Physics Notes by Jakob Schwichtenberg

Sidebar

unification_of_spacetime_symmetries:desitter_group

The deSitter Group

The symmetry group that we use when we only consider the principle that all inertial frames of reference should be the same, called Galilean relativity, is the Galilean group.

If we add to this principle that there should be an invariant velocity $c$, we end up with special relativity and the corresponding symmetry group is the Poincare group.

If we then add a second principle that says there should be an invariant length scale $R$ (= an invariant energy scale $\Lambda$), the deSitter group is the group that we must use.

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 cosmological constant $\Lambda \propto \frac{1}{R^2}$ instead. Analogously, the Poincare group becomes the Galilean group in the $c \rightarrow \infty$ limit.

Why is it interesting?

The simple existence of a non-vanishing cosmological constant in the universe means that Poincare is no longer the kinematic group of spacetime; this is a largely overlooked point.

To the extent that* ~$74 \simeq 100$ %, we can say that our universe is observed to be almost maximally symmetric and de Sitter.(*Dark energy amounts to ~74% of the universe […]). [..] Our universe might very well be described by $dS^4$ to a good approximation, as discussed in chapter VIII.2 and in the preceding section.

Einstein Gravity in a Nutshell - A. Zee

We will confront certain recent astronomical observations suggesting that, even in an empty universe, the event world may possess properties not reflected in the structure of Minkowski spacetime, at least on the cosmological scale. Remarkably, there is a viable alternative [the deSitter spacetime], nearly 100 years old, that has precisely these properties and we will devote a little time to becoming acquainted with it.

The Geometry of Minkowski Spacetime - Naber

The present experimental value for the cosmological constant is tiny, but nonzero: $\Lambda \approx 1.19·10^{-52}$ $1/m^2$.

The Poincare group is the contraction of the deSitter group in the limit $\Lambda \rightarrow 0$, analogous to how the Galilean group is the contraction of the Poincare group in the limit $c\rightarrow \infty$.

Thus a non-zero $\Lambda$ means that the exact spacetime symmetry group is the deSitter group and the Poincare group is only a good approximation because $\Lambda$ is so small.

“Apart from the current accelerating cosmic expansion, further motivation for the interest in de Sitter gravity comes from the inflationary era, during which one assumes that the universe was also described by a de Sitter phase.” Aspects of Quantum Gravity in de Sitter Spaces Dietmar Klemm and Luciano Vanzo

In quantum field theory (in its present imperfect form) the Minkowski metric is the vacuum expectation value of the Riemannian metric. It seems unsafe to restrict the attention arbitrarily to the special case of vanishing cosmological constant, for this ease is unstable to deformations; a nonzero cosmological constant may, for example, appear spontaneously through renormalization. In that case the zeroth approximation (or vacuum expectation value) of the metric cannot be Minkowski, but must be de Sitter. Our previous analysis must therefore be modified by the substitution of the de Sitter group for the Poincare group” from Massless particles, conformal group, and de Sitter Universe by E. Angelopoulos

General relativity can be formulated as a gauge theory of the deSitter group, see http://journals.aps.org/prd/pdf/10.1103/PhysRevD.21.1466.

Since the renormalized value of the cosmological constant is experimentally quite small, $ρ ∼ 10^{-122}M_{Pl} \sim 10^{-3}$ eV (and positive – we live in de Sitter space), we can ignore it for terrestrial experiments. To account for a non- zero cosmological constant in quantum field theory requires field theory in curved space, a topic beyond the scope of this text.

Page 415 in Schwartz - Quantum Field Theory and the Standard Model

Interpretation

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, $H=1/L$.

page 627 in Einstein Gravity in a Nutshell - A. Zee

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 a phase when the cosmological constant was much higher.

Possible Points of Confusion

Q: 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?

A 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 Symmetries and Group Theory in Particle Physics by Costa and Fogly.

Aspects of De Sitter Spacetime

De Sitter Space has Finite Spatial Volume

This can be understood by embedding de Sitter space in 5-dimensional Minkowski space and then performing a Wick rotation: $x_0 \to i x_5$. Then, de Sitter space simply becomes the sphere $S^4$, which has finite volume.

Viewed the other way around, we can start with the sphere $S^4$ in 5-dimensional Minkowski space, which has finite volume, and get de Sitter space by Wick rotating: $x_5 \to i x_0$. This only changes the time coordinate $x_0$, but leaves the spatial structure unchanged which therefore still has a finite volume.

See also the footnote at page 4 in http://www.hartmanhep.net/topics2015/6-GravityPathIntegral.pdf and https://arxiv.org/abs/hep-th/0407217, where the authors write: “However, de Sitter space has finite volume, and this leads to a non-zero tunneling amplitude.”