# Definition of twisted geometries and existence of coordinate transformation for twisted $AdS_2 \times S^2$

In the paper Multiply Twisted Products by Yong Wang, general definitions for so called warped and twisted products are given:

A (singly) warped product $$B \times_b F$$ of two pseudo-Riemannian manifolds $$\left(B\,,g_B\right)$$ and $$\left(F\,,g_F\right)$$ with a smooth function $$b\,:\,B \to \left(0\,, \infty \right)$$ is the product manifold $$B \times F$$ with the metric tensor $$g = g_B \oplus b^2 g_F$$. We call $$\left(B\,,g_B\right)$$ the base manifold, $$\left(F\,,g_F\right)$$ the fiber manifold and $$b$$ the warping function.

A twisted product $$\left(M\,,g\right)$$ is a product manifold $$M = B \times_b F$$, with a smooth function $$b\,:\,B \times F \to \left(0\,, \infty \right)$$ and the metric tensor $$g = g_B \oplus b^2 g_F$$.

The notion of warped products seems to be a rather standard definition, first introduced by O'Neill. I wonder if the definition of twisted products (and the generalizations of warped an twisted products) in the paper given above is also standard. Many theoretical physics papers sometimes talk about warped and twisted spacetimes in a rather handwavy way, so I am never really sure if they are talking about a mathematical definition or they just want to sound fancy.

For example, Compère talks in an overview paper about the Kerr/CFT correspondence about the following metric

\begin{align} \mathrm{d}s^2 = J\left(1+\cos^2\theta\right)\left[-r^2dt^2 + \frac{dr^2}{r^2} + d\theta^2 \right] + \frac{4 \sin^2\theta}{1+\cos^2\theta} \left(d\phi + rdt\right)^2\,, \end{align}

being a " warped and twisted product of $$AdS_2 \times S^2$$ " (J is just scaling the metric). This seems to be a different definition of a twisted product, as in Wangs paper twisted is a generalization of warped. Furthermore I can't neither make out the above metric to be $$AdS_2 \times S^2$$, nor that it is for $$\theta = \frac{\pi}{2}$$ a "twisted product of $$AdS_2$$ and a circle of constant radius" (as this paper suggests), as the suggested geometry should not have off-diagonal terms like $$\mathrm{d}\phi \mathrm{d}t$$ in the metric. For clarity, the metric of $$AdS_2$$ in Poincare coordinates is \begin{align} \label{eq:poincarepatch} ds^2 = - r^2dt^2 + \frac{dr^2}{r^2}\,, \end{align} where we set the curvature constant to $$1$$ to match the above expression.

My questions are:

• Am I missing coordinate transformations that make this obvious? As mentioned below this would have to make the metric diagonal.
• What is a standard definition for a twisted product of Pseudo-Riemannian manifolds?

EDIT: So as some comments say, the definition of a twisted product seems to be canonical. I need a coordinate transformation that gets rid of the off-diagonal term or at least a reference to a theorem that ensures the existence of such a coordinate transformation.

• What I've seen in relativity is the notion of a twisted tensor. I don't know if this relates at all to the idea a twisted spacetime. Re twisted tensors, see Burke, Applied differential geometry, p. 183ff. "Twisted tensors were introduced by Hermann Weyl (1922). Schouten (1951) developed them, and called them Weyl tensors or W-tensors. Synge and Schild (1949) refer to them as oriented tensors, and de Rham (1960) called them tensors of odd kind. Nowadays they are usually called twisted tensors." – Ben Crowell Sep 20 '19 at 21:35
• I have seen something similar, maybe the same, in the relativity literature, (e.g. Stephani Exact solutions of Einsteins field equations), where it is connected to null geodesic congruences and thus probably to the Weyl tensor (e.g. p. 73: Diverging vacuum type D solutions are twistfree exactly if the Weyl tensor is purely electric). I am pretty sure though that the term "twisted" is related to the term "warped" and is usually used in context of the latter. – horropie Sep 20 '19 at 22:31
• I have seen the notion of twisted product the first time here: M. Gutiérrez, B. Olea; Totally umbilic null hypersurfaces in generalized Robertson-Walker spaces. Differential Geom. Appl. 42 (2015), 15–30 and I think the standard definition should be as follows: $(M,g), (N,h)$ two semi-Riemannian manifolds, $f: M\times N\rightarrow (0,\infty)$ smooth, then $(M\times N, g+f h)$ is the twisted product of $M$ and $N$. So as compared to the warped product $f$ is also allowed to depend on the fiber. However, I do not know how much that is used. – Clemens Sämann Sep 20 '19 at 22:58
• @ClemensSämann This seems to coincide with the definition in the paper by Wang. I added them in the text for clarity. So as this seems to be a rather standard definition, can you make out why the above is a twisted (and warped?) product of $AdS_2 \times S^2$? – horropie Sep 20 '19 at 23:47
• To be a twisted product the metric has to be diagonal, so you have to get rid of the $d\phi dt$-term, you get by expanding $(𝑑𝜙+𝑟 𝑑𝑡)^2$. But I don't see a clever coordinate transformation to do that. – Clemens Sämann Sep 22 '19 at 0:42

