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I'm not a topologist or a group theorist and I need a clarification about some notations.

Consider the Bertotti-Robinson metric in General Relativity (relativity students should study this metric, by the way, it's a really nice one!): \begin{equation}\tag{1} ds^2 = dt^2 - a^2 \, (d\vartheta^2 + \sin^2 \vartheta \, d\varphi^2) - \sin^2{\!\omega t} \: dz^2, \end{equation} where $a$ and $\omega$ are some constants. This metric is often described as the direct product of an ordinary sphere ($S_2$) and a 2 dimensional Anti-deSitter spacetime ($AdS_2$). It's usually described as $AdS_2 \otimes S_2$. I have three simple questions:

  1. In this example, is the direct product $\otimes$ the same as a cartesian product $\times$ ? Does it make sense to write $AdS_2 \times S_2$ instead? While I know what is the direct product of matrices and cartesian product of vector spaces, I'm a bit confused here!

  2. If the whole 4D spacetime manifold is $\mathcal{M}^4$, does it make sense to write $\mathcal{M}^4 = AdS_2 \otimes S_2$ ? What about $\mathcal{M}^4 = AdS_2 \times S_2$ ?

  3. Is $S_2 \otimes AdS_2$ the same thing as $AdS_2 \otimes S_2$? I know that the direct product of matrices isn't commutative ($A \otimes B \ne B \otimes A$), but I wonder if this is pertinent to the description of manifolds, not matrices.

Cham
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2 Answers2

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It is actually the direct product, not the tensor product (physicists frequently get too sloppy and end up using one or another without acknowledging the difference between the two). It is trivial to show that a direct product of manifolds is a manifold.

Direct product is commutative, so $S_2 \times AdS_2 = AdS_2 \times S_2$ as manifolds in the sense that there exists a 1:1 diffeomorphism.

Also, tensor product of vector spaces is commutative as well (in the sense that the two resulting vector spaces are isomorphic as vector spaces, but as pointed out by Accidental@ in the comments, this doesn't hold for the elements of vector spaces which are also usually tensored together by abuse of notation). For every two vector spaces $U$ and $V$, there exists a canonical isomorphism between $U \otimes V$ and $V \otimes U$: $$ u \otimes v \mapsto v \otimes u.$$

Prof. Legolasov
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Take a look at:

Extremal limits and black hole entropy Sean M. Carroll, Matthew C. Johnson, Lisa Randall

In this paper they show the extremal limit of a Kerr black hole where $r_\pm=m\pm\sqrt{m^2-a^2cos^2\theta}$ and $r_+\rightarrow r_-$ discontinuously pushes the spacelike region between the horizons into $AdS_2\times\mathbb S^2$.

I illustrate something about $AdS/black~hole$ correspondence by showing how the near horizon condition for an accelerated observer is approximately $AdS_2\times\mathbb S^2$.

AdS Black holes

Lawrence B. Crowell
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