Wsrtjtyk

In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, …, n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations.

The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis:

εi1i2…in{displaystyle varepsilon _{i_{1}i_{2}dots i_{n}}}

where each index i₁, i₂, ..., i_n takes values 1, 2, ..., n. There are nⁿ indexed values of ε_i1i2…in, which can be arranged into an n-dimensional array. The key defining property of the symbol is total antisymmetry in all the indices. When any two indices are interchanged, equal or not, the symbol is negated:

ε…ip…iq…=−ε…iq…ip….{displaystyle varepsilon _{dots i_{p}dots i_{q}dots }=-varepsilon _{dots i_{q}dots i_{p}dots }.}

If any two indices are equal, the symbol is zero. When all indices are unequal, we have:

εi1i2…in=(−1)pε12…n,{displaystyle varepsilon _{i_{1}i_{2}dots i_{n}}=(-1)^{p}varepsilon _{1,2,dots n},}

where p (called the parity of the permutation) is the number of pairwise interchanges of indices necessary to unscramble i₁, i₂, ..., i_n into the order 1, 2, ..., n, and the factor (−1)^p is called the sign or signature of the permutation. The value ε_{1 2 ... n} must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose ε_{1 2 ... n} = +1, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article.

The term "n-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol n matches the dimensionality of the vector space in question, which may be Euclidean or non-Euclidean, for example, ℝ³ or Minkowski space. The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate systems; however it can be interpreted as a tensor density.

The Levi-Civita symbol allows the determinant of a square matrix, and the cross product of two vectors in three-dimensional Euclidean space, to be expressed in Einstein index notation.

Definition

The Levi-Civita symbol is most often used in three and four dimensions, and to some extent in two dimensions, so these are given here before defining the general case.

Two dimensions

In two dimensions, Levi-Civita symbol is defined by:

εij={+1if (i,j)=(1,2)−1if (i,j)=(2,1)0if i=j{displaystyle varepsilon _{ij}={begin{cases}+1&{text{if }}(i,j)=(1,2)\-1&{text{if }}(i,j)=(2,1)\;;,0&{text{if }}i=jend{cases}}}

The values can be arranged into a 2 × 2 antisymmetric matrix:

(ε11ε12ε21ε22)=(01−10){displaystyle {begin{pmatrix}varepsilon _{11}&varepsilon _{12}\varepsilon _{21}&varepsilon _{22}end{pmatrix}}={begin{pmatrix}0&1\-1&0end{pmatrix}}}

Use of the two-dimensional symbol is relatively uncommon, although in certain specialized topics like supersymmetry^[1] and twistor theory^[2] it appears in the context of 2-spinors. The three- and higher-dimensional Levi-Civita symbols are used more commonly.

Three dimensions

For the indices (i, j, k) in ε_ijk, the values 1, 2, 3 occurring in the      cyclic order (1, 2, 3) correspond to ε = +1, while occurring in the      reverse cyclic order correspond to ε = −1, otherwise ε = 0.

In three dimensions, the Levi-Civita symbol is defined by:^[3]

εijk={+1if (i,j,k) is (1,2,3),(2,3,1), or (3,1,2),−1if (i,j,k) is (3,2,1),(1,3,2), or (2,1,3),0if i=j, or j=k, or k=i{displaystyle varepsilon _{ijk}={begin{cases}+1&{text{if }}(i,j,k){text{ is }}(1,2,3),(2,3,1),{text{ or }}(3,1,2),\-1&{text{if }}(i,j,k){text{ is }}(3,2,1),(1,3,2),{text{ or }}(2,1,3),\;;,0&{text{if }}i=j,{text{ or }}j=k,{text{ or }}k=iend{cases}}}

That is, ε_ijk is 1 if (i, j, k) is an even permutation of (1, 2, 3), −1 if it is an odd permutation, and 0 if any index is repeated. In three dimensions only, the cyclic permutations of (1, 2, 3) are all even permutations, similarly the anticyclic permutations are all odd permutations. This means in 3d it is sufficient to take cyclic or anticyclic permutations of (1, 2, 3) and easily obtain all the even or odd permutations.

Analogous to 2-dimensional matrices, the values of the 3-dimensional Levi-Civita symbol can be arranged into a 3 × 3 × 3 array:

where i is the depth (blue: i = 1; red: i = 2; green: i = 3), j is the row and k is the column.

