Derivation of the metric tensor, Christoffel symbols, and covariant derivative from first principles

Originally published at 狗和留美者不得入内. You can comment here or there.

I am going to actually do the “simple calculation” for change of coordinates of Christoffel symbols, aware that simply following texts without actively doing some calculations oneself can be rather self-deceiving. In the process I’ll jot down some notes pertaining to the covariant derivative.

First I shall explain some intuition behind curvilinear coordinates.

Image from the Wikipedia page for "curvilinear coordinates"

The example which L&L Classical Theory of Fields used to illustrate curvilinear coordinates was a rotating coordinate axis, with the [公式] -axis as the axis of rotation. Moving at uniform velocity with respect to the Galiliean spacetime coordinates, after an infinitesimal change in time, the [公式] and [公式] components of the rotating axis will have changed infinitesimally in their directions. From this, it is apparent that parallel transport of a velocity vector on a spacetime manifold must change the value of the velocity vector.

Take any sufficiently differentiable bijective self-map on [公式] , the domain corresponding to spacetime. To visualize this, one can superimpose the blue and red images on the Euclidean metric corresponding black one (of course this would only be on [公式] in a such a way that the origin goes to itself. At any point we can make infinitesimal displacements [公式] and see how much the superimposed coordinates are displaced in each of the two directions, the values of which correspond to the Jacobian of course.

The existence of an invertible Jacobian [公式] of curvilinear coordinates with respect to Galilean spacetime corresponds to an open [公式] of the spacetime manifold [公式] that is diffeomorphic to some subset of [公式] . The metric tensor [公式] defining an inner product on the space of one-forms at point [公式] enables us to define infinitesimal distances on the manifold by assigning a scalar to every pair of elements in the standard basis of [公式] and the angle at the point of intersection of two curves on the manifold is also defined as in the Euclidean case, via [公式] , but with a different inner product.

1. The relationship between the metric tensor and the Christoffel symbols

We can on the tangent space at a given point define orthonormal tangent vectors [公式] , here [公式] indexed, with [公式] being the all ones vector. Associated with that point is of course a chart on the which with distance defining metric tensor is that such that

[公式] which is basically saying that the value of same inner product of two basis vectors (which by linearity induces inner products on entire vector space) does not change upon transport along any direction on the manifold. We let

[公式] which defines the Christoffel symbols. How to interpret this? Well, in curvilinear coordinates, if one moves along solely along one coordinate in a Galilean coordinate frame at the same unit speed, after an infinitesimal displacement, the coordinates of one’s direction in the curvilinear frame can change by some amount along every direction. Since the differential operator is linear, we can assume the change function to be linear over the tangent space, and at a different point on the manifold, the tangent space the dimension of which does not change still has an orthonormal basis. Thus, we can express the rate of change of the basis elements as a linear combination of the basis elements themselves.

Substitution of [公式] into [公式] yields

[公式] Seeing that the contracted out indices can be renamed arbitrarily separately in each of the sums, we rename such that the expression becomes, viewing [公式] as fixed, the application of the sum of three covariant rank 2 tensors to [公式] . Then, [公式] transforms to

[公式] which equates to the following relation between the metric tensor and the Christoffel symbols:

[公式]

2. Metric tensor and Christoffel symbols in the Riemannian manifold defined via polar coordinates

The (bijective and smooth) relationship between rectangular and polar coordinates is

[公式] We define a real, smooth manifold the points of which are [公式] and to satisfy the second countable and Hausdorff condition of a topological manifold, we note that any subspace of [公式] inherit its second countable and Hausdorff properties. As for the atlas, we need only one chart, the definition of which is given by [公式] .

As for the tangent space at [公式] with basis elements [公式] and the associated metric tensor, we first note that we want the metric to satisfy

[公式] gives us [公式] for the metric, wherein the metric varies with [公式] but not with [公式] . Thus, the Christoffel symbols, which are determined by the metric, will also vary in the same way.

Take some differentiable curve [公式] on our manifold passing through a given point [公式] , chart coordinates of which we denote as [公式]. The rate of change of [公式] on our chart

[公式] One who has studied manifolds and actually understood it should be able to easily recall that class of curves passing through the same point [公式] such that the rate of change with respect to the curve’s parameter of the coordinates local at [公式] is equal form a equivalence class well-defined with respect to addition and multiplication by a scalar, which is used to define the velocity of a curve on a manifold at a point.

If partial derivatives of chart coordinates with respect to local coordinates on the manifolds are prescribed at every point, which is done implicitly by assigning a metric tensor to every point, then the partial derivative operators of first order ( [公式] in our concrete example), which form a basis of covariant vectors, also span the partial derivative operators of second order via the Christoffel symbols. Being a two dimensional vectors, each of

[公式] is spanned by

[公式] Here we note that the basis vectors have to non-zero determinant everywhere in order to linearly independent everywhere, and moreover, that the symmetry of partial derivatives suggests symmetry in the Christoffel symbols. As for explicit computation of the Christoffel symbols for our polar coordinate manifold, it is very straightforward and will be left to the reader.

3. Transformation of Christoffel symbols under change of coordinates

In section 1, we used [公式] to represent basis elements of the tangent space, except these are contravariant vectors when basis elements of the tangent space as partial derivative operators are supposed to be covariant. I personally did so, then being influenced by [公式] for the distance induced by infinitesimal displacements along each of the coordinate axes. If our basis here spanning the space of infinitesimal displacement is [公式] , then the above [公式] is simply the inner product of the all ones vectors with itself. As for this inner product, it is uniquely determined by [公式] or more like [公式] inner products due to symmetry, which are pairs of values of projections of the all one vectors onto some basis element.

The reader might feel of course as did myself that using [公式] in that case to represent the all ones contravariant vector and operating along that was somewhat a bastardization of notation. From now on, I will define the covariant basis [公式] and a velocity field along a Riemannian manifold as represented by [公式] . At every point on the manifold, there exists by definition a local chart that is smooth. We are interested in the rate of change of the velocity field at a given point along each of the local chart directions, which we shall denote via [公式] . Per the metric tensor for the given chart at the given point, we can find Christoffel symbols compatible with [公式] (with a tweak that alters a contravariant component to a covariant one) at that point, satisfying

[公式] This is the covariant derivative with respect to an arbitrary basis tangent vector as applied to any arbitrary basis element of [公式] , which lies in the dual space of the tangent space that is denoted as [公式] . The magitude of the projection of tangent vector [公式] onto the [公式] th basis element is given by the scalar [公式] , from which is it easy to see that the elements of [公式] are linear maps from tangent vectors to scalars.

From the partial derivative in [公式] , which exhibits linearity, it is easy to see that the covariant derivative along a specified direction must also be linear. Thus if we assume that the tangent vector [公式] does not vary when transported along [公式] , the formal statement of which is [公式] , then

[公式] In the general case, this becomes

[公式] Now, we proceed to derive the transformation of the Christoffel symbols under a change of coordinates wherein [公式] . We will in the transformed frame use [公式] in place of [公式] and a bar above the tensor. [公式] in the transformed frame is then written as

[公式]

This is a covariant vector, as is the value of [公式] , and covariant vectors should transform as follows:

[公式] Thus we can establish the relation between the values of [公式] and [公式] .

[公式]

Taking out the same partial derivative operator on both sides gives followed by some evaluations on the left hand side gives[公式] In the product of four partial derivatives on the left hand side wherein includes the product of the Jacobian with its inverse, we have

[公式]

Thus, we have

[公式]

which then equates to

[公式] the formula for transformation of the Christoffel symbols under change of coordinates. One notices that because of the second term with the second derivative, which violates the law of transformation of tensors, the Christoffel symbols do not constitute a tensor.

Rearranging [公式] , one obtains

[公式] which is the same as the formula given in L&L Classical Theory of Fields ([1]). Sometime to note here is that [公式], which involve two different coordinates bases certainly does not transform as a tensor. Therefore, the “argument” that if one rearranges [公式] to express [公式] in terms of [公式] , one would get a negative sign for the second order term that makes the transformation formula ill-defined is clearly invalid.

When the coordinate change is linear or the partial derivatives of second order associated with it are all zero, [公式] does transform like a tensor. [公式] , which cancels out the second term visibly transforms like a tensor as well, and this is called the “curvature tensor” of space.

By the equivalence principle, there must be at the given point a “galilean” coordinate system at each in point in which the Christoffel symbols and thus also [公式] are zero. Since [公式] transforms as a tensor, it must be zero in any coordinate system if it is in some specific coordinate system. This shows that the Christoffel symbols are symmetric with respect to the lower indices. As for this final paragraph, it relates the physics or more specifically the general relativity to the math. I will, hopefully, write in more detail about the physics of this once I gain a deeper understanding of it. In any case, I believe that the mathematics and differential geometry behind the covariant derivative I have explained quite thoroughly and intuitively in this very article.

References

A list of mathematicians who did Fields Medal level work but did not win the prize

Originally published at 狗和留美者不得入内. You can comment here or there.

作者:Zeldovich Yakov
链接:https://www.zhihu.com/question/436551699/answer/1708511621
来源:知乎
著作权归作者所有。商业转载请联系作者获得授权,非商业转载请注明出处。</p>

