gr01 Gravity and Gauge Theory S Weinstein differmorphism 三維中的二維拉來拉去，看長得什麼樣 拓樸流形是局部與歐氏空間 induce the same gauge transformation.

https://uwaterloo.ca/philosophy/people-profiles/steven-Weinstein
gauge-invariance (or for that matter diffeomorphism-invariance) as a property of the equations of motion. I.e., a theory is gauge- (diffeomorphism-) invariant if a solution of the equations of motion, when acted on by a gauge transformation (diffeomorphism) yields another solution

Saturday, August 15, 2015

"低速情况下坐标时固有时", wolfking97 常的欧式平面，是个平直的空间。现在我考虑以原点为圆心的单位圆。什么是上面的诱导度规呢？就是普通的弧长。诱导度规是个局部的概念，是局部用切线来逼近子流形上的曲线来定义长度，或者用形象的语言就是局部把曲线拉直了来量长度

Re: [分析] 想問微分幾何的問題- 看板Math - 批踢踢實業坊

https://www.ptt.cc/bbs/Math/M.1316517842.A.194.html

• 頁庫存檔

2011年9月20日 - 14 篇文章 - ‎5 位作者

如果要講diffeomorphism中文應該完整的表達微分同胚且不能省略"微分"。微分流形的範疇與拓樸空間的範疇有很大的差異，通常拓樸空間並不具有 ...

Special Relativity - Google Books Result

https://books.google.com/books?isbn=1447100832

N.M.J. Woodhouse - 2012 - ‎Mathematics

with the energy density of the electromagnetic field and to identify the vector with the energy flux (it is called the Poynting vector). Then we can interpretour ...

PDF]Gravity and Gauge Theory∗ - PhilSci-Archive - University of ...

philsci-archive.pitt.edu/834/1/gr_gauge.pdf

• Cached
• Similar

by S Weinstein - ‎1998 - ‎Cited by 24 - ‎Related articles

General relativity is invariant under transformations of the diffeomorphism group. .... local gauge transformations by operating on a single particle wavefunction ...

[PDF]Symmetry Transformations, the Einstein-Hilbert Action ... - MIT

web.mit.edu/edbert/GR/gr5.pdf

• Cached
• Similar

Massachusetts Institute of Technology

Loading...

stand gauge symmetry and the parallels between gravity, electromagnetism, and .... is always taken to be a scalar in order to ensure local Lorentz invariance (no.

[PDF]Local Gauge Invariance - science.uu.nl project csg

www.staff.science.uu.nl/~wit00103/ftip/Ch11.pdf

• Cached
• Similar

Utrecht University

Loading...

by a gauge theory, as the theory of general relativity is invariant under con- ... in view of the fact that the invariance under local gauge transformations still.

[PDF]Gravity and Gauge Theory - Philosophy of Science ...

www.philsci.org/.../weinstein.pdf

• Cached
• Similar

Philosophy of Science Association

Loading...

by S Weinstein - ‎1998 - ‎Cited by 24 - ‎Related articles

Jul 31, 1998 - der a group of local transformations, i.e., transformations Which may vary ... that general relativity is not a gauge theory at all, in the specific ...

Gauge invariance - Scholarpedia

www.scholarpedia.org/article/Gauge_invariance

• Cached
• Similar

Scholarpedia

Loading...

Jump to General relativity - Einstein's relativistic theory of gravitation, also known as General ... A gauge transformation corresponds to a change of local ...

[PDF]Noether's Theorems and Gauge Symmetries

arxiv.org/pdf/hep-th/0009058

• Cached
• Similar

arXiv

Loading...

by K Brading - ‎2000 - ‎Cited by 15 - ‎Related articles

Sep 8, 2000 - invariant under global and/or local gauge transformations leads to ..... here for local gauge symmetry have analogues in General Relativity,.

Chemistry, Quantum Mechanics and Reductionism: ...

https://books.google.com/books?isbn=3662113147

H. Primas - 2013 - ‎Science

The requirement of invariance under local phase transformations ... special relativity while local gauge invariance is analogous to Einsteinian general relativity.

