musical isomorphism

In linear algebra, a finite-dimensional vector space is isomorphic to its dual space, but not canonically isomorphic to it. On the other hand, a finite-dimensional vector space $V$ endowed with a non-degenerate bilinear form $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$, is canonically isomorphic to its dual. The canonical isomorphism $V\; \backslash to\; V^*$ is given by

: $v\; \backslash mapsto\; \backslash langle\; v,\; \backslash cdot\; \backslash rangle$.

The non-degeneracy of $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ means exactly that the above map is an isomorphism.

An example is where $V\; =\; \backslash mathbb\; R^n$, and $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ is the dot product.

The musical isomorphisms are the global version of this isomorphism and its inverse for the tangent bundle and cotangent bundle of a (pseudo-)Riemannian manifold $(M,g)$. They are canonical isomorphisms of vector bundles which are at any point {{math|p}} the above isomorphism applied to the tangent space of {{math|M}} at {{math|p}} endowed with the inner product $g\_p$.

Because every paracompact manifold can be (non-canonically) endowed with a Riemannian metric, the musical isomorphisms show that a vector bundle on a paracompact manifold is (non-canonically) isomorphic to its dual.

Let {{math|(M, g)}} be a (pseudo-)Riemannian manifold. At each point {{mvar|p}}, the map {{math|g{{sub|p}}}} is a non-degenerate bilinear form on the tangent space {{math|T{{sub|p}}M}}. If {{mvar|v}} is a vector in {{math|T{{sub|p}}M}}, its flat is the covector

: $v^\backslash flat\; =\; g\_p(v,\backslash cdot)$

in {{math|T{{su|lh=1em|p=∗|b=p }}M}}. Since this is a smooth map that preserves the point {{mvar|p}}, it defines a morphism of smooth vector bundles $\backslash flat\; :\; \backslash mathrm\{T\}M\; \backslash to\; \backslash mathrm\{T\}^*M$. By non-degeneracy of the metric, $\backslash flat$ has an inverse $\backslash sharp$ at each point, characterized by

: $g\_p(\backslash alpha^\backslash sharp,\; v)\; =\; \backslash alpha(v)$

for {{mvar|α}} in {{math|T{{su|lh=1em|p=∗|b=p }}M}} and {{mvar|v}} in {{math|T{{sub|p}}M}}. The vector $\backslash alpha^\backslash sharp$ is called the sharp of {{mvar|α}}. The sharp map is a smooth bundle map $\backslash sharp\; :\; \backslash mathrm\{T\}^*M\; \backslash to\; \backslash mathrm\{T\}M$.

Flat and sharp are mutually inverse isomorphisms of smooth vector bundles, hence, for each {{mvar|p}} in {{mvar|M}}, there are mutually inverse vector space isomorphisms between {{math|T{{sub|p }}M}} and {{math|T{{su|lh=1em|p=∗|b=p }}M}}.

The flat and sharp maps can be applied to vector fields and covector fields by applying them to each point. Hence, if {{mvar|X}} is a vector field and {{mvar|ω}} is a covector field,

: $X^\backslash flat\; =\; g(X,\backslash cdot)$

and

: $g(\backslash omega^\backslash sharp,\; X)\; =\; \backslash omega(X)$.

Suppose {{math|{e{{sub|i}}}{{null}}}} is a moving tangent frame (see also smooth frame) for the tangent bundle {{math|TM}} with, as dual frame (see also dual basis), the moving coframe (a moving tangent frame for the cotangent bundle $\backslash mathrm\{T\}^*M$; see also coframe) {{math|{e{{sup|i}}}{{null}}}}. Then the pseudo-Riemannian metric, which is a symmetric and nondegenerate {{math|2}}-covariant tensor field can be written locally in terms of this coframe as {{math|g {{=}} g{{sub|ij }}e{{sup|i}} ⊗ e{{i sup|j}}}} using Einstein summation notation.

Given a vector field {{math|X {{=}} X{{i sup|i }}e{{sub|i}}}} and denoting {{math|1=g{{sub|ij}} X{{i sup|i}} = X{{sub|j}}}}, its flat is

: $X^\backslash flat\; =\; g\_\{ij\}\; X^i\; \backslash mathbf\{e\}^j\; =\; X\_j\; \backslash mathbf\{e\}^j$.

