# Can black hole complementarity be described in general relativistic language?

If my understanding is correct, Black hole complementarity says that from the perspective of those not falling into a black hole, those falling into a black hole is seen as never crossing the event horizon.

As far as my understanding goes, this can be made consistent with general relativity, as this can be understood in terms of coordinate change and that some spacetime simply is not "visible" - or non-actual coordinate singularity (but to an observer, it effectively is a singularity).

Ignoring that singularity, we can picture spacetime as usual in general relativity.

Is this correct understanding, of course ignoring necessity of quantum gravity rising right out of existence of singularities?

• I have edited in the information tag and a link to the Wikipedia article because that on article on BH complementary seems to be almost completely based on information and entropy and this edit ( if approved), may more accurately reflect your question. – user163104 Jul 24 '17 at 17:51
• I'm not sure if I understand this question. Have you looked at this sort of thing in a spacetime diagram? The resolution of any sort of paradox should be clear there. – Jerry Schirmer Jul 24 '17 at 18:55

If my understanding is correct, Black hole complementarity says that from the perspective of those not falling into a black hole, those falling into a black hole is seen as never crossing the event horizon.

Black hole complementarity (BHC) does not say that. This is a conclusion drawn from general relativity alone, that an asymptotic observer records infinite time watching an observer cross the event horizon, and that the proper time of an infalling observer is finite.

There are 3 postulates proposed in the paper by Susskind, Thorlacius and Uglum for their description of BHC. I will paraphrase them briefly below:

1. From the perspective of an outside observer, the formation and evaporation of a black hole is consistent with standard quantum theory, whereby the evolution proceeds through a unitary S matrix.

2. Physics outside the stretched horizon can be described by a semiclassical theory to a good approximation.

3. From the perspective of a distant observer, the black hole looks like a quantum system (see 1. above), where the number of states of the black hole is $e^{S_{BH}}$ and its thermodynamic properties arise because of coarse-graining of this hugely complex quantum system.

Here, $S_{BH}$ is the Bekenstein-Hawking entropy. The system is hugely complex because, for instance, a black hole of solar mass has $e^{S_{BH}} \sim 10^{10^{77}}$ (see this paper by Mathur). And the stretched horizon is a timelike surface slightly above the global event horizon $r=2M$ which displays thermal, electric and mechanical properties consistent with the earlier derived black hole thermodynamics incorporating mass, charge and angular momentum. It is timelike because we want to be able to observe such properties of black holes as a distant observer, which is otherwise not possible for a global event horizon as it is nulllike.

Now, there are two kinds of descriptions when observing a black hole: one from the perspective of a distant observer and the other from the perspective of an infalling observer. The outside observer sees the stretched horizon getting heated up as matter falls into it, displays thermodynamic properties and ultimately evaporates via Hawking radiation. On the other hand, an infalling observer does not encounter a stretched horizon, crosses the global event horizon, and hits the singularity in finite proper time. The infalling observer does not encounter any 'barrier' when falling in (in today's parlance, this is to say that there is no drama at the horizon), in conformity with general relativity's equivalence principle.

The matter that falls in has a quantum state description, and also one that undergoes unitary evolution because we have assumed the first BHC postulate to be true. The quantum state inside the black hole is independent of the initial infalling state. This will mean that all differences between the initial states of infalling matter must be destroyed before it crosses the global horizon. This means that something drastic is happening at the global horizon, which is contrary to the conclusion we just saw above that an infalling observer encounters nothing special at the horizon. I have just given the gist of the argument in this paragraph; the detailed analysis with a Penrose diagram is already given clearly on pages 4-5 of the Susskind, Thorlacius and Uglum paper cited above.

So we now see that if we assume a unitary quantum mechanical evolution for a black hole, it also necessarily leads to a violation of equivalence principle. What BHC really says is that although there are two copies of information, one outside the horizon from the perspective of a distant observer, and another inside the horizon from the perspective of an infalling observer, they can never compare their results with each other - an observer who has crossed the horizon will never be able to send information back to the distant observer outside the black hole. In this sense, there is a complementarity between the two descriptions, and although each description is correct within its own regime, both cannot be correct simultaneously. Only a 'superobserver' outside our universe can have access to both the Hilbert spaces of states outside the black hole $and$ inside. And so going by this scheme of logic, there is no contradiction between unitarity and equivalence principle.