In the paper Anti-de Sitter space, squashed and stretched Bengtsson and Sandin introduce the Lorentzian analogue of the squashed 3-sphere. After looking up Berger spheres, it seems what is meant with "squashing" in the case of $S^3$ is the following:

Take the Lie algebra $L$ spanned by $z_1$,$z_2$ and $z_3$, which fulfill the necessary relations so $L$ can generate $S^3$. Then take the quotient group $B(\beta) = G/H$ of the group $G = S^3 \bigoplus \mathbb{R}$ by the one-parameter subgroup $H$ generated by (e.g.) $\alpha z_1 + \beta z_2$, where $\alpha^2 + \beta^2 = 1$, $z_1 \in L$ and $z_4$ a left invariant vector field tangent to $\mathbb{R}$. $B(\beta)$ is then called a Berger sphere. I interpret this as "squashing" $S^3$ along $\mathbb{R}$. Why is the condition $\alpha z_1 + \beta z_2$ necessary?

Coming back to the mentioned paper, the authors state that Taub-NUT cosmologies can be described by squashing $S^3$ along Hopf fibres, i.e. $S^1$. This confuses me as the topology is just $S^3 \times \mathbb{R}$, is there a covering space involved?

In the Lorentzian analogue of squashed $S^3$, $AdS_3$ has the topology of $\mathbb{R}^2\times S^1$ and $z_4$ can be either spacelike ($z_4^i z_{4,i} = 1$) or timelike ($z_4^i z_{4,i} = -1$). Consequently there are two ways of of squashing, but along which fibre does one squash in this case? Furthermore, what is meant with "Stretching"?


So I figured out at least the "stretching" part: For example at one stage the introduce the metric of $AdS_3$, squashed/stretched along a spacelike fibre: \begin{align} \mathrm{d}s^2_\lambda = \frac{1}{4}\left(-\mathrm{cosh}^2(\omega)\mathrm{d}\tau^2 + \mathrm{d}\omega^2 + \lambda^2 \left(\mathrm{d}\sigma + \mathrm{sinh}(\omega)\mathrm{d}\tau\right)\right)\,, \end{align} where $\lambda$ is a real stretching parameter. They simply call the case $\lambda^2 < 1$ "squashing" and $\lambda^2 > 1$ "stretching".

Furthermore, the Hopf-fibres, along which one squashes $AdS_3$, are the geodesic congruences of the spacetime, i.e. $\partial_t$ and $\partial_{\phi}$, which both are compactified do to identifications like e.g. $\phi \sim \phi + 2 \pi$.

The two standing questions are:

  • Along which fibres is one squashing $S^3$ to get Taub-NUT space?
  • Is in the case of the Berger sphere the condition $\alpha^2 + \beta^2 = 1$ necessary as a identification condition, if not why is it necessary?
  • Where are the Hopf fibres in the calculation, along which we are squashing?

The subgroup $H$ is not a normal subgroup (or invariant subgroup, in physicist's language), so the quotient $B(\beta)=G/H$ is not naturally a quotient group, only a quotient space. The condition $\alpha^2+\beta^2=1$ is not necessary. Indeed any vector with a nonzero component of $z_4$ will do. Better: any one-dimensional linear subspace of the Lie algebra of $G$ generates a Lie group $H$ which gives a Berger sphere quotient space. Cheeger and Ebin choose to normalize that linear subspace by automorphisms of the 3-sphere, so that they just get finitely many Berger spheres in each isometry class. The Hopf fibers are the curves on which $\omega$ and $\tau$ are constant, parameterized by $\sigma$.

The description in terms of a 4-dimensional Lie group $G$ is not very natural, but fits into a bigger picture in the book of Cheeger and Ebin. It is easier to simply write that the Hopf fibration is given by taking any connected 1-dimensional subgroup $K$ of $S^3$, and then the quotient $S^3\to S^3/K$ is a Hopf fibration. Since the adjoint action of $S^3$ acts transitively on 1-dimensional subspaces of its Lie algebra (i.e. rotations of 3-dimensional Euclidean space act transitively on lines through the origin), $S^3$ acts transitively, by conjugation, on all connected 1-dimensional Lie subgroups. Hence the choice of $K$ is arbitrary. In your basis $z_1,z_2,z_3$ for the Lie algebra of $S^3$, you could just take dual basis, say $\omega_1,\omega_2,\omega_3$, and then your Hopf fibration can have fibers given by setting two linearly independent linear combinations of these two zero, say $0=\omega_1=\omega_2$, so the subgroup $K$ is then tangent to $z_3$. We can make the Berger metric then be anything of the form $\omega_1^2+\omega_2^3+\lambda \omega_3^2$, carried around $S^3$ by left (or right, if you prefer) translation.

In the Lorentzian picture, there are 3 different connected 1-dimensional Lie subgroups of $\operatorname{SL}_2\mathbb{R}$, so there are two different ways to carry out a 1-parameter family of Hopf fibrations.

  • $\begingroup$ Thank you for your helpful answer! Can you maybe elaborate on why one says squashing along "Hopf fibers", when any one-dimensional linear subspace will do? I thought that one had to compactify the fibers so that they count as Hopf fibers, as $S^3$ is (roughly) locally a $S^2$ product with $S^1$. In the case of squashed $AdS_3$ this is done by identifying $\phi \sim \phi + 2\pi$, where can I see this in the Berger sphere or the Taub-NUT case? $\endgroup$ – horropie Sep 13 '19 at 10:15
  • $\begingroup$ Sorry, in the notation of the paper we have $\sigma \sim \sigma + 2\pi$. $\endgroup$ – horropie Sep 13 '19 at 10:25
  • $\begingroup$ Thank you again for the very helpful edit. Do you have any literature where I could read the things up you posted above? $\endgroup$ – horropie Sep 13 '19 at 14:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.