This is referred to as lowering an index.

In the same way, given a covector field {{math|1=ω = ω{{sub|i}} e{{sup|i}}}} and denoting {{math|1=g{{sup|ij}} ω{{sub|i}} = ω{{sup|j}}}}, its sharp is

: $\backslash omega^\backslash sharp\; =\; g^\{ij\}\; \backslash omega\_i\; \backslash mathbf\{e\}\_j\; =\; \backslash omega^j\; \backslash mathbf\{e\}\_j$,

where {{math|g{{i sup|ij}}}} are the components of the inverse metric tensor (given by the entries of the inverse matrix to {{math|g{{sub|ij}}}}). Taking the sharp of a covector field is referred to as raising an index.

The musical isomorphisms may also be extended to the bundles

$$\backslash bigotimes\; ^k\; \{\backslash rm\; T\}\; M,\; \backslash qquad\; \backslash bigotimes\; ^k\; \{\backslash rm\; T\}^*\; M\; .$$

Which index is to be raised or lowered must be indicated. For instance, consider the {{nowrap|(0, 2)}}-tensor field {{math|X {{=}} X{{sub|ij }}e{{sup|i}} ⊗ e{{i sup|j}}}}. Raising the second index, we get the {{nowrap|(1, 1)}}-tensor field

$$X^\backslash sharp\; =\; g^\{jk\}\; X\_\{ij\}\; \backslash ,\; \{\backslash rm\; e\}^i\; \backslash otimes\; \{\backslash rm\; e\}\_k\; .$$

In the context of exterior algebra, an extension of the musical operators may be defined on {{math|⋀V}} and its dual {{math|⋀{{su|lh=1em|b=|p=∗}}V}}, which with minor abuse of notation may be denoted the same, and are again mutual inverses:{{sfn|Vaz|da Rocha|2016|loc=pp. 48, 50}}

$$\backslash flat\; :\; \{\backslash bigwedge\}\; V\; \backslash to\; \{\backslash bigwedge\}^*\; V\; ,\; \backslash qquad\; \backslash sharp\; :\; \{\backslash bigwedge\}^*\; V\; \backslash to\; \{\backslash bigwedge\}\; V\; ,$$

defined by

$$(X\; \backslash wedge\; \backslash ldots\; \backslash wedge\; Z)^\backslash flat\; =\; X^\backslash flat\; \backslash wedge\; \backslash ldots\; \backslash wedge\; Z^\backslash flat\; ,\; \backslash qquad$$

(\alpha \wedge \ldots \wedge \gamma)^\sharp = \alpha^\sharp \wedge \ldots \wedge \gamma^\sharp .

In this extension, in which {{math|{{music|flat}}}} maps p-vectors to p-covectors and {{math|{{music|sharp}}}} maps p-covectors to p-vectors, all the indices of a totally antisymmetric tensor are simultaneously raised or lowered, and so no index need be indicated:

$$Y^\backslash sharp\; =\; (\; Y\_\{i\; \backslash dots\; k\}\; \backslash mathbf\{e\}^i\; \backslash otimes\; \backslash dots\; \backslash otimes\; \backslash mathbf\{e\}^k)^\backslash sharp\; =\; g^\{ir\}\; \backslash dots\; g^\{kt\}\; \backslash ,\; Y\_\{i\; \backslash dots\; k\}\; \backslash ,\; \backslash mathbf\{e\}\_r\; \backslash otimes\; \backslash dots\; \backslash otimes\; \backslash mathbf\{e\}\_t\; .$$

More generally, musical isomorphisms always exist between a vector bundle endowed with a bundle metric and its dual.

Given a type {{nowrap|(0, 2)}} tensor field {{math|X {{=}} X{{sub|ij }}e{{sup|i}} ⊗ e{{i sup|j}}}}, we define the trace of {{mvar|X}} through the metric tensor {{mvar|g}} by

$$\backslash operatorname\{tr\}\_g\; (\; X\; )\; :=\; \backslash operatorname\{tr\}\; (\; X^\backslash sharp\; )\; =\; \backslash operatorname\{tr\}\; (\; g^\{jk\}\; X\_\{ij\}\; \backslash ,\; \{\backslash bf\; e\}^i\; \backslash otimes\; \{\backslash bf\; e\}\_k\; )\; =\; g^\{ij\}\; X\_\{ij\}\; .$$

Observe that the definition of trace is independent of the choice of index to raise, since the metric tensor is symmetric.

