times \mathrm{T}_pM \to \mathbf{R}.</math>

The mapping (Template:EquationNote) is required to be continuous, and often continuously differentiable, smooth, or real analytic, depending on the case of interest, and whether Template:Mvar can support such a structure.

Metric as a section of a bundleEdit

By the universal property of the tensor product, any bilinear mapping (Template:EquationNote) gives rise naturally to a section Template:Math of the dual of the tensor product bundle of Template:Math with itself

<math>g_\otimes \in \Gamma\left((\mathrm{T}M \otimes \mathrm{T}M)^*\right).</math>

The section Template:Math is defined on simple elements of Template:Math by

<math>g_\otimes(v \otimes w) = g(v, w)</math>

and is defined on arbitrary elements of Template:Math by extending linearly to linear combinations of simple elements. The original bilinear form Template:Mvar is symmetric if and only if

<math>g_\otimes \circ \tau = g_\otimes</math>

where

<math>\tau : \mathrm{T}M \otimes \mathrm{T}M \stackrel{\cong}{\to} TM \otimes TM</math>

is the braiding map.

Since Template:Mvar is finite-dimensional, there is a natural isomorphism

<math>(\mathrm{T}M \otimes \mathrm{T}M)^* \cong \mathrm{T}^*M \otimes \mathrm{T}^*M,</math>

so that Template:Math is regarded also as a section of the bundle Template:Math of the cotangent bundle Template:Math with itself. Since Template:Mvar is symmetric as a bilinear mapping, it follows that Template:Math is a symmetric tensor.

Metric in a vector bundleEdit

Template:See also More generally, one may speak of a metric in a vector bundle. If Template:Mvar is a vector bundle over a manifold Template:Mvar, then a metric is a mapping

<math>g : E\times_M E\to \mathbf{R}</math>

from the fiber product of Template:Mvar to Template:Math which is bilinear in each fiber:

<math>g_p : E_p \times E_p\to \mathbf{R}.</math>

Using duality as above, a metric is often identified with a section of the tensor product bundle Template:Math.

Tangent–cotangent isomorphismEdit

Template:See also The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called the musical isomorphism.<ref>For the terminology "musical isomorphism", see Template:Harvtxt. See also Template:Harvtxt</ref> This isomorphism is obtained by setting, for each tangent vector Template:Math,

<math>S_gX_p\, \stackrel\text{def}{=}\, g(X_p, -),</math>

the linear functional on Template:Math which sends a tangent vector Template:Math at Template:Mvar to Template:Math. That is, in terms of the pairing Template:Math between Template:Math and its dual space Template:Math,

<math>[S_gX_p, Y_p] = g_p(X_p, Y_p)</math>

for all tangent vectors Template:Math and Template:Math. The mapping Template:Math is a linear transformation from Template:Math to Template:Math. It follows from the definition of non-degeneracy that the kernel of Template:Math is reduced to zero, and so by the rank–nullity theorem, Template:Math is a linear isomorphism. Furthermore, Template:Math is a symmetric linear transformation in the sense that

<math>[S_gX_p, Y_p] = [S_gY_p, X_p] </math>

for all tangent vectors Template:Math and Template:Math.

Conversely, any linear isomorphism Template:Math defines a non-degenerate bilinear form on Template:Math by means of

<math>g_S(X_p, Y_p) = [SX_p, Y_p]\,.</math>

This bilinear form is symmetric if and only if Template:Mvar is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on Template:Math and symmetric linear isomorphisms of Template:Math to the dual Template:Math.

As Template:Mvar varies over Template:Mvar, Template:Math defines a section of the bundle Template:Math of vector bundle isomorphisms of the tangent bundle to the cotangent bundle. This section has the same smoothness as Template:Mvar: it is continuous, differentiable, smooth, or real-analytic according as Template:Mvar. The mapping Template:Math, which associates to every vector field on Template:Mvar a covector field on Template:Mvar gives an abstract formulation of "lowering the index" on a vector field. The inverse of Template:Math is a mapping Template:Math which, analogously, gives an abstract formulation of "raising the index" on a covector field.

