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Metric tensor
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In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of, and angle between, tangent vectors.
A metric tensor is defined to be a nondegenerate symmetric bilinear form on each tangent space, which varies smoothly from point to point.
It is an example of a tensor field.
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In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of, and angle between, tangent vectors.
A metric tensor is defined to be a nondegenerate symmetric bilinear form on each tangent space, which varies smoothly from point to point.
It is an example of a tensor field. Relative to a local coordinate system, a metric tensor takes on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system.
Introduction
Carl Friedrich Gauss in his 1827 Disquisitiones generales circa superficies curvas (General investigations of curved surfaces) considered a surface parametrically, with the Cartesian coordinates x, y, and z of points on the surface depend on two auxiliary variables u and v. Thus a parametric surface is (in contemporary terms) a vector valued function
depending on an ordered pair of real variables (u,v), and defined in an open set D in the uv-plane. One of the chief aims of Gauss' 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, or tangent vectors at the same point of the surface. 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.
Arclength
If the variables u and v are taken to depend on a third variable, t, taking values in an interval [a,b], then will trace out a parametric curve in M. The arclength of that curve is given by the integral
-
(the subscripts indicate partial derivatives). The integrand is the restriction to the curve of the square root of the (quadratic) differential
where
The quantity ds2 in is called the line element or first fundamental form of M. Intuitively, it represents the principal part of the square of the displacement undergone by when u is increased by du units, and v is increased by dv units.
Suppose now that a different parameterization is selected, by allowing u and v to depend on another pair of variables u′ and v′. Then the analog of for the new variables is
The chain rule relates E′, F′, and G′ to E,F, and G via the matrix equation
where the superscript T denotes the matrix transpose. The matrix with the coefficients E, F, and G arranged in this way therefore transforms by the Jacobian matrix of the coordinate change
A matrix which transforms in this way is one kind of what is called a tensor. The matrix
with the transformation law is known as the metric tensor of the surface.
first observed the significance of a system of coefficients E, F, and G, that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form 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,
so that
Angle
Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the angle between two tangent vectors to the surface. 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 M can be written in the form
for suitable real numbers a and b. 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. It is more profitably viewed, however, as a function that takes a pair of arguments a = [a1 b1] and b = [a2 b2] which are vectors in the uv-plane. That is, put
This is a symmetric function in a and b, meaning that
It is also bilinear meaning that it is linear in each variable a and b separately. That is,
for any vectors a, a′, b, and b′ in the uv plane, and any real numbers μ and λ.
Area
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 M is parameterized by the function over the domain D in the uv-plane, then the surface area of M is given by the integral
where × denotes the cross product, and the absolute value denotes the length of a vector in Euclidean space. By Lagrange's identity for the cross product, the integral can be written
where det is the determinant.
Definition
Let M be a smooth manifold of dimension n; for instance a surface (in the case n = 2) or hypersurface in the Cartesian space Rn+1. 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 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:
gp is bilinear. A function of two vector arguments is bilinear if it is linear separately in each argument. Thus if Up, Vp, Yp are two tangent vectors at p and a and b are real numbers, then
-
gp is symmetric. A function of two vector arguments is symmetric provided that for all vectors Xp and Yp,
-
gp is nondegenerate. A bilinear function is nondegenerate provided that, for every tangent vector Xp ? 0, the function
- obtained by holding
Xp constant and allowing Yp to vary is not identically zero. That is, for every Xp ? 0 there exists a Yp such that gp(Xp,Yp) ? 0.
A metric tensor g on M assigns to each point p of M a metric gp in the tangent space at p such that in a way that varies smoothly with p. More precisely, given any open set U ? M and any vector fields X and Y on U, the function
is a smooth function of p.
Components of the metric
The components of the metric in any basis of vector fields, or frame, f = (X1, …, Xn) are given by
The n2 functions gij[f] form the entries of an n×n symmetric matrix, G[f]. If
are two vectors at p ? U, then value of the metric applied to v and w is determined by the coefficients by bilinearity:
Denoting the matrix (gij[f]) by G[f] and arranging the components of the vectors v and w into column vectors v[f] and w[f],
where v[f]T and w[f]T denote the transpose of the vectors v[f] and w[f], respectively. Under a change of basis of the form
for some invertible n×n matrix A, the matrix of components of the metric changes by A as well. That is,
or, in terms of the entries of this matrix,
For this reason, the system of quantities gij[f] is said to transform covariantly with respect to changes in the frame f.
Metric in coordinates
A system of n real valued functions (x1, …, xn), giving a local coordinate system on an open set U in M, determines a basis of vector fields on U
The metric g has components relative to this frame given by
Relative to a new system of local coordinates, say
the 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
so that
Or, in terms of the matrices G[f] = (gij[f]) and G[f′] = (gij[f′]),
where Dy denotes the Jacobian matrix of the coordinate change.