Andrey Kolmogorov 俄罗斯

Israel Gelfand 俄罗斯

Ivan Petrovsky 俄罗斯 偏微分方程

Eugene Dynkin 俄罗斯 表示论、随机过程

Sergei Sobolev 俄罗斯 偏微分方程

Igor Shafarecvich 俄罗斯 代数数论、代数几何

Vladimir Arnold 俄罗斯 动力系统、辛几何、代数几何

柏原正树 (Masaki Kashiwara) 日本 代数分析

Jacques-Louis Lions 法国 偏微分方程

吉田耕作 (Kōsaku Yosida) 日本 泛函分析

伊藤清 (Kiyosi Itô) 日本 概率论

佐藤干夫 (Mikio Sato) 日本 代数分析

冈洁 (Kiyoshi Oka) 日本 多复分析

志村五郎 (Goro Shimura) 日本 代数数论

岩泽健吉 (Kenkichi Iwasawa) 日本 代数数论

深谷贤治 (Kenji Fukaya) 日本 辛几何

中岛启 (Tadashi Nakayama) 日本 表示论

Mikhail Gromov 俄罗斯 度量几何与辛几何

André Weil 法国 代数数论、代数几何

Jean Leray 法国 代数拓扑、偏微分方程

Andreas Floer 德国 辛几何

Jurgen Moser 德国 动力系统、偏微分方程

Friedrich Hirzebruch 德国 代数拓扑、代数几何

Stefan Muller 德国 数学物理、变分法、偏微分方程

Lev Pontryagin 俄罗斯 代数拓扑、最优控制

Alexandr Alexandroff 俄罗斯 度量几何

Victor Maslov 俄罗斯 数学物理、流形上的分析

Vladimir Marchenko 乌克兰 数学物理、反问题

Olga Ladyzenskaya 俄罗斯 偏微分方程

Olga Oleinik 俄罗斯 偏微分方程

Ludwig Faddeev 俄罗斯 数学物理

Yuri Manin 俄罗斯 代数几何、代数数论、数学物理

Aleksei Pogorelov 乌克兰 微分几何、偏微分方程

Louis Nirenberg 加拿大 微分几何、偏微分方程

Robert Langlands 加拿大 代数几何、代数数论、表示论

Nikolai Efimov 俄罗斯 微分几何

Andrey Tikhonov 俄罗斯 拓扑学、偏微分方程、不适定问题、计算数学

Yuri Linnik 俄罗斯 解析数论、概率极限理论、数理统计

Vladimir Platonov 俄罗斯 代数几何、代数数论

Mark Krein 乌克兰 泛函分析

Dimitry Anosov 俄罗斯 微分几何、动力系统

Anatoly Maltsev 俄罗斯 一般代数学、数理逻辑

Yakov Eliashberg 俄罗斯 辛几何

Vasily Vladimirov 俄罗斯 数学物理、泛函分析

Ilya Piatetski-Shapiro 俄罗斯 表示论、代数数论

Alexandre Gelfond 俄罗斯 超越数论

Oleg Besov 俄罗斯 调和分析

Sergey Nikolsky 俄罗斯 调和分析、逼近论

Nikolay Bogoliubov 俄罗斯 数学物理

Victor Vassiliev 俄罗斯 几何拓扑

Claude Chevalley 法国 一般代数学、代数几何、李群

Paul Malliavin 法国 概率论

Jean-François Le Gall 概率论

Yves Meyer 法国 调和分析

Jean-Michel Bismut 法国 微分几何

Vincent Lafforgue 法国 几何表示论、非交换几何

Gérard Laumon 法国 代数几何、代数数论、表示论

Haïm Brezis 法国 泛函分析、偏微分方程

Jean-Pierre Demailly 法国 多复分析与复几何

Michèle Vergne 法国 表示论、流形上的分析

Harish-Chandra 印度 表示论

陈省身 (Shiing-shen Chern) 中国 微分几何

冯康 (Feng Kang) 中国 计算数学

Raoul Bott 匈牙利 代数拓扑

Peter Lax 匈牙利 偏微分方程

George Lusztig 罗马尼亚 表示论

Antoni Zygmund 波兰 调和分析

Richard Bellman 美国 控制论

How to compute the volume of an arbitrary parallelotope embedded in R^n

Originally published at 狗和留美者不得入内. You can comment here or there.

[公式] linearly independent vectors in [公式] result in a [公式] -dimensional parallelotope (a generalization of parallelogram or parallelpiped in higher dimensions). We wish to determine its generalized volume with orientation in [公式] dimensions. The determinant of an [公式] matrix gives the (signed) volume of the parallelotope induced by the column vectors, the order of which affects the sign. (The proof of this is rather straightforward and will be left to the reader.) Independent of coordinates, a linear transformation [公式] between two [公式] -dimensional vector spaces [公式] the positively oriented orthogonal bases of which are [公式] respectively assigns each element in [公式] to a linear combination of the elements of [公式] , and the determinant is the volume of the parallelotope generated by elements of [公式] [公式] .

Let [公式] be the vectors for our [公式] -dimensional parallelotope in [公式] . Together they define a map [公式] from [公式] to [公式] that by the rank-nullity theorem is onto, with [公式] . Let [公式] be a positively oriented orthogonal basis of [公式] , the subspace of [公式] perpendicular to the kernel of [公式] with [公式] . We then define the same map on a restricted domain as

[公式]We then have

[公式]

The coordinate free definition adjoint operator (or transpose) and its determinant

If [公式] is a linear map from [公式] and inner products are defined on [公式] -dimensional vector space [公式] with respect to their bases such that

[公式] Then the adjoint of [公式] , which we denote as [公式] is the map from [公式] such that

[公式] In the language of transposes, we have

[公式] Note how the left hand side of [公式] gives for every element in an ordered set of vectors [公式] in [公式] with respect to an orthonormal basis of [公式] , and then prescribes an ordered set of vectors [公式] in [公式] the coordinates of the [公式] th of which, with respect to an orthonormal basis of [公式] , is the [公式] th coordinate of the elements of [公式] , and vice versa.

We’ve essentially defined an [公式] matrix the elements of which are [公式] , and then the elements of transpose matrix [公式] . Applying the permutation based determinant formula gives us

[公式]

Computing the volume of the parallelotope

The matrix formed by linearly independent [公式] column vectors in [公式] , [公式] , corresponds directly to a linear isomorphism from [公式] , the codomain of which is of course a [公式] dimensional subspace of [公式] . We showed in the previous section that the adjoint of [公式] , namely [公式] , has the same determinant. Since determinant is a multiplicative function, [公式] thus gives us the square of the volume of the parallelotope.

Per the rule of matrix multiplication, given a linear isomorphism [公式] which takes the elements of the orthonormal basis of its domain to vectors [公式] , regardless of basis or coordinates in the range, its composition with its adjoint is represented with respect to aforementioned basis by

[公式] wherein the invoked inner product is, of course, per the properties of inner product invariant with respect to coordinate transformations, which means [公式] is well-defined. This corresponds to the matrix formed from our [公式] vectors in [公式] ,

[公式]

mutiplied by its transpose on the left, which gives us the result prescribed by [公式] , wherein the inner product is the Euclidean inner product in [公式] restricted to a [公式] dimensional subspace within it. The resulting Gramian matrix

[公式]

is, as we’ve already explained, such that

[公式] or in words the square of the volume of the parallelotope generated by [公式] .

Decomposing a [公式]volume element into orthogonal components in [公式] -form space

There is also intimate connection here with differential forms, exterior products, and Hodge dual. In [公式] of [1], we defined an inner product on the [公式] th exterior product space with the Gramian determinant invoked in [公式] such that

[公式] Let [公式] be an orthonormal positively oriented basis for our vector space, which by definition results in the equivalence relation

[公式] as far as [公式] -dimensional volume is concerned, with which each of our [公式] s decomposes to [公式] . Substitution of this decomposition into [公式] yields for the coefficient of [公式] the determinant of the matrix assembled from the [公式] rows of our [公式] column vectors, with an appropriate sign adjustment. This coefficient is necessarily also an anti-symmetric [公式] th rank tensor.

Geometrically this is the oriented [公式] -dimensional volume obtained if only the components corresponding to indices [公式] are considered. The result in [公式] of [1], which was calculated via the Hodge star in a way that equates to the Gramian matrix definition of the inner product, yields for the value of the inner product given in [公式] the sum of squares of the coefficients with respect to our [公式] basis elements [公式] . Essentially there is a basis of [公式] of dimension [公式] volume elements of equivalent volume in [公式] space represented by [公式] -forms, which are mutually orthogonal with respect to the inner product we defined on the space of [公式] -forms, and we have projected our arbitrary [公式] dimensional volume element onto each of them. In this sense it is natural that the square of its norm or size would be the sum of the squares of its components.


I am dedicating this article to Seki Takakazu (1642-1708), who based on no more than 13th century Chinese algebra and arithmetic obtained results regarding determinants and resultants decades before the West and who discovered Bernoulli numbers (or Takakazu numbers) in connection to the closed formula for sum of the first [公式] th powers around the same time as did Jacob Bernoulli.

References

How to naturally construct and compute explicitly the Hodge dual of a differential form

Originally published at 狗和留美者不得入内. You can comment here or there.

There is antisymmetry in the wedge product of differential forms, i.e. [公式] . Take the decomposable [公式] -form in [公式] -dimensional space denoted by

[公式] We can take [公式] and the corresponding [公式] -form

[公式] as the Hodge dual with signs yet to be determined. As for how to reasonably prescribe the sign, we first note that a [公式] -form wedged producted with its Hodge dual gives

[公式] The right hand side of this is equal to

[公式] We let [公式] be the standard ordering of the basis of the underlying vector space, which we regard as positively oriented. Moreover, we let [公式] be the permutation such that

[公式] It is reasonable to impose for symmetry that

[公式] To satisfy this, we choose an ordering of remaining indices [公式] such that the sign of the corresponding permutation [公式] is positive.

[公式] , the space of [公式] -forms over finite vector space [公式], has dimension [公式] . The standard basis for it we represent as

[公式]

[公式] has the same dimension [公式] . It is easy to see that the Hodge star operator maps the standard basis of [公式] to that of [公式] , and moreover that

[公式] Imposing linearity on the Hodge star operator then gives us

[公式] Now we wish to in Einstein notation in terms of [公式] indices represent the Hodge star of an arbitrary [公式] -form. In doing so, we will use the complete antisymmetric tensor of rank [公式] , which in addition to being zero when not all indices are different, satisfies

[公式] In the basis above, we assumed [公式] , and the coefficients [公式] . If we impose no restriction on the ordering of the indices and for each combination pick its associated basis element arbitrarily, then for the sum of our linear combination of the basis elements to remain invariant [公式]must be a completely antisymmetric tensor. That is, when we swap adjacent elements in our wedge product which flips signs, we must flip the sign of the coefficient as well. Given this, we can also simply sum through all permutations of [公式] , the set of first [公式] integers, in which case the set [公式] is mapped to fixed combination [公式] in [公式] of the [公式] permutations. Thus, we have for any arbitrary [公式] -form, implicitly summing across all [公式] ,

[公式] Essentially, for each combination or basis element, we count it [公式] times, which we also divide by in the result to normalize. In calculating its Hodge dual, we note how

[公式] In replacing the parenthesis with a subscript for [公式], it is to emphasize that [公式] are different indices each of which are iterated across [公式] per Einstein notation. Complete antisymmetry means that a duplication of index results in [公式] . From [公式] , we get

[公式]

Some function isomorphisms between exterior product spaces

The Hodge star is a function of signature

[公式] that is clearly also a linear isomorphism, which means that

[公式] We also showed via the above calculation that wedge product is a bilinear function of signature

[公式]

We denote the volume corresponding to the wedge product of our [公式] basis vectors of real vector space [公式] in an order of positive orientation, [公式] , as [公式] and observe that every elements of [公式] is of the form [公式] , [公式] , or

[公式] Moreover, [公式] tells us that plugging in a [公式] -form on the left results in a function of signature

[公式] which is a functional over the [公式] -forms, which we denote with [公式] , and no different different [公式] -form inputs result in the same functional, which means we have induced an injective function of signature

[公式] We now prove that it is also surjective. Every element in [公式] is, by definition of the vector space underlying the functional, uniquely defined by a collection of [公式] mappings of combinations to real numbers, or in other words, each[公式] in our basis is assigned to a real number [公式] . This corresponds uniquely to [公式] in our wedge product.

Bijectivity means

[公式]

Inducing an inner product on [公式] via the Hodge star

We now examine the function described by

[公式]We now verify that it satisfies the properties of an inner product. Below, we per [公式] set

[公式]

and given this, by [公式] ,

[公式] We will for simplicity also omit the constant factor in subsequent calculations.

  1. Linearity in the first argument is satisfied because the wedge product is bilinear.
  2. The calculation below gives conjugate symmetry.