Mathematically vectors are elements of a vector space $V$ over a field $K$, and for use in physics $V$ is usually defined with $K=\backslash mathbb\{R\}$ or $\backslash mathbb\{C\}$. Concretely, if the dimension $n=\backslash text\{dim\}(V)$ of $V$ is finite, then, after making a choice of basis, we can view such vector spaces as $\backslash mathbb\{R\}^n$ or $\backslash mathbb\{C\}^n$.

The dual space is the space of linear functionals mapping $V\backslash rightarrow\; K$. Concretely, in the case where the vector space has an inner product, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by $V^*:=\; \backslash text\{Hom\}(V,K)$, so that $\backslash alpha\; \backslash in\; V^*$ is a linear map $\backslash alpha:V\backslash rightarrow\; K$.

Then under a choice of basis $\backslash \{e\_i\backslash \}$, we can view vectors $v\backslash in\; V$ as a $K^n$ vector with components $v^i$ (vectors are taken by convention to have indices up). This picks out a choice of basis $\backslash \{e^i\backslash \}$ for $V^*$, defined by the set of relations $e^i(e\_j)\; =\; \backslash delta^i\_j$.

For applications, raising and lowering is done using a structure known as the (pseudo{{nbh}})metric tensor (the 'pseudo-' refers to the fact we allow the metric to be indefinite). Formally, this is a non-degenerate, symmetric bilinear form

:$g:V\backslash times\; V\backslash rightarrow\; K\; \backslash text\{\; a\; bilinear\; form\}$

:$g(u,v)\; =\; g(v,u)\; \backslash text\{\; for\; all\; \}u,v\backslash in\; V\; \backslash text\{\; (Symmetric)\}$

:$\backslash forall\; v\backslash in\; V\; \backslash text\{\; s.t.\; \}\; v\backslash neq\; \backslash vec\{0\},\; \backslash exists\; u\backslash in\; V\; \backslash text\{\; such\; that\; \}\; g(v,u)\backslash neq\; 0\; \backslash text\{\; (Non-degenerate)\}$

In this basis, it has components $g(e\_i,e\_j)\; =\; g\_\{ij\}$, and can be viewed as a symmetric matrix in $\backslash text\{Mat\}\_\{n\backslash times\; n\}(K)$ with these components. The inverse metric exists due to non-degeneracy and is denoted $g^\{ij\}$, and as a matrix is the inverse to $g\_\{ij\}$.

Raising and lowering is then done in coordinates. Given a vector with components $v^i$, we can contract with the metric to obtain a covector:

:$g\_\{ij\}v^j\; =\; v\_i$

and this is what we mean by lowering the index. Conversely, contracting a covector with the inverse metric gives a vector:

:$g^\{ij\}\backslash alpha\_j=\backslash alpha^i.$

This process is called raising the index.

Raising and then lowering the same index (or conversely) are inverse operations, which is reflected in the metric and inverse metric tensors being inverse to each other (as is suggested by the terminology):

:$g^\{ij\}g\_\{jk\}=g\_\{kj\}g^\{ji\}=\{\backslash delta^i\}\_k=\{\backslash delta\_k\}^i$

where $\backslash delta^i\_j$ is the Kronecker delta or identity matrix.

Finite-dimensional real vector spaces with (pseudo-)metrics are classified up to signature, a coordinate-free property which is well-defined by Sylvester's law of inertia. Possible metrics on real space are indexed by signature $(p,q)$. This is a metric associated to $n=p+q$ dimensional real space. The metric has signature $(p,q)$ if there exists a basis (referred to as an orthonormal basis) such that in this basis, the metric takes the form $(g\_\{ij\})\; =\; \backslash text\{diag\}(+1,\; \backslash cdots,\; +1,\; -1,\; \backslash cdots,\; -1)$ with $p$ positive ones and $q$ negative ones.

The concrete space with elements which are $n$-vectors and this concrete realization of the metric is denoted $\backslash mathbb\{R\}^\{p,q\}=(\backslash mathbb\{R\}^n,g\_\{ij\})$, where the 2-tuple $(\backslash mathbb\{R\}^n,\; g\_\{ij\})$ is meant to make it clear that the underlying vector space of $\backslash mathbb\{R\}^\{p,q\}$ is $\backslash mathbb\{R\}^n$: equipping this vector space with the metric $g\_\{ij\}$ is what turns the space into $\backslash mathbb\{R\}^\{p,q\}$.