The short answer is "what physicists mean by 'warped' and 'twisted' geometry" is not the same as "what differential geometers mean by 'warped' and 'twisted' geometry". The use is a lot more qualitative and a little loosey-goosey, but on the other hand very easy to visualize.

From the physicists' point of view, a twisted geometry is basically something that is not diagonal. The idea is basically that you can consider "nice" manifolds that admit local fibrations with the fiber $$F$$ orthogonal to the base $$B$$, so locally the geometry looks like $$g_B \oplus g_F$$, where here we do allow both $$g_B$$ and $$g_F$$ to depend on both $$B$$ and $$F$$ coordinates.

A "twisted" version of this geometry would be one that still has the fibration, but you now "twist the fibers around the base", so now that the fibers are no longer orthogonal to the base.

A classic example in relativity is that of the Kerr(-Newman) space-times. The idea being that an "untwisted" black hole is something like the Schwarzschild solution, which has nice factorization as some base manifold that is orthogonal to the integral curves of the time-like Killing vector field. The rotating black holes on the other hand have the time-like Killing vector field "twisted" around the base, losing the orthogonality.

The nomenclature of "twist" came from the examination of geodesic congruences. The "vorticity" tensor is also called the "twist" tensor, and its vanishing is (essentially by Frobenius' theorem) equivalent to the local integrability of the orthogonal hyperplane distribution to a vector field.

Next, from the physicists' point of view, a warped geometry is something that is deformed from a reference geometry by squashing/stretching factors. (So the notion of warping is closed to the notion of a "warped product", but not the same.) Basically you start again with a nice manifold that admit local fibrations as $$B\times F$$, with the geometry $$g_B \oplus g_F$$. A "warping deformation" of this geometry is one that replaces the geometry by $$\lambda g_B \oplus \lambda^{-1} g_F$$ so that the new geometry looks like the old one but stretched in some directions but squashed in some other directions.

As an example: from the paper of Compere that you linked to (pg. 15), you see that he (like many physicists) think of $$\mathrm{AdS}_3$$ as the fibered manifold with base $$\mathrm{AdS}_2$$ and fiber $$\mathbb{S}^1$$, but with the fiber "twisted around" the base.

Insofar as "warping": you can see this survey which uses the common physics terminology to discuss what is meant by "warped $$\mathrm{AdS}_3$$".

• Thank you so much, this really helped me a lot! But I still wonder about a few things, If you dont mind. When you talk about "$g_F$ and $g_B$ depending on coordinates of both $B$ and $F$, you mean that that prefactors can both depend on these coordinates? Similarly, does $\lambda$ in the warping deformation depend on only one of the coordinates (so one could make a warped product by rescaling), or is this term meant even looser? Furthermore, do you have literature where Kerr manifolds are characterized in the above way? – horropie Sep 24 '19 at 12:10
• The sentence "$g_F$ and $g_B$ depending on coordinates of both $B$ and $F$" just mean that we do not assume that $B\times F$ is given with the product geometry of (pseudo)Riemannian manifolds $(B,g_B)$ and $(F,g_F)$. All we care about is that there exists a local factorization of the manifold so that the metric becomes block diagonal with respect to the factorization. – Willie Wong Sep 24 '19 at 13:22
• For the "original" studies of the warped $\mathrm{AdS}_3$ manifolds, the assumptions are actually that $\lambda$ are constants $\neq 1$, so the manifolds are globally warped the same way. However, the meaning has evolved to allow $\lambda$ to be essentially an arbitrary function on the total manifold. // As to Kerr: nothing very handy, but you can probably see all this if you read some history-of-black-holes papers specifically those that talk about the discovery of Kerr solution. – Willie Wong Sep 24 '19 at 13:24