The inverse Template:Math defines a linear mapping

<math>S_g^{-1} : \mathrm{T}^*M \to \mathrm{T}M</math>

which is nonsingular and symmetric in the sense that

<math>\left[S_g^{-1}\alpha, \beta\right] = \left[S_g^{-1}\beta, \alpha\right]</math>

for all covectors Template:Mvar, Template:Mvar. Such a nonsingular symmetric mapping gives rise (by the tensor-hom adjunction) to a map

<math>\mathrm{T}^*M \otimes \mathrm{T}^*M \to \mathbf{R}</math>

or by the double dual isomorphism to a section of the tensor product

<math>\mathrm{T}M \otimes \mathrm{T}M.</math>

Arclength and the line elementEdit

Suppose that Template:Mvar is a Riemannian metric on Template:Mvar. In a local coordinate system Template:Math, Template:Math, the metric tensor appears as a matrix, denoted here by Template:Math, whose entries are the components Template:Math of the metric tensor relative to the coordinate vector fields.

Let Template:Math be a piecewise-differentiable parametric curve in Template:Mvar, for Template:Math. The arclength of the curve is defined by

<math>L = \int_a^b \sqrt{ \sum_{i,j=1}^n g_{ij}(\gamma(t)) \left(\frac{d}{dt}x^i \circ \gamma(t)\right) \left(\frac{d}{dt} x^j \circ \gamma(t)\right)}\,dt \,.</math>

In connection with this geometrical application, the quadratic differential form

<math>ds^2 = \sum_{i,j=1}^n g_{ij}(p) dx^i dx^j</math>

is called the first fundamental form associated to the metric, while Template:Mvar is the line element. When Template:Math is pulled back to the image of a curve in Template:Mvar, it represents the square of the differential with respect to arclength.

For a pseudo-Riemannian metric, the length formula above is not always defined, because the term under the square root may become negative. We generally only define the length of a curve when the quantity under the square root is always of one sign or the other. In this case, define

<math>L = \int_a^b \sqrt{ \left|\sum_{i,j=1}^ng_{ij}(\gamma(t)) \left(\frac{d}{dt}x^i \circ \gamma(t)\right)\left(\frac{d}{dt}x^j \circ \gamma(t)\right)\right|}\,dt \, .</math>

While these formulas use coordinate expressions, they are in fact independent of the coordinates chosen; they depend only on the metric, and the curve along which the formula is integrated.

The energy, variational principles and geodesicsEdit

Given a segment of a curve, another frequently defined quantity is the (kinetic) energy of the curve:

<math>E = \frac{1}{2} \int_a^b \sum_{i,j=1}^ng_{ij}(\gamma(t)) \left(\frac{d}{dt}x^i \circ \gamma(t)\right)\left(\frac{d}{dt}x^j \circ \gamma(t)\right)\,dt \,. </math>

This usage comes from physics, specifically, classical mechanics, where the integral Template:Mvar can be seen to directly correspond to the kinetic energy of a point particle moving on the surface of a manifold. 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.

In many cases, whenever a calculation calls for the length to be used, a similar calculation using the energy may be done as well. This often leads to simpler formulas by avoiding the need for the square-root. Thus, for example, the geodesic equations may be obtained by applying variational principles to either the length or the energy. 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.<ref>Template:Harvnb</ref>

Canonical measure and volume formEdit

In analogy with the case of surfaces, a metric tensor on an Template:Mvar-dimensional paracompact manifold Template:Mvar gives rise to a natural way to measure the Template:Mvar-dimensional volume of subsets of the manifold. 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.

A measure can be defined, by the Riesz representation theorem, by giving a positive linear functional Template:Mvar on the space Template:Math of compactly supported continuous functions on Template:Mvar. More precisely, if Template:Mvar is a manifold with a (pseudo-)Riemannian metric tensor Template:Mvar, then there is a unique positive Borel measure Template:Math such that for any coordinate chart Template:Math, <math display="block">\Lambda f = \int_U f \, d\mu_g = \int_{\varphi(U)} f \circ \varphi^{-1}(x) \sqrt{\left|\det g\right|}\,dx</math> for all Template:Mvar supported in Template:Mvar. Here Template:Math is the determinant of the matrix formed by the components of the metric tensor in the coordinate chart. That Template:Math is well-defined on functions supported in coordinate neighborhoods is justified by Jacobian change of variables. It extends to a unique positive linear functional on Template:Math by means of a partition of unity.