Examples:

• $\backslash mathbb\{R\}^3$ is a model for 3-dimensional space. The metric is equivalent to the standard dot product.
• $\backslash mathbb\{R\}^\{n,0\}\; =\; \backslash mathbb\{R\}^n$, equivalent to $n$ dimensional real space as an inner product space with $g\_\{ij\}\; =\; \backslash delta\_\{ij\}$. In Euclidean space, raising and lowering is not necessary due to vectors and covector components being the same.
• $\backslash mathbb\{R\}^\{1,3\}$ is Minkowski space (or rather, Minkowski space in a choice of orthonormal basis), a model for spacetime with weak curvature. It is common convention to use greek indices when writing expressions involving tensors in Minkowski space, while Latin indices are reserved for Euclidean space.

Well-formulated expressions are constrained by the rules of Einstein summation: any index may appear at most twice and furthermore a raised index must contract with a lowered index. With these rules we can immediately see that an expression such as

:$g\_\{ij\}v^iu^j$

is well formulated while

:$g\_\{ij\}v\_iu\_j$

is not.

The covariant 4-position is given by

:$X\_\backslash mu\; =\; (-ct,\; x,\; y,\; z)$

with components:

:$X\_0\; =\; -ct,\; \backslash quad\; X\_1\; =\; x,\; \backslash quad\; X\_2\; =\; y,\; \backslash quad\; X\_3\; =\; z$

(where {{mvar|x}},{{mvar|y}},{{mvar|z}} are the usual Cartesian coordinates) and the Minkowski metric tensor with metric signature (− + + +) is defined as

:$\backslash eta\_\{\backslash mu\; \backslash nu\}\; =\; \backslash eta^\{\backslash mu\; \backslash nu\}\; =\; \backslash begin\{pmatrix\}$

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix}

in components:

:$\backslash eta\_\{00\}\; =\; -1,\; \backslash quad\; \backslash eta\_\{i0\}\; =\; \backslash eta\_\{0i\}\; =\; 0,\backslash quad\; \backslash eta\_\{ij\}\; =\; \backslash delta\_\{ij\}\backslash ,(i,j\; \backslash neq\; 0).$

To raise the index, multiply by the tensor and contract:

:$X^\backslash lambda\; =\; \backslash eta^\{\backslash lambda\backslash mu\}X\_\backslash mu\; =\; \backslash eta^\{\backslash lambda\; 0\}X\_0\; +\; \backslash eta^\{\backslash lambda\; i\}X\_i$

then for {{math|λ {{=}} 0}}:

:$X^0\; =\; \backslash eta^\{00\}X\_0\; +\; \backslash eta^\{0i\}X\_i\; =\; -X\_0$

and for {{math|λ {{=}} j {{=}} 1, 2, 3}}:

:$X^j\; =\; \backslash eta^\{j0\}X\_0\; +\; \backslash eta^\{ji\}X\_i\; =\; \backslash delta^\{ji\}X\_i\; =\; X\_j\; \backslash ,.$

So the index-raised contravariant 4-position is:

:$X^\backslash mu\; =\; (ct,\; x,\; y,\; z)\backslash ,.$

This operation is equivalent to the matrix multiplication

:$\backslash begin\{pmatrix\}$

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix} \begin{pmatrix}

-ct \\

x \\

y \\

z

\end{pmatrix} = \begin{pmatrix}

ct \\

x \\

y \\

z

\end{pmatrix}.

Given two vectors, $X^\backslash mu$ and $Y^\backslash mu$, we can write down their (pseudo-)inner product in two ways:

:$\backslash eta\_\{\backslash mu\backslash nu\}X^\backslash mu\; Y^\backslash nu.$

By lowering indices, we can write this expression as

:$X\_\backslash mu\; Y^\backslash mu.$

In matrix notation, the first expression can be written as

:

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix} \begin{pmatrix}

Y^0 \\

Y^1 \\

Y^2 \\

Y^3\end{pmatrix}

while the second is, after lowering the indices of $X^\backslash mu$,

:

Y^0 \\

Y^1 \\

Y^2 \\

Y^3\end{pmatrix}.