Some examples:

ε132=−ε123=−1ε312=−ε213=−(−ε123)=1ε231=−ε132=−(−ε123)=1ε232=−ε232=0{displaystyle {begin{aligned}varepsilon _{color {BrickRed}{1}color {Violet}{3}color {Orange}{2}}=-varepsilon _{color {BrickRed}{1}color {Orange}{2}color {Violet}{3}}&=-1\varepsilon _{color {Violet}{3}color {BrickRed}{1}color {Orange}{2}}=-varepsilon _{color {Orange}{2}color {BrickRed}{1}color {Violet}{3}}&=-(-varepsilon _{color {BrickRed}{1}color {Orange}{2}color {Violet}{3}})=1\varepsilon _{color {Orange}{2}color {Violet}{3}color {BrickRed}{1}}=-varepsilon _{color {BrickRed}{1}color {Violet}{3}color {Orange}{2}}&=-(-varepsilon _{color {BrickRed}{1}color {Orange}{2}color {Violet}{3}})=1\varepsilon _{color {Orange}{2}color {Violet}{3}color {Orange}{2}}=-varepsilon _{color {Orange}{2}color {Violet}{3}color {Orange}{2}}&=0end{aligned}}}

Four dimensions

In four dimensions, the Levi-Civita symbol is defined by:

εijkl={+1if (i,j,k,l) is an even permutation of (1,2,3,4)−1if (i,j,k,l) is an odd permutation of (1,2,3,4)0otherwise{displaystyle varepsilon _{ijkl}={begin{cases}+1&{text{if }}(i,j,k,l){text{ is an even permutation of }}(1,2,3,4)\-1&{text{if }}(i,j,k,l){text{ is an odd permutation of }}(1,2,3,4)\;;,0&{text{otherwise}}end{cases}}}

These values can be arranged into a 4 × 4 × 4 × 4 array, although in 4 dimensions and higher this is difficult to draw.

Some examples:

ε1432=−ε1234=−1ε2134=−ε1234=−1ε4321=−ε1324=−(−ε1234)=1ε3243=−ε3243=0{displaystyle {begin{aligned}varepsilon _{color {BrickRed}{1}color {RedViolet}{4}color {Violet}{3}color {Orange}{color {Orange}{2}}}=-varepsilon _{color {BrickRed}{1}color {Orange}{color {Orange}{2}}color {Violet}{3}color {RedViolet}{4}}&=-1\varepsilon _{color {Orange}{color {Orange}{2}}color {BrickRed}{1}color {Violet}{3}color {RedViolet}{4}}=-varepsilon _{color {BrickRed}{1}color {Orange}{color {Orange}{2}}color {Violet}{3}color {RedViolet}{4}}&=-1\varepsilon _{color {RedViolet}{4}color {Violet}{3}color {Orange}{color {Orange}{2}}color {BrickRed}{1}}=-varepsilon _{color {BrickRed}{1}color {Violet}{3}color {Orange}{color {Orange}{2}}color {RedViolet}{4}}&=-(-varepsilon _{color {BrickRed}{1}color {Orange}{color {Orange}{2}}color {Violet}{3}color {RedViolet}{4}})=1\varepsilon _{color {Violet}{3}color {Orange}{color {Orange}{2}}color {RedViolet}{4}color {Violet}{3}}=-varepsilon _{color {Violet}{3}color {Orange}{color {Orange}{2}}color {RedViolet}{4}color {Violet}{3}}&=0end{aligned}}}

Generalization to n dimensions

More generally, in n dimensions, the Levi-Civita symbol is defined by:^[4]

εa1a2a3…an={+1if (a1,a2,a3,…,an) is an even permutation of (1,2,3,…,n)−1if (a1,a2,a3,…,an) is an odd permutation of (1,2,3,…,n)0otherwise{displaystyle varepsilon _{a_{1}a_{2}a_{3}ldots a_{n}}={begin{cases}+1&{text{if }}(a_{1},a_{2},a_{3},ldots ,a_{n}){text{ is an even permutation of }}(1,2,3,dots ,n)\-1&{text{if }}(a_{1},a_{2},a_{3},ldots ,a_{n}){text{ is an odd permutation of }}(1,2,3,dots ,n)\;;,0&{text{otherwise}}end{cases}}}

Thus, it is the sign of the permutation in the case of a permutation, and zero otherwise.