[公式] 3. If in [公式] , we set [公式] in order to equate [公式] , we find that

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<p><small>Originally published at <a href="https://gmachine1729.wpcomstaging.com/2021/04/04/how-to-naturally-construct-and-compute-explicitly-the-hodge-dual-of-a-differential-form/">狗和留美者不得入内</a>. You can comment here or <a href="https://gmachine1729.wpcomstaging.com/2021/04/04/how-to-naturally-construct-and-compute-explicitly-the-hodge-dual-of-a-differential-form/#comments">there</a>.</small></p><div class="RichText ztext Post-RichText"> <p>There is antisymmetry in the wedge product of differential forms, i.e. <img src="https://www.zhihu.com/equation?tex=dx%5E%5Cmu+%5Cwedge+dx%5E%5Cnu+%3D+-+dx%5E%5Cnu+%5Cwedge+dx%5E%5Cmu+" alt="[公式]" eeimg="1" data-formula="dx^\mu \wedge dx^\nu = - dx^\nu \wedge dx^\mu "> . Take the decomposable <img src="https://www.zhihu.com/equation?tex=p" alt="[公式]" eeimg="1" data-formula="p"> -form in <img src="https://www.zhihu.com/equation?tex=n" alt="[公式]" eeimg="1" data-formula="n"> -dimensional space denoted by</p> <p><img src="https://www.zhihu.com/equation?tex=dx%5E%7B%5Cmu_1%7D%5Cwedge+dx%5E%7B%5Cmu_2%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_p%7D.%5C%5C" alt="[公式]" eeimg="1" data-formula="dx^{\mu_1}\wedge dx^{\mu_2}\wedge \ldots \wedge dx^{\mu_p}.\\"> We can take <img src="https://www.zhihu.com/equation?tex=%5C%7B%5Cnu_1%2C%5Cnu_2%2C%5Cldots%2C%5Cnu_%7Bn-p%7D%5C%7D+%3D+%5C%7B1%2C2%2C%5Cldots%2Cn%5C%7D+%5Csetminus+%5C%7B%5Cmu_1%2C%5Cmu_2%2C%5Cldots%2C%5Cmu_p%5C%7D" alt="[公式]" eeimg="1" data-formula="\{\nu_1,\nu_2,\ldots,\nu_{n-p}\} = \{1,2,\ldots,n\} \setminus \{\mu_1,\mu_2,\ldots,\mu_p\}"> and the corresponding <img src="https://www.zhihu.com/equation?tex=%28n-p%29" alt="[公式]" eeimg="1" data-formula="(n-p)"> -form</p> <p><img src="https://www.zhihu.com/equation?tex=%28dx%5E%7B%5Cmu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_p%7D%29%5E%2A+%3D+%5Cpm+dx%5E%7B%5Cnu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cnu_%7Bn-p%7D%7D.%5C%5C" alt="[公式]" eeimg="1" data-formula="(dx^{\mu_1}\wedge \ldots \wedge dx^{\mu_p})^* = \pm dx^{\nu_1}\wedge \ldots \wedge dx^{\nu_{n-p}}.\\"> as the Hodge dual with signs yet to be determined. As for how to reasonably prescribe the sign, we first note that a <img src="https://www.zhihu.com/equation?tex=p" alt="[公式]" eeimg="1" data-formula="p"> -form wedged producted with its Hodge dual gives</p> <p><img src="https://www.zhihu.com/equation?tex=%28dx%5E%7B%5Cmu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_p%7D%29%5Cwedge+%28dx%5E%7B%5Cmu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_p%7D%29%5E%2A+%3D+%5Cpm+dx%5E%7B%5Cmu_1%7D%5Cwedge+dx%5E%7B%5Cmu_p%7D+%5Cwedge+dx%5E%7B%5Cnu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cnu_%7Bn-p%7D%7D.+%5Cqquad+%281%29%5C%5C" alt="[公式]" eeimg="1" data-formula="(dx^{\mu_1}\wedge \ldots \wedge dx^{\mu_p})\wedge (dx^{\mu_1}\wedge \ldots \wedge dx^{\mu_p})^* = \pm dx^{\mu_1}\wedge dx^{\mu_p} \wedge dx^{\nu_1}\wedge \ldots \wedge dx^{\nu_{n-p}}. \qquad (1)\\"> The right hand side of this is equal to</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cpm+dx%5E%7B1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7Bn%7D.%5C%5C" alt="[公式]" eeimg="1" data-formula="\pm dx^{1}\wedge \ldots \wedge dx^{n}.\\"> We let <img src="https://www.zhihu.com/equation?tex=dx%5E1%2Cdx%5E2%2C%5Cldots+dx%5En" alt="[公式]" eeimg="1" data-formula="dx^1,dx^2,\ldots dx^n"> be the standard ordering of the basis of the underlying vector space, which we regard as positively oriented. Moreover, we let <img src="https://www.zhihu.com/equation?tex=%5Csigma" alt="[公式]" eeimg="1" data-formula="\sigma"> be the permutation such that</p> <p><img src="https://www.zhihu.com/equation?tex=%5Csigma%281%2C2%2C%5Cldots%2C+n%29+%3D+%28%5Csigma%281%29%2C%5Csigma%282%29%2C%5Cldots%2C+%5Csigma%28n%29%29+%3D+%28%5Cmu_1%2C%5Cmu_2%2C%5Cldots%2C+%5Cmu_p%2C%5Cnu_1%2C%5Cldots%2C+%5Cnu_%7Bn-p%7D%29.%5C%5C" alt="[公式]" eeimg="1" data-formula="\sigma(1,2,\ldots, n) = (\sigma(1),\sigma(2),\ldots, \sigma(n)) = (\mu_1,\mu_2,\ldots, \mu_p,\nu_1,\ldots, \nu_{n-p}).\\"> It is reasonable to impose for symmetry that</p> <p><img src="https://www.zhihu.com/equation?tex=%28dx%5E%7B%5Cmu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_p%7D%29%5Cwedge+%28dx%5E%7B%5Cmu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_%7Bp%7D%7D%29%5E%2A+%3D+dx%5E1%5Cwedge+%5Cldots+%5Cwedge+dx%5En.%5C%5C" alt="[公式]" eeimg="1" data-formula="(dx^{\mu_1}\wedge \ldots \wedge dx^{\mu_p})\wedge (dx^{\mu_1}\wedge \ldots \wedge dx^{\mu_{p}})^* = dx^1\wedge \ldots \wedge dx^n.\\"> To satisfy this, we choose an ordering of remaining indices <img src="https://www.zhihu.com/equation?tex=%28%5Cnu_1%2C%5Cldots%2C+%5Cnu_%7Bn-p%7D%29" alt="[公式]" eeimg="1" data-formula="(\nu_1,\ldots, \nu_{n-p})"> such that the sign of the corresponding permutation <img src="https://www.zhihu.com/equation?tex=%5Csigma" alt="[公式]" eeimg="1" data-formula="\sigma"> is positive.</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5Ep%28V%29" alt="[公式]" eeimg="1" data-formula="\bigwedge^p(V)"> , the space of <img src="https://www.zhihu.com/equation?tex=p" alt="[公式]" eeimg="1" data-formula="p"> -forms over finite vector space <img src="https://www.zhihu.com/equation?tex=V" alt="[公式]" eeimg="1" data-formula="V">, has dimension <img src="https://www.zhihu.com/equation?tex=%5Cbinom%7Bn%7D%7Bp%7D" alt="[公式]" eeimg="1" data-formula="\binom{n}{p}"> . The standard basis for it we represent as</p> <p><img src="https://www.zhihu.com/equation?tex=%28e%5E1%2C+%5Cldots%2C+e%5E%7B%5Cbinom%7Bn%7D%7Bp%7D%7D%29+%3D+%5C%7Bdx%5E%7B%5Cmu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_p%7D+%3A+1+%5Cleq+%5Cmu_1+%3C+%5Cmu_2+%3C+%5Cldots+%3C+%5Cmu_p+%5Cleq+n%5C%7D.%5C%5C" alt="[公式]" eeimg="1" data-formula="(e^1, \ldots, e^{\binom{n}{p}}) = \{dx^{\mu_1}\wedge \ldots \wedge dx^{\mu_p} : 1 \leq \mu_1 < \mu_2 < \ldots < \mu_p \leq n\}.\\"> </p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5E%7Bn-p%7D%28V%29" alt="[公式]" eeimg="1" data-formula="\bigwedge^{n-p}(V)"> has the same dimension <img src="https://www.zhihu.com/equation?tex=%5Cbinom%7Bn%7D%7Bn-p%7D+%3D+%5Cbinom%7Bn%7D%7Bp%7D" alt="[公式]" eeimg="1" data-formula="\binom{n}{n-p} = \binom{n}{p}"> . It is easy to see that the Hodge star operator maps the standard basis of <img src="https://www.zhihu.com/equation?tex=%5CLambda%5Ep%28V%29" alt="[公式]" eeimg="1" data-formula="\Lambda^p(V)"> to that of <img src="https://www.zhihu.com/equation?tex=%5CLambda%5E%7Bn-p%7D%28V%29" alt="[公式]" eeimg="1" data-formula="\Lambda^{n-p}(V)"> , and moreover that</p> <p><img src="https://www.zhihu.com/equation?tex=e%5Ei+%5Cwedge+%28e%5Ej%29%5E%2A+%3D+%5Cdelta_i%5Ej+dx%5E1%5Cwedge+dx%5E2%5Cwedge+%5Cldots+%5Cwedge+dx%5En.%5C%5C" alt="[公式]" eeimg="1" data-formula="e^i \wedge (e^j)^* = \delta_i^j dx^1\wedge dx^2\wedge \ldots \wedge dx^n.\\"> Imposing linearity on the Hodge star operator then gives us</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%28A_ie%5Ei%29%5Cwedge+%5B%28B_j+e%5Ej%29%5E%2A%5D+%26%3D%26+%28A_ie%5Ei%29%5Cwedge+%5BB_j+%28e%5Ej%29%5E%2A%5D%5C%5C+%26%3D%26+%28A_ie%5Ei%29%5Cwedge+%5BB_i+%28e%5Ei%29%5E%2A%5D%5C%5C+%26%3D%26+A_iB_i+dx_1+%5Cwedge+dx_2+%5Cldots+%5Cwedge+dx_n.+%5Cqquad+%282%29+%5Cend%7Beqnarray%7D%5C%5C" alt="[公式]" eeimg="1" data-formula="\begin{eqnarray} (A_ie^i)\wedge [(B_j e^j)^*] &amp;=&amp; (A_ie^i)\wedge [B_j (e^j)^*]\\ &amp;=&amp; (A_ie^i)\wedge [B_i (e^i)^*]\\ &amp;=&amp; A_iB_i dx_1 \wedge dx_2 \ldots \wedge dx_n. \qquad (2) \end{eqnarray}\\"> Now we wish to in Einstein notation in terms of <img src="https://www.zhihu.com/equation?tex=p" alt="[公式]" eeimg="1" data-formula="p"> indices represent the Hodge star of an arbitrary <img src="https://www.zhihu.com/equation?tex=p" alt="[公式]" eeimg="1" data-formula="p"> -form. In doing so, we will use the complete antisymmetric tensor of rank <img src="https://www.zhihu.com/equation?tex=n" alt="[公式]" eeimg="1" data-formula="n"> , which in addition to being zero when not all indices are different, satisfies</p> <p><img src="https://www.zhihu.com/equation?tex=e%5E%7B%5Csigma%281%29%5Csigma%282%29%5Cldots+%5Csigma%28n%29%7D+%3D+%5Cmathrm%7Bsgn%7D%28%5Csigma%29e%5E%7B12%5Cldots+n%7D+%3D+%5Cmathrm%7Bsgn%7D%28%5Csigma%29.%5C%5C" alt="[公式]" eeimg="1" data-formula="e^{\sigma(1)\sigma(2)\ldots \sigma(n)} = \mathrm{sgn}(\sigma)e^{12\ldots n} = \mathrm{sgn}(\sigma).