It is instructive to consider what raising and lowering means in the abstract linear algebra setting.

We first fix definitions: $V$ is a finite-dimensional vector space over a field $K$. Typically $K=\backslash mathbb\{R\}$ or $\backslash mathbb\{C\}$.

$\backslash phi$ is a non-degenerate bilinear form, that is,

$\backslash phi:V\backslash times\; V\backslash rightarrow\; K$

is a map which is linear in both arguments, making it a bilinear form.

By $\backslash phi$ being non-degenerate we mean that for each $v\backslash in\; V$ such that $v\backslash neq\; 0$, there is a $u\backslash in\; V$ such that

:$\backslash phi(v,u)\backslash neq\; 0.$

In concrete applications, $\backslash phi$ is often considered a structure on the vector space, for example an inner product or more generally a metric tensor which is allowed to have indefinite signature, or a symplectic form $\backslash omega$. Together these cover the cases where $\backslash phi$ is either symmetric or anti-symmetric, but in full generality $\backslash phi$ need not be either of these cases.

There is a partial evaluation map associated to $\backslash phi$,

:$\backslash phi(\backslash cdot,\; -\; ):V\backslash rightarrow\; V^*;\; v\backslash mapsto\; \backslash phi(v,\backslash cdot)$

where $\backslash cdot$ denotes an argument which is to be evaluated, and $-$ denotes an argument whose evaluation is deferred. Then $\backslash phi(v,\backslash cdot)$ is an element of $V^*$, which sends $u\backslash mapsto\; \backslash phi(v,u)$.

We made a choice to define this partial evaluation map as being evaluated on the first argument. We could just as well have defined it on the second argument, and non-degeneracy is also independent of argument chosen. Also, when $\backslash phi$ has well defined (anti-)symmetry, evaluating on either argument is equivalent (up to a minus sign for anti-symmetry).

Non-degeneracy shows that the partial evaluation map is injective, or equivalently that the kernel of the map is trivial. In finite dimension, the dual space $V^*$ has equal dimension to $V$, so non-degeneracy is enough to conclude the map is a linear isomorphism. If $\backslash phi$ is a structure on the vector space sometimes call this the canonical isomorphism $V\backslash rightarrow\; V^*$.

It therefore has an inverse, $\backslash phi^\{-1\}:V^*\backslash rightarrow\; V,$ and this is enough to define an associated bilinear form on the dual:

:$\backslash phi^\{-1\}:V^*\backslash times\; V^*\backslash rightarrow\; K,\; \backslash phi^\{-1\}(\backslash alpha,\backslash beta)\; =\; \backslash phi(\backslash phi^\{-1\}(\backslash alpha),\backslash phi^\{-1\}(\backslash beta)).$

where the repeated use of $\backslash phi^\{-1\}$ is disambiguated by the argument taken. That is, $\backslash phi^\{-1\}(\backslash alpha)$ is the inverse map, while $\backslash phi^\{-1\}(\backslash alpha,\backslash beta)$ is the bilinear form.

Checking these expressions in coordinates makes it evident that this is what raising and lowering indices means abstractly.

We will not develop the abstract formalism for tensors straightaway. Formally, an $(r,s)$ tensor is an object described via its components, and has $r$ components up, $s$ components down. A generic $(r,s)$ tensor is written

:$T^\{\backslash mu\_1\backslash cdots\; \backslash mu\_r\}\{\}\_\{\backslash nu\_1\backslash cdots\; \backslash nu\_s\}.$

We can use the metric tensor to raise and lower tensor indices just as we raised and lowered vector indices and raised covector indices.

• A (0,0) tensor is a number in the field $\backslash mathbb\{F\}$.
• A (1,0) tensor is a vector.
• A (0,1) tensor is a covector.
• A (0,2) tensor is a bilinear form. An example is the metric tensor $g\_\{\backslash mu\backslash nu\}.$
• A (1,1) tensor is a linear map. An example is the delta, $\backslash delta^\backslash mu\{\}\_\backslash nu$, which is the identity map, or a Lorentz transformation $\backslash Lambda^\backslash mu\{\}\_\backslash nu.$

