25. Geometry and causal structure of de Sitter space (Inflationary perturbations 4)

Posted by Dmitry
April 23, 2008

Since we are interested so much to understand physics in de Sitter space, let us take a closer look on its geometry and causal structure.

Geometry of d-dimensional de Sitter space (spacetime of constant positive curvature with maximal allowed symmetry which is O(d,1)) can be understood as the geometry of the hypersurface

-X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+\ldots+X_{d}^{2}=1

embedded into (d+1)-dimensional Minkowski space with metric given by

ds^{2}=-dX_{0}^{2}+dX_{1}^{2}+dX_{2}^{2}+\ldots+dX_{d}^{2}=
=-dX_{0}^{2}+dX_{1}^{2}+\ldots
+dX_{d-1}^{2}+
\left(d(1+X_{0}^{2}-X_{1}^{2}-\ldots-X_{d-1}^{2})^{-1/2}\right). (1)

The latter expression for the linear element in de Sitter space is horrable and in order to simplify it one can introduce several coordinate parametrizations of the de Sitter space. Let us focus on the case d=2 (so that there are only two independent coordinates, one is timelike and another is spacelike) and review them.

A. Global coordinates and Penrose diagram

Global coordinates (\tau,\phi) are found by choosing the following parametrization for X_{0,1,2}:

X_{0}={\rm sinh}\,\tau, (2.1)
X_{1}=\cos\phi{\rm cosh}\tau (2.2)
X_{2}=\sin\phi{\rm cosh}\tau, (2.3)

where 0\le\phi\le2\pi and -\infty<\tau<+\infty. The linear element (1) for this parametrization is given by

ds^{2}=-d\tau^{2}+{\rm cosh}^{2}\tau\cdot d\phi^{2}.

Let us take a closer look on the following picture representing 2-dimensional de Sitter space
dS_2:

de Sitter in global coordinates

As you can see, surfaces \tau={\rm Const} are flat, and the crossection of dS_{2} by such a plane is given by a circle with \phi parametrizing it.

The global coordinate system covers the whole de Sitter space (d-dimensional generalization of the parametrization (2) is straightforward), that is why this coordinate system is called “global” :-)

Causal structure of the de Sitter space can be understood after making a substitution

{\rm cosh}\tau=\frac{1}{{\rm cos}T},

where -\frac{\pi}{2}<\frac{\pi}{2}. In new coordinates the linear element acquires the form

ds^{2}=\frac{1}{{\rm cos}^{2}T}\left(dT^{2}+d\phi^{2}\right),

so that de Sitter space becomes conformally equivalent to Minkowski space in new coordinates (T,\phi). Null geodesics are defined by the equation

T={\rm Const}\pm\phi

so that the Penrose diagram has the form represented in the following picture:

Penrose diagram of de Sitter space

(note that the light ray starting at \tau=-\infty at \phi=\pi only reaches \phi=0 at \tau=+\infty, so that the actual, complete, Penrose diagram of dS_{2} consists of two squares like the one presented above: one with 0<\phi<\pi and another — with \pi<\phi<2\pi).

What’s important on the picture above for good understanding of the QFT in de Sitter space?

B. Static coordinates

This is the most favorite string theorist’s coordinate system; in a moment you will understand why.

de Sitter in static coordinates

Static coordinate system covering dS_{2} is obtained by setting the parametrization

X_{0}=\sqrt{1-r^{2}}{\rm sinh}t, (3.1)
X_{1}=r, (3.2)
X_{2}=\sqrt{1-r^{2}}{\rm cosh}t, (3.3)

so that the linear element acquires the form

ds^{2}=-(1-r^{2})dt^{2}+\frac{dr^{2}}{1-r^{2}}.

The name “static” comes from the fact that the vectors \frac{\partial}{\partial t} are Killing vectors of the dS_{2} in this coordinate system, and it looks like nothing intersting dynamically happens with QFT in dS_{2} in this coordinate system (and that is why string theorists like it :-) we will see that life is much more complicated though that this static picture).

As one can see from the representation (3) and the Fig. above, static coordinate system covers only quarter of the de Sitter space.

As it follows from the Penrose diagram, lines r=1 represent horizon for an observer living at \phi=\pi, and the modes of quantum fields in static coordinate system strongly oscillate in the vicinity of r=1. However, geometrically, we see that r=1 is just a single point on the dS_{2} hyperboloid corresponding to X_{0}=X_{2}=0, so the singularity of the QFT Green functions here is unphysical (it is due to the fact that a single point X_{0}=X_{2}=0,\, X_{1}=1 corresponds to a single value of r and arbitrary value of t).

C. Planar coordinates

This one is in turn the cosmologist’s most favorite patch of de Sitter space. Planar coordinates are determined as parametrization

X_{0}=\frac{1}{2}\left(e^{t}-e^{-t}(1+x^{2})\right),

X_{1}=xe^{-t},

X_{2}=\frac{1}{2}\left(e^{t}+e^{-t}(1-x^{2})\right),

so that the linear element in planar patch is given by

ds^{2}=dt^{2}-e^{-2t}dx^{2}. (4)

This coordinate system covers only half of de Sitter (on the Penrose diagram it corresponds to the past triangle + southern diamond). Another half is described by the metric (4) with t\to.

t={\rm Const} hypersurfaces are planes, t=+\infty is again a coordinate singularity where quantum modes diverge:

de Sitter in planar coordinates

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Cosmological perturbations and large scale structure, Cosmology, Fellow Craft, Lectures on de Sitter spacetime, Quantum field theory, String theory

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