What is the connection between DeSitter group SO(4,1) and Minkowski spacetime?

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Discussion Overview

The discussion centers around the connection between the DeSitter group SO(4,1) and Minkowski spacetime, exploring theoretical aspects of DeSitter space, its symmetries, and mathematical representations. Participants delve into the implications of these concepts in physics, particularly in relation to cosmological models and the structure of spacetime.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that understanding the DeSitter group SO(4,1) requires familiarity with its mathematical foundations, such as the hyperboloid representation in a 5D Euclidean space.
  • Others argue that DeSitter space can be viewed as a slight modification of Minkowski spacetime, introducing a cosmological constant to allow for expansion while maintaining a flat structure.
  • A participant proposes that the symmetries of DeSitter spacetime, represented by SO(4,1), are analogous to the symmetries of Minkowski spacetime, which are described by the Lorentz or Poincaré groups.
  • There is a discussion on the DeSitter algebra so(4,1) and its physical interpretation, with some participants suggesting that understanding this algebra through Clifford algebras provides a more intuitive grasp of its properties.
  • Some participants express uncertainty about the implications of recent research papers related to DeSitter space and its symmetries, indicating that the connections between various theories and models remain to be fully understood.

Areas of Agreement / Disagreement

Participants express a range of views on the relationship between DeSitter space and Minkowski spacetime, with no clear consensus on the implications of these connections or the best methods for studying them. Multiple competing interpretations and models are presented, indicating an unresolved discussion.

Contextual Notes

Limitations include the dependence on specific mathematical frameworks and the potential for differing interpretations of the implications of DeSitter space in physical theories. Some mathematical steps and assumptions remain unresolved within the discussion.

  • #91
Yep, Derek's written an excellent expository paper that fills in a lot of geometric details and background behind BF gravity. It's also quite readable.

Perhaps now that he's let his cat out of its bag he'll come talk to us again on this thread, or on his.
 
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  • #92
this thread has a lot of conversation between Garrett Lisi and John Baez about things that might be relevant to Garrett's recent paper, so i thought I'd bring it back from the limbo of Forgotten Threads
 
  • #93
This thread needs updating. Quite a lot of research has come out in the wake of what John Baez and Garrett Lisi were discussing here. I will try to catch it up a little. Others' help woud be very welcome.

In fact, JB and Garrett were talking about some work by Derek Wise and as it happens Derek published a paper just last month (August 2009) which continues the line of research they were discussing.

http://arxiv.org/abs/0904.1738
Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions
Derek K. Wise
Article prepared for special journal issue dedicated to Elie Cartan
SIGMA 5 (2009), 080, 18 pages
(Submitted on 10 Apr 2009, revised for publication 3 Aug 2009)
"Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formulation of topologically massive gravity, with arbitrary cosmological constant, as a single constrained Chern-Simons action. In 4 dimensions the main model of interest is MacDowell-Mansouri gravity, generalized to include the Immirzi parameter in a natural way. I formulate these theories in Cartan geometric language, emphasizing also the role played by the symmetric space structure of the model. I also explain how, from the perspective of these Cartan-geometric formulations, both the topological mass in 3d and the Immirzi parameter in 4d are the result of non-simplicity of the Lorentz Lie algebra so(3,1) and its relatives. Finally, I suggest how the language of Cartan geometry provides a guiding principle for elegantly reformulating any 'gauge theory of geometry'."

Derek will be giving a talk next week in Corfu, on Saturday 19 September. John Baez is giving a series of 5 lectures at the Corfu QG School.
 
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