Authors:
(1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute.
Table of Links
Brief Review of Wedge Holography
Emerging Multiverse from Wedge Holography
Application to Information Paradox
Application to Grandfather Paradox
Acknowledgements and References
3 Emerging Multiverse from Wedge Holography
In this section, we discuss how one can describe multiverse from wedge holography.
3.1 Anti de-Sitter Background
In this subsection, we construct a multiverse from AdS spacetimes. Let us first start with the simplest case discussed in 2. To describe multiverse, we need multiple Karch-Randall branes located at r = ±nρ such that bulk metric should satisfy Neumann boundary condition at the aforementioned locations. Extrinsic curvature on the Karch-Randall brane and its trace is computed as:
Three descriptions of our setup are as follows:
• Boundary description: d-dimensional boundary conformal field theory with (d − 1)- dimensional boundary.
• Intermediate description: All 2n gravitating systems are connected at the interface point by transparent boundary condition.
• Bulk description: Einstein gravity in the (d + 1)-dimensional bulk.
We see that in the intermediate description, there is a transparent boundary condition at the defect; therefore multiverse constructed in this setup consists of communicative universes localized on Karch-Randall branes (see Figs. 2,3). Wedge holography dictionary for “multiverse” with 2n
AdS branes can be stated as follows.
3.2 de-Sitter Background
In this subsection, we study the realization of the multiverse in such a way that the geometry of Karch-Randall branes is of de-Sitter spacetime. Wedge holography with de-Sitter metric on Karch-Randall branes was discussed in [42] where the bulk spacetime is AdS spacetime and in [52] with flat space bulk metric. Before going into the details of construction of “multiverse” with de-Sitter geometry on Karch-Randall branes, first let us summarise some key points of [52].
Authors in [52] constructed wedge holography in (d + 1)-dimensional flat spacetime with Lorentzian signature. Karch-Randall branes in their construction have either geometry of d dimensional hyperbolic space or de-Sitter space. Since our interest lies in the de-Sitter space therefore we only discuss the results related to the same. Geometry of the defect is S d−1 . Wedge holography states that
Third line in the above duality is coming from dS/CFT correspondence [53, 54]. Authors in [52] explicitly calculated the central charge of dual CFT which was imaginary and hence CFT living at the defect is non-unitary.
The above discussion also applies to the AdS bulk as well. In this case one can state the wedge holographic dictionary as:
branes are obtained as:
• Boundary description: d-dimensional BCFT with (d − 1)-dimensional defect.
• Intermediate description: 2n gravitating systems with de-Sitter geometry connected to each other at the (d − 1)-dimensional defect.
• Bulk description: (d + 1)-dimensional Einstein gravity with negative cosmological constant in the bulk.
First and third description are related to each other via AdS/BCFT correspondence and (d−1)- dimensional defect which is non-unitary CFT exists because of dS/CFT correspondence [53, 54]. de-Sitter space exists for finite time and then disappear. Another de-Sitter space born after the disappearance of previous one [55]. Therefore it is possible to have a “multiverse” (say M1) with de-Sitter branes provided all of them should be created at the same “creation time” [7] but this will exist for finite time and then M1 disappears. After disappearance of M1, other multiverse (say M2) consