[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/general-covariant-transformations-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/general-covariant-transformations-wikipedia\/","headline":"General covariant transformations – Wikipedia","name":"General covariant transformations – Wikipedia","description":"From Wikipedia, the free encyclopedia Symmetries in a gravitational theory In physics, general covariant transformations are symmetries of gravitation theory","datePublished":"2017-06-13","dateModified":"2017-06-13","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/68baa052181f707c662844a465bfeeb135e82bab","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/68baa052181f707c662844a465bfeeb135e82bab","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/general-covariant-transformations-wikipedia\/","about":["Wiki"],"wordCount":6753,"articleBody":"From Wikipedia, the free encyclopediaSymmetries in a gravitational theoryIn physics, general covariant transformations are symmetries of  gravitation theory on a world manifold X{displaystyle X}. They are gauge transformations whose parameter functions are vector fields on X{displaystyle X}. From the physical viewpoint, general covariant transformations are treated as particular (holonomic) reference frame transformations in general relativity. In mathematics, general covariant transformations are defined as particular automorphisms of so-called natural fiber bundles.Mathematical definition[edit]Let \u03c0:Y\u2192X{displaystyle pi :Yto X} be a fibered manifold with local fibered coordinates (x\u03bb,yi){displaystyle (x^{lambda },y^{i}),}. Every automorphism of Y{displaystyle Y} is projected onto a diffeomorphism of its base X{displaystyle X}. However, the converse is not true. A diffeomorphism of X{displaystyle X} need not give rise to an automorphism of Y{displaystyle Y}.In particular, an infinitesimal generator of a one-parameter Lie group of automorphisms of Y{displaystyle Y} is a projectable vector fieldu=u\u03bb(x\u03bc)\u2202\u03bb+ui(x\u03bc,yj)\u2202i{displaystyle u=u^{lambda }(x^{mu })partial _{lambda }+u^{i}(x^{mu },y^{j})partial _{i}}on Y{displaystyle Y}. This vector field is projected onto a vector field \u03c4=u\u03bb\u2202\u03bb{displaystyle tau =u^{lambda }partial _{lambda }} on X{displaystyle X}, whose flow is a one-parameter group of diffeomorphisms of X{displaystyle X}. Conversely, let \u03c4=\u03c4\u03bb\u2202\u03bb{displaystyle tau =tau ^{lambda }partial _{lambda }} be a vector field on X{displaystyle X}. There is a problem of constructing its lift to a projectable vector field on Y{displaystyle Y} projected onto \u03c4{displaystyle tau }. Such a lift always exists, but it need not be canonical. Given a connection \u0393{displaystyle Gamma } on Y{displaystyle Y}, every vector field \u03c4{displaystyle tau } on X{displaystyle X} gives rise to the horizontal vector field\u0393\u03c4=\u03c4\u03bb(\u2202\u03bb+\u0393\u03bbi\u2202i){displaystyle Gamma tau =tau ^{lambda }(partial _{lambda }+Gamma _{lambda }^{i}partial _{i})}on Y{displaystyle Y}. This horizontal lift \u03c4\u2192\u0393\u03c4{displaystyle tau to Gamma tau } yields a monomorphism of the C\u221e(X){displaystyle C^{infty }(X)}-module of vector fields on X{displaystyle X} to the C\u221e(Y){displaystyle C^{infty }(Y)}-module of vector fields on Y{displaystyle Y}, but this monomorphisms is not a Lie algebra morphism, unless \u0393{displaystyle Gamma } is flat.However, there is a category of above mentioned natural bundles T\u2192X{displaystyle Tto X} which admit the functorial lift \u03c4~{displaystyle {widetilde {tau }}} onto T{displaystyle T} of any vector field \u03c4{displaystyle tau } on X{displaystyle X} such that \u03c4\u2192\u03c4~{displaystyle tau to {widetilde {tau }}} is a Lie algebra monomorphism[\u03c4~,\u03c4~\u2032]=[\u03c4,\u03c4\u2032]~.{displaystyle [{widetilde {tau }},{widetilde {tau }}’]={widetilde {[tau ,tau ‘]}}.}This functorial lift \u03c4~{displaystyle {widetilde {tau }}} is an infinitesimal general covariant transformation of T{displaystyle T}.In a general setting, one considers a monomorphism f\u2192f~{displaystyle fto {widetilde {f}}} of a group of diffeomorphisms of X{displaystyle X} to a group of bundle automorphisms of a natural bundle T\u2192X{displaystyle Tto X}. Automorphisms f~{displaystyle {widetilde {f}}} are called the general covariant transformations of T{displaystyle T}. For instance, no vertical automorphism of T{displaystyle T} is a general covariant transformation.Natural bundles are exemplified by tensor bundles. For instance, the tangent bundle TX{displaystyle TX} of X{displaystyle X} is a natural bundle. Every diffeomorphism f{displaystyle f} of X{displaystyle X} gives rise to the tangent automorphism f~=Tf{displaystyle {widetilde {f}}=Tf} of TX{displaystyle TX} which is a general covariant transformation of TX{displaystyle TX}. With respect to the holonomic coordinates (x\u03bb,x\u02d9\u03bb){displaystyle (x^{lambda },{dot {x}}^{lambda })} on TX{displaystyle TX}, this transformation readsx\u02d9\u2032\u03bc=\u2202x\u2032\u03bc\u2202x\u03bdx\u02d9\u03bd.{displaystyle {dot {x}}’^{mu }={frac {partial x’^{mu }}{partial x^{nu }}}{dot {x}}^{nu }.}A frame bundle FX{displaystyle FX} of linear tangent frames in TX{displaystyle TX} also is a natural bundle. General covariant transformations constitute a subgroup of holonomic automorphisms of FX{displaystyle FX}. All bundles associated with a frame bundle are natural. However, there are natural bundles which are not associated with FX{displaystyle FX}.See also[edit]References[edit]Kol\u00e1\u0159, I., Michor, P., Slov\u00e1k, J., Natural operations in differential geometry.  Springer-Verlag: Berlin Heidelberg, 1993.  ISBN\u00a03-540-56235-4, ISBN\u00a00-387-56235-4.Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing: Saarbr\u00fccken, 2013. ISBN\u00a0978-3-659-37815-7; arXiv:0908.1886Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN\u00a00-521-36948-7  "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/general-covariant-transformations-wikipedia\/#breadcrumbitem","name":"General covariant transformations – Wikipedia"}}]}]