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Since the observer is in the center of the cosmic event horizon sphere (of an approximately de Sitter space) and according the Holographic principle the event horizon has mappings of all inside objects on its surface, where the observer is then mapped?

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The holographic mapping is non-local: a "point" in the bulk is not mapped to any single point on the "boundary."

(I put "point" in quotes because the idea of a "point" in the bulk is only an approximation of limited validity, and I put "boundary" in quotes because the "boundary" isn't a subset of the bulk. It's a different, lower-dimensional way of encoding the same physics as the bulk.)

Lessons from AdS/CFT

The best-understood family of examples of the holographic principle is the AdS/CFT correspondence, where a quantum theory with gravity in asymptotically-anti-de Sitter spacetime is (almost certainly) equivalent to a lower-dimensional quantum theory without gravity. Anti-de Sitter spacetime is homogeneous: all of its points are on equal footing, so it has no distinguished center.

(The OP is asking about cosmic horizons, which is different than AdS/CFT. Still, it's an important example of how holography can work. The fact that AdS is homogeneous presents an even stronger version of the paradox described in the OP, so we can hope to get some insight into that question by considering AdS/CFT.)

The way objects in the higher-dimensional bulk are encoded in the lower-dimensional theory is analogous to the theory of quantum error-correcting codes (QEC). The relation between the higher-dimensional bulk and the lower-dimensional theory is analogous to the relationship between information encoded using logical qubits versus physical qubits in a QEC. In a QEC, the information encoded in a set of logical qubits can be recovered from any one of a number of different subsets of the physical qubits. That's why it works as an error-correcting code. In the AdS/CFT case, the information that defines an "object" of sufficiently low energy in the bulk is redundantly encoded in the lower-dimensional theory: the same object's existence and properties can be "seen" by looking any one of a number of different subsets of the "boundary."

With that in mind, a rough answer to the question is that an object anywhere in the bulk is mapped to everywhere in the lower-dimensional theory. The relationship is non-local.

This all assumes that we're talking about an "object" of sufficiently low energy in the bulk. That's because the bulk only approximately has the qualities of a spacetime continuum, and that approximation is only valid for a tiny subset of the possible states in the lower-dimensional theory — just like a QEC uses only a tiny subset of the possible states of the system of physical qubits.

This is reviewed in TASI Lectures on the Emergence of the Bulk in AdS/CFT.

Holography in more realistic spacetimes

As far as I know, we don't yet have a similar level of understanding in the case of asymptotically-de Sitter (dS) spacetime, which is a better model of the real universe. The basic message is the same, though: de Sitter spacetime also doesn't have any distinguished "center," and a holographic mapping must be non-local (not point-to-point).

The paper Quantum Gravity In De Sitter Space explains some special challenges with extending the holographic principle to asymptotically de Sitter spacetimes.

What about observer-dependent horizons?

The OP asked about the case of a cosmic horizon, which is a special case of an observer-dependent horizon. As far as I know, we don't yet have a good understanding (compared to AdS/CFT) of how holography works in that case. Bousso has some papers about how to formulate a the holographic principle for general holographic screens at a thermodynamic-like level, but an understanding of how it works in terms of microstates — how the holographic principle actually works — is not yet availalbe in that level of generality, as far as I know.

One thing is safe to say, though: the mapping must be non-local, not point-to-point.

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  • $\begingroup$ 1. If anywhere is mapped everywhere, then why we call it "boundary"? Boundary is something enclosing the bulk so there is some spatial relation. $\endgroup$ – Anixx Aug 29 at 13:23
  • $\begingroup$ 2. If there is redunancy in different parts of the boundary, then how it comes that those peces keep agreement with each other in the 2D theory being so far separated? $\endgroup$ – Anixx Aug 29 at 13:24
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    $\begingroup$ @Anixx The "boundary" language is concise but not necessarily accurate. In AdS/CFT, the "boundary" is not any subset of the bulk. It is simply a different way of encoding the same information. $\endgroup$ – Chiral Anomaly Aug 29 at 13:24
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    $\begingroup$ @Anixx The dS case isn't well-understood yet. There's a paper by Witten about why it's difficult. If I can find it, all add a link to the answer. $\endgroup$ – Chiral Anomaly Aug 29 at 13:40
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    $\begingroup$ @Anixx I don't think it's quite correct to say that the observer will see himself anywhere else. The holographic principle is an equivalence between two different ways of describing the same thing, one higher-dimensional and one lower-dimensional. But we're exceeding the limits of my understanding, and we're also exceeding the limits of how many comments are allowed without moving them to chat. :) $\endgroup$ – Chiral Anomaly Aug 29 at 13:53

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