Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Metric tensor

From Wikipedia, the free encyclopedia
Structure defining distance on a manifold
This article is about metric structures on manifolds. For the specific case of spacetime of relativity, seeMetric tensor (general relativity).

In themathematical field ofdifferential geometry, ametric tensor (or simplymetric) is an additionalstructure on amanifoldM (such as asurface) that allows defining distances and angles, just as theinner product on aEuclidean space allows defining distances and angles there. More precisely, a metric tensor at a pointp ofM is abilinear form defined on thetangent space atp (that is, abilinear function that maps pairs oftangent vectors toreal numbers), and a metric field onM consists of a metric tensor at each pointp ofM that varies smoothly withp.

A metric tensorg ispositive-definite ifg(v,v) > 0 for every nonzero vectorv. A manifold equipped with a positive-definite metric tensor is known as aRiemannian manifold. Such a metric tensor can be thought of as specifyinginfinitesimal distance on the manifold. On a Riemannian manifoldM, the length of a smooth curve between two pointsp andq can be defined by integration, and thedistance betweenp andq can be defined as theinfimum of the lengths of all such curves; this makesM ametric space. Conversely, the metric tensor itself is thederivative of the distance function (taken in a suitable manner).[citation needed]

While the notion of a metric tensor was known in some sense to mathematicians such asGauss from the early 19th century, it was not until the early 20th century that its properties as atensor were understood by, in particular,Gregorio Ricci-Curbastro andTullio Levi-Civita, who first codified the notion of a tensor. The metric tensor is an example of atensor field.

The components of a metric tensor in acoordinate basis take on the form of asymmetric matrix whose entries transformcovariantly under changes to the coordinate system. Thus a metric tensor is a covariantsymmetric tensor. From thecoordinate-independent point of view, a metric tensor field is defined to be anondegeneratesymmetric bilinear form on each tangent space that variessmoothly from point to point.

Introduction

[edit]

Carl Friedrich Gauss in his 1827Disquisitiones generales circa superficies curvas (General investigations of curved surfaces) considered a surfaceparametrically, with theCartesian coordinatesx,y, andz of points on the surface depending on two auxiliary variablesu andv. Thus a parametric surface is (in today's terms) avector-valued function

r→(u,v)=(x(u,v),y(u,v),z(u,v)){\displaystyle {\vec {r}}(u,\,v)={\bigl (}x(u,\,v),\,y(u,\,v),\,z(u,\,v){\bigr )}}

depending on anordered pair of real variables(u,v), and defined in anopen setD in theuv-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 thelength of a curve drawn along the surface. Another is theangle between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is thearea 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[EFFG]{\textstyle {\begin{bmatrix}E&F\\F&G\end{bmatrix}}} in the description below; E, F, and G in the matrix can contain any number as long as the matrix is positive definite.

Arc length

[edit]

If the variablesu andv are taken to depend on a third variable,t, taking values in aninterval[a,b], thenr→(u(t),v(t)) will trace out aparametric curve in parametric surfaceM. Thearc length of that curve is given by theintegral

s=∫ab‖ddtr→(u(t),v(t))‖dt=∫abu′(t)2r→u⋅r→u+2u′(t)v′(t)r→u⋅r→v+v′(t)2r→v⋅r→vdt,{\displaystyle {\begin{aligned}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{aligned}}}

where‖⋅‖{\displaystyle \left\|\cdot \right\|} represents theEuclidean norm. Here thechain rule has been applied, and the subscripts denotepartial derivatives:

r→u=∂r→∂u,r→v=∂r→∂v.{\displaystyle {\vec {r}}_{u}={\frac {\partial {\vec {r}}}{\partial u}}\,,\quad {\vec {r}}_{v}={\frac {\partial {\vec {r}}}{\partial v}}\,.}

The integrand is the restriction[1] to the curve of the square root of the (quadratic)differential

(ds)2=E(du)2+2Fdudv+G(dv)2,{\displaystyle (ds)^{2}=E\,(du)^{2}+2F\,du\,dv+G\,(dv)^{2},}1

where

E=r→u⋅r→u,F=r→u⋅r→v,G=r→v⋅r→v.{\displaystyle E={\vec {r}}_{u}\cdot {\vec {r}}_{u},\quad F={\vec {r}}_{u}\cdot {\vec {r}}_{v},\quad G={\vec {r}}_{v}\cdot {\vec {r}}_{v}.}2

The quantityds in (1) is called theline element, whileds2 is called thefirst fundamental form ofM. Intuitively, it represents theprincipal part of the square of the displacement undergone byr→(u,v) whenu is increased bydu units, andv is increased bydv units.

Using matrix notation, the first fundamental form becomes

ds2=[dudv][EFFG][dudv]{\displaystyle ds^{2}={\begin{bmatrix}du&dv\end{bmatrix}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}du\\dv\end{bmatrix}}}

Coordinate transformations

[edit]

Suppose now that a different parameterization is selected, by allowingu andv to depend on another pair of variablesu′ andv′. Then the analog of (2) for the new variables is

E′=r→u′⋅r→u′,F′=r→u′⋅r→v′,G′=r→v′⋅r→v′.{\displaystyle E'={\vec {r}}_{u'}\cdot {\vec {r}}_{u'},\quad F'={\vec {r}}_{u'}\cdot {\vec {r}}_{v'},\quad G'={\vec {r}}_{v'}\cdot {\vec {r}}_{v'}.}2'

