Abstract
The AdS spacetime is one of spacetimes with constant curvature. To be familiar with the AdS spacetime, we first discuss spaces with constant curvature such as the sphere and then discuss spacetimes with constant curvature.
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- 1.
One can have closed timelike curves, where causal curves are closed.
- 2.
The name “conformal coordinates” is not a standard one.
- 3.
As in Eq. (6.67), it takes an infinite affine time from \(r=\infty \) to \(r=0\), but it takes a finite amount of time from finite \(R\) to \(r=0\).
- 4.
Otherwise, the spacetime would be geodesically incomplete which signals a spacetime singularity. The geodesic incompleteness means that there is at least one geodesic which is inextensible in a finite “proper time” (in a finite affine parameter). This is the definition of a spacetime singularity.
- 5.
For example, in the near-horizon limit, the extreme RN-AdS\(_4\) black hole becomes the AdS\(_2\) spacetime in Poincaré coordinates (Sect. 14.3.3). In this case, \(r=0\) corresponds to the true horizon of the black hole. So, whether \(r=0\) is a true horizon or not depends on the context.
- 6.
So far we took dimensionless coordinates, but we take dimensionful ones below.
- 7.
This equation looks similar to the gravitational redshift (6.63), but the interpretations are different. The gravitational redshift compares proper energies at different radial positions. The UV/IR relation compares the proper energy and the energy conjugate to the coordinate \(t\) at one radial position.
- 8.
This should coincide with the surface gravity (for the asymptotic observer). When we discussed the surface gravity, we assumed that the spacetime is asymptotically flat [in the last expression of Eq. (3.12), we used \(f(\infty )=1\)]. For AdS black holes, the surface gravity vanishes by taking \(f(\infty )\rightarrow \infty \) into account. The expression \(f'(r_0)/2\) corresponds to the “surface gravity” in the \(t\)-coordinate.
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Natsuume, M. (2015). The AdS Spacetime. In: AdS/CFT Duality User Guide. Lecture Notes in Physics, vol 903. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55441-7_6
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DOI: https://doi.org/10.1007/978-4-431-55441-7_6
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