For a (0,2) tensor,{{cite book |title=Tensor Calculus |first=D. C. |last=Kay |series=Schaum’s Outlines |publisher=McGraw Hill |location=New York |year=1988 |isbn=0-07-033484-6 }} twice contracting with the inverse metric tensor and contracting in different indices raises each index:

:$A^\{\backslash mu\backslash nu\}=g^\{\backslash mu\backslash rho\}g^\{\backslash nu\backslash sigma\}A\_\{\backslash rho\; \backslash sigma\}.$

Similarly, twice contracting with the metric tensor and contracting in different indices lowers each index:

:$A\_\{\backslash mu\backslash nu\}=g\_\{\backslash mu\backslash rho\}g\_\{\backslash nu\backslash sigma\}A^\{\backslash rho\backslash sigma\}$

Let's apply this to the theory of electromagnetism.

The contravariant electromagnetic tensor in the {{math|(+ − − −)}} signature is given byNB: Some texts, such as: {{cite book | author=Griffiths, David J. | authorlink = David J. Griffiths | title=Introduction to Elementary Particles | publisher=Wiley, John & Sons, Inc | year=1987 | isbn=0-471-60386-4}}, will show this tensor with an overall factor of −1. This is because they used the negative of the metric tensor used here: {{math|(− + + +)}}, see metric signature. In older texts such as Jackson (2nd edition), there are no factors of {{mvar|c}} since they are using Gaussian units. Here SI units are used.

:$F^\{\backslash alpha\backslash beta\}\; =\; \backslash begin\{pmatrix\}$

0 & -\frac{E_x}{c} & -\frac{E_y}{c} & -\frac{E_z}{c} \\

\frac{E_x}{c} & 0 & -B_z & B_y \\

\frac{E_y}{c} & B_z & 0 & -B_x \\

\frac{E_z}{c} & -B_y & B_x & 0

\end{pmatrix}.

In components,

:$F^\{0i\}\; =\; -F^\{i0\}\; =\; -\; \backslash frac\{E^i\}\{c\}\; ,\backslash quad\; F^\{ij\}\; =\; -\; \backslash varepsilon^\{ijk\}\; B\_k$

To obtain the covariant tensor {{mvar|F_αβ}}, contract with the inverse metric tensor:

:$\backslash begin\{align\}$

F_{\alpha\beta} & = \eta_{\alpha\gamma} \eta_{\beta\delta} F^{\gamma\delta} \\

& = \eta_{\alpha 0} \eta_{\beta 0} F^{0 0} + \eta_{\alpha i} \eta_{\beta 0} F^{i 0}

+ \eta_{\alpha 0} \eta_{\beta i} F^{0 i} + \eta_{\alpha i} \eta_{\beta j} F^{i j}

\end{align}

and since {{math|F⁰⁰ {{=}} 0 }} and {{math|F⁰ⁱ {{=}} − Fⁱ⁰}}, this reduces to

:$F\_\{\backslash alpha\backslash beta\}\; =\; \backslash left(\backslash eta\_\{\backslash alpha\; i\}\; \backslash eta\_\{\backslash beta\; 0\}\; -\; \backslash eta\_\{\backslash alpha\; 0\}\; \backslash eta\_\{\backslash beta\; i\}\; \backslash right)\; F^\{i\; 0\}\; +\; \backslash eta\_\{\backslash alpha\; i\}\; \backslash eta\_\{\backslash beta\; j\}\; F^\{i\; j\}$

Now for {{math|α {{=}} 0}}, {{math|β {{=}} k {{=}} 1, 2, 3}}:

:$\backslash begin\{align\}$

F_{0k} & = \left(\eta_{0i} \eta_{k0} - \eta_{00} \eta_{ki} \right) F^{i0} + \eta_{0i} \eta_{kj} F^{ij} \\

& = \bigl(0 - (-\delta_{ki}) \bigr) F^{i0} + 0 \\

& = F^{k0} = - F^{0k} \\

\end{align}

and by antisymmetry, for {{math|α {{=}} k {{=}} 1, 2, 3}}, {{math|β {{=}} 0}}:

:$F\_\{k0\}\; =\; -\; F^\{k0\}$

then finally for {{math|α {{=}} k {{=}} 1, 2, 3}}, {{math|β {{=}} l {{=}} 1, 2, 3}};