想問微分幾何的問題

時間Tue Sep 20 19:23:59 2011

※ 引述《moon0815 (阿呆￾ )》之銘言： : 我想請問一下 : 微分幾何裡面的同構 同胚 : 代表的是什麼意思呢? : 我看了書 但是不太能領會他的意思... : 希望高手能解釋 提點一下 : 謝謝各位大大 Two topological spaces are homeomorphic (同胚) if there exists a continuous bijection between them whose inverse is also continuous。 假如你有兩個拓樸空間X與Y，如果你可以找到一個X與Y之間的一一對應f並且 f與f^-1均是連續函數，則稱X與Y同胚。同胚在拓樸空間的意義下指的是兩個 拓樸空間可以視為一樣的，儘管兩個看起來(在幾何上)是很不一樣的東西。 舉例來說，橢圓形跟圓形幾何上看起來是不相同，但以拓樸空間這個範疇來 說，可以視為同樣(同胚)的拓樸空間。 因為f與f^-1均是連續，如果U是X中的開集合，那麼f(U)=V會是Y中的開集合。 反之，如果V是Y中的開集合f^-1(V)會是U中的開集合。假如我令Top(X)， Top(Y) 分別表示X與Y中的拓樸。那麼利用f與f^-1我們可以建構出 Top(X) <-> Top(Y) 一個一一對應的關係。也就是說同胚的拓樸空間，你看不出來他們拓樸之間的 差別在哪。 同構的話看你指的是哪一種同構關係。 在任何的範疇(Category)中，同構(isomorphic)的物件(objects)指的是具有 相同結構的。在拓樸空間的範疇裡，同構等同於同胚。在微分流形的範疇裡，同 構是微分同胚。在代數裡，群環體有各自同構的概念。例如說f:G-> G'是群同構，指 的是f保持群運算，並且是一個G與G'間的一一對應(one-to-one correspondence)。 觀於範疇相關的概念可以查閱維基網站 http://0rz.tw/TaadM -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 195.37.209.182

→ sleep123 :這邊定義C^0，有些要求到C^1 09/20 20:12

推 jacky7987 :C^1是differmorphism 為了保持流形的好XD 09/20 20:20

→ jacky7987 : (吧 09/20 20:21

→ sleep123 :樓上正解 不過我不知道diffeomorphism中文是什麼 09/20 20:44

微分同胚 通常講同胚的話單純指的是拓樸那個層級。如果要講diffeomorphism中文應該完整的表達 微分同胚且不能省略"微分"。微分流形的範疇與拓樸空間的範疇有很大的差異，通常拓樸 空間並不具有微分結構的。所以當他提同胚的時候，我只會回答，拓樸空間的範疇。

→ sleep123 :初學微分幾何通常不會特別講manifold 09/20 20:45

如果是古典的微分幾何會從曲線曲面開始談。如果是學微分流形的，有些書會從 拓樸流形(topological manifold)開始，由此就會從拓樸流形是局部與歐氏空間 同胚的定義開始。

→ sleep123 :大抵上就三維中的二維拉來拉去，看長得什麼樣子 09/20 20:46

推 WINDHEAD :翔爸翔爸翔爸翔爸翔爸翔爸翔爸翔爸翔爸翔爸翔爸翔爸 09/20 20:48

推 xgcj :YA! 09/20 22:07

※ 編輯: herstein 來自: 195.37.209.182 (09/20 23:39)