Thechain rule relatesE′,F′, andG′ toE,F, andG via thematrix equation

[E′F′F′G′]=[∂u∂u′∂u∂v′∂v∂u′∂v∂v′]T[EFFG][∂u∂u′∂u∂v′∂v∂u′∂v∂v′]{\displaystyle {\begin{bmatrix}E'&F'\\F'&G'\end{bmatrix}}={\begin{bmatrix}{\frac {\partial u}{\partial u'}}&{\frac {\partial u}{\partial v'}}\\{\frac {\partial v}{\partial u'}}&{\frac {\partial v}{\partial v'}}\end{bmatrix}}^{\mathsf {T}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}{\frac {\partial u}{\partial u'}}&{\frac {\partial u}{\partial v'}}\\{\frac {\partial v}{\partial u'}}&{\frac {\partial v}{\partial v'}}\end{bmatrix}}}3

where the superscript T denotes thematrix transpose. The matrix with the coefficientsE,F, andG arranged in this way therefore transforms by theJacobian matrix of the coordinate change

J=[∂u∂u′∂u∂v′∂v∂u′∂v∂v′].{\displaystyle 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}}\,.}

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

[EFFG]{\displaystyle {\begin{bmatrix}E&F\\F&G\end{bmatrix}}}

with the transformation law (3) is known as the metric tensor of the surface.

Invariance of arclength under coordinate transformations

[edit]

Ricci-Curbastro & Levi-Civita (1900) first observed the significance of a system of coefficientsE,F, andG, that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form (1) isinvariant under changes in the coordinate system, and that this follows exclusively from the transformation properties ofE,F, andG. Indeed, by the chain rule,

[dudv]=[∂u∂u′∂u∂v′∂v∂u′∂v∂v′][du′dv′]{\displaystyle {\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}}}

so that

ds2=[dudv][EFFG][dudv]=[du′dv′][∂u∂u′∂u∂v′∂v∂u′∂v∂v′]T[EFFG][∂u∂u′∂u∂v′∂v∂u′∂v∂v′][du′dv′]=[du′dv′][E′F′F′G′][du′dv′]=(ds′)2.{\displaystyle {\begin{aligned}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{aligned}}}

Length and angle

[edit]

Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length oftangent vectors to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute thedot product of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surfaceM can be written in the form

p=p1r→u+p2r→v{\displaystyle \mathbf {p} =p_{1}{\vec {r}}_{u}+p_{2}{\vec {r}}_{v}}

for suitable real numbersp1 andp2. If two tangent vectors are given:

a=a1r→u+a2r→vb=b1r→u+b2r→v{\displaystyle {\begin{aligned}\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{aligned}}}

then using thebilinearity of the dot product,

a⋅b=a1b1r→u⋅r→u+a1b2r→u⋅r→v+a2b1r→v⋅r→u+a2b2r→v⋅r→v=a1b1E+a1b2F+a2b1F+a2b2G.=[a1a2][EFFG][b1b2].{\displaystyle {\begin{aligned}\mathbf {a} \cdot \mathbf {b} &=a_{1}b_{1}{\vec {r}}_{u}\cdot {\vec {r}}_{u}+a_{1}b_{2}{\vec {r}}_{u}\cdot {\vec {r}}_{v}+a_{2}b_{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_{1}b_{2}F+a_{2}b_{1}F+a_{2}b_{2}G.\\[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{aligned}}}

This is plainly a function of the four variablesa1,b1,a2, andb2. It is more profitably viewed, however, as a function that takes a pair of argumentsa = [a1a2] andb = [b1b2] which are vectors in theuv-plane. That is, put

g(a,b)=a1b1E+a1b2F+a2b1F+a2b2G.{\displaystyle g(\mathbf {a} ,\mathbf {b} )=a_{1}b_{1}E+a_{1}b_{2}F+a_{2}b_{1}F+a_{2}b_{2}G\,.}

This is asymmetric function ina andb, meaning that

g(a,b)=g(b,a).{\displaystyle g(\mathbf {a} ,\mathbf {b} )=g(\mathbf {b} ,\mathbf {a} )\,.}

It is alsobilinear, meaning that it islinear in each variablea andb separately. That is,

g(λa+μa′,b)=λg(a,b)+μg(a′,b),andg(a,λb+μb′)=λg(a,b)+μg(a,b′){\displaystyle {\begin{aligned}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{aligned}}}

for any vectorsa,a′,b, andb′ in theuv plane, and any real numbersμ andλ.

In particular, the length of a tangent vectora is given by

‖a‖=g(a,a){\displaystyle \left\|\mathbf {a} \right\|={\sqrt {g(\mathbf {a} ,\mathbf {a} )}}}

and the angleθ between two vectorsa andb is calculated by

cos⁡(θ)=g(a,b)‖a‖‖b‖.{\displaystyle \cos(\theta )={\frac {g(\mathbf {a} ,\mathbf {b} )}{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}\,.}

Area

[edit]

Thesurface area is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surfaceM is parameterized by the functionr→(u,v) over the domainD in theuv-plane, then the surface area ofM is given by the integral

∬D|r→u×r→v|dudv{\displaystyle \iint _{D}\left|{\vec {r}}_{u}\times {\vec {r}}_{v}\right|\,du\,dv}

where× denotes thecross product, and the absolute value denotes the length of a vector in Euclidean space. ByLagrange's identity for the cross product, the integral can be written

∬D(r→u⋅r→u)(r→v⋅r→v)−(r→u⋅r→v)2dudv=∬DEG−F2dudv=∬Ddet[EFFG]dudv{\displaystyle {\begin{aligned}&\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{aligned}}}

wheredet is thedeterminant.