:$\backslash begin\{align\}$

F_{kl} & = \left(\eta_{ k i} \eta_{ l 0} - \eta_{ k 0} \eta_{ l i} \right) F^{i 0} + \eta_{ k i} \eta_{ l j} F^{i j} \\

& = 0 + \delta_{ k i} \delta_{ l j} F^{i j} \\

& = F^{k l} \\

\end{align}

The (covariant) lower indexed tensor is then:

:$F\_\{\backslash alpha\backslash beta\}\; =\; \backslash begin\{pmatrix\}$

0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\

-\frac{E_x}{c} & 0 & -B_z & B_y \\

-\frac{E_y}{c} & B_z & 0 & -B_x \\

-\frac{E_z}{c} & -B_y & B_x & 0

\end{pmatrix}

This operation is equivalent to the matrix multiplication

:$\backslash begin\{pmatrix\}$

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix}

\begin{pmatrix}

0 & -\frac{E_x}{c} & -\frac{E_y}{c} & -\frac{E_z}{c} \\

\frac{E_x}{c} & 0 & -B_z & B_y \\

\frac{E_y}{c} & B_z & 0 & -B_x \\

\frac{E_z}{c} & -B_y & B_x & 0

\end{pmatrix}

\begin{pmatrix}

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix}

=\begin{pmatrix}

0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\

-\frac{E_x}{c} & 0 & -B_z & B_y \\

-\frac{E_y}{c} & B_z & 0 & -B_x \\

-\frac{E_z}{c} & -B_y & B_x & 0

\end{pmatrix}.

For a tensor of order {{mvar|n}}, indices are raised by (compatible with above):

:$g^\{j\_1i\_1\}g^\{j\_2i\_2\}\backslash cdots\; g^\{j\_ni\_n\}A\_\{i\_1i\_2\backslash cdots\; i\_n\}\; =\; A^\{j\_1j\_2\backslash cdots\; j\_n\}$

and lowered by:

:$g\_\{j\_1i\_1\}g\_\{j\_2i\_2\}\backslash cdots\; g\_\{j\_ni\_n\}A^\{i\_1i\_2\backslash cdots\; i\_n\}\; =\; A\_\{j\_1j\_2\backslash cdots\; j\_n\}$

and for a mixed tensor:

:$g\_\{p\_1i\_1\}g\_\{p\_2i\_2\}\backslash cdots\; g\_\{p\_ni\_n\}g^\{q\_1j\_1\}g^\{q\_2j\_2\}\backslash cdots\; g^\{q\_mj\_m\}\{A^\{i\_1i\_2\backslash cdots\; i\_n\}\}\_\{j\_1j\_2\backslash cdots\; j\_m\}\; =\; \{A\_\{p\_1p\_2\backslash cdots\; p\_n\}\}^\{q\_1q\_2\backslash cdots\; q\_m\}$

We need not raise or lower all indices at once: it is perfectly fine to raise or lower a single index. Lowering an index of an $(r,s)$ tensor gives a $(r-1,s+1)$ tensor, while raising an index gives a $(r+1,s-1)$ (where $r,s$ have suitable values, for example we cannot lower the index of a $(0,2)$ tensor.)

• Duality (mathematics)
• {{section link|Dual space|Bilinear products and dual spaces}}
• Einstein notation
• Falling and rising factorials
• Flat (music) and Sharp (music) about the signs {{music|flat}} and {{music|sharp}}
• Hodge star operator
• Metric tensor
• Vector bundle

{{Reflist}}

• {{cite book|last=Lee|first=J. M.|title=Introduction to Smooth manifolds|year=2003|series=Springer Graduate Texts in Mathematics|isbn=0-387-95448-1|volume=218}}
• {{cite book|last=Lee|first=J. M.|title=Riemannian Manifolds – An Introduction to Curvature|year=1997|publisher=Springer Verlag|series=Springer Graduate Texts in Mathematics|volume=176|isbn=978-0-387-98322-6}}
• {{cite book|last1=Vaz|first1=Jayme|last2=da Rocha|first2=Roldão|year=2016|title=An Introduction to Clifford Algebras and Spinors |publisher=Oxford University Press|isbn=978-0-19-878-292-6}}

{{Riemannian geometry}}

{{tensors}}

{{Manifolds}}

Category:Differential geometry

Category:Riemannian geometry

Category:Riemannian manifolds

Category:Symplectic geometry