\\"> In the basis above, we assumed <img src="https://www.zhihu.com/equation?tex=%5Cmu_1+%3C+%5Cldots+%3C+%5Cmu_n+" alt="[公式]" eeimg="1" data-formula="\mu_1 < \ldots < \mu_n "> , and the coefficients <img src="https://www.zhihu.com/equation?tex=A_i+%5Cequiv+A_%7B%5Cmu_1%5Cldots+%5Cmu_p%7D" alt="[公式]" eeimg="1" data-formula="A_i \equiv A_{\mu_1\ldots \mu_p}"> . If we impose no restriction on the ordering of the indices and for each combination pick its associated basis element arbitrarily, then for the sum of our linear combination of the basis elements to remain invariant <img src="https://www.zhihu.com/equation?tex=A_%7B%5Cmu_1%5Cldots+%5Cmu_p%7D" alt="[公式]" eeimg="1" data-formula="A_{\mu_1\ldots \mu_p}">must be a completely antisymmetric tensor. That is, when we swap adjacent elements in our wedge product which flips signs, we must flip the sign of the coefficient as well. Given this, we can also simply sum through all permutations of <img src="https://www.zhihu.com/equation?tex=%5Bn%5D" alt="[公式]" eeimg="1" data-formula="[n]"> , the set of first <img src="https://www.zhihu.com/equation?tex=n" alt="[公式]" eeimg="1" data-formula="n"> integers, in which case the set <img src="https://www.zhihu.com/equation?tex=%5C%7B1%2C+%5Cldots%2C+p%5C%7D" alt="[公式]" eeimg="1" data-formula="\{1, \ldots, p\}"> is mapped to fixed combination <img src="https://www.zhihu.com/equation?tex=%5C%7B%5Cmu_1%2C%5Cldots%2C+%5Cmu_p%5C%7D" alt="[公式]" eeimg="1" data-formula="\{\mu_1,\ldots, \mu_p\}"> in <img src="https://www.zhihu.com/equation?tex=p%21%28n-p%29%21" alt="[公式]" eeimg="1" data-formula="p!(n-p)!"> of the <img src="https://www.zhihu.com/equation?tex=n%21" alt="[公式]" eeimg="1" data-formula="n!"> permutations. Thus, we have for any arbitrary <img src="https://www.zhihu.com/equation?tex=p" alt="[公式]" eeimg="1" data-formula="p"> -form, implicitly summing across all <img src="https://www.zhihu.com/equation?tex=%5Csigma+%5Cin+S_n" alt="[公式]" eeimg="1" data-formula="\sigma \in S_n"> ,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Comega+%5Cequiv+%5Cfrac%7B1%7D%7Bp%21%28n-p%29%21%7DA_%7B%5Csigma%281%29%5Cldots+%5Csigma%28p%29%7Ddx%5E%7B%5Csigma%281%29%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma%28p%29%7D.+%5Cqquad+%283%29%5C%5C" alt="[公式]" eeimg="1" data-formula="\omega \equiv \frac{1}{p!(n-p)!}A_{\sigma(1)\ldots \sigma(p)}dx^{\sigma(1)}\wedge \ldots \wedge dx^{\sigma(p)}. \qquad (3)\\"> Essentially, for each combination or basis element, we count it <img src="https://www.zhihu.com/equation?tex=p%21%28n-p%29%21" alt="[公式]" eeimg="1" data-formula="p!(n-p)!"> times, which we also divide by in the result to normalize. In calculating its Hodge dual, we note how</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%28dx%5E%7B%5Csigma%281%29%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma%28p%29%7D%29%5E%2A+%26%3D%26+%5Cmathrm%7Bsgn%7D%28%5Csigma%29+dx%5E%7B%5Csigma%28p%2B1%29%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma%28n%29%7D%5C%5C+%26%3D%26+e%5E%7B%5Csigma_1%5Csigma_2%5Cldots+%5Csigma_n%7Ddx%5E%7B%5Csigma_%7Bp%2B1%7D%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_n%7D.+%5Cqquad+%284%29+%5Cend%7Beqnarray%7D%5C%5C" alt="[公式]" eeimg="1" data-formula="\begin{eqnarray} (dx^{\sigma(1)}\wedge \ldots \wedge dx^{\sigma(p)})^* &amp;=&amp; \mathrm{sgn}(\sigma) dx^{\sigma(p+1)}\wedge \ldots \wedge dx^{\sigma(n)}\\ &amp;=&amp; e^{\sigma_1\sigma_2\ldots \sigma_n}dx^{\sigma_{p+1}}\wedge \ldots \wedge dx^{\sigma_n}. \qquad (4) \end{eqnarray}\\"> In replacing the parenthesis with a subscript for <img src="https://www.zhihu.com/equation?tex=%5Csigma" alt="[公式]" eeimg="1" data-formula="\sigma">, it is to emphasize that <img src="https://www.zhihu.com/equation?tex=%5Csigma_1%2C%5Csigma_2%2C%5Cldots+%5Csigma_n" alt="[公式]" eeimg="1" data-formula="\sigma_1,\sigma_2,\ldots \sigma_n"> are different indices each of which are iterated across <img src="https://www.zhihu.com/equation?tex=%5B1%5D" alt="[公式]" eeimg="1" data-formula="[1]"> per Einstein notation. Complete antisymmetry means that a duplication of index results in <img src="https://www.zhihu.com/equation?tex=e_%7B%5Csigma_1%5Csigma_2%5Cldots+%5Csigma_n%7D+%3D+0" alt="[公式]" eeimg="1" data-formula="e_{\sigma_1\sigma_2\ldots \sigma_n} = 0"> . From <img src="https://www.zhihu.com/equation?tex=%284%29" alt="[公式]" eeimg="1" data-formula="(4)"> , we get</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%5Comega%5E%2A+%26%3D%26+%5Cfrac%7B1%7D%7Bp%21%28n-p%29%21%7DA_%7B%5Csigma_1%5Cldots+%5Csigma_p%7D%28dx%5E%7B%5Csigma_%7B1%7D%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_p%7D%29%5E%2A%5C%5C+%26%3D%26+%5Cfrac%7B1%7D%7Bp%21%28n-p%29%21%7DA_%7B%5Csigma_1%5Cldots+%5Csigma_p%7D+e%5E%7B%5Csigma_1%5Csigma_2%5Cldots+%5Csigma_n%7Ddx%5E%7B%5Csigma_%7Bp%2B1%7D%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_n%7D.+%5Cqquad+%285%29+%5Cend%7Beqnarray%7D+%5C%5C" alt="[公式]" eeimg="1" data-formula="\begin{eqnarray} \omega^* &amp;=&amp; \frac{1}{p!(n-p)!}A_{\sigma_1\ldots \sigma_p}(dx^{\sigma_{1}}\wedge \ldots \wedge dx^{\sigma_p})^*\\ &amp;=&amp; \frac{1}{p!(n-p)!}A_{\sigma_1\ldots \sigma_p} e^{\sigma_1\sigma_2\ldots \sigma_n}dx^{\sigma_{p+1}}\wedge \ldots \wedge dx^{\sigma_n}. \qquad (5) \end{eqnarray} \\"> </p> <h3>Some function isomorphisms between exterior product spaces</h3> <p>The Hodge star is a function of signature</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cstar_p+%3A+%5Cbigwedge%5Ep%28V%29+%5Cto+%5Cbigwedge%5E%7Bn-p%7D%28V%29%5C%5C" alt="[公式]" eeimg="1" data-formula="\star_p : \bigwedge^p(V) \to \bigwedge^{n-p}(V)\\"> that is clearly also a linear isomorphism, which means that</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5Ep%28V%29+%5Csimeq+%5Cbigwedge%5E%7Bn-p%7D%28V%29.+%5Cqquad+%286%29%5C%5C" alt="[公式]" eeimg="1" data-formula="\bigwedge^p(V) \simeq \bigwedge^{n-p}(V). \qquad (6)\\"> We also showed via the above calculation that wedge product is a bilinear function of signature</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cwedge+%3A+%5Cbigwedge%5Ep%28V%29+%5Ctimes+%5Cbigwedge%5E%7Bn-p%7D%28V%29+%5Cto+%5Cbigwedge%5E%7Bn%7D%28V%29.%5C%5C+" alt="[公式]" eeimg="1" data-formula="\wedge : \bigwedge^p(V) \times \bigwedge^{n-p}(V) \to \bigwedge^{n}(V).\\ "> </p> <p>We denote the volume corresponding to the wedge product of our <img src="https://www.zhihu.com/equation?tex=n" alt="[公式]" eeimg="1" data-formula="n"> basis vectors of real vector space <img src="https://www.zhihu.com/equation?tex=V" alt="[公式]" eeimg="1" data-formula="V"> in an order of positive orientation, <img src="https://www.zhihu.com/equation?tex=dx%5E1%5Cwedge+dx%5E2%5Cwedge+%5Cldots+%5Cwedge+dx%5En" alt="[公式]" eeimg="1" data-formula="dx^1\wedge dx^2\wedge \ldots \wedge dx^n"> , as <img src="https://www.zhihu.com/equation?tex=%5Cmathrm%7BVol%7D_%5Cmu" alt="[公式]" eeimg="1" data-formula="\mathrm{Vol}_\mu"> and observe that every elements of <img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5E%7Bn%7D%28V%29" alt="[公式]" eeimg="1" data-formula="\bigwedge^{n}(V)"> is of the form <img src="https://www.zhihu.com/equation?tex=c%5Ccdot+%5Cmathrm%7BVol%7D_%5Cmu" alt="[公式]" eeimg="1" data-formula="c\cdot \mathrm{Vol}_\mu"> , <img src="https://www.zhihu.com/equation?tex=c+%5Cin+%5Cmathbb%7BR%7D" alt="[公式]" eeimg="1" data-formula="c \in \mathbb{R}"> , or</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5E%7Bn%7D%28V%29+%5Csimeq+%5Cmathbb%7BR%7D%5Cmathrm%7BVol%7D_%5Cmu+%5Csimeq+%5Cmathbb%7BR%7D.+%5Cqquad+%287%29%5C%5C" alt="[公式]" eeimg="1" data-formula="\bigwedge^{n}(V) \simeq \mathbb{R}\mathrm{Vol}_\mu \simeq \mathbb{R}. \qquad (7)\\"> Moreover, <img src="https://www.zhihu.com/equation?tex=%282%29" alt="[公式]" eeimg="1" data-formula="(2)"> tells us that plugging in a <img src="https://www.zhihu.com/equation?tex=p" alt="[公式]" eeimg="1" data-formula="p"> -form on the left results in a function of signature</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5E%7Bn-p%7D%28V%29+%5Cto+%5Cleft%28%5Cbigwedge%5E%7Bn%7D%28V%29+%5Csimeq+%5Cmathbb%7BR%7D%5Cright%29%2C%5C%5C+" alt="[公式]" eeimg="1" data-formula="\bigwedge^{n-p}(V) \to \left(\bigwedge^{n}(V) \simeq \mathbb{R}\right),\\ "> which is a functional over the <img src="https://www.zhihu.com/equation?tex=%28n-p%29" alt="[公式]" eeimg="1" data-formula="(n-p)"> -forms, which we denote with <img src="https://www.zhihu.com/equation?tex=%5Cleft%28%5Cbigwedge%5E%7Bn-p%7D%28V%29%5Cright%29%5E%5Cvee" alt="[公式]" eeimg="1" data-formula="\left(\bigwedge^{n-p}(V)\right)^\vee"> , and no different different <img src="https://www.zhihu.com/equation?tex=p" alt="[公式]" eeimg="1" data-formula="p"> -form inputs result in the same functional, which means we have induced an injective function of signature</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5E%7Bp%7D%28V%29+%5Cto+%5Cleft%28%5Cbigwedge%5E%7Bn-p%7D%28V%29%5Cright%29%5E%5Cvee.%5C%5C" alt="[公式]" eeimg="1" data-formula="\bigwedge^{p}(V) \to \left(\bigwedge^{n-p}(V)\right)^\vee.