Definition

[edit]

LetM be asmooth manifold of dimensionn; for instance asurface (in the casen = 2) orhypersurface in theCartesian spaceRn+1{\displaystyle \mathbb {R} ^{n+1}}. At each pointp ∈M there is avector spaceTpM, called thetangent space, consisting of all tangent vectors to the manifold at the pointp. A metric tensor atp is a functiongp(Xp,Yp) which takes as inputs a pair of tangent vectorsXp andYp atp, and produces as an output areal number (scalar), so that the following conditions are satisfied:

A metric tensor fieldg onM assigns to each pointp ofM a metric tensorgp in the tangent space atp in a way that variessmoothly withp. More precisely, given anyopen subsetU of manifoldM and any (smooth)vector fieldsX andY onU, the real functiong(X,Y)(p)=gp(Xp,Yp){\displaystyle g(X,Y)(p)=g_{p}(X_{p},Y_{p})}is a smooth function ofp.

Components of the metric

[edit]
This section assumes some familiarity withcoordinate vectors.

The components of the metric in anybasis ofvector fields, orframe,f = (X1, ...,Xn) are given by[3]

gij[f]=g(Xi,Xj).{\displaystyle g_{ij}[\mathbf {f} ]=g\left(X_{i},X_{j}\right).}4

Then2 functionsgij[f] form the entries of ann ×nsymmetric matrix,G[f]. If

v=∑i=1nviXi,w=∑i=1nwiXi{\displaystyle v=\sum _{i=1}^{n}v^{i}X_{i}\,,\quad w=\sum _{i=1}^{n}w^{i}X_{i}}

are two vectors atp ∈U, then the value of the metric applied tov andw is determined by the coefficients (4) by bilinearity:

g(v,w)=∑i,j=1nviwjg(Xi,Xj)=∑i,j=1nviwjgij[f]{\displaystyle g(v,w)=\sum _{i,j=1}^{n}v^{i}w^{j}g\left(X_{i},X_{j}\right)=\sum _{i,j=1}^{n}v^{i}w^{j}g_{ij}[\mathbf {f} ]}

Denoting thematrix(gij[f]) byG[f] and arranging the components of the vectorsv andw intocolumn vectorsv[f] andw[f],

g(v,w)=v[f]TG[f]w[f]=w[f]TG[f]v[f]{\displaystyle 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} ]}

wherev[f]T andw[f]T denote thetranspose of the vectorsv[f] andw[f], respectively. Under achange of basis of the form

f↦f′=(∑kXkak1,…,∑kXkakn)=fA{\displaystyle \mathbf {f} \mapsto \mathbf {f} '=\left(\sum _{k}X_{k}a_{k1},\dots ,\sum _{k}X_{k}a_{kn}\right)=\mathbf {f} A}

for someinvertiblen ×n matrixA = (aij), the matrix of components of the metric changes byA as well. That is,

G[fA]=ATG[f]A{\displaystyle G[\mathbf {f} A]=A^{\mathsf {T}}G[\mathbf {f} ]A}

or, in terms of the entries of this matrix,

gij[fA]=∑k,l=1nakigkl[f]alj.{\displaystyle g_{ij}[\mathbf {f} A]=\sum _{k,l=1}^{n}a_{ki}g_{kl}[\mathbf {f} ]a_{lj}\,.}

For this reason, the system of quantitiesgij[f] is said to transform covariantly with respect to changes in the framef.

Metric in coordinates

[edit]

A system ofn real-valued functions(x1, ...,xn), giving a localcoordinate system on anopen setU inM, determines a basis of vector fields onU

f=(X1=∂∂x1,…,Xn=∂∂xn).{\displaystyle \mathbf {f} =\left(X_{1}={\frac {\partial }{\partial x^{1}}},\dots ,X_{n}={\frac {\partial }{\partial x^{n}}}\right)\,.}

The metricg has components relative to this frame given by

gij[f]=g(∂∂xi,∂∂xj).{\displaystyle g_{ij}\left[\mathbf {f} \right]=g\left({\frac {\partial }{\partial x^{i}}},{\frac {\partial }{\partial x^{j}}}\right)\,.}

Relative to a new system of local coordinates, say

yi=yi(x1,x2,…,xn),i=1,2,…,n{\displaystyle y^{i}=y^{i}(x^{1},x^{2},\dots ,x^{n}),\quad i=1,2,\dots ,n}

the metric tensor will determine a different matrix of coefficients,

gij[f′]=g(∂∂yi,∂∂yj).{\displaystyle g_{ij}\left[\mathbf {f} '\right]=g\left({\frac {\partial }{\partial y^{i}}},{\frac {\partial }{\partial y^{j}}}\right).}

This new system of functions is related to the originalgij(f) by means of thechain rule

∂∂yi=∑k=1n∂xk∂yi∂∂xk{\displaystyle {\frac {\partial }{\partial y^{i}}}=\sum _{k=1}^{n}{\frac {\partial x^{k}}{\partial y^{i}}}{\frac {\partial }{\partial x^{k}}}}

so that

gij[f′]=∑k,l=1n∂xk∂yigkl[f]∂xl∂yj.{\displaystyle 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}}}.}

Or, in terms of the matricesG[f] = (gij[f]) andG[f′] = (gij[f′]),

G[f′]=((Dy)−1)TG[f](Dy)−1{\displaystyle G\left[\mathbf {f} '\right]=\left((Dy)^{-1}\right)^{\mathsf {T}}G\left[\mathbf {f} \right](Dy)^{-1}}

whereDy denotes theJacobian matrix of the coordinate change.

