فضاء دي سيتر المضاد
في الرياضيات و الفيزياء، فضاء ده سيتر المضاد anti de Sitter space نوني-الأبعاد، أحياناً يـُكتـَب ، هو مشعب لورنتسي ذو تماثل أقصى مع انحناء عددي سالب ثابت. وهو النظير اللورنتسي ل للفراغ زائدي المقطع نوني-الأبعاد، تماماً مثلما فضاء منكوڤسكي و فضاء ده سيتر هما نظيرا الفضائين الاقليدي و البيضاوي بالترتيب. وأكثر ما يـُعرف عنه هو دوره في تراسل AdS/CFT.
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التعريف والخصائص
![](/w/images/thumb/7/7f/HyperboloidOfOneSheet.png/300px-HyperboloidOfOneSheet.png)
The anti de Sitter space of signature (p,q) can then be isometrically embedded in the space with coordinates (x1, ..., xp, t1, ..., tq+1) and the pseudometric
as the sphere
where is a nonzero constant with dimensions of length (the radius of curvature). Note that this is a sphere in the sense that it is a collection of points at constant metric distance from the origin, but visually it is a hyperboloid, as in the image shown.
ده سيتر المضاد كفضاء متجانس ومتماثل
In the same way that the sphere , anti de Sitter with parity aka reflectional symmetry and time reversal symmetry can be seen as a quotient of two groups whereas AdS without P or C can be seen as
This quotient formulation gives to a homogeneous space structure. The Lie algebra of is given by matrices
- ,
where is a skew-symmetric matrix. A complementary in the Lie algebra of is
These two fulfil . Then explicit matrix computation shows that . So anti de Sitter is a reductive homogeneous space, and a non-Riemannian symmetric space.
المصادر
- Bengtsson, Ingemar: Anti-de Sitter space. Lecture notes.
- Qingming Cheng (2001), "Anti de Sitter space", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Ellis, G. F. R.; Hawking, S. W. The large scale structure of space-time. Cambridge university press (1973). (see pages 131-134).
- Frances, C: The conformal boundary of anti-de Sitter space-times. AdS/CFT correspondence: Einstein metrics and their conformal boundaries, 205--216, IRMA Lect. Math. Theor. Phys., 8, Eur. Math. Soc., Zürich, 2005.
- Matsuda, H. A note on an isometric imbedding of upper half-space into the anti de Sitter space. Hokkaido Mathematical Journal Vol.13 (1984) p. 123-132.
- Wolf, Joseph A. Spaces of constant curvature. (1967) p. 334.