Does the Topology of AdS4 Affect Global Hyperbolicity?

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In summary, the conversation discusses the global hyperbolicity of the metric ##g = -\left(1+ \dfrac{r^2}{l^2} \right)dt^2 + \dfrac{dr^2}{1+ \dfrac{r^2}{l^2}}+ r^2d\Omega^2## and its relation to the metric ##\tilde{g} = -dt^2 + l^2 (d\chi^2 + \sin^2 \chi d\Omega^2)##, which has the topology ##\mathbb{R} \times S^3##. The conversation also touches on the use of hyperspherical coordinates on the
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ergospherical
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Is ##\mathrm{AdS_4}## globally hyperbolic?$$g = -\left(1+ \dfrac{r^2}{l^2} \right)dt^2 + \dfrac{dr^2}{1+ \dfrac{r^2}{l^2}}+ r^2d\Omega^2$$Letting ##r = l \tan \chi## then defining ##\tilde{g} = g \cos^2 \chi##\begin{align*}
g &= \sec^2 \chi (-dt^2 + l^2 d\chi^2) + l^2 \tan^2 \chi d\Omega^2 \\ \\

\tilde{g} &= -dt^2 + l^2 (d\chi^2 + \sin^2 \chi d\Omega^2) \\
\tilde{g} &= -dt^2 + l^2 d\omega^2\end{align*}the topology is ##\mathbb{R} \times S^3##. Does global hyperbolicity of ##\tilde{g}## ##\iff## global hyperbolicity of ##g##?
 
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I don't know anything about string theory, but isn't that coordinate system a mapping to ##\mathbb{R^2} \times S^2## not ##\mathbb{R^1} \times S^3##? It looks like only two angles and a radius to me (plus time). Am I missing something?

If I'm missing something obvious, ignore me. I just chimed in because the thread went unanswered.
 
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My thinking was that ##l^2(d\chi^2 + \sin^2 \chi d\Omega^2)## is the round metric on a ##3##-sphere of radius ##l## (or in fact since ##\chi \in \bigg{[} 0, \dfrac{\pi}{2} \bigg{)}## it'll only be half of the 3-sphere...)
 
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  • #4
Just to make sure we're talking about the same thing, when you say ##S^3##, you mean a surface embedded in ##\mathbb{R}^4## given by ##x^2 + y^2 + z^2 + w^2 = 1##, right? As in, the sphere that is diffeomorphic to the special orthogonal group ##\mathrm{SO}(3)## and the special unitary group ##\mathrm{SU}(2)##, right?
 
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Yeah, and ##(\chi, \theta, \varphi)## would be the hyperspherical coordinates on the 3-sphere of radius ##l##.
 
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I'm a dope and I only just caught that ##d\Omega^2## was a total solid angle over ##S^2##. Now I'm on board with your claim about the topology. Sorry for derailing ya
 

FAQ: Does the Topology of AdS4 Affect Global Hyperbolicity?

What is AdS4?

AdS4 stands for Anti-de Sitter space in four dimensions. It is a mathematical model used in theoretical physics to describe the geometry of spacetime.

What does it mean for AdS4 to be globally hyperbolic?

Globally hyperbolic means that the spacetime described by AdS4 has a well-defined causal structure and all points in the spacetime can be reached by a unique time-like curve from a given initial surface.

Why is it important to study the global hyperbolicity of AdS4?

Understanding the global hyperbolicity of AdS4 is important for theoretical physicists because it helps to determine the stability and predictability of the spacetime. It also has implications for the behavior of matter and energy in this model.

How is the global hyperbolicity of AdS4 determined?

The global hyperbolicity of AdS4 is determined by analyzing the causal structure of the spacetime using mathematical tools such as the Cauchy surface and the Cauchy development. These tools help to identify whether the spacetime is globally hyperbolic or not.

Are there any real-world applications of studying the global hyperbolicity of AdS4?

While AdS4 is primarily used in theoretical physics, the study of its global hyperbolicity has potential applications in understanding the behavior of black holes and the evolution of the universe. It may also have implications for future developments in general relativity and quantum gravity.

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