Signature of a metric

[edit]
Main article:Metric signature

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

qm(Xm)=gm(Xm,Xm),Xm∈TmM.{\displaystyle q_{m}(X_{m})=g_{m}(X_{m},X_{m})\,,\quad X_{m}\in T_{m}M.}

Ifqm is positive for all non-zeroXm, then the metric ispositive-definite atm. If the metric is positive-definite at everym ∈M, theng is called aRiemannian metric. More generally, if the quadratic formsqm have constantsignature independent ofm, then the signature ofg is this signature, andg is called apseudo-Riemannian metric.[4] IfM isconnected, then the signature ofqm does not depend onm.[5]

BySylvester's law of inertia, a basis of tangent vectorsXi can be chosen locally so that the quadratic form diagonalizes in the following manner

qm(∑iξiXi)=(ξ1)2+(ξ2)2+⋯+(ξp)2−(ξp+1)2−⋯−(ξn)2{\displaystyle q_{m}\left(\sum _{i}\xi ^{i}X_{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}}

for somep between 1 andn. Any two such expressions ofq (at the same pointm ofM) will have the same numberp of positive signs. The signature ofg is the pair of integers(p,n −p), signifying that there arep positive signs andn −p negative signs in any such expression. Equivalently, the metric has signature(p,n −p) if the matrixgij of the metric hasp positive andn −p negativeeigenvalues.

Certain metric signatures which arise frequently in applications are:

  • Ifg has signature(n, 0), theng is a Riemannian metric, andM is called aRiemannian manifold. Otherwise,g is a pseudo-Riemannian metric, andM is called apseudo-Riemannian manifold (the term semi-Riemannian is also used).
  • IfM is four-dimensional with signature(1, 3) or(3, 1), then the metric is calledLorentzian. More generally, a metric tensor in dimensionn other than 4 of signature(1,n − 1) or(n − 1, 1) is sometimes also called Lorentzian.
  • IfM is2n-dimensional andg has signature(n,n), then the metric is calledultrahyperbolic.

Inverse metric

[edit]

Letf = (X1, ...,Xn) be a basis of vector fields, and as above letG[f] be the matrix of coefficients

gij[f]=g(Xi,Xj).{\displaystyle g_{ij}[\mathbf {f} ]=g\left(X_{i},X_{j}\right)\,.}

One can consider theinverse matrixG[f]−1, which is identified with theinverse metric (orconjugate ordual metric). The inverse metric satisfies a transformation law when the framef is changed by a matrixA via

G[fA]−1=A−1G[f]−1(A−1)T.{\displaystyle G[\mathbf {f} A]^{-1}=A^{-1}G[\mathbf {f} ]^{-1}\left(A^{-1}\right)^{\mathsf {T}}.}5

The inverse metric transformscontravariantly, or with respect to the inverse of the change of basis matrixA. 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 oflinear functionals.

To see this, suppose thatα is a covector field. To wit, for each pointp,α determines a functionαp defined on tangent vectors atp so that the followinglinearity condition holds for all tangent vectorsXp andYp, and all real numbersa andb:

αp(aXp+bYp)=aαp(Xp)+bαp(Yp).{\displaystyle \alpha _{p}\left(aX_{p}+bY_{p}\right)=a\alpha _{p}\left(X_{p}\right)+b\alpha _{p}\left(Y_{p}\right)\,.}

Asp varies,α is assumed to be asmooth function in the sense that

p↦αp(Xp){\displaystyle p\mapsto \alpha _{p}\left(X_{p}\right)}

is a smooth function ofp for any smooth vector fieldX.

Any covector fieldα has components in the basis of vector fieldsf. These are determined by

αi=α(Xi),i=1,2,…,n.{\displaystyle \alpha _{i}=\alpha \left(X_{i}\right)\,,\quad i=1,2,\dots ,n\,.}

Denote therow vector of these components by

α[f]=[α1α2…αn].{\displaystyle \alpha [\mathbf {f} ]={\big \lbrack }{\begin{array}{cccc}\alpha _{1}&\alpha _{2}&\dots &\alpha _{n}\end{array}}{\big \rbrack }\,.}

Under a change off by a matrixA,α[f] changes by the rule

α[fA]=α[f]A.{\displaystyle \alpha [\mathbf {f} A]=\alpha [\mathbf {f} ]A\,.}

That is, the row vector of componentsα[f] transforms as acovariant vector.

For a pairα andβ of covector fields, define the inverse metric applied to these two covectors by

g~(α,β)=α[f]G[f]−1β[f]T.{\displaystyle {\tilde {g}}(\alpha ,\beta )=\alpha [\mathbf {f} ]G[\mathbf {f} ]^{-1}\beta [\mathbf {f} ]^{\mathsf {T}}.}6

The resulting definition, although it involves the choice of basisf, does not actually depend onf in an essential way. Indeed, changing basis tofA gives

α[fA]G[fA]−1β[fA]T=(α[f]A)(A−1G[f]−1(A−1)T)(ATβ[f]T)=α[f]G[f]−1β[f]T.{\displaystyle {\begin{aligned}&\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{aligned}}}

So that the right-hand side of equation (6) is unaffected by changing the basisf to any other basisfA whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrixG[f] are denoted bygij, where the indicesi andj have been raised to indicate the transformation law (5).

Raising and lowering indices

[edit]
See also:Raising and lowering indices

In a basis of vector fieldsf = (X1, ...,Xn), any smooth tangent vector fieldX can be written in the form

X=v1[f]X1+v2[f]X2+⋯+vn[f]Xn=f[v1[f]v2[f]⋮vn[f]]=fv[f]{\displaystyle X=v^{1}[\mathbf {f} ]X_{1}+v^{2}[\mathbf {f} ]X_{2}+\dots +v^{n}[\mathbf {f} ]X_{n}=\mathbf {f} {\begin{bmatrix}v^{1}[\mathbf {f} ]\\v^{2}[\mathbf {f} ]\\\vdots \\v^{n}[\mathbf {f} ]\end{bmatrix}}=\mathbf {f} v[\mathbf {f} ]}7

for some uniquely determined smooth functionsv1, ...,vn. Upon changing the basisf by a nonsingular matrixA, the coefficientsvi change in such a way that equation (7) remains true. That is,

X=fAv[fA]=fv[f].{\displaystyle X=\mathbf {fA} v[\mathbf {fA} ]=\mathbf {f} v[\mathbf {f} ]\,.}

Consequently,v[fA] =A−1v[f]. In other words, the components of a vector transformcontravariantly (that is, inversely or in the opposite way) under a change of basis by the nonsingular matrixA. The contravariance of the components ofv[f] is notationally designated by placing the indices ofvi[f] in the upper position.

A frame also allows covectors to be expressed in terms of their components. For the basis of vector fieldsf = (X1, ...,Xn) define thedual basis to be thelinear functionals(θ1[f], ...,θn[f]) such that

θi[f](Xj)={1if i=j0if i≠j.{\displaystyle \theta ^{i}[\mathbf {f} ](X_{j})={\begin{cases}1&\mathrm {if} \ i=j\\0&\mathrm {if} \ i\not =j.\end{cases}}}

That is,θi[f](Xj) =δji, theKronecker delta. Let

θ[f]=[θ1[f]θ2[f]⋮θn[f]].{\displaystyle \theta [\mathbf {f} ]={\begin{bmatrix}\theta ^{1}[\mathbf {f} ]\\\theta ^{2}[\mathbf {f} ]\\\vdots \\\theta ^{n}[\mathbf {f} ]\end{bmatrix}}.}

Under a change of basisf ↦fA for a nonsingular matrixA,θ[f] transforms via

θ[fA]=A−1θ[f].{\displaystyle \theta [\mathbf {f} A]=A^{-1}\theta [\mathbf {f} ].}

Any linear functionalα on tangent vectors can be expanded in terms of the dual basisθ

α=a1[f]θ1[f]+a2[f]θ2[f]+⋯+an[f]θn[f]=[a1[f]a2[f]…an[f]]θ[f]=a[f]θ[f]{\displaystyle {\begin{aligned}\alpha &=a_{1}[\mathbf {f} ]\theta ^{1}[\mathbf {f} ]+a_{2}[\mathbf {f} ]\theta ^{2}[\mathbf {f} ]+\cdots +a_{n}[\mathbf {f} ]\theta ^{n}[\mathbf {f} ]\\[8pt]&={\big \lbrack }{\begin{array}{cccc}a_{1}[\mathbf {f} ]&a_{2}[\mathbf {f} ]&\dots &a_{n}[\mathbf {f} ]\end{array}}{\big \rbrack }\theta [\mathbf {f} ]\\[8pt]&=a[\mathbf {f} ]\theta [\mathbf {f} ]\end{aligned}}}8

wherea[f] denotes therow vector[a1[f] ...an[f] ]. The componentsai transform when the basisf is replaced byfA in such a way that equation (8) continues to hold. That is,

α=a[fA]θ[fA]=a[f]θ[f]{\displaystyle \alpha =a[\mathbf {f} A]\theta [\mathbf {f} A]=a[\mathbf {f} ]\theta [\mathbf {f} ]}

whence, becauseθ[fA] =A−1θ[f], it follows thata[fA] =a[f]A. That is, the componentsa transformcovariantly (by the matrixA rather than its inverse). The covariance of the components ofa[f] is notationally designated by placing the indices ofai[f] in the lower position.

Now, the metric tensor gives a means to identify vectors and covectors as follows. HoldingXp fixed, the function

gp(Xp,−):Yp↦gp(Xp,Yp){\displaystyle g_{p}(X_{p},-):Y_{p}\mapsto g_{p}(X_{p},Y_{p})}

of tangent vectorYp defines alinear functional on the tangent space atp. This operation takes a vectorXp at a pointp and produces a covectorgp(Xp, −). In a basis of vector fieldsf, if a vector fieldX has componentsv[f], then the components of the covector fieldg(X, −) in the dual basis are given by the entries of the row vector

a[f]=v[f]TG[f].{\displaystyle a[\mathbf {f} ]=v[\mathbf {f} ]^{\mathsf {T}}G[\mathbf {f} ].}

Under a change of basisf ↦fA, the right-hand side of this equation transforms via

v[fA]TG[fA]=v[f]T(A−1)TATG[f]A=v[f]TG[f]A{\displaystyle 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}

so thata[fA] =a[f]A:a transforms covariantly. The operation of associating to the (contravariant) components of a vector fieldv[f] = [v1[f]v2[f] ...vn[f] ]T the (covariant) components of the covector fielda[f] = [a1[f]a2[f] …an[f] ], where

ai[f]=∑k=1nvk[f]gki[f]{\displaystyle a_{i}[\mathbf {f} ]=\sum _{k=1}^{n}v^{k}[\mathbf {f} ]g_{ki}[\mathbf {f} ]}

is calledlowering the index.

Toraise the index, one applies the same construction but with the inverse metric instead of the metric. Ifa[f] = [a1[f]a2[f] ...an[f] ] are the components of a covector in the dual basisθ[f], then the column vector

v[f]=G−1[f]a[f]T{\displaystyle v[\mathbf {f} ]=G^{-1}[\mathbf {f} ]a[\mathbf {f} ]^{\mathsf {T}}}9

has components which transform contravariantly:

v[fA]=A−1v[f].{\displaystyle v[\mathbf {f} A]=A^{-1}v[\mathbf {f} ].}

Consequently, the quantityX =fv[f] does not depend on the choice of basisf in an essential way, and thus defines a vector field onM. The operation (9) associating to the (covariant) components of a covectora[f] the (contravariant) components of a vectorv[f] given is calledraising the index. In components, (9) is

vi[f]=∑k=1ngik[f]ak[f].{\displaystyle v^{i}[\mathbf {f} ]=\sum _{k=1}^{n}g^{ik}[\mathbf {f} ]a_{k}[\mathbf {f} ].}

Induced metric

[edit]

LetU be anopen set inℝn, and letφ be acontinuously differentiable function fromU into theEuclidean spaceℝm, wherem >n. The mappingφ is called animmersion if its differential isinjective at every point ofU. The image ofφ is called animmersed submanifold. More specifically, form = 3, which means that the ambient Euclidean space isℝ3, the induced metric tensor is called thefirst fundamental form.