Signature of a metric
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. More generally, if the quadratic forms qmhave constant signature independent of m, then the signature of g is this signature, and g is called a pseudo-Riemannian metric. If M is connected, then the signature of qm does not depend on m.
By Sylvester's law of inertia, a basis of tangent vectors Xi can be chosen locally so that the quadratic form diagonalizes in the following manner
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. The signature of g is the pair of integers (p, n − p), signifying that there are p positive signs and n − p negative signs in any such expression. Equivalently, the metric has signature (p,n − p) if the matrix gij of the metric has p positive and n − p negative eigenvalues.
Certain metric signatures which arise frequently in applications are:
- If g has signature (n, 0), then g is a Riemannian metric, and M is called a Riemannian manifold. Otherwise, g is a pseudo-Riemannian metric, and M is called a pseudo-Riemannian manifold (the term semi-Riemannian is also used).
- If M is four-dimensional with signature (1,3) or (3,1), then the metric is called Lorentzian. More generally, a metric tensor in dimension n other than 4 of signature (1,n − 1) or (n − 1, 1) is sometimes also called Lorentzian.
- If M is 2n-dimensional and g has signature (n,n), then the metric is called ultrahyperbolic.
Inverse metric
Let f = (X1, …, Xn) be a basis of vector fields, and as above let G[f] be the matrix of coeffients
One can consider the inverse matrix G[f]-1, which is identified with the inverse metric (or conjugate or dual metric). The inverse metric satisfies a transformation law when the frame f is changed by a matrix A via
The inverse metric transforms contravariantly, or with respect to the inverse of the change of basis matrix A. 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 α is a covector field. To wit, for each point p, α determines a function αp defined on tangent vectors at p so that the following linearity condition holds for all tangent vectors Xp and Yp, and all real numbers a and b:
As p varies, α is assumed to be a smooth function in the sense that
is a smooth function of p for any smooth vector field X.
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
That is, the row vector of components α[f] transforms as a covariant vector.
For a pair α and β of covector fields, define the inverse metric applied to these two covectors by
The resulting definition, although it involves the choice of basis f, does not actually depend on f in an essential way. Indeed, changing basis to fA gives
So that the right-hand side of equation is unaffected by changing the basis f to any other basis fA whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrix G[f] are denoted by gij, where the indices i and j have been raised to indicate the transformation law .
Raising and lowering indices
In a basis of vector fields f = (X1, …, Xn), any smooth tangent vector field X can be written in the form
for some uniquely determined smooth functions v1, …, vn. Upon changing the basis f by a nonsingular matrix A, the coefficients vi change in such a way that equation remains true. That is,
Consequently, v[fA] = A-1v[f]. In other words, the components of a vector transform contravariantly (with respect to the inverse) under a change of basis by the nonsingular matrix A. The contravariance of the components of v[f] is notationally designated by placing the indices of vi[f] in the upper position.
A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields f = (X1, …, Xn) define the dual basis to be the linear functionals (θ1[f], …, θn[f]) such that
That is, θi[f](Xj) = δji, the Kronecker delta. Let
Under a change of basis f → fA for a nonsingular matrix A, θ[f] transforms via
Any linear functional α on tangent vectors can be expanded in terms of the dual basis θ
where a[f] denotes the row vector [a1[f] … an[f] ]. The components ai transform when the basis f is replaced by fA in such a way that equation continues to hold. That is,
whence, because θ[fA] = A-1θ[f], it follows that a[fA] = a[f]A. That is, the components a transform covariantly (by the matrix A rather than its inverse). The covariance of the components of a[f] is notationally designated by placing the indices of ai[f] in the lower position.
Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding Xp fixed, the function
of tangent vector Yp defines a linear functional on the tangent space at p. This operation takes a vector Xp at a point p and produces a covector gp(Xp, −). 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. The operation of associating to the (contravariant) components of a vector field v[f] = [v1[f] v2[f] … vn[f]]T the (covariant) components of the covector field a[f] = [a1[f] a2[f] … an[f]] where
is called lowering the index.
To raise the index, one applies the same construction but with the inverse metric instead of the metric. If a[f] = [a1[f] a2[f] … an[f]] are the components of a covector in the dual basis θ[f], then the column vector
has components which transform contravariantly:
Consequently, the quantity X = fv[f] does not depend on the choice of basis f in an essential way, and thus defines a vector field on M. The operation associating to the (covariant) components of a covector a[f] the (contravariant) components of a vector v[f] given is called raising the index. In components, is
Induced metric
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 φ is an injective function and the Jacobian matrix of φ has rank n at every point of U. The image of φ is called an immersed submanifold.
Suppose that φ is an immersion onto the submanifold M ⊂ Rm. The image The usual Euclidean dot product in Rm is a metric which, when restricted to vectors tangent to M, gives a means for taking the dot product of these tangent vectors. This is called the induced metric.