Using the capital pi notation ∏ for ordinary multiplication of numbers, an explicit expression for the symbol is:

εa1a2a3…an=∏1≤i<j≤nsgn⁡(aj−ai)=sgn⁡(a2−a1)sgn⁡(a3−a1)…sgn⁡(an−a1)sgn⁡(a3−a2)sgn⁡(a4−a2)…sgn⁡(an−a2)…sgn⁡(an−an−1){displaystyle {begin{aligned}varepsilon _{a_{1}a_{2}a_{3}ldots a_{n}}&=prod _{1leq i<jleq n}operatorname {sgn} (a_{j}-a_{i})\&=operatorname {sgn} (a_{2}-a_{1})operatorname {sgn} (a_{3}-a_{1})dots operatorname {sgn} (a_{n}-a_{1})operatorname {sgn} (a_{3}-a_{2})operatorname {sgn} (a_{4}-a_{2})dots operatorname {sgn} (a_{n}-a_{2})dots operatorname {sgn} (a_{n}-a_{n-1})end{aligned}}}

where the signum function (denoted sgn) returns the sign of its argument while discarding the absolute value if nonzero. The formula is valid for all index values, and for any n (when n = 0 or n = 1, this is the empty product). However, computing the formula above naively has a time complexity of O(n²), whereas the sign can be computed from the parity of the permutation from its disjoint cycles in only O(n log(n)) cost.

Properties

A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called a permutation tensor.

Under the ordinary transformation rules for tensors the Levi-Civita symbol is unchanged under pure rotations, consistent with that it is (by definition) the same in all coordinate systems related by orthogonal transformations. However, the Levi-Civita symbol is a pseudotensor because under an orthogonal transformation of Jacobian determinant −1, for example, a reflection in an odd number of dimensions, it should acquire a minus sign if it were a tensor. As it does not change at all, the Levi-Civita symbol is, by definition, a pseudotensor.

As the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector.^[5]

Under a general coordinate change, the components of the permutation tensor are multiplied by the Jacobian of the transformation matrix. This implies that in coordinate frames different from the one in which the tensor was defined, its components can differ from those of the Levi-Civita symbol by an overall factor. If the frame is orthonormal, the factor will be ±1 depending on whether the orientation of the frame is the same or not.^[5]

In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual.

Summation symbols can be eliminated by using Einstein notation, where an index repeated between two or more terms indicates summation over that index. For example,

εijkεimn≡∑i=1,2,3εijkεimn{displaystyle varepsilon _{ijk}varepsilon ^{imn}equiv sum _{i=1,2,3}varepsilon _{ijk}varepsilon ^{imn}}.

In the following examples, Einstein notation is used.

Two dimensions

In two dimensions, when all i, j, m, n each take the values 1 and 2,^[3]

εijεmn=δimδjn−δinδjm{displaystyle varepsilon _{ij}varepsilon ^{mn}={delta _{i}}^{m}{delta _{j}}^{n}-{delta _{i}}^{n}{delta _{j}}^{m}}

( 1 )

εijεin=δjn{displaystyle varepsilon _{ij}varepsilon ^{in}={delta _{j}}^{n}}

( 2 )

εijεij=2.{displaystyle varepsilon _{ij}varepsilon ^{ij}=2.}

( 3 )

Three dimensions

Index and symbol values

In three dimensions, when all i, j, k, m, n each take values 1, 2, and 3:^[3]

εijkεimn=δjmδkn−δjnδkm{displaystyle varepsilon _{ijk}varepsilon ^{imn}=delta _{j}{}^{m}delta _{k}{}^{n}-delta _{j}{}^{n}delta _{k}{}^{m}}

( 4 )

εjmnεimn=2δji{displaystyle varepsilon _{jmn}varepsilon ^{imn}=2{delta _{j}}^{i}}

( 5 )

εijkεijk=6.{displaystyle varepsilon _{ijk}varepsilon ^{ijk}=6.}

( 6 )

Product

The Levi-Civita symbol is related to the Kronecker delta. In three dimensions, the relationship is given by the following equations (vertical lines denote the determinant):^[4]

