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 advanced_tools:group_theory:desitter [2018/03/24 11:08]jakobadmin [Why is it interesting?] advanced_tools:group_theory:desitter [2018/03/24 11:10] (current)jakobadmin [FAQ] Both sides previous revision Previous revision 2018/03/24 11:10 jakobadmin [FAQ] 2018/03/24 11:10 jakobadmin 2018/03/24 11:09 jakobadmin [Layman] 2018/03/24 11:09 jakobadmin [Student] 2018/03/24 11:08 jakobadmin [Why is it interesting?] 2018/03/24 11:08 jakobadmin [Why is it interesting?] 2018/03/24 11:05 jakobadmin [Layman] 2018/03/24 11:01 jakobadmin created Next revision Previous revision 2018/03/24 11:10 jakobadmin [FAQ] 2018/03/24 11:10 jakobadmin 2018/03/24 11:09 jakobadmin [Layman] 2018/03/24 11:09 jakobadmin [Student] 2018/03/24 11:08 jakobadmin [Why is it interesting?] 2018/03/24 11:08 jakobadmin [Why is it interesting?] 2018/03/24 11:05 jakobadmin [Layman] 2018/03/24 11:01 jakobadmin created Line 70: Line 70: The deSitter group becomes the Poincare group in the contraction limit $R \rightarrow \infty$, where $R$ is the so-called deSitter radius. Oftentimes, people prefer to work with the [[open_problems:​cosmological_constant|cosmological constant]] $\Lambda \propto \frac{1}{R^2}$ instead. Analogously,​ the Poincare group becomes the Galilean group in the $c \rightarrow \infty$ limit. ​ The deSitter group becomes the Poincare group in the contraction limit $R \rightarrow \infty$, where $R$ is the so-called deSitter radius. Oftentimes, people prefer to work with the [[open_problems:​cosmological_constant|cosmological constant]] $\Lambda \propto \frac{1}{R^2}$ instead. Analogously,​ the Poincare group becomes the Galilean group in the $c \rightarrow \infty$ limit. ​ + + The fact that the deSitter group contracts to the Poincare group in the $R\rightarrow \infty$ limit, means that the Poincare group is a good approximation as long as we consider systems with a length scale that is small compared to $R$. This is analogous to how the Galilean group is good enough as long as we are only dealing with velocities much smaller than the invariant velocity $c$. + + Expressed differently:​ the deSitter group is only important for cosmological systems, which have a length scale comparable to $R$. + + Alternatively,​ we can talk about the invariant energy scale $\Lambda$. The deSitter group contracts to the Poincare group in the $\Lambda \rightarrow 0$ limit. Thus the deSitter structure is not important, as long as we are dealing with energies much larger than $\Lambda$. In systems with energies much larger than $\Lambda$ such a small constant energy has no effect. The present day value for the cosmological constant is $\Lambda \approx 10^{-56} \mathrm{m^{-2}}$ and this means that present day effects of the deSitter group structure are tiny. This means, the Poincare group is a great approximate symmetry nowadays, because $\Lambda$ is almost zero. + + However, the deSitter group could be very important in the early universe, too. For example, because it seems plausible that there was [[https://​en.wikipedia.org/​wiki/​Inflation_(cosmology)|a phase when the cosmological constant was much higher]]. ​ ​  ​  ​ - +
- In this section things should be explained by analogy and with pictures ​and, if necessary, some formulas. + The group $SO(d,1)$ moves points on $dS^d$ around. We thus conclude that, just like the sphere, deSitter spacetime is maximally symmetric. So, according to the general theory of maximally symmetric spaces ​explained ​in chapter IX.6, the Riemann curvature tensor $R_{\mu\nu\lambda\sigma}$ must be equal to $(g_{\mu\lambda}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\lambda})$ up to an overall constant. [...] Then, by dimensional analysis, we must have - + $$R_{\mu\nu\lambda\sigma} =\frac{1}{L} (g_{\mu\lambda}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\lambda})$$ + [...] de Sitter spacetime is a solution of Einstein'​s field equation $R_{\mu \nu}=8\pi G\Lambda g_{\mu\nu}$, ​with a positive cosmological constant given by + $$8\pi G\Lambda = \frac{3}{L}$$ + [...] + Topologically,​ de Sitter spacetime is $R \times S^3$, with a spatial section given by $S^3$, as just explained. In contrast, we know from chapters VI.2 and VI.5 that Einstein'​s field equation with a positive cosmological constant leads to $ds^2=-dt^2+e^{2Ht}(dx^2+dy^2+dz^2)$ with the Hubble constant given by $H= \left( \frac{8\pi G}{3}\Lambda\right)^{1/​3}$, or in terms of the de Sitter length, $H=1/L$. + + <​cite>​page 627 in Einstein Gravity in a Nutshell - A. Zee​ +
​  ​ Line 83: Line 98: ​ - ​ -  ​ - --> Example1# - +  ​ - <-- + - --> ​Example2:# + -->It ist confusing what a "​five-dimensional"​ group like $SO(4,1)$ has to say about our four-dimensional world. Does this mean deSitter theories predict a fifth dimension?# + + No, recall that an explicit representation of the Poincare group is given by $(5 \times 5)$ matrices, too! This is, because the Lorentz group transformations are $(4 \times 4)$ matrices themselves and in oredr to describe translations,​ we need to make these matrices into $(5 \times 5)$ matrices. See, for example, chapter 4 "The Poincaré transformations"​ in [[http://​www.springer.com/​us/​book/​9783642154812|Symmetries and Group Theory in Particle Physics]] by Costa and Fogly. - <-- <-- - -  ​ ​ ​  ​  ​ 