\\"> We now prove that it is also surjective. Every element in <img src="https://www.zhihu.com/equation?tex=%5Cleft%28%5Cbigwedge%5E%7Bn-p%7D%28V%29%5Cright%29%5E%5Cvee" alt="[公式]" eeimg="1" data-formula="\left(\bigwedge^{n-p}(V)\right)^\vee"> is, by definition of the vector space underlying the functional, uniquely defined by a collection of <img src="https://www.zhihu.com/equation?tex=%5Cbinom%7Bn%7D%7Bn-p%7D" alt="[公式]" eeimg="1" data-formula="\binom{n}{n-p}"> mappings of combinations to real numbers, or in other words, each<img src="https://www.zhihu.com/equation?tex=dx%5E%7B%5Cnu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cnu_%7Bn-p%7D%7D" alt="[公式]" eeimg="1" data-formula="dx^{\nu_1}\wedge \ldots \wedge dx^{\nu_{n-p}}"> in our basis is assigned to a real number <img src="https://www.zhihu.com/equation?tex=C%5E%7B%5Cnu_1%5Cldots+%5Cnu_%7Bn-p%7D%7D" alt="[公式]" eeimg="1" data-formula="C^{\nu_1\ldots \nu_{n-p}}"> . This corresponds uniquely to <img src="https://www.zhihu.com/equation?tex=e%5E%7B%5Cmu_1%5Cldots+%5Cmu_p%5Cnu_1%5Cldots+%5Cnu_%7Bn-p%7D%7DC%5E%7B%5Cnu_1%5Cldots+%5Cnu_%7Bn-p%7D%7D+dx%5E%7B%5Cmu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_p%7D+%5Cin+%5Cbigwedge%5Ep%28V%29" alt="[公式]" eeimg="1" data-formula="e^{\mu_1\ldots \mu_p\nu_1\ldots \nu_{n-p}}C^{\nu_1\ldots \nu_{n-p}} dx^{\mu_1}\wedge \ldots \wedge dx^{\mu_p} \in \bigwedge^p(V)"> in our wedge product.</p> <p>Bijectivity means</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5E%7Bp%7D%28V%29+%5Csimeq+%5Cleft%28%5Cbigwedge%5E%7Bn-p%7D%28V%29%5Cright%29%5E%5Cvee.+%5Cqquad+%288%29%5C%5C" alt="[公式]" eeimg="1" data-formula="\bigwedge^{p}(V) \simeq \left(\bigwedge^{n-p}(V)\right)^\vee. \qquad (8)\\"> </p> <h3>Inducing an inner product on <img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5Ep%28V%29" alt="[公式]" eeimg="1" data-formula="\bigwedge^p(V)"> via the Hodge star</h3> <p>We now examine the function described by</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbegin%7Balign%7D+%5Clangle%5Ccdot+%2C%5Ccdot+%5Crangle+%5Cbigwedge%5Ep%28V%29+%5Ctimes+%5Cbigwedge%5Ep%28V%29+%5Cto+%5Cmathbb%7BR%7D%2C%5C%5C+%5C%5C+%5Clangle+%5Comega%2C+%5Ceta+%5Crangle+%3D+%5Comega+%5Cwedge+%28%5Cstar_p+%5Ceta%29%2C%5C%5C%5C%5C++dx%5E1%5Cwedge+dx%5E2%5Cwedge+%5Cldots+%5Cwedge+dx%5En+%5Csim+%5Cmathrm%7BVol%7D_%5Cmu.+%5Cend%7Balign%7D%5Cqquad+%289%29%5C%5C" alt="[公式]" eeimg="1" data-formula="\begin{align} \langle\cdot ,\cdot \rangle \bigwedge^p(V) \times \bigwedge^p(V) \to \mathbb{R},\\ \\ \langle \omega, \eta \rangle = \omega \wedge (\star_p \eta),\\\\ dx^1\wedge dx^2\wedge \ldots \wedge dx^n \sim \mathrm{Vol}_\mu. \end{align}\qquad (9)\\">We now verify that it satisfies the properties of an inner product. Below, we per <img src="https://www.zhihu.com/equation?tex=%283%29" alt="[公式]" eeimg="1" data-formula="(3)"> set</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%5Comega+%26%3D%26+%5Cfrac%7B1%7D%7Bp%21%28n-p%29%21%7DA_%7B%5Csigma_1%5Cldots+%5Csigma_p%7Ddx%5E%7B%5Csigma_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_p%7D%5C%5C+%5Ceta+%26%3D%26+%5Cfrac%7B1%7D%7Bp%21%28n-p%29%21%7DB_%7B%5Csigma_1%5Cldots+%5Csigma_p%7Ddx%5E%7B%5Csigma_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_p%7D%5C%5C+%5Cend%7Beqnarray%7D+%5Cqquad+%2810%29%5C%5C" alt="[公式]" eeimg="1" data-formula="\begin{eqnarray} \omega &amp;=&amp; \frac{1}{p!(n-p)!}A_{\sigma_1\ldots \sigma_p}dx^{\sigma_1}\wedge \ldots \wedge dx^{\sigma_p}\\ \eta &amp;=&amp; \frac{1}{p!(n-p)!}B_{\sigma_1\ldots \sigma_p}dx^{\sigma_1}\wedge \ldots \wedge dx^{\sigma_p}\\ \end{eqnarray} \qquad (10)\\"> </p> <p>and given this, by <img src="https://www.zhihu.com/equation?tex=%285%29" alt="[公式]" eeimg="1" data-formula="(5)"> ,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%5Cstar_p%5Comega+%26%3D%26+%5Cfrac%7B1%7D%7Bp%21%28n-p%29%21%7DA_%7B%5Csigma_1%5Cldots+%5Csigma_p%7D+e%5E%7B%5Csigma_1%5Csigma_2%5Cldots+%5Csigma_n%7Ddx%5E%7B%5Csigma_%7Bp%2B1%7D%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_n%7D%5C%5C+%5Cstar_p%5Ceta+%26%3D%26+%5Cfrac%7B1%7D%7Bp%21%28n-p%29%21%7DB_%7B%5Csigma_1%5Cldots+%5Csigma_p%7D+e%5E%7B%5Csigma_1%5Csigma_2%5Cldots+%5Csigma_n%7Ddx%5E%7B%5Csigma_%7Bp%2B1%7D%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_n%7D.%5C%5C+%5Cend%7Beqnarray%7D+%5Cqquad+%2811%29%5C%5C" alt="[公式]" eeimg="1" data-formula="\begin{eqnarray} \star_p\omega &amp;=&amp; \frac{1}{p!(n-p)!}A_{\sigma_1\ldots \sigma_p} e^{\sigma_1\sigma_2\ldots \sigma_n}dx^{\sigma_{p+1}}\wedge \ldots \wedge dx^{\sigma_n}\\ \star_p\eta &amp;=&amp; \frac{1}{p!(n-p)!}B_{\sigma_1\ldots \sigma_p} e^{\sigma_1\sigma_2\ldots \sigma_n}dx^{\sigma_{p+1}}\wedge \ldots \wedge dx^{\sigma_n}.\\ \end{eqnarray} \qquad (11)\\"> We will for simplicity also omit the constant factor in subsequent calculations.</p> <ol> <li>Linearity in the first argument is satisfied because the wedge product is bilinear.</li> <li>The calculation below gives conjugate symmetry.</li> </ol> <p><img src="https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%5Clangle+%5Comega%2C+%5Ceta+%5Crangle+%26%3D%26+%5Comega+%5Cwedge+%28%5Cstar_p+%5Ceta%29%5C%5C+%26%3D%26%28A_%7B%5Csigma_1%5Cldots+%5Csigma_p%7Ddx%5E%7B%5Csigma_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_p%7D%29%5Cwedge+%28B_%7B%5Csigma_1%5Cldots+%5Csigma_p%7D+e%5E%7B%5Csigma_1%5Csigma_2%5Cldots+%5Csigma_n%7Ddx%5E%7B%5Csigma_%7Bp%2B1%7D%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_n%7D%29%5C%5C+%26%3D%26+A_%7B%5Csigma_1%5Cldots+%5Csigma_p%7DB_%7B%5Csigma_1%5Cldots+%5Csigma_p%7De%5E%7B%5Csigma_1%5Csigma_2%5Cldots+%5Csigma_n%7Ddx%5E%7B%5Csigma_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_p%7D%5Cwedge+dx%5E%7B%5Csigma_%7Bp%2B1%7D%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_n%7D%5C%5C+%26%3D%26A_%7B%5Csigma_1%5Cldots+%5Csigma_p%7DB_%7B%5Csigma_1%5Cldots+%5Csigma_p%7Ddx%5E1%5Cwedge+dx%5E2%5Cldots+%5Cwedge+dx%5En%5C%5C+%26%3D%26%28B_%7B%5Csigma_1%5Cldots+%5Csigma_p%7Ddx%5E%7B%5Csigma_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_p%7D%29%5Cwedge+%28A_%7B%5Csigma_1%5Cldots+%5Csigma_p%7D+e%5E%7B%5Csigma_1%5Csigma_2%5Cldots+%5Csigma_n%7Ddx%5E%7B%5Csigma_%7Bp%2B1%7D%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Csigma_n%7D%29%5C%5C+%26%3D%26%5Ceta+%5Cwedge+%28%5Cstar_p+%5Comega%29%5C%5C+%26%3D%26%5Clangle+%5Ceta%2C+%5Comega+%5Crangle.+%5Cend%7Beqnarray%7D%5Cqquad+%2812%29%5C%5C+" alt="[公式]" eeimg="1" data-formula="\begin{eqnarray} \langle \omega, \eta \rangle &amp;=&amp; \omega \wedge (\star_p \eta)\\ &amp;=&amp;(A_{\sigma_1\ldots \sigma_p}dx^{\sigma_1}\wedge \ldots \wedge dx^{\sigma_p})\wedge (B_{\sigma_1\ldots \sigma_p} e^{\sigma_1\sigma_2\ldots \sigma_n}dx^{\sigma_{p+1}}\wedge \ldots \wedge dx^{\sigma_n})\\ &amp;=&amp; A_{\sigma_1\ldots \sigma_p}B_{\sigma_1\ldots \sigma_p}e^{\sigma_1\sigma_2\ldots \sigma_n}dx^{\sigma_1}\wedge \ldots \wedge dx^{\sigma_p}\wedge dx^{\sigma_{p+1}}\wedge \ldots \wedge dx^{\sigma_n}\\ &amp;=&amp;A_{\sigma_1\ldots \sigma_p}B_{\sigma_1\ldots \sigma_p}dx^1\wedge dx^2\ldots \wedge dx^n\\ &amp;=&amp;(B_{\sigma_1\ldots \sigma_p}dx^{\sigma_1}\wedge \ldots \wedge dx^{\sigma_p})\wedge (A_{\sigma_1\ldots \sigma_p} e^{\sigma_1\sigma_2\ldots \sigma_n}dx^{\sigma_{p+1}}\wedge \ldots \wedge dx^{\sigma_n})\\ &amp;=&amp;\eta \wedge (\star_p \omega)\\ &amp;=&amp;\langle \eta, \omega \rangle. \end{eqnarray}\qquad (12)\\ "> 3. If in <img src="https://www.zhihu.com/equation?tex=%2812%29" alt="[公式]" eeimg="1" data-formula="(12)"> , we set <img src="https://www.zhihu.com/equation?tex=A_%7B%5Csigma_1%5Cldots+%5Csigma_p%7D+%3D+B_%7B%5Csigma_1%5Cldots+%5Csigma_p%7D" alt="[公式]" eeimg="1" data-formula="A_{\sigma_1\ldots \sigma_p} = B_{\sigma_1\ldots \sigma_p}"> in order to equate <img src="https://www.zhihu.com/equation?tex=%5Comega+%3D+%5Ceta" alt="[公式]" eeimg="1" data-formula="\omega = \eta"> , we find that</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clangle%5Comega%2C+%5Comega%5Crangle+%3D+A_%7B%5Csigma_1%5Cldots+%5Csigma_p%7DA_%7B%5Csigma_1%5Cldots+%5Csigma_p%7D%5Cmathrm%7BVol%7D_%5Cmu+%3E+0%2C%5C%5C" alt="[公式]" eeimg="1" data-formula="\langle\omega, \omega\rangle = A_{\sigma_1\ldots \sigma_p}A_{\sigma_1\ldots \sigma_p}\mathrm{Vol}_\mu > 0,\\&#8221;> being the product of a positive volume and a sum of squares holds iff only <img src="https://www.zhihu.com/equation?tex=%5Comega+%5Cneq+0" alt="[公式]" eeimg="1" data-formula="\omega \neq 0"> , which shows positive definiteness.