εijkεlmn=|δilδimδinδjlδjmδjnδklδkmδkn|=δil(δjmδkn−δjnδkm)−δim(δjlδkn−δjnδkl)+δin(δjlδkm−δjmδkl).{displaystyle {begin{aligned}varepsilon _{ijk}varepsilon _{lmn}&={begin{vmatrix}delta _{il}&delta _{im}&delta _{in}\delta _{jl}&delta _{jm}&delta _{jn}\delta _{kl}&delta _{km}&delta _{kn}\end{vmatrix}}\[6pt]&=delta _{il}left(delta _{jm}delta _{kn}-delta _{jn}delta _{km}right)-delta _{im}left(delta _{jl}delta _{kn}-delta _{jn}delta _{kl}right)+delta _{in}left(delta _{jl}delta _{km}-delta _{jm}delta _{kl}right).end{aligned}}}

A special case of this result is (4):

∑i=13εijkεimn=δjmδkn−δjnδkm{displaystyle sum _{i=1}^{3}varepsilon _{ijk}varepsilon _{imn}=delta _{jm}delta _{kn}-delta _{jn}delta _{km}}

sometimes called the "contracted epsilon identity".

In Einstein notation, the duplication of the i index implies the sum on i. The previous is then denoted ε_ijkε_imn = δ_jmδ_kn − δ_jnδ_km.

∑i=13∑j=13εijkεijn=2δkn{displaystyle sum _{i=1}^{3}sum _{j=1}^{3}varepsilon _{ijk}varepsilon _{ijn}=2delta _{kn}}

n dimensions

Index and symbol values

In n dimensions, when all i₁, …,i_n, j₁, ..., j_n take values 1, 2, ..., n:

εi1…inεj1…jn=n!δ[i1j1…δin]jn=δi1…inj1…jn{displaystyle varepsilon _{i_{1}dots i_{n}}varepsilon ^{j_{1}dots j_{n}}=n!delta _{[i_{1}}^{j_{1}}dots delta _{i_{n}]}^{j_{n}}=delta _{i_{1}dots i_{n}}^{j_{1}dots j_{n}}}

( 7 )

εi1…ik ik+1…inεi1…ik jk+1…jn=k!(n−k)! δ[ik+1jk+1…δin]jn=k! δik+1…injk+1…jn{displaystyle varepsilon _{i_{1}dots i_{k}~i_{k+1}dots i_{n}}varepsilon ^{i_{1}dots i_{k}~j_{k+1}dots j_{n}}=k!(n-k)!~delta _{[i_{k+1}}^{j_{k+1}}dots delta _{i_{n}]}^{j_{n}}=k!~delta _{i_{k+1}dots i_{n}}^{j_{k+1}dots j_{n}}}

( 8 )

εi1…inεi1…in=n!{displaystyle varepsilon _{i_{1}dots i_{n}}varepsilon ^{i_{1}dots i_{n}}=n!}

( 9 )

where the exclamation mark (!) denotes the factorial, and δ^α…
_β… is the generalized Kronecker delta. For any n, the property

∑i,j,k,⋯=1nεijk…εijk…=n!{displaystyle sum _{i,j,k,dots =1}^{n}varepsilon _{ijkdots }varepsilon _{ijkdots }=n!}

follows from the facts that

• every permutation is either even or odd,


• 
  (+1)² = (−1)² = 1, and


• the number of permutations of any n-element set number is exactly n!.

Product

In general, for n dimensions, one can write the product of two Levi-Civita symbols as:

εi1i2…inεj1j2…jn=|δi1j1δi1j2…δi1jnδi2j1δi2j2…δi2jn⋮⋮⋱⋮δinj1δinj2…δinjn|{displaystyle varepsilon _{i_{1}i_{2}dots i_{n}}varepsilon _{j_{1}j_{2}dots j_{n}}={begin{vmatrix}delta _{i_{1}j_{1}}&delta _{i_{1}j_{2}}&dots &delta _{i_{1}j_{n}}\delta _{i_{2}j_{1}}&delta _{i_{2}j_{2}}&dots &delta _{i_{2}j_{n}}\vdots &vdots &ddots &vdots \delta _{i_{n}j_{1}}&delta _{i_{n}j_{2}}&dots &delta _{i_{n}j_{n}}\end{vmatrix}}}.

Proofs

For (1), both sides are antisymmetric with respect of ij and mn. We therefore only need to consider the case i ≠ j and m ≠ n. By substitution, we see that the equation holds for ε₁₂ε¹², that is, for i = m = 1 and j = n = 2. (Both sides are then one). Since the equation is antisymmetric in ij and mn, any set of values for these can be reduced to the above case (which holds). The equation thus holds for all values of ij and mn.

