It is necessary for the path of the probe to be smooth that the probe remembers what happened before encountering the shockwave, otherwise one may argue that the probe was killed earlier and regenerated by the shockwave. We consider the following state to study this:
|ΨT(t)i=e−i(HL+HR)teiHRT|Ψtfdi t < tin (A.17)
|ΨT(t)i=e−i(HL+HR)(t−tin)eiφLe−i(HL+HR)tineiHRT|Ψtfdi tin< t < ts (A.18)
|ΨT(t)i=e−i(HL+HR)(t−ts)eiaϕRe−i(HL+HR)(ts−tin)eiφLe−i(HL+HR)tineiHRT|Ψtfdi ts < t <0 (A.19)
|ΨT(t)i=e−i(HL+HR)teigOLXRei(HL+HR)tseiaϕR (A.20)
×e−i(HL+HR)(ts−tin)eiφLe−i(HL+HR)tineiHRT|Ψtfdi 0< t (A.21) Where eiaϕR generates a shockwave coming from the right, at some early time ts. It is important that this shockwave is extremely weak, otherwise it will kill the probe. The double trace perturbation should still be able to extract the probe after the interaction of the additional shockwave. Moreover we want to use the same double trace perturbation, whether we would send this additional shockwave or not. An alternative would be to modify the double trace perturbation to remove this additional shockwave. That setup, however, cannot answer whether the memories of the probe are genuine. We rewrite the state, for t >0, in such a way that it is clear that it corresponds to the same experiment in the thermoeld state. The correlator follows directly from that.
|ΨT(t)i=e−i(HL+HR)teigOLXRei(HL+HR)(ts)eiaϕRe−i(HL+HR)(ts−tin)eiφLe−i(HL+HR)tineiHRT|Ψtfdi (A.22)
=e−i(HL+HR)(t−T)eigOLORei(HL+HR)(ts−T)eiaϕRe−i(HL+HR)((ts−T)−tin)eiφLe−i(HL+HR)tin|Ψtfdi (A.23)
If we calculate hφRi in this state we get the same response as in the thermoeld state, with both the response and the additional shockwave being shifted by T. Thus we could extract information about the additional shockwave from the response.
Appendix B
Appendix to Chapter 3
B.1 The exterior geometry of typical black hole microstates
Here we briey review some arguments which show that at large N the exterior geometry dual to a typical microstate should be the AdS-Schwarzschild geometry. The most conser-vative version of this statement is that, to leading order at largeN and for time separations that are not too large, the boundary two-point function of light single trace operators on a typical pure state is close to the two-point function that would be computed from analytic continuation starting with the geometry of the Euclidean AdS black hole. By large N factorization this also implies that the leading large N disconnected part of higher point functions is also the same between the typical pure state and the eternal black hole.
We will start with the assumption that
Z−1 Tr[e−βHO(τ, ~x)O(0, ~0)] =hφ(τ, ~x)φ(0, ~0)iEBH+O(1/N). (B.1) where the RHS is the boundary-limit of the two-point function of the dual bulk eld that would be computed by a Witten diagram on the Euclidean black hole geometry (the subscript EBH refers to Euclidean Black Hole). We can not prove this statement, but we will take it as being true given that it is a basic prediction of the AdS/CFT correspondence at large N.
The rst step is to analytically continue both sides to real time. The leading term on the RHS decays exponentially in time. For time scales of the order of scrambling time and longer, the leading term becomes comparable to the subleading term and we will not be able to control the approximations. Hence we restrict to time scales which are smaller than that. We can also restrict the correlators by considering only frequencies which do not scale with N, similar to the ω < ω∗ approximation used in the main text. For short time scales, and ltering out high frequencies, we have that the subleading1/N corrections in the Euclidean computation will remain small. Hence we conclude that
Z−1 Tr[e−βHO(t, ~x)O(0, ~0)] =hφ(t, ~x)φ(0, ~0)iEBH+O(1/N) , tβlogN (B.2) The next step is to replace the canonical with the microcanonical ensemble. Since the RHS depends on the energy of the black hole only via the temperature, we expect that the saddle point approximation will be reliable and we conclude that
Tr[ρmO(t, ~x)O(0, ~0)] =hφ(t, ~x)φ(0, ~0)iEBH+O(1/N) , tβlogN (B.3)
The next step is to replace the microcanonical with the typical pure state. Here we expect the approximation of the LHS to be good up to exponentially small corrections, as discussed in section 3.4. Hence we nd
hΨ|O(t, ~x)O(0, ~0)|Ψi=hφ(t, ~x)φ(0, ~0)iEBH+O(1/N) , tβlogN (B.4) as claimed.
In general we expect that the time separation on the boundary has an inverse relation to the distance from the horizon. The argument above suggests that there cannot be any modications to the exterior geometry (or the state of the quantum eld on top of the geometry) to anyO(N0)distance from the horizon. The argument above does not exclude the possibility that there are modications at say Planckian distance from the horizon, which would be detectable by an observer hovering very near the horizon.
However, for the infalling observer these possible modications do not aect low point functions computed at macroscopic space and time separations in the frame of the infaller.
For example, if we compute a two-point function between a point in the interior and a point in the exterior, separated by a distance of horizon scale, the two point function is robust even if we introduce cutos in the frequencies of the modes involved, as well as on the time scales over which we have access on the boundary [6, 68].
The argument above does not directly apply to the connected part of higher point functions, since that is suppressed by powers of1/N and hence the perturbative expansion mixes up with the corrections of the order of 1/N from comparing the ensembles. It might be possible to disentangle the two types of corrections by taking the limit where the temperature of the black hole is very large. However, we will not explore this possibility further in this chapter.
Notice that similar approximations for time scales of the order of scrambling time, and for theO(1)part of higher point functions, would follow from the conjecture of section 3.4 for this system, which remains currently unproven.