</p> <p>We note how</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clangle+dx%5E%7B%5Cmu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_p%7D%2C+dx%5E%7B%5Cnu_1%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cnu_p%7D+%5Crangle+%3D+%5Cpm+dx%5E1%5Cwedge+dx%5E2%5Cldots+%5Cwedge+dx%5En+%5C%5C" alt="[公式]" eeimg="1" data-formula="\langle dx^{\mu_1}\wedge \ldots \wedge dx^{\mu_p}, dx^{\nu_1}\wedge \ldots \wedge dx^{\nu_p} \rangle = \pm dx^1\wedge dx^2\ldots \wedge dx^n \\"> if and only if <img src="https://www.zhihu.com/equation?tex=%5C%7B%5Cmu_1%2C%5Cmu_2%5Cldots%2C+%5Cmu_p%5C%7D+%3D+%5C%7B%5Cnu_1%2C%5Cnu_2%5Cldots%2C+%5Cnu_p%5C%7D" alt="[公式]" eeimg="1" data-formula="\{\mu_1,\mu_2\ldots, \mu_p\} = \{\nu_1,\nu_2\ldots, \nu_p\}"> and that if such is not satisfied, the inner product is necessarily <img src="https://www.zhihu.com/equation?tex=0" alt="[公式]" eeimg="1" data-formula="0"> . Thus this inner product evaluates to <img src="https://www.zhihu.com/equation?tex=%5Cmathrm%7BVol%7D_%5Cmu" alt="[公式]" eeimg="1" data-formula="\mathrm{Vol}_\mu"> whenever both its inputs are the same basis vector. There is of course an inner product defined on the vector space spanned by <img src="https://www.zhihu.com/equation?tex=%5C%7Bdx%5E%5Cmu%5C%7D" alt="[公式]" eeimg="1" data-formula="\{dx^\mu\}"> , which is the inner product defined above for the case <img src="https://www.zhihu.com/equation?tex=p%3D1" alt="[公式]" eeimg="1" data-formula="p=1"> on<img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5E1%28V%29" alt="[公式]" eeimg="1" data-formula="\bigwedge^1(V)"> . We can use this to induce an inner product expressed in terms of the determinant for arbitrary <img src="https://www.zhihu.com/equation?tex=p" alt="[公式]" eeimg="1" data-formula="p"> . Treating <img src="https://www.zhihu.com/equation?tex=%5Cmathrm%7BVol%7D_%5Cmu" alt="[公式]" eeimg="1" data-formula="\mathrm{Vol}_\mu"> as a pre-set value representing an <img src="https://www.zhihu.com/equation?tex=n" alt="[公式]" eeimg="1" data-formula="n"> -dimensional volume corresponding to <img src="https://www.zhihu.com/equation?tex=n" alt="[公式]" eeimg="1" data-formula="n"> differentials. We can modify the inner product on <img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5E1%28V%29" alt="[公式]" eeimg="1" data-formula="\bigwedge^1(V)"> by changing to <img src="https://www.zhihu.com/equation?tex=dx%5E1%5Cwedge+dx%5E2%5Cwedge+%5Cldots+%5Cwedge+dx%5En+%5Csim+%28%5Cmathrm%7BVol%7D_%5Cmu%29%5E%7B1%2Fp%7D" alt="[公式]" eeimg="1" data-formula="dx^1\wedge dx^2\wedge \ldots \wedge dx^n \sim (\mathrm{Vol}_\mu)^{1/p}"> in that inner product, which we denote as <img src="https://www.zhihu.com/equation?tex=%5Clangle+%5Ccdot+%2C+%5Ccdot+%5Crangle_1" alt="[公式]" eeimg="1" data-formula="\langle \cdot , \cdot \rangle_1"> . Then, noting that the determinant of a diagonal matrix is the product of the diagonal entries, it is easy to observe that</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clangle+dx%5E%7B%5Cmu_1%7D%5Cwedge+dx%5E%7B%5Cmu_2%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_p%7D%2C+dx%5E%7B%5Cnu_1%7D%5Cwedge+dx%5E%7B%5Cnu_2%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cnu_p%7D%5Crangle+%3D+%5Cdet+%28%5Clangle+dx%5E%7B%5Cmu_i%7D%2C++dx%5E%7B%5Cnu_j%7D%5Crangle_1%29%2C+%5Cqquad+%2813%29%5C%5C" alt="[公式]" eeimg="1" data-formula="\langle dx^{\mu_1}\wedge dx^{\mu_2}\wedge \ldots \wedge dx^{\mu_p}, dx^{\nu_1}\wedge dx^{\nu_2}\wedge \ldots \wedge dx^{\nu_p}\rangle = \det (\langle dx^{\mu_i}, dx^{\nu_j}\rangle_1), \qquad (13)\\"> with <img src="https://www.zhihu.com/equation?tex=i%2Cj" alt="[公式]" eeimg="1" data-formula="i,j"> as the row and column indices of a <img src="https://www.zhihu.com/equation?tex=p%5Ctimes+p" alt="[公式]" eeimg="1" data-formula="p\times p"> matrix the elements of which are inner products. The determinant in <img src="https://www.zhihu.com/equation?tex=%2810%29" alt="[公式]" eeimg="1" data-formula="(10)"> is called the <i>Gram determinant</i> or <i>Gramian</i>. In this case <img src="https://www.zhihu.com/equation?tex=%5Clangle+dx%5Ei%2C+dx%5Ei+%5Crangle+%3D+%28%5Cmathrm%7BVol%7D_%5Cmu%29%5E%7B1%2Fp%7D" alt="[公式]" eeimg="1" data-formula="\langle dx^i, dx^i \rangle = (\mathrm{Vol}_\mu)^{1/p}"> for all <img src="https://www.zhihu.com/equation?tex=i" alt="[公式]" eeimg="1" data-formula="i"> , which results in</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clangle+dx%5E%7B%5Cmu_1%7D%5Cwedge+dx%5E%7B%5Cmu_2%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_p%7D%2C+dx%5E%7B%5Cmu_1%7D%5Cwedge+dx%5E%7B%5Cmu_2%7D%5Cwedge+%5Cldots+%5Cwedge+dx%5E%7B%5Cmu_p%7D%5Crangle+%3D+%5Cmathrm%7BVol%7D_%5Cmu%2C+%5Cqquad+%2814%29%5C%5C" alt="[公式]" eeimg="1" data-formula="\langle dx^{\mu_1}\wedge dx^{\mu_2}\wedge \ldots \wedge dx^{\mu_p}, dx^{\mu_1}\wedge dx^{\mu_2}\wedge \ldots \wedge dx^{\mu_p}\rangle = \mathrm{Vol}_\mu, \qquad (14)\\"> which is consistent with <img src="https://www.zhihu.com/equation?tex=%2812%29" alt="[公式]" eeimg="1" data-formula="(12)"> . This inner product applied to pairs of basis vectors yield <img src="https://www.zhihu.com/equation?tex=0" alt="[公式]" eeimg="1" data-formula="0"> , when the two inputs are not the same. Linearity naturally extends this inner product to arbitrary vectors in <img src="https://www.zhihu.com/equation?tex=%5Cbigwedge%5Ep%28V%29" alt="[公式]" eeimg="1" data-formula="\bigwedge^p(V)"> . We will leave to the reader to check that <img src="https://www.zhihu.com/equation?tex=%2813%29" alt="[公式]" eeimg="1" data-formula="(13)"> is well-defined, that is, for <img src="https://www.zhihu.com/equation?tex=%5C%7Bu_1%2Cu_2%2C%5Cldots%2C+u_p%2C+v_1%2C+v_2%2C%5Cldots+v_p%5C%7D+%5Csubset+%5Cbigwedge%5E1%28V%29" alt="[公式]" eeimg="1" data-formula="\{u_1,u_2,\ldots, u_p, v_1, v_2,\ldots v_p\} \subset \bigwedge^1(V)"> ,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clangle+u_1%5Cwedge+u_2%5Cwedge+%5Cldots+%5Cwedge+u_p%2C+v_1%5Cwedge+v_2%5Cwedge+%5Cldots+%5Cwedge+v_p%5Crangle+%3D+%5Cdet+%28%5Clangle+u_i%2C++v_j%5Crangle_1%29.+%5Cqquad+%2815%29%5C%5C" alt="[公式]" eeimg="1" data-formula="\langle u_1\wedge u_2\wedge \ldots \wedge u_p, v_1\wedge v_2\wedge \ldots \wedge v_p\rangle = \det (\langle u_i, v_j\rangle_1). \qquad (15)\\"> </p> <hr> <p>The above was written after reading section 6 on four-vectors in [2] and then <a href="https://link.zhihu.com/?target=http%3A//math.stanford.edu/~conrad/diffgeomPage/handouts/star.pdf" class=" wrap external" target="_blank" rel="noreferrer noopener">[1]</a>. [2] gave an explicit formula for the Hodge dual in the case of two forms in 4 dimensions without explicitly mentioned &#8220;Hodge&#8221; using completely antisymmetric tensor (or pseudotensor) of rank 4. I wrote about this pseudotensor at <a href="https://zhuanlan.zhihu.com/p/361583379" class="internal" rel="noreferrer">[3]</a>. As for Brian Conrad&#8217;s notes, they did not present an explicit formula for Hodge dual though in the examples section, it was noted tht computation of them is mostly a matter of being careful with signs. I do remember that one of examples was under the Minkowski inner product in which case one would need to make an appropriate adjustment for the signs. Prof Conrad first gave the inner product induced by the Gramian determinant and he proceeded to in pure mathematician formalism to define the Hodge dual as the function satisfying <img src="https://www.zhihu.com/equation?tex=%289%29" alt="[公式]" eeimg="1" data-formula="(9)"> , the well-defined-ness of which is guaranteed implicitly by the chain of isomorphisms.</p> <p>I was quite pleased that while typing this up, I did not look up any sources directly, aside from a few peeks at Prof Conrad&#8217;s notes, which were entirely meant to ensure more notational consistency, which would be more convenient if one refers to both his notes and mine. I used to believe pessimistically that differential forms and the Hodge dual might actually be somewhat beyond my level of g (general intelligence factor, or IQ). When reading about them, I could always passively follow and feel like it all makes sense, but I never was able to comfortably write or talk about them in detail independent from another text. Now, after having written this without difficulty, I am much more confident about my prospect of learning differential geometry and general relativity and my mathematical and intellectual ability in general.</p> <p><b>References</b></p> <ul> <li>[1] Professor Brian Conrad&#8217;s notes on the Hodge star operator: <a href="https://link.zhihu.com/?target=http%3A//math.stanford.edu/~conrad/diffgeomPage/handouts/star.pdf" class=" external" target="_blank" rel="noreferrer noopener"><span class="invisible">http://</span><span class="visible">math.stanford.edu/~conr</span><span class="invisible">ad/diffgeomPage/handouts/star.pdf</span><span class="ellipsis"></span></a></li> <li>[2] Landau Lifshitz classical theory of fields (I downloaded it on libgenesis)</li> <li>[3] <a href="https://zhuanlan.zhihu.com/p/361583379" class="internal" rel="noreferrer">gmachine1729:On the completely antisymmetric unit rank 4 tensor (or pseudotensor) over spacetime coordinates</a></li> </ul> </div>