Using (1), we have for (2)

εijεin=δiiδjn−δinδji=2δjn−δjn=δjn.{displaystyle varepsilon _{ij}varepsilon ^{in}=delta _{i}{}^{i}delta _{j}{}^{n}-delta _{i}{}^{n}delta _{j}{}^{i}=2delta _{j}{}^{n}-delta _{j}{}^{n}=delta _{j}{}^{n},.}

Here we used the Einstein summation convention with i going from 1 to 2. Next, (3) follows similarly from (2).

To establish (5), notice that both sides vanish when i ≠ j. Indeed, if i ≠ j, then one can not choose m and n such that both permutation symbols on the left are nonzero. Then, with i = j fixed, there are only two ways to choose m and n from the remaining two indices. For any such indices, we have

εjmnεimn=(εimn)2=1{displaystyle varepsilon _{jmn}varepsilon ^{imn}=left(varepsilon ^{imn}right)^{2}=1}

(no summation), and the result follows.

Then (6) follows since 3! = 6 and for any distinct indices i, j, k taking values 1, 2, 3, we have

εijkεijk=1{displaystyle varepsilon _{ijk}varepsilon ^{ijk}=1} (no summation, distinct i, j, k)

Applications and examples

Determinants

In linear algebra, the determinant of a 3 × 3 square matrix A = [a_ij] can be written^[6]

det(A)=∑i=13∑j=13∑k=13εijka1ia2ja3k{displaystyle det(mathbf {A} )=sum _{i=1}^{3}sum _{j=1}^{3}sum _{k=1}^{3}varepsilon _{ijk}a_{1i}a_{2j}a_{3k}}

Similarly the determinant of an n × n matrix A = [a_ij] can be written as^[5]

det(A)=εi1…ina1i1…anin,{displaystyle det(mathbf {A} )=varepsilon _{i_{1}dots i_{n}}a_{1i_{1}}dots a_{ni_{n}},}

where each i_r should be summed over 1, …, n, or equivalently:

det(A)=1n!εi1…inεj1…jnai1j1…ainjn,{displaystyle det(mathbf {A} )={frac {1}{n!}}varepsilon _{i_{1}dots i_{n}}varepsilon _{j_{1}dots j_{n}}a_{i_{1}j_{1}}dots a_{i_{n}j_{n}},}

where now each i_r and each j_r should be summed over 1, …, n. More generally, we have the identity^[5]

∑i1,i2,…εi1…inai1j1…ainjn=det(A)εj1…jn{displaystyle sum _{i_{1},i_{2},dots }varepsilon _{i_{1}dots i_{n}}a_{i_{1},j_{1}}dots a_{i_{n},j_{n}}=det(mathbf {A} )varepsilon _{j_{1}dots j_{n}}}

Vector cross product

Cross product (two vectors)

If a = (a¹, a², a³) and b = (b¹, b², b³) are vectors in ℝ³ (represented in some right-handed coordinate system using an orthonormal basis), their cross product can be written as a determinant:^[5]

a×b=|e1e2e3a1a2a3b1b2b3|=∑i=13∑j=13∑k=13εijkeiajbk{displaystyle mathbf {atimes b} ={begin{vmatrix}mathbf {e_{1}} &mathbf {e_{2}} &mathbf {e_{3}} \a^{1}&a^{2}&a^{3}\b^{1}&b^{2}&b^{3}\end{vmatrix}}=sum _{i=1}^{3}sum _{j=1}^{3}sum _{k=1}^{3}varepsilon _{ijk}mathbf {e} _{i}a^{j}b^{k}}

hence also using the Levi-Civita symbol, and more simply:

(a×b)i=∑j=13∑k=13εijkajbk.{displaystyle (mathbf {atimes b} )^{i}=sum _{j=1}^{3}sum _{k=1}^{3}varepsilon _{ijk}a^{j}b^{k}.}

In Einstein notation, the summation symbols may be omitted, and the ith component of their cross product equals^[4]

(a×b)i=εijkajbk.{displaystyle (mathbf {atimes b} )^{i}=varepsilon _{ijk}a^{j}b^{k}.}

