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February 4, 2006

Behind the Horizon

I never did write about Fidkowski et al’s work on attempting to probe the blackhole singularity in AdS 5\text{AdS}_5, by studying correlation functions of the boundary theory. Hong Liu visited us this week, and gave a nice talk about his work with Festuccia, which gives me an excuse to catch up.

The maximally-extended spacetime for an AdS blackhole has two asymptotic regions: I, III.
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×S^3\times \mathbb{R}. Setting the radius of AdS to 1, the metric, in Schwarzchild coordinates, is ds 2=f(r)dt 2+f(r) 1dr 2+r 2dΩ 3 2 ds^2 = -f(r) d t^2 +f(r)^{-1} d r^2 +r^2 d\Omega_3^2 with f(r)=1+r 2r 0 2r 1 2r 2,r 1 2=1+r 0 2 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 0r=r_0 and Schwarzschild time, tt, is identified with time in the boundary theory.

It’s convenient to replace rr by the tortoise coordinate z=β4πlog(rr 0r+r 0)+β˜2πtan 1(r 1/r) 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 β=2πr 0r 0 2+r 1 2,β˜=2πr 1r 0 2+r 1 2 \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 <t<-\infty \lt t\lt \infty, 0<z<0\lt z\lt \infty.

The Schwarzschild coordinates only cover region I. But if one complexifies (t,z)(t,z), subject to the identifications tt+iβn+n2,zz+iβ˜nn2,n,n 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 β\beta is the inverse Hawking temperature), one can identify region III as Im(t)=iβ/2,Im(z)=0 Im(t)= -i\beta/2,\qquad Im(z)=0

The idea is to study real-time thermal Wightman functions G(t,x)=Tr[e βH𝒪(t,x)𝒪(0,0)] 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[e βH𝒪(tiβ/2)𝒪(0)]=G(tiβ/2) G_{12}(t) = Tr\left[e^{-\beta H} \mathcal{O}(t-i\beta/2) \mathcal{O}(0)\right]= G(t-i\beta/2)

G 12G_{12} is, in turn, related to a bulk 2-point function. For large dimension of the operator, 𝒪\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|t c=β˜/2|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(ω,l)G(\omega,\vec{l}), where l\vec{l} are spherical harmonics on the S 3S^3, they can isolate the contribution of the desired saddle-point and probe the region of the singularity.

Unsuprisingly, at large-NN 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 NN (quantum corrections). Hong and his collaborator are hard at work …


1 Compared to the AdS cosmological constant. One still has mM plm\ll M_{pl}, of course.

Posted by distler at February 4, 2006 3:26 PM

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Re: Behind the Horizon

This sounds really interesting, thanks for mentioning it.

I also want to say that I really appreciate the paper summaries you post here.

Posted by: anon on February 7, 2006 10:27 AM | Permalink | Reply to this

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