推 Lindemann :推很詳細那可以請大大補充一下為何後來需Categor概念 09/21 03:44

推 Lindemann :我的問題是是否這個概念把所有的數學態射都簡化出來? 09/21 03:55

推 Lindemann :應該是說範疇所想定義的大學就一直再用了何必用這詞? 09/21 04:01

→ Lindemann :因為我覺得通常不是念代數幾何看到這類字也會有點怕? 09/21 04:02

推 Lindemann :homeomorphism跟diffeomorphism天差地遠吧Milnor怪

The gauge group versus the diffeomorphism group of a manifold

up vote1down votefavorite 2

Let M be an m dimensional differentiable manifold. Define Gauge(M):=C^{\infty}(M, Aut(TM)) to be the group of all (smooth) fiberwise linear transformations of the tangent bundle. This is the natural gauge group of the manifold. If (U, x_1,...,x_m) is a local coordinate system with induced frame on TU then an element of Gauge(U) looks like an invetable matrix g_{ij}(x_1,...,x_m) (with i,j=1,...,m) depending smoothly on the point. If we take a diffeomorphism of M interpreted as a coordinate transformation i.e., taking (U,x_1,...,x_m) into (U,y_1,...,y_m) with y_i(x_1,...,x_m) (with i=1,...,m) smooth functions then the corresponding Jacobi matrix gives rise to an element of Gauge(U) by putting locally g_{ij}(x_1,...,x_m):=dy_i/dx_j. Hence among gauge transformations there are those which stem from a diffeomorphism hence we get a natural embedding Diff(M) < Gauge(M). The question is: (after appropriate topologies considered) can we say something about the quotient Gauge(M)/Diff(M) i.e., in what extent is the gauge group "bigger" than the diffeomorhism group of a manifold? I would expect that the answer splits into a local answer and then a global one (involving the topology of M). The motivation comes from Kodaira-Spencer deformation theory of complex structures. In this theory two almost complex operators are considered to be equivalent if they differ by a diffeomorphism. However apparently gauge equivalence would be also a natural equivalence relation. Is this beacause simply Kodaira-Spencer theory historically preceded gauge theory? Thanks! dg.differential-geometry share|cite|improve this question edited Sep 20 '10 at 12:10 asked Sep 20 '10 at 11:14 E von Tuzzenthaler 152

8   The diffeomorphism group is not a subgroup of the gauge group, because a diffeomorphism f induces maps T x M→T f(x) M  , rather than from T x M  to itself. In other words, Df is not a map of bundles over X  . – Lucas Culler Sep 20 '10 at 12:05 4   Something seems a little odd about your map from Diff(M) to Gauge(M). An element of Diff(M) defines an isomorphism T_xM -> T_yM (where x -> y) but an element of Gauge(M) can only define an isomorphism T_xM -> T_xM. – Loop Space Sep 20 '10 at 12:09 1   The Diff(M) group can be viewed either in an "active" way carrying point x to y or in a passive way changing the coordinate system about a point (the group of coordinate transformations). I use this second picture. – E von Tuzzenthaler Sep 20 '10 at 13:48 1   Even if you work in coordinates, as you do, observe that your map which associates to a diffeomorphism a gauge transformation is not injective. For example the identity and the shift x↦x+1  on R  induce the same gauge transformation. – Michael Bächtold Sep 20 '10 at 17:27 1   Repeating what Andrew Stacey and Lucas Culler have said in more physics-y language: the Jacobi matrix does not transform as a tensor. So it does not define a section of GL(TM). As a trivial example, let M be the disjoint union of two lines. Pick a coordinate x on one of the lines and a coordinate y on the other one. Then there is a diffeomorphism of the form y(x) = x, x(y) = y. The Jacobi matrix near x=0 is 1 in these coordinates. But under the change of coordinates Y = Y(y), which does not change the x coordinates at all, the Jacobi matrix near x=0 changes to Y'(x). – Theo Johnson-Freyd Sep 20 '10 at 18:03  |  show 1 more comment

1 Answer 1

activeoldestvotes

up vote2down vote

What you are trying to express, is the following, imho. For the sake of clarity let us split M  into two manifolds, M  , N  . Consider the 1-jet bundle π M×N :J 1 (M,N)→M×N  , which is bundle isomorphic to L(TM,TN)  . Given smooth f:M→N  , we get the 1-jet section j 1 f:M→J 1 (M,N)  of π M :J 1 (M,N)→M  which satisfies π N ∘j 1 f=f:M→N  . Now your question is: Given a section s:M→J 1 (M,N)  of π M :J 1 (M,N)→M  , can you recognize when s=j 1 (π N ∘s)  . Answer: In fact you can. There is a module (over C ∞ (M)  ) of canonical 1-forms (called contact forms or Lepage forms) on J 1 (M,N)  , (edited) locally generated by dy j −k j i dx i   in terms of coordinates (x i ,y j ,k j i )  on J 1 (M,N)  induced by coordinates (x i )  on M  and (y j )  on N  . We have s=j 1 (π N ∘s)  if and only if s ∗ ω=0  for each contact form. See Wikipedia. Note that the gauge group Gau(M)  acts from the right on J 1 (M,N)  , and Gau(N)  acts from the left. share|cite|improve this answer edited Dec 12 '14 at 6:49 answered Dec 11 '14 at 21:41 Peter Michor 15.1k13162