The first component is

(a×b)1=a2b3−a3b2,{displaystyle (mathbf {atimes b} )^{1}=a^{2}b^{3}-a^{3}b^{2},,}

then by cyclic permutations of 1, 2, 3 the others can be derived immediately, without explicitly calculating them from the above formulae:

(a×b)2=a3b1−a1b3,(a×b)3=a1b2−a2b1.{displaystyle {begin{aligned}(mathbf {atimes b} )^{2}&=a^{3}b^{1}-a^{1}b^{3},,\(mathbf {atimes b} )^{3}&=a^{1}b^{2}-a^{2}b^{1},.end{aligned}}}

Triple scalar product (three vectors)

From the above expression for the cross product, we have:

a×b=−b×a{displaystyle mathbf {atimes b} =-mathbf {btimes a} }.

If c = (c¹, c², c³) is a third vector, then the triple scalar product equals

a⋅(b×c)=εijkaibjck.{displaystyle mathbf {a} cdot (mathbf {btimes c} )=varepsilon _{ijk}a^{i}b^{j}c^{k}.}

From this expression, it can be seen that the triple scalar product is antisymmetric when exchanging any pair of arguments. For example,

a⋅(b×c)=−b⋅(a×c){displaystyle mathbf {a} cdot (mathbf {btimes c} )=-mathbf {b} cdot (mathbf {atimes c} )}.

Curl (one vector field)

If F = (F¹, F², F³) is a vector field defined on some open set of ℝ³ as a function of position x = (x¹, x², x³) (using Cartesian coordinates). Then the ith component of the curl of F equals^[4]

(∇×F)i(x)=εijk∂∂xjFk(x),{displaystyle (nabla times mathbf {F} )^{i}(mathbf {x} )=varepsilon ^{ijk}{frac {partial }{partial x^{j}}}F_{k}(mathbf {x} ),}

which follows from the cross product expression above, substituting components of the gradient vector operator (nabla).

Tensor density

In any arbitrary curvilinear coordinate system and even in the absence of a metric on the manifold, the Levi-Civita symbol as defined above may be considered to be a tensor density field in two different ways. It may be regarded as a contravariant tensor density of weight +1 or as a covariant tensor density of weight −1. In n dimensions using the generalized Kronecker delta,^[7][8]

εμ1…μn=δ1…nμ1…μnεν1…νn=δν1…νn1…n.{displaystyle {begin{aligned}varepsilon ^{mu _{1}dots mu _{n}}&=delta _{,1,dots ,n}^{mu _{1}dots mu _{n}},\varepsilon _{nu _{1}dots nu _{n}}&=delta _{nu _{1}dots nu _{n}}^{,1,dots ,n},.end{aligned}}}

Notice that these are numerically identical. In particular, the sign is the same.

Levi-Civita tensors

On a pseudo-Riemannian manifold, one may define a coordinate-invariant covariant tensor field whose coordinate representation agrees with the Levi-Civita symbol wherever the coordinate system is such that the basis of the tangent space is orthonormal with respect to the metric and matches a selected orientation. This tensor should not be confused with the tensor density field mentioned above. The presentation in this section closely follows Carroll 2004.

The covariant Levi-Civita tensor (also known as the Riemannian volume form) in any coordinate system that matches the selected orientation is

Ea1…an=|det[gab]|εa1…an,{displaystyle E_{a_{1}dots a_{n}}={sqrt {left|det[g_{ab}]right|}},varepsilon _{a_{1}dots a_{n}},,}

where g_ab is the representation of the metric in that coordinate system. We can similarly consider a contravariant Levi-Civita tensor by raising the indices with the metric as usual,

Ea1…an=Eb1…bn∏i=1ngaibi,{displaystyle E^{a_{1}dots a_{n}}=E_{b_{1}dots b_{n}}prod _{i=1}^{n}g^{a_{i}b_{i}},,}

but notice that if the metric signature contains an odd number of negatives q, then the sign of the components of this tensor differ from the standard Levi-Civita symbol:

Ea1…an=sgn⁡(det[gab])|det[gab]|εa1…an,{displaystyle E^{a_{1}dots a_{n}}={frac {operatorname {sgn} left(det[g_{ab}]right)}{sqrt {left|det[g_{ab}]right|}}},varepsilon ^{a_{1}dots a_{n}},}

where sgn(det[g_ab]) = (−1)^q, and εa1…an{displaystyle varepsilon ^{a_{1}dots a_{n}}} is the usual Levi-Civita symbol discussed in the rest of this article. More explicitly, when the tensor and basis orientation are chosen such that E01…n=+|det[gab]|{displaystyle E_{01dots n}=+{sqrt {left|det[g_{ab}]right|}}}, we have that E01…n=sgn⁡(det[gab])|det[gab]|{displaystyle E^{01dots n}={frac {operatorname {sgn}(det[g_{ab}])}{sqrt {left|det[g_{ab}]right|}}}}.