Suppose thatφ is an immersion onto the submanifoldM ⊂Rm. The usual Euclideandot product inℝm is a metric which, when restricted to vectors tangent toM, gives a means for taking the dot product of these tangent vectors. This is called theinduced metric.

Suppose thatv is a tangent vector at a point ofU, say

v=v1e1+⋯+vnen{\displaystyle v=v^{1}\mathbf {e} _{1}+\dots +v^{n}\mathbf {e} _{n}}

whereei are the standard coordinate vectors inℝn. Whenφ is applied toU, the vectorv goes over to the vector tangent toM given by

φ∗(v)=∑i=1n∑a=1mvi∂φa∂xiea.{\displaystyle \varphi _{*}(v)=\sum _{i=1}^{n}\sum _{a=1}^{m}v^{i}{\frac {\partial \varphi ^{a}}{\partial x^{i}}}\mathbf {e} _{a}\,.}

(This is called thepushforward ofv alongφ.) Given two such vectors,v andw, the induced metric is defined by

g(v,w)=φ∗(v)⋅φ∗(w).{\displaystyle g(v,w)=\varphi _{*}(v)\cdot \varphi _{*}(w).}

It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fieldse is given by

G(e)=(Dφ)T(Dφ){\displaystyle G(\mathbf {e} )=(D\varphi )^{\mathsf {T}}(D\varphi )}

whereDφ is the Jacobian matrix:

Dφ=[∂φ1∂x1∂φ1∂x2…∂φ1∂xn∂φ2∂x1∂φ2∂x2…∂φ2∂xn⋮⋮⋱⋮∂φm∂x1∂φm∂x2…∂φm∂xn].{\displaystyle 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}}.}

Intrinsic definitions of a metric

[edit]

The notion of a metric can be definedintrinsically using the language offiber bundles andvector bundles. In these terms, ametric tensor is a function

g:TM×MTM→R{\displaystyle g:\mathrm {T} M\times _{M}\mathrm {T} M\to \mathbf {R} }10

from thefiber product of thetangent bundle ofM with itself toR such that the restriction ofg to each fiber is a nondegenerate bilinear mapping

gp:TpM×TpM→R.{\displaystyle g_{p}:\mathrm {T} _{p}M\times \mathrm {T} _{p}M\to \mathbf {R} .}

The mapping (10) is required to becontinuous, and oftencontinuously differentiable,smooth, orreal analytic, depending on the case of interest, and whetherM can support such a structure.

Metric as a section of a bundle

[edit]

By theuniversal property of the tensor product, any bilinear mapping (10) gives risenaturally to asectiong⊗ of thedual of thetensor product bundle ofTM with itself

g⊗∈Γ((TM⊗TM)∗).{\displaystyle g_{\otimes }\in \Gamma \left((\mathrm {T} M\otimes \mathrm {T} M)^{*}\right).}

The sectiong⊗ is defined on simple elements ofTM ⊗ TM by

g⊗(v⊗w)=g(v,w){\displaystyle g_{\otimes }(v\otimes w)=g(v,w)}

and is defined on arbitrary elements ofTM ⊗ TM by extending linearly to linear combinations of simple elements. The original bilinear formg is symmetric if and only if

g⊗∘τ=g⊗{\displaystyle g_{\otimes }\circ \tau =g_{\otimes }}

where

τ:TM⊗TM→≅TM⊗TM{\displaystyle \tau :\mathrm {T} M\otimes \mathrm {T} M{\stackrel {\cong }{\to }}TM\otimes TM}

is thebraiding map.

SinceM is finite-dimensional, there is anatural isomorphism

(TM⊗TM)∗≅T∗M⊗T∗M,{\displaystyle (\mathrm {T} M\otimes \mathrm {T} M)^{*}\cong \mathrm {T} ^{*}M\otimes \mathrm {T} ^{*}M,}

so thatg⊗ is regarded also as a section of the bundleT*M ⊗ T*M of thecotangent bundleT*M with itself. Sinceg is symmetric as a bilinear mapping, it follows thatg⊗ is asymmetric tensor.

Metric in a vector bundle

[edit]
See also:metric (vector bundle)

More generally, one may speak of a metric in avector bundle. IfE is a vector bundle over a manifoldM, then a metric is a mapping

g:E×ME→R{\displaystyle g:E\times _{M}E\to \mathbf {R} }

from thefiber product ofE toR which is bilinear in each fiber:

gp:Ep×Ep→R.{\displaystyle g_{p}:E_{p}\times E_{p}\to \mathbf {R} .}

Using duality as above, a metric is often identified with asection of thetensor product bundleE* ⊗E*.