If Template:Mvar is also oriented, then it is possible to define a natural volume form from the metric tensor. In a positively oriented coordinate system Template:Math the volume form is represented as <math display="block">\omega = \sqrt{\left|\det g\right|} \, dx^1 \wedge \cdots \wedge dx^n</math> where the Template:Math are the coordinate differentials and Template:Math denotes the exterior product in the algebra of differential forms. The volume form also gives a way to integrate functions on the manifold, and this geometric integral agrees with the integral obtained by the canonical Borel measure.

ExamplesEdit

Euclidean metricEdit

The most familiar example is that of elementary Euclidean geometry: the two-dimensional Euclidean metric tensor. In the usual Cartesian Template:Math coordinates, we can write

<math>g = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} \,. </math>

The length of a curve reduces to the formula:

<math>L = \int_a^b \sqrt{ (dx)^2 + (dy)^2} \,. </math>

The Euclidean metric in some other common coordinate systems can be written as follows.

Polar coordinates Template:Math:

<math>\begin{align}
 x &= r \cos\theta \\
 y &= r \sin\theta \\
 J &= \begin{bmatrix}\cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta\end{bmatrix} \,.

\end{align}</math>

So

<math>g = J^\mathsf{T}J =
 \begin{bmatrix}
     \cos^2\theta + \sin^2\theta                  & -r\sin\theta \cos\theta + r\sin\theta\cos\theta \\
   -r\cos\theta\sin\theta + r\cos\theta\sin\theta & r^2 \sin^2\theta + r^2\cos^2\theta
 \end{bmatrix} = \begin{bmatrix}
   1 & 0 \\
   0 & r^2
 \end{bmatrix}

</math> by trigonometric identities.

In general, in a Cartesian coordinate system Template:Math on a Euclidean space, the partial derivatives Template:Math are orthonormal with respect to the Euclidean metric. Thus the metric tensor is the Kronecker delta δij in this coordinate system. The metric tensor with respect to arbitrary (possibly curvilinear) coordinates Template:Math is given by

<math>g_{ij} =
 \sum_{kl}\delta_{kl}\frac{\partial x^k}{\partial q^i} \frac{\partial x^l}{\partial q^j} =
 \sum_k\frac{\partial x^k}{\partial q^i}\frac{\partial x^k}{\partial q^j}.

</math>

The round metric on a sphereEdit

The unit sphere in Template:Math comes equipped with a natural metric induced from the ambient Euclidean metric, through the process explained in the induced metric section. In standard spherical coordinates Template:Math, with Template:Math the colatitude, the angle measured from the Template:Mvar-axis, and Template:Mvar the angle from the Template:Mvar-axis in the Template:Mvar-plane, the metric takes the form

<math>g = \begin{bmatrix} 1 & 0 \\ 0 & \sin^2 \theta\end{bmatrix} \,.</math>

This is usually written in the form

<math>ds^2 = d\theta^2 + \sin^2\theta\,d\varphi^2\,.</math>

Lorentzian metrics from relativityEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In flat Minkowski space (special relativity), with coordinates

<math>r^\mu \rightarrow \left(x^0, x^1, x^2, x^3\right) = (ct, x, y, z) \, ,</math>

the metric is, depending on choice of metric signature,

<math>g = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \quad \text{or} \quad g = \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \,. </math>

For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. For a timelike curve, the length formula gives the proper time along the curve.

In this case, the spacetime interval is written as

<math>ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 = dr^\mu dr_\mu = g_{\mu \nu} dr^\mu dr^\nu\,. </math>

The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. With coordinates

<math>\left(x^0, x^1, x^2, x^3\right) = (ct, r, \theta, \varphi) \,,</math>

we can write the metric as

<math>g_{\mu\nu} =
 \begin{bmatrix}
   \left(1 - \frac{2GM}{rc^2}\right) & 0 & 0 & 0 \\
   0 & -\left(1 - \frac{2GM}{r c^2}\right)^{-1} & 0 & 0 \\
   0 & 0 & -r^2 & 0 \\
   0 & 0 & 0 & -r^2 \sin^2 \theta
 \end{bmatrix}\,,

</math>

where Template:Mvar (inside the matrix) is the gravitational constant and Template:Mvar represents the total mass–energy content of the central object.

See alsoEdit

NotesEdit

Template:Reflist

ReferencesEdit

Template:Riemannian geometry Template:Tensors Template:Manifolds

Retrieved from "https://zalansite.site/mediawiki/index.php?title=Metric_tensor&oldid=137696"