On the completely antisymmetric unit rank 4 tensor (or pseudotensor) over spacetime coordinates

Originally published at 狗和留美者不得入内. You can comment here or there.

Over the past month, I re-derived much of special relativity and its relation to electromagnetism in four-vector form. I thought perhaps I was ready to move on to general relativity. I did learn a bit about it, including writing a piece on Christoffel symbols; however, from the combined experience of reading Sean Carroll’s lecture notes and L&L Classical Theory of Fields, surfing relevant Wikipedia pages, and trying to re-derive the theory of differential forms on manifolds on my own (with mixed success), it became somewhat clear that my foundation on tensors and differential forms is not solid enough. I was quickly a bit lost reading the first chapter on general relativity especially when they went into transformations of rank four tensors in relation to determinants that referred to section 6, the title of which is “four-vectors”. So I will, here, go through that section in detail, at least the parts of it which I do not already know pretty well.

The tensors [公式] are special in the sense that their components are the same in all coordinate systems (This was demonstrated via [公式] in [2]). We want that the completely antisymmetric unit tensor of fourth rank, [公式] , which changes sign under interchange of any pair of indices, has the same property. We set

[公式] Now, how do we lower all the indices?

[公式] In the above, we notice that because [公式] is diagonal, the only non-zero term in the sum across [公式] permutations is given by [公式] , thus three negatives and one positive gives a sign reversal ( [公式] ). Moreover,

[公式] We notice that for any [公式] ,

In the above, we viewed the contravariant rank four tensor [公式] as a linear map of a covariant 4-vector to a scalar, and this should hold in any coordinate system. If we Lorentz transform this four tensor, the result is

[公式] wherein we simply applied the same Lorentz transform to each of the indices. In [公式] , if [公式] , then we have

[公式] We note that the Lorentz group actually has four connected components (parity inversion flips the sign of the determinant, and one can then partition again corresponding to time reversal). When we speak of Lorentz group, we actually mean the proper orthochronous subgroup of the Lorentz group, which is its own connected component. Since interchanging two rows in a matrix results in a flip of sign of the determinant, the above result generalizes to [公式] .

Moreover, by [公式] of [3] we have

[公式] from which follows that

[公式] namely invariance under change of coordinates per some Lorentz rotation.

Let [公式] denote a parity inversion transformation, one which flips one or three of the coordinate axes. In this case, the values of

[公式] would be signed flipped relative to [公式] and thus if [公式] were strictly a tensor, it would not be the same in all coordinate systems which preserve the Minkowski metric. If we enforce that condition already shown impossible for a tensor, then [公式] is a pseudotensor, which is a quantity that behaves the same in all coordinate transformations not reducible to rotations.

The products [公式] form a four-tensor of rank [公式] . One observes how in the case of parity inversion, it is done twice here, thereby no change. Relatedly, for any permutation [公式] ,

[公式] We now determine the value of [公式] . We note that because [公式] , [公式] . In doing so, we first take [公式] to be some permutation of [公式]. As for [公式] , for the corresponding tensor to be non-zero, it must be a permutation. The sign of the value is determined by the relative sign of [公式] with respect to [公式] . This feature is provided to us by the permutation based definition of determinant. We have accordingly

[公式]

Contracting the last index, we get

[公式] We note that in the above, [公式] and if the two sets of three indices (one covariant, one contravariant in the [公式] ) are not equal, we must have some row all zero and some column all zero, thereby the minor determinant must also be zero, thus the reduction. Moreover, we did not explicitly write [公式] to emphasize that we are summing across all [公式] . However, for fixed [公式] , there is only only one value of [公式] that yields a non-zero result.

We next set [公式] and compute [公式] . We notice that for this tensor to be non-zero, [公式] is a must. Upon fixing [公式] , there are two choices for the value of [公式] . Thus,

[公式]

Similarly, we obtain

[公式] and [公式] upon further contraction.

We furthermore observe that

[公式] where [公式] is the determinant formed from the quantities [公式] .

The fact that the components of the four-tensor [公式] are unchanged under rotations of a four-dimensional coordinate system, and that the components of the three-tensor [公式] are unchanged by rotations of the space axes are special cases of a general rule: any completely antisymmetric tensor of rank equal to the number of dimensions of the space in which it is defined is invariant under rotations of the coordinate system in the space. This general rule holds because:

  • Antisymmetric implies zero if the indices are not all equal, restricting to permutations.
  • Antisymmetric with respect to permutation swap implies a single absolute value for all non-zero entries.
  • Application of this rank [公式] antisymmetric tensor to [公式] th tensor power of any rotation results in the permutation based formula for determinant to be applied to the rotation matrix, which has determinant [公式] .

References

A note regarding the inverse of a Lorentz transformation and its representation in Einstein notation

Originally published at 狗和留美者不得入内. You can comment here or there.

I can remember the velocity addition formula corresponding to Lorentz transformations along a single dimension, [公式] , off the top of my head. It might mistakenly give the impression that the Lorentz group in general is abelian. It is not for four dimension. A Lorentz boost along one direction and one along another direction do not commute. The Lorentz boost matrix is given by

[公式]

Moreover, [公式] tells us that to obtain the inverse, we flip the sign of the entries in the boost matrix such that one index corresponds to time coordinate and the other index corresponds to a space coordinate. This is given formulaically by

[公式] wherein [公式] is the matrix corresponding to Minkowski metric. In Einstein notation, we denote [公式] with [公式] and raising the lower index and then lowering the upper index in it gives

[公式] Raising an index corresponds to applying Minkowski matrix from the left, in which case for each entry in the result, we iterate along the entries of some column of the Lorentz matrix, which is along the row index. Lowering an index then corresponds to applying Minkowski matrix from the right, which demonstrates the equivalence of [公式] and [公式], with [公式] . [公式] equates to[公式]which if appearing in expression involving Lorentz transformations of tensors of higher rank conveniently cancels out to the identity matrix. Letting [公式] , we obtain

[公式] where [公式] is the matrix corresponding to the transform of the contravariant four-vector for a given Lorentz transformation.

An intuitive, visual explanation of gravitational redshift

Originally published at 狗和留美者不得入内. You can comment here or there.

I was enticed to learn a bit general relativity after more awareness of the likes of the LIGO detector and gravitational waves, as well as of how astrophysics and cosmology has advanced much more than high energy or particle physics over the past four decades. Moreover, I learned about what appears to be Soviet pre-eminence in astrophysics and cosmology, fields wherein the likes of Sakharov, Zeldovich, Starobinsky, and Linde made seminal contributions. In fact, it seems like the Big Bang Theory is considerably more Soviet than Anglo-American despite the hype of Stephen Hawking and that idiotic TV series with Sheldon Cooper as protagonist.

I never would have considered general relativity (GR) to be worth the time investment, much because I am by the standard of a guy into theoretical physics rather practical and pragmatic actually. But this might actually be a good reason why I would choose general relativity over high energy physics. After all, the LIGO detector as well as detectors of cosmic microwave background radiation which were realized in the 60s are significantly cheaper than the post-80s particle accelerators.

As for deciding what textbook to use, I first downloaded Landau-Lifshitz classical theory of fields. It did not cover very directly or concretely the gravitional redshift and more or less dove straight into the formalism that is heavy on tensors and Riemannian geometry. I wanted to begin by first learning the most concrete ideas behind GR before diving into the mathematics. So in addition to some scattered sources on the internet, I thought of Gerard ‘t Hooft and his page on how to become a good theoretical physicist. From his lecture notes [1], I learned about Rindler coordinates which describe accelerating frames of reference. The way it was presented was more formal and in terms of infinitesimal changes of the Lorentz transform over infinitesimal changes of time. I was able to more or less follow it with a bit of thinking.

Then I looked at Sean M. Carroll’s lecture notes [2], wherein it was explained more visually with diagrams of accelerating boxes, with reference to “Doppler shift”. After reading the Wikipedia page on relativistic Doppler shift, gravitational redshift made much sense. It was from that that I grasped the idea behind inertial references frames local to an infinitesimal volume of spacetime. Before I had realized that taking an inertial reference frame that matches the instantaneous speed of an object at some point of time to model it was a common theme in relativity. It is in some sense obvious that over a [公式] , in that frame, the object’s displacement of [公式] , being second-order, is often negligible.

Understanding of gravitational redshift, or equivalently by Einstein equivalence principle, enlargement of wavelength of light emitted in an accelerating frame in the direction of acceleration requires a correct or appropriate modeling of wavelength. A plane wave has a wavefront, which is some extreme point of the wave in space that is attained one per the period of the wave at the same point on the axis of propagation of the wave. We consider an accelerating boxes as below.

As for the bottom box’s emitting a photon of wavelength [公式] we regard it as emitting a wavefront at [公式] and then the next wavefront at [公式] . As for the detector on the upper box, it detects the wavelength by measuring with its clock the difference in time of arrival of the two wavefronts and multipling it by the speed of light [公式] .

We choose the inertial reference frame [公式] at point of emission the first wavefront such that the lower box is instantaneously at rest. We note that [公式] is necessarily extremely small. The wavelengths of radio waves, the electromagnetic waves with the highest wavelengths, range from [公式] to [公式] . The speed of light is about [公式] . Even the highest wavelength radio wave has a period of only about [公式] . The displacement in that case if the accelerating boxes simulate earth’s gravity would be as small as [公式] , which we can assume to be negligible compared to the distance between the boxes [公式] . Thus we simply assume the next wavefront to be emitted from the same point in frame [公式] .

In frame [公式] , the initial wavefront reaches the top box at the time [公式] such that [公式] . If [公式] as we assume, [公式] is more or less an infinitesimal change and we again ignore the second order term, which gives [公式] . By then, the upper box has accumulated a velocity of [公式] and thus the next wavefront being [公式] behind it takes [公式] amount of time to reach it, according to the clock of frame [公式] . Due to time dilation though, the clock at the upper box would measure it as

[公式]

Assuming [公式] , we neglect the [公式] which is a second order term of [公式] , which gives the wavelength redshift factor as [公式] , which is linear in the distance between the boxes (or the height of the elevator simulating gravity). One can interpret this of course also as the redshift rate with respect to distance.