Tangent–cotangent isomorphism

[edit]
See also:Musical isomorphism

The metric tensor gives anatural isomorphism from thetangent bundle to thecotangent bundle, sometimes called themusical isomorphism.[6] This isomorphism is obtained by setting, for each tangent vectorXp ∈ TpM,

SgXp=defg(Xp,−),{\displaystyle S_{g}X_{p}\,{\stackrel {\text{def}}{=}}\,g(X_{p},-),}

thelinear functional onTpM which sends a tangent vectorYp atp togp(Xp,Yp). That is, in terms of the pairing[−, −] betweenTpM and itsdual spaceT∗
pM,

[SgXp,Yp]=gp(Xp,Yp){\displaystyle [S_{g}X_{p},Y_{p}]=g_{p}(X_{p},Y_{p})}

for all tangent vectorsXp andYp. The mappingSg is alinear transformation fromTpM toT∗
pM. It follows from the definition of non-degeneracy that thekernel ofSg is reduced to zero, and so by therank–nullity theorem,Sg is alinear isomorphism. Furthermore,Sg is asymmetric linear transformation in the sense that

[SgXp,Yp]=[SgYp,Xp]{\displaystyle [S_{g}X_{p},Y_{p}]=[S_{g}Y_{p},X_{p}]}

for all tangent vectorsXp andYp.

Conversely, any linear isomorphismS : TpM → T∗
pM defines a non-degenerate bilinear form onTpM by means of

gS(Xp,Yp)=[SXp,Yp].{\displaystyle g_{S}(X_{p},Y_{p})=[SX_{p},Y_{p}]\,.}

This bilinear form is symmetric if and only ifS is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms onTpM and symmetric linear isomorphisms ofTpM to the dualT∗
pM.

Asp varies overM,Sg defines a section of the bundleHom(TM, T*M) ofvector bundle isomorphisms of the tangent bundle to the cotangent bundle. This section has the same smoothness asg: it is continuous, differentiable, smooth, or real-analytic according asg. The mappingSg, which associates to every vector field onM a covector field onM gives an abstract formulation of "lowering the index" on a vector field. The inverse ofSg is a mappingT*M → TM which, analogously, gives an abstract formulation of "raising the index" on a covector field.

The inverseS−1
g defines a linear mapping

Sg−1:T∗M→TM{\displaystyle S_{g}^{-1}:\mathrm {T} ^{*}M\to \mathrm {T} M}

which is nonsingular and symmetric in the sense that

[Sg−1α,β]=[Sg−1β,α]{\displaystyle \left[S_{g}^{-1}\alpha ,\beta \right]=\left[S_{g}^{-1}\beta ,\alpha \right]}

for all covectorsα,β. Such a nonsingular symmetric mapping gives rise (by thetensor-hom adjunction) to a map

T∗M⊗T∗M→R{\displaystyle \mathrm {T} ^{*}M\otimes \mathrm {T} ^{*}M\to \mathbf {R} }

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

TM⊗TM.{\displaystyle \mathrm {T} M\otimes \mathrm {T} M.}

Arclength and the line element

[edit]

Suppose thatg is a Riemannian metric onM. In a local coordinate systemxi,i = 1, 2, …,n, the metric tensor appears as amatrix, denoted here byG, whose entries are the componentsgij of the metric tensor relative to the coordinate vector fields.

Letγ(t) be a piecewise-differentiableparametric curve inM, fora ≤t ≤b. Thearclength of the curve is defined by

L=∫ab∑i,j=1ngij(γ(t))(ddtxi∘γ(t))(ddtxj∘γ(t))dt.{\displaystyle 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\,.}

In connection with this geometrical application, thequadraticdifferential form

ds2=∑i,j=1ngij(p)dxidxj{\displaystyle ds^{2}=\sum _{i,j=1}^{n}g_{ij}(p)dx^{i}dx^{j}}

is called thefirst fundamental form associated to the metric, whileds is theline element. Whends2 ispulled back to the image of a curve inM, 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

L=∫ab|∑i,j=1ngij(γ(t))(ddtxi∘γ(t))(ddtxj∘γ(t))|dt.{\displaystyle L=\int _{a}^{b}{\sqrt {\left|\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)\right|}}\,dt\,.}

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 geodesics

[edit]

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

E=12∫ab∑i,j=1ngij(γ(t))(ddtxi∘γ(t))(ddtxj∘γ(t))dt.{\displaystyle E={\frac {1}{2}}\int _{a}^{b}\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\,.}

This usage comes fromphysics, specifically,classical mechanics, where the integralE can be seen to directly correspond to thekinetic energy of apoint particle moving on the surface of a manifold. Thus, for example, in Jacobi's formulation ofMaupertuis' 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, thegeodesic equations may be obtained by applyingvariational principles to either the length or the energy. In the latter case, the geodesic equations are seen to arise from theprinciple 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]

Canonical measure and volume form

[edit]

In analogy with the case of surfaces, a metric tensor on ann-dimensional paracompact manifoldM gives rise to a natural way to measure then-dimensionalvolume of subsets of the manifold. The resulting natural positiveBorel measure allows one to develop a theory of integrating functions on the manifold by means of the associatedLebesgue integral.

A measure can be defined, by theRiesz representation theorem, by giving apositive linear functionalΛ on the spaceC0(M) ofcompactly supportedcontinuous functions onM. More precisely, ifM is a manifold with a (pseudo-)Riemannian metric tensorg, then there is a unique positiveBorel measureμg such that for anycoordinate chart(U,φ),Λf=∫Ufdμg=∫φ(U)f∘φ−1(x)|detg|dx{\displaystyle \Lambda f=\int _{U}f\,d\mu _{g}=\int _{\varphi (U)}f\circ \varphi ^{-1}(x){\sqrt {\left|\det g\right|}}\,dx}for allf supported inU. Heredetg is thedeterminant of the matrix formed by the components of the metric tensor in the coordinate chart. ThatΛ is well-defined on functions supported in coordinate neighborhoods is justified byJacobian change of variables. It extends to a unique positive linear functional onC0(M) by means of apartition of unity.

