Musings

Penrose diagram of the maximally-extended spacetime for an AdS blackhole.

The maximally-extended spacetime for a “large” AdS Schwarzschild blackhole has two asymptotic regions, each of which has the topology of $S^3\times \mathbb{R}$. Setting the radius of AdS to 1, the metric, in Schwarzchild coordinates, is $$ds^2 = -f(r) d t^2 +f(r)^{-1} d r^2 +r^2 d\Omega_3^2$$ with $$f(r)= 1+r^2 - \frac{r_0^2 r_1^2}{r^2}, \qquad r_1^2 = 1+r_0^2$$ The horizon is at $r=r_0$ and Schwarzschild time, $t$, is identified with time in the boundary theory.

It’s convenient to replace $r$ by the tortoise coordinate $$z= - \frac{\beta}{4\pi} \log\left(\frac{r-r_0}{r+r_0}\right)+\frac{\tilde{\beta}}{2\pi} \tan^{-1}(r_1/r)$$ where $$\beta =\frac{2\pi r_0}{r_0^2 +r_1^2},\qquad \tilde{\beta} = \frac{2\pi r_1}{r_0^2 +r_1^2}$$ so that region I is $-\infty \lt t\lt \infty$, $0\lt z\lt \infty$.

The Schwarzschild coordinates only cover region I. But if one complexifies $(t,z)$, subject to the identifications $$t\sim t+ i\beta \frac{n+n'}{2},\qquad z\sim z+ i\tilde{\beta} \frac{n-n'}{2},\qquad n,n'\in\mathbb{Z}$$ (so that $\beta$ is the inverse Hawking temperature), one can identify region III as $$Im(t)= -i\beta/2,\qquad Im(z)=0$$

The idea is to study real-time thermal Wightman functions $$G(t,\vec{x}) = Tr\left[e^{-\beta H} \mathcal{O}(t,\vec{x}) \mathcal{O}(0,0)\right]$$ of the boundary gauge theory. If we analytically continue these Wightman functions to complex time, we compute the correlation function of operators inserted on opposite asymptotic boundaries $$G_{12}(t) = Tr\left[e^{-\beta H} \mathcal{O}(t-i\beta/2) \mathcal{O}(0)\right]= G(t-i\beta/2)$$

$G_{12}$ is, in turn, related to a bulk 2-point function. For large dimension of the operator, $\mathcal{O}$, the mass of the corresponding bulk particle is large¹, and one expects the correlation function to be dominated by a saddle-point, corresponding to a spacelike geodesic, connecting the two points on opposite boundaries, and which passes behind the horizon. As $|t|\to t_c= \tilde{\beta}/2$, this geodesic passes closer and closer to the singularity.

So, by studying the gauge theory correlation function for complex time, one can probe the geometry behind the horizon, and near the singularity. Or so they thought. As it turns out, however, this real-space geodesic is not the only saddle-point. And before it gets close to the singularity, it ceases to be the dominant saddle-point. So the gauge theory correlation functions don’t end up probing the geometry near the singularity.

At least, that’s what happens in the case of Fidkowski et al. What Festuccia and Liu found was that, by working with the momentum-space Wightman functions, $G(\omega,\vec{l})$, where $\vec{l}$ are spherical harmonics on the $S^3$, they can isolate the contribution of the desired saddle-point and probe the region of the singularity.

Unsuprisingly, at large-$N$ and large 't Hooft coupling, the gauge theory momentum-space Wightman functions have a singularity, reproducing what one would expect from classical supergravity. The challenge, now, is to see what happens at finite 't Hooft coupling (stringy corrections to supergravity) and at finite $N$ (quantum corrections). Hong and his collaborator are hard at work …