As illustrated from the above diagram, a photon of a given wavelength emitted from upward the ground will also be detected to be redshifted on a sufficiently high tower. This was demonstrated by the Pound-Rebka experiment in 1960.

Recall that by the equivalence principle, a person standing on the ground feels the same as a person standing on a box in free space accelerating at [公式] relative to a true inertial reference frame. Note how the ground on earth is not actually an inertial reference frame due to gravity though it can be regarded as one in the two dimensions orthogonal to the line between the ground the certain of the earth. In contrast, a skydiver in free fall would be in an inertial reference frame. He is, if you neglect the effects of air resistance, accelerating at [公式] towards the center of the earth and if he throws a ball upwards during his sky dive, he will view it as moving upwards with no change in velocity, as Newton’s first law would predict.

If one has difficulty visualizing the diagram with the tower, one can an extremely deep pit dug in the vicinity of tower and a person jumping into it at the time of emission of the photon in which case the ground and the top of the tower as both accelerating upwards with magnitude [公式] . That person in free fall would be in an inertial frame of reference analogous to [公式] in the case of the accelerating boxes.

We conclude with a few notes.

  • Simulation of gravity of a planet per an elevator of constant acceleration is only valid if the distance traversed radially along the planet is negligible as far as the change in gravity is concerned.
  • Clocks tick faster at places where the strength of gravity is weaker. In the case of an accelerating elevator [公式] in one direction, one can place another elevator [公式] accelerating in the same direction with a different (positive) magnitude that begins with the same speed relative to an inertial reference frame [公式] . If [公式] accelerates faster, then its speed after an infinitesimal time change is higher, and thus the factor of time dilation of [公式] relative to that of the clock at the top of [公式] would be higher than the factor of time dilation of [公式] relative to that of the clock at the top of [公式] , since [公式] has attained a higher velocity. Thus the clock corresponding to stronger gravity ticks slower. A clock ticking on Mount Everest or on an airplane would tick faster than a clock on the ground. This phenomenon is called gravitational time dilation.
  • We deduced in this article gravitational redshift. Gravitational blueshift, which corresponds to a decrease in a wavelength upon emission of a photon from a tower to the ground, can be deduced analogously.

References

I agree with Duke of Qin that Asian Americans are disgusting slavish cucks who deserve to be discrim

Originally published at 狗和留美者不得入内. You can comment here or there.

Asian Americans are disgusting slavish cucks and they deserve to be discriminated against. Tons of talented and wealthy individuals and the only organized activism they can manage is to secure token slots for some sluts in Hollywood. They get shat on by Harvard and the Ivy’s and their response is to kiss their feet and give them billions in donations. If that isn’t fucking cuck behavior, nothing else is. Instead of padding Harvard’s investments, they could pool their money to start their own damn elite schools where they aren’t gas lit by the fags in the admissions office. Seriously stop giving Stanford, Harvard, Princeton, Yale any money and start a new school in some safe cheap location in the Midwest and staff it an all star faculty paid dirty tech money and provide guaranteed employment for graduates at your companies. It would overnight become a go to recruiting location for tech giants, allow young adults to start from secure economic footing and start early families. On a local level, can do impromptu private schools to keep kids out of the garbage public schools instead of surrending to insane local educational bureaucrats determined to fucking torture your children and the wealthier parents could provide financial aid to the poorer working class parents and give them a safe quality education at a fraction of the cost of some fancy White private school. The role of the state in the West is descending into anarcho-tyranny and is totally useless to protect your interests so you need to bypass it and forge alternate community institutions where you can organize for collective benefit and protection. This is good advice for everyone really, and not just Asians alone. As Western state-society becomes more dysfunctional overall, functional people determined to not get dragged down with it will need to rescind any loyalty to it’s institutions in lieu of institutions under their control that represent their interests.


我对此的看法是:

绝大多美国亚裔不是家在亚洲无背景,就是政治败者,如国民党后代和逃到港台的有钱人。是有不少才理工科高端人才和一些商人,尤其在科技领域,但是那帮人的政治组织能力和对社会的认知实在是搞笑的差。当然,中产读书人和买办亡国奴的后代混合而成的畸形无国家的群体出现这种现象大多也是理所当然的。

我的翻译:

美国亚裔是一帮恶心的奴才,遭到歧视活该。那么多有才华的或有钱的人,但他们能实现的群体政治组织限于为给几个荡妇在好莱坞安排几个少数民族位置。他们遭到哈佛和常春藤贬低拒绝,反应确是跪舔以及上亿的捐款。如果这不算奴才的表现,那没有什么别的算了。而非附加哈佛的投资,他们可以把钱合起来创办自己的没录取办公室王八蛋审核他们的精英学校。真的,停止给斯坦福,哈佛,普林斯顿,耶鲁任何钱,在美国中部某个安全便宜的地方创办新的学校,用互联网公司的钱资助一群明星教授并给毕业生提供在你们公司的职业机会的保证。它不久会成为巨头公司的招聘地点,并使得年轻人从一个安稳的经济基础开始,早年成家。在地区级别,可以搞一些的私立学校让孩子逃避垃圾公立学校,而非投降给无头脑坚决要害你们孩子的教育官僚,更有钱的父母可以给穷工人阶层孩子提供些资助,以豪华白人私立学校学费的几分之一给他们提供既安全又质量的教育。

Memories of racist white American exceptionalist teachers I had in primary and secondary school

Originally published at 狗和留美者不得入内. You can comment here or there.

In response to the following comments at https://infoproc.blogspot.com/2021/03/academic-freedom-alliance.html:


A map documenting Austronesian historical migration.

Looks like they very likely reached Chile from Easter Islands, given the distance they traveled to Hawaii from the nearest island is around the distance from Easter Islands to Chile. I vaguely remember reading some DNA evidence for that too. As for Columbus, one can say that Vikings/Erik the Red/Leif Erikson predated him by several centuries. Columbus mattered way more of course the Vikings didn’t really do anything in the Americas and the overall impact of their explorations, as with Zheng He’s, marginal in comparison.

Having re-read about Columbus, Vasco de Gama, and other explorers of that age, I realize there was some luck aspect to it as well. The route to India across the tip of Africa by Vasco de Gama was much catalyzed by the severing of trade route by Ottomans from 1450s on. Columbus in some sense was close to not receiving funding for his expedition, as it was done after the Africa route had already been discovered; I believe he was funded mostly out of fear that he would employ his talents and skills for a competitor nation instead. The discovery of Americas by Columbus was an accident, and I remember reading that he misestimated the distance to Asia along that other direction.

One year in grade school, each student in the class had to do an “explorer project”, where one would write about an explorer. I remember the teacher was a female white American exceptionalist, who would evoke the negative “German/Nazi” stereotype and also go,

The Japanese also wanted to take over other countries. It’s called “imperialism”.

That year, we learned a fair bit about Native Americans too. I remember this (white) girl in the class basically cried after learning about their dispossession by those “evil powerful white men”. That teacher was literally so American exceptionalist that had anyone picked Yuri Gagarin for his explorer project (and some kid did pick Neil Armstrong), she may well have even objected. Nobody picked Genghis Khan or Zheng He either; that might also have been somewhat inappropriate. xD

Someone else is that last year, I read a bit about the Russian explorers who conquered Siberia between 1500-1750. Nobody in America knows about them either. Speaking of which, I’m sure the Russians regret selling Alaska a lot, especially since it would have been an ideal Cold War base.

White European supremacy and American exceptionalism coupled with a worship of blacks in some sense was already being drilled down our throats in grade school. That teacher spoke of how though blacks used to be discriminated against in sports, the best home run hitters in baseball are pretty much all black, and how they were simply too good. On the other hand, some white American teachers visibly treated me, a PRC immigrant kid, with a prejudiced attitude. They didn’t say it very openly of course but it was more or less obvious how they felt inside based on their decisions and tone of voice. I could tell they especially disliked East Asian scientists or intellectuals. When I once mentioned Qian Xuesen to a history teacher in high school, he/she basically expressed outrage that Qian was allowed to play such a major role in American rocket development. Of course, the truth is that the cognitive elite from China, Japan, and Korea that came to America between 1945-1980 made enormous contribution to science and technology for America, and post-1980 you had another wave much from PRC. They were overall under-recognized, especially by the media, for their contributions. America would be less competitive economically and technologically if not for such far tail brain drain. Theoretical physics post-war in America was very much a Jewish and East Asian affair. Yet, I remember Stephen Hawking in A Brief History of Time explicitly regarded Chen-Ning Yang and Tsung-Dao Lee as American physicists even though they held Chinese citizenship up through at least 1960, when they did their most groundbreaking work. About half the top scientists in America are foreign born, and while it’s okay to label a white European immigrant as American, I don’t think it’s okay to do so for an East Asian. Something else is that the international math, physics, and computing olympiad teams now in America are mostly Asian, but let’s just say the general attitude towards that is very different from the general attitude towards the NFL/NBA/MLB composition. We can call it an Asian PR failure in America.

Predicting International Olympiad in Informatics (IOI) results from contestant photos

Originally published at 狗和留美者不得入内. You can comment here or there.

The 0.3-0.4 correlation between brain size and IQ is well known among the neuroscience community. I had also long realized that brain size could be very crudely approximated from photos. It had occurred to me that given photo data labeled with IQ or some proxy for it, one would likely find a statistically significant correlation. After looking at International Olympiad in Informatics (which selects 4 high schoolers each year from each country) results on the official site, which includes photos of the contestants, it occurred to me that I could do a data science project based on this.

I initially hesitated due to the controversy that such a project might generate, but after seeing that the face recognition service of this Chinese company even estimates “attractiveness”, which like it or not, does objectively exist in a statistical sense, I decided it would be okay. Besides that, two non-Chinese friends of mine had both showed me similar services of non Chinese origin: both based on photo and face recognition, one was an ethnicity estimator and the other estimated attractiveness, IQ, etc.

I wrote code to scrape the data, along with code using perhaps the most popular open source face recognition library in Python and OpenCV to pick out locations of top of head, eyebrows, and chin and then vertically align the image per a calibrated standard. Face recognition and especially top of head recognition by a relative computer vision noob like myself was imperfect, so I also made some manual adjustments. The bad photos were excluded, and those with an excess of hair were also manually tuned in their feature parameters, or excluded in some cases. And it turned out that

[公式]

a very crude estimate for brain size correlated [公式] with percentile represented via Z score on IOI at sample size almost [公式] . The p-value of this was less than [公式] , which is pretty statistically significant. Though a small correlation, it is much higher than the IQ correlation between unrelated children reared together, which is only [公式] at adulthood. And midway though the project, I realized that eyes would very likely make a better predictor than eyebrows, given that there is actually quite some variation in eyebrow to eye distance among the entire human population. I would not be surprised the use of eyes in place of eyebrows yielded a correlation higher than [公式] .

I won’t release the code and data publicly at this point (and might never will) though it can be requested via email to gmachine1729 at foxmail.com. I know/knew personally some IOI contestants myself and would also be interested in feedback from them on this. I will release publicly here a few graphs.

I told a biologist about this project too, and he was somewhat to my relief not surprised at all, regarding the genes, shared-environment, non-shared environment stuff to be pretty basic knowledge for genetics research, and also the correlation between brain size and intelligence to be well known. To be fair, “big brains” as a metaphor for smarts I have heard used multiple times in America too. This is not observed only across people but also in each individual person, like there is a reason why many things I found difficult at age 20 I now find pretty routine. And I also am like Professor Robert Plomin in accordance with the idea that most parents overestimate the effect of what they did for the education of their children.