IfM is alsooriented, then it is possible to define a naturalvolume form from the metric tensor. In apositively oriented coordinate system(x1, ...,xn) the volume form is represented asω=|detg|dx1∧⋯∧dxn{\displaystyle \omega ={\sqrt {\left|\det g\right|}}\,dx^{1}\wedge \cdots \wedge dx^{n}}where thedxi are thecoordinate differentials and∧ denotes theexterior product in the algebra ofdifferential 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.

Examples

[edit]

Euclidean metric

[edit]

The most familiar example is that of elementaryEuclidean geometry: the two-dimensionalEuclidean metric tensor. In the usualCartesian(x,y) coordinates, we can write

g=[1001].{\displaystyle g={\begin{bmatrix}1&0\\0&1\end{bmatrix}}\,.}

The length of a curve reduces to the formula:

L=∫ab(dx)2+(dy)2.{\displaystyle L=\int _{a}^{b}{\sqrt {(dx)^{2}+(dy)^{2}}}\,.}

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

Polar coordinates(r,θ):

x=rcos⁡θy=rsin⁡θJ=[cos⁡θ−rsin⁡θsin⁡θrcos⁡θ].{\displaystyle {\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta \\J&={\begin{bmatrix}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \end{bmatrix}}\,.\end{aligned}}}

So

g=JTJ=[cos2⁡θ+sin2⁡θ−rsin⁡θcos⁡θ+rsin⁡θcos⁡θ−rcos⁡θsin⁡θ+rcos⁡θsin⁡θr2sin2⁡θ+r2cos2⁡θ]=[100r2]{\displaystyle 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}}}

bytrigonometric identities.

In general, in aCartesian coordinate systemxi on aEuclidean space, the partial derivatives∂ / ∂xi areorthonormal with respect to the Euclidean metric. Thus the metric tensor is theKronecker delta δij in this coordinate system. The metric tensor with respect to arbitrary (possibly curvilinear) coordinatesqi is given by

gij=∑klδkl∂xk∂qi∂xl∂qj=∑k∂xk∂qi∂xk∂qj.{\displaystyle 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}}}.}

The round metric on a sphere

[edit]

The unit sphere inℝ3 comes equipped with a natural metric induced from the ambient Euclidean metric, through the process explained in theinduced metric section. In standard spherical coordinates(θ,φ), withθ thecolatitude, the angle measured from thez-axis, andφ the angle from thex-axis in thexy-plane, the metric takes the form

g=[100sin2⁡θ].{\displaystyle g={\begin{bmatrix}1&0\\0&\sin ^{2}\theta \end{bmatrix}}\,.}

This is usually written in the form

ds2=dθ2+sin2⁡θdφ2.{\displaystyle ds^{2}=d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\,.}

Lorentzian metrics from relativity

[edit]
Main article:Metric tensor (general relativity)

In flatMinkowski space (special relativity), with coordinates

rμ→(x0,x1,x2,x3)=(ct,x,y,z),{\displaystyle r^{\mu }\rightarrow \left(x^{0},x^{1},x^{2},x^{3}\right)=(ct,x,y,z)\,,}

the metric is, depending on choice ofmetric signature,

g=[10000−10000−10000−1]org=[−1000010000100001].{\displaystyle 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}}\,.}

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

In this case, thespacetime interval is written as

ds2=c2dt2−dx2−dy2−dz2=drμdrμ=gμνdrμdrν.{\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}=dr^{\mu }dr_{\mu }=g_{\mu \nu }dr^{\mu }dr^{\nu }\,.}

TheSchwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or ablack hole. With coordinates

(x0,x1,x2,x3)=(ct,r,θ,φ),{\displaystyle \left(x^{0},x^{1},x^{2},x^{3}\right)=(ct,r,\theta ,\varphi )\,,}

we can write the metric as

gμν=[(1−2GMrc2)0000−(1−2GMrc2)−10000−r20000−r2sin2⁡θ],{\displaystyle g_{\mu \nu }={\begin{bmatrix}\left(1-{\frac {2GM}{rc^{2}}}\right)&0&0&0\\0&-\left(1-{\frac {2GM}{rc^{2}}}\right)^{-1}&0&0\\0&0&-r^{2}&0\\0&0&0&-r^{2}\sin ^{2}\theta \end{bmatrix}}\,,}

whereG (inside the matrix) is thegravitational constant andM represents the totalmass–energy content of the central object.

See also

[edit]

Notes

[edit]
  1. ^More precisely, the integrand is thepullback of this differential to the curve.
  2. ^In several formulations ofclassical unified field theories, the metric tensor was allowed to be non-symmetric; however, the antisymmetric part of such a tensor plays no role in the contexts described here, so it will not be further considered.
  3. ^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 ofWells (1980). Typically, such explicit dependence on the basis is entirely suppressed.
  4. ^Dodson & Poston 1991, Chapter VII §3.04
  5. ^Vaughn 2007, §3.4.3
  6. ^For the terminology "musical isomorphism", seeGallot, Hulin & Lafontaine (2004, p. 75). See alsoLee (1997, pp. 27–29)
  7. ^Sternberg 1983

References

[edit]
Basic concepts
Types of manifolds
Main results
Generalizations
Applications
Scope
Mathematics
Notation
Tensor
definitions
Operations
Related
abstractions
Notable tensors
Mathematics
Physics
Mathematicians
Basic concepts
Main theorems(list)
Maps
Types of
manifolds
Tensors
Vectors
Covectors
Bundles
Connections
Related
Generalizations
Retrieved from "https://en.wikipedia.org/w/index.php?title=Metric_tensor&oldid=1314503942"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp