Let's clarify some common misconceptions here.

Suppose we have a spherically symmetric black hole. Let's perform a mode analysis here. For simplicity, work with l=0 spherical harmonics for massless fields first. The same conclusion still applies to higher harmonics or massive fields, but the analysis is more complicated.

Work in Eddington-Finkelstein coordinates. The modes are complex oscillatory modes of the form $A(r,t)\exp[-i\theta(r,t)]$. Even if a quantum field is real, we still need a complex mode basis for a Bogoliubov analysis using creation and annihilation operators. Roughly speaking, we call a mode infalling if the wavefronts of constant phase $\theta$ are infalling. Ditto for outgoing modes. In the eikonal approximation, the wavefronts are null. Outgoing null wavefronts can either lie outside the event horizon, or inside it. If inside, even though they are locally outgoing, they will still hit the singularity.

If $\theta$ increases with time, we say it's a positive frequency mode. If it decreases, it's a negative frequency mode. The former is associated with creation operators, while the latter is associated with annihilation operators. **The positive/negative frequency analysis is independent of whether a mode is infalling/outgoing.**

Suppose $\theta$ goes exponentially in time $t/R$. Then, even though $\theta$ increases monotonically with time, the Fourier transform of $A(t)\exp[-i\theta(t)]$ will contain negative frequency components. This is the case for outgoing modes for black holes in the near horizon region where t is the Eddington-Finkelstein time, for locally "natural" looking modes according to freefalling observers.

In general, there are a couple of *distinct* cases of entangled Hawkng pairs:

- Both Hawking quanta are locally outgoing, but one lies outside the horizon while the other lies inside.
- Both Hawking quanta are locally outgoing, and both lie outside the horizon. So, the entanglement is between two outgoing external Hawking quanta.
- A Hawking quanta of the pair is locally outgoing and lies outsiide the horizon, while the other is locally infalling.
- Other miscellaneous cases, like both infalling, or both locally outgoing but inside the horizon.

These cases must not be confused with each other! Some authors have claimed entanglement between Hawking pairs have transplankian origins at the event horizon. This is the case for case 1. The origins for case 1 and case 2 entanglements come from frequency mixing due to the Fourier transform of $A(t)\exp[-i\theta(t)]$ with exponentially varying $\theta$ in time.

Of course, we can ask why a freefalling observer ought to detect no local excitations outside the horizon. This isn't the case for a nonblack hole metric which is Schwarzschild up till a short distance $d < h$ above where the horizon ought to be, but with a massive shell at that height so that the shell's interior contains no black hole. Then, it's the external observer static relative to the shell who detects no quanta while freefalling observers will detect quanta.

For a black hole which formed a finite time in the past, we can trace locally outgoing modes outside the event horizon to transplankian modes present during the formation of the black hole a long long time in the past in semiclassical analysis. If such transplankian analyses are valid, we can conclude a freefalling observer will detect no local quanta outside the horizon. However, we ought to make transplankian cutoffs. In that case, we still need to justify why a freefalling observer ought to detect no locally outgoing quanta at a height of h above the horizon. The equivalence principle by itself isn't enough.

Case 3 can only come about from mixings between outgoing and infalling modes, which is distinct from positive/negative frequency mixings. This doesn't happen in 1+1D for CGHS models with a massless scalar/fermion. Such fields in 1+1D only couple to the conformal metric modulo surface Lagrangian terms. So, **there are no firewalls in the CGHS model.**

However, let's look at spherically symmetric 3+1D black hole metrics. Then, a near horizon mode which is locally outgoing is backscattered by the curved metric when it is redshifted to around the black hole radius R so that it becomes a linear combination of far away outgoing modes and a backscattered infalling mode. The mode goes as
$$e^{-i\omega t}\frac{1}{r} e^{i\omega r},\;r\gg R$$
and
$$e^{-i\omega t}\left[ A(r)e^{i\theta_1(r)} + B(r)e^{-i\theta_2(r)} \right],\; 0<r-R\ll R$$
A gives the initial outgoing near horizon mode. B gives the backscattered mode. Both $\theta_1,\theta_2$ increase monotonically with r. No significant backscattering occurs during the period when the locally outgoing wavelength is still much less than R.

What happens is due to frequency mixing among the locally outgoing modes when we move from freefalling proper time to EF time while counting the number of wavefronts intersected, we initially get case 2 entanglement between two near horizon locally outgoing modes after a Bogoliubov transformation. Then, when these modes are redshifted to the R scale, one of the quanta escapes far away while the other backscatters. We end up with case 3 entanglement.

So, if we have an outgoing external Hawking quanta, it could be entangled with another locally outgoing Hawking quanta behind the horizon which is eventually dragged to the singularity (case 1), or another outgoing external Hawking quanta (case 2), or a locally infalling Hawking quanta (case 3).

Suppose a freefalling observer at a height of $\ell_P \ll h \ll R$ above the horizon observes no excited modes with wavelength around $h$. Then, a Bogoliubov transform from freefalling local time to EF external time means locally, there ought to be case 2 entanglements at a height of h according to an external observer. Then, after a time of order $R\ln(R/h)$ according to distant clocks, some of the outgoing quanta get backscattered.

Suppose the information contained in the totality of all outgoing external Hawking quanta which never get backscattered is a very scrambled version of all the infalling information which ever enter the black hole throughout its lifetime. Wait until more that 1/2 of all the outgoing Hawking quanta has been emitted. Then, the totality of all the external Hawking quanta radiated after then has to be maximally entangled with the totality of all the external Hawking quanta emitted before then. This is a debatable assumption. If so, by the monogamy of entanglement, any external Hawking quanta can't be entangled at all with any nonexternal Hawking quanta. There's no way an external observer can test for case 1 entanglement. However, he can test for case 3 backscattered entanglement before the backscattered infalling mode crosses the horizon. In fact, he can do so while the backscattered quanta is still above the horizon by much more than the Planck scale.

So, if we wish to avoid a violation of the monogamy of entanglement between modes which are all outside the horizon by far more than the Planck scale, and are hence accessible to an external observer, we have to conclude there can't be any case 3 entanglement according to an external observer, not even those dynamically generated from backscattering. This means, evolving backward in external time by $R\ln(R/h)$, if a *freefalling* probe beams the results of measuring the presence of excited modes with local wavelength of order h at a height of h above the horizon, the beamed results ought to report the detection of excited quanta. This is the firewall. To translate back from the external frame to the freefalling frame requires another inverse Bogoliubov transformation.

Evolving further backward in external time by $R\ln(h/\ell_P)$, this firewall is at a Planck height above the horizon. We expect transplanckian cutoffs in quantum gravity, so we should not evolve any further backward according to semiclassical gravity. The best we can say is that this firewall came from a stretched horizon hovering a Planck distance above where the horizon ought to be.

This post imported from StackExchange Physics at 2014-03-09 15:48 (UCT), posted by SE-user Gidom Mera