advanced_tools:group_theory:desitter
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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] |
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| 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. | ||
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| + | 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 $c$. | ||
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| + | Expressed differently: the deSitter group is only important for cosmological systems, which have a length scale comparable to $R$. | ||
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| + | 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. | ||
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| + | 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]]. | ||
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| <tabbox Student> | <tabbox Student> | ||
| - | <note tip> | + | <blockquote> |
| - | 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 |
| - | </note> | + | $$ R_{\mu\nu\lambda\sigma} =\frac{1}{L} (g_{\mu\lambda}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\lambda}) $$ |
| + | [...] 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, $H=1/L$. | ||
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| + | <cite>page 627 in Einstein Gravity in a Nutshell - A. Zee</cite> | ||
| + | </blockquote> | ||
| <tabbox Researcher> | <tabbox Researcher> | ||
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| </note> | </note> | ||
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| - | <tabbox Examples> | ||
| - | --> Example1# | ||
| - | + | <tabbox FAQ> | |
| - | <-- | + | |
| - | --> 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?# |
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| + | 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. | ||
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| <-- | <-- | ||
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| - | <tabbox FAQ> | ||
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| <tabbox History> | <tabbox History> | ||
advanced_tools/group_theory/desitter.1521886097.txt.gz · Last modified: 2018/03/24 10:08 (external edit)
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