Suppose that v is a tangent vector at a point of U, say
where ei are the standard coordinate vectors in Rn. When φ is applied to U, the vector v goes over to the vector tangent to M given by
(this is called the pushforward of v along φ.) Given two such vectors, v and w, the induced metric is defined by
It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields e is given by
where Dφ is the Jacobian matrix:
Intrinsic definitions of a metric
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
from the fiber product of the tangent bundle of M to R such that the restriction of g to each fiber is a nondegenerate bilinear mapping
The mapping is required to be continuous, and often continuously differentiable, smooth, or real analytic, depending on the case of interest, and whether M can support such a structure.
Metric as a section of a bundle
By the universal property of the tensor product, any bilinear mapping gives rise naturally to a section g⊗ of the dual of the tensor product bundle of TM with itself
The section g⊗ is defined on simple elements of TM⊗TM by
and is defined on arbitrary elements of TM⊗TM by extending linearly to linear combinations of simple elements. The original bilinear form g is symmetric if and only if
where
is the braiding map.
Since M is finite-dimensional, there is a natural isomorphism
so that g⊗ is regarded also as a section of the bundle T*M⊗T*M of the cotangent bundle T*M with itself. Since g is symmetric as a bilinear mapping, it follows that g⊗ is a symmetric tensor.
Metric in a vector bundle
More generally, one may speak of a metric in a vector bundle. If E is a vector bundle over a manifold M, then a metric is a mapping
from the fiber product of E to R which is bilinear in each fiber:
Using duality as above, a metric is often identified with a section of the tensor product bundle E* ? E*. (See metric (vector bundle).)
Tangent-cotangent isomorphism
The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle. This isomorphism is obtained by setting, for each tangent vector Xp ∈ TpM,
the linear functional on TpM which sends a tangent vector Yp at p to gp(Xp,Yp). That is, in terms of the pairing [•,•] between TpM and its dual space Tp*M,
for all tangent vectors Xp and Yp. The mapping Sg is a linear transformation from TpM to Tp*M. It follows from the definition of non-degeneracy that the kernel of Sg is reduced to zero, and so by the rank-nullity theorem, Sg is a linear isomorphism. Furthermore, Sg is a symmetric linear transformation in the sense that
for all tangent vectors Xp and Yp.
Conversely, any linear isomorphism S : TpM → TpM defines a non-degenerate bilinear form on TpM by means of
This bilinear form is symmetric if and only if S is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on TpM and symmetric linear isomorphisms of TpM to the dual Tp*M.
As p varies over M, Sg defines a section of the bundle Hom(TM,T*M) of vector bundle isomorphisms of the tangent bundle to the cotangent bundle. 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. The inverse of Sg is a mapping T*M → TM which, analogously, gives an abstract formulation of "raising the index" on a covector field.
The inverse Sg-1 defines a linear mapping
which is nonsingular and symmetric in the sense that
for all covectors α, β. Such a nonsingular symmetric mapping gives rise (by the tensor-hom adjunction) to a map
or by the double dual isomorphism to a section of the tensor product
Arclength and the line element
Suppose that g is a Riemannian metric on M. In a local coordinate system xi, i = 1,2,…,n, the metric tensor appears as a matrix, denoted here by G, whose entries are the components gij of the metric tensor relative to the coordinate vector fields.
Let ?(t) be a piecewise differentiable parametric curve in M, for a =t = b. The arclength of the curve is defined by
In connection with this geometrical application, the quadratic differential form
is called the line element or first fundamental form associated to the metric. When ds2 is pulled back to the image of a curve in M, 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
Note that, 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
Given a segment of a curve, another frequently defined quantity is the (kinetic) energy of the curve:
This usage comes from physics, specifically, classical mechanics, where the integral E can be seen to directly correspond to the kinetic energy of a point particle moving on the surface of a manifold. 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 later 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.
Other properties
- The angle between two tangent vectors, and , is defined as:
-
Examples
The Euclidean metric
The most familiar example is that of basic high-school geometry: the two-dimensional Euclidean metric tensor. In the usual - coordinates, we can write
The length of a curve reduces to the familiar calculus formula:
The Euclidean metric in some other common coordinate systems can be written as follows.
Polar coordinates:
So
by trigonometric identities.
In general, if the are Cartesian (i.e. orthogonal) coordinates on a Euclidean space, the metric tensor with respect to arbitrary (possibly curvilinear) coordinates is given by:
-
The round metric on a sphere
The unit sphere in R3 comes equipped with a natural metric induced from the ambient Euclidean metric. In standard spherical coordinates the metric takes the form
This is usually written in the form
Lorentzian metrics from relativity
In flat Minkowski space (special relativity), with coordinates the metric is
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
The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. With coordinates , we can write the metric as
See also
External links
- — A simple introduction to the basics of metrics in the context of relativity.
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