From this we can infer the identity,

Eμ1…μpα1…αn−pEμ1…μpβ1…βn−p=(−1)qp!δβ1…βn−pα1…αn−p,{displaystyle E^{mu _{1}dots mu _{p}alpha _{1}dots alpha _{n-p}}E_{mu _{1}dots mu _{p}beta _{1}dots beta _{n-p}}=(-1)^{q}p!delta _{beta _{1}dots beta _{n-p}}^{alpha _{1}dots alpha _{n-p}},,}

where

δβ1…βn−pα1…αn−p=(n−p)!δβ1[α1…δβn−pαn−p]{displaystyle delta _{beta _{1}dots beta _{n-p}}^{alpha _{1}dots alpha _{n-p}}=(n-p)!delta _{beta _{1}}^{lbrack alpha _{1}}dots delta _{beta _{n-p}}^{alpha _{n-p}rbrack }}

is the generalized Kronecker delta.

Example: Minkowski space

In Minkowski space (the four-dimensional spacetime of special relativity), the covariant Levi-Civita tensor is

Eαβγδ=±|det[gμν]|εαβγδ,{displaystyle E_{alpha beta gamma delta }=pm {sqrt {|det[g_{mu nu }]|}},varepsilon _{alpha beta gamma delta },,}

where the sign depends on the orientation of the basis. The contravariant Levi-Civita tensor is

Eαβγδ=gαζgβηgγθgδιEζηθι.{displaystyle E^{alpha beta gamma delta }=g^{alpha zeta }g^{beta eta }g^{gamma theta }g^{delta iota }E_{zeta eta theta iota },.}

The following are examples of the general identity above specialized to Minkowski space (with the negative sign arising from the odd number of negatives in the signature of the metric tensor in either sign convention):

EαβγδEρσμν=−gαζgβηgγθgδιδρσμνζηθιEαβγδEρσμν=−gαζgβηgγθgδιδζηθιρσμνEαβγδEαβγδ=−24EαβγδEρβγδ=−6δραEαβγδEρσγδ=−2δρσαβEαβγδEρσθδ=−δρσθαβγ.{displaystyle {begin{aligned}E_{alpha beta gamma delta }E_{rho sigma mu nu }&=-g_{alpha zeta }g_{beta eta }g_{gamma theta }g_{delta iota }delta _{rho sigma mu nu }^{zeta eta theta iota }\E^{alpha beta gamma delta }E^{rho sigma mu nu }&=-g^{alpha zeta }g^{beta eta }g^{gamma theta }g^{delta iota }delta _{zeta eta theta iota }^{rho sigma mu nu }\E^{alpha beta gamma delta }E_{alpha beta gamma delta }&=-24\E^{alpha beta gamma delta }E_{rho beta gamma delta }&=-6delta _{rho }^{alpha }\E^{alpha beta gamma delta }E_{rho sigma gamma delta }&=-2delta _{rho sigma }^{alpha beta }\E^{alpha beta gamma delta }E_{rho sigma theta delta }&=-delta _{rho sigma theta }^{alpha beta gamma },.end{aligned}}}

See also

• List of permutation topics


• Symmetric tensor

Notes

References

• 
  Wheeler, J. A.; Misner, C.; Thorne, K. S. (1973). Gravitation. W. H. Freeman & Co. pp. 85–86, §3.5. ISBN 0-7167-0344-0.
  


• 
  Neuenschwander, D. E. (2015). Tensor Calculus for Physics. Johns Hopkins University Press. pp. 11, 29, 95. ISBN 978-1-4214-1565-9.
  


• 
  Carroll, Sean M. (2004), Spacetime and Geometry, Addison-Wesley, ISBN 0-8053-8732-3
  

External links

This article incorporates material from Levi-Civita permutation symbol on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

• Weisstein, Eric W. "Permutation Tensor". MathWorld.