Just noticed that your condition, π ∗ M ω=0  , on contact forms can't be correct (the arrows point the wrong way). I think what you wanted was to characterize contact forms in a way that easier to check than the defining condition s ∗ ω=0  for all s=j 1 f  . Perhaps the quickest way to do that is to use adapted coordinates on J 1 (M,N)  , say (x i ,y j ,k j i )  . Then, contact forms are all those that are locally generated by the forms dy j −k j i dx i   , as you well know of course. – Igor Khavkine Dec 11 '14 at 23:47 回复：【探讨】相对论中的“刚体”_相对论吧_百度贴吧 tieba.baidu.com/p/1810706956?pn=2 頁庫存檔 轉為繁體網頁 此时大家都意识到问题的关键是对做加速运动的物体如何定义刚性。就是说 ... 波恩的刚性要求就是，这个诱导度规关于固有时这个参量的导数处处为零。 好了说了一 ... phymath999: 这种诱导度规其实就是直接指向随动系宁信度 ... phymath999.blogspot.com/2015/08/blog-post_69.html 頁庫存檔 轉為繁體網頁 2015年8月13日 - 这种诱导度规其实就是直接指向随动系宁信度，无自信“固有时”作为量度物理进程的参量，这个概念只能针对质点或物质场定义。 时空流形是唯一的， ... [PDF]（＝） 2．7 attach3.bdwm.net/attach/boards/PhysicsReview/M.../gx.pdf 頁庫存檔 轉為繁體網頁 由 高思杰 著作 - ‎1997 - ‎被引用 1 次 - ‎相關文章 在相对论中，观者是一条以固有时为参数的类时曲线，参考系是一个类时线汇，其中每. 一. 类时线代表 ... 时空度规g 在三上的诱导度规h 就描述该三维空间的几何．然. 关于时钟佯谬 - 卢昌海个人主页 www.changhai.org/articles/science/.../clock_paradox.php 頁庫存檔 類似內容 轉為繁體網頁 2011年5月15日 - 正是这种度规改变抵消了“曲”、 “直” 互换的影响， 使得长度不变， 从而保证了时钟佯 .... 对于类时曲线来说， 则常被称为“原时” 或“固有时” (proper time)。 .... 是等效原理 的体现)， 其度规则是可以局部地由Minkowski 度规诱导出来的。 [PDF]刚性参考系与转盘几何* www.paper.edu.cn/.../journal-0476-0301(1997)02-0211-0... 頁庫存檔 轉為繁體網頁 在相对论中，观者是一条以固有时为参数的类时曲线，参考系必是一个类时线汇，其中每. 一类时线 ... 时空度规gab 在Z 上的诱导度规hab 就描述该三维空间的几何.然. [PDF]Beltrami_de Sitter时空和de Sitter不变的狭义相对论 metc.gdut.edu.cn/ddlx/xgzs/xslw/xyxdllw/1.pdf 頁庫存檔 轉為繁體網頁 由 郭汉英 著作 - ‎2005 - ‎被引用 11 次 - ‎相關文章 把狭义相对性原理推广到非零常曲率时空,在具有Beltrami 度规的de SitterΠ反de Sitter 时空中建立 ... 除了Beltrami 坐标时同时性之外,对于共动观测, 还可以取固有时同时性;此时,Beltrami 度规成为 ...... 延迟函数、位移矢量和在3 维超曲面∑c 给出诱导. 讲座信息-广东天文学会 www.gdtw.org.cn/NewsDetail.aspx?code=0906&id=757 頁庫存檔 轉為繁體網頁 2012年6月12日 - 1.3.1 世界线与时空图，1.3.2 观者和固有时，1.3.3 参考系和坐标 ... 的诱导度规；超曲面上的投影映射；微分形式及其外微分；适配体元；诱导体元；对偶 ... [PDF]Beltrami-de Sitter 时空和de Sitter 不变的狭义相对论! - 物理学报 wulixb.iphy.ac.cn/EN/article/downloadArticleFile.do?... 頁庫存檔 轉為繁體網頁 由 G Han-Ying 著作 - ‎2005 - ‎被引用 37 次 - ‎相關文章 2005年6月6日 - 把狭义相对性原理推广到非零常曲率时空，在具有Beltrami 度规的de Sitter/反de ... 除了Beltrami 坐标时同时性之外，对于共动观测，还可以取固有时同时性；此时，Beltrami 度规成为 ...... 延迟函数、位移矢量和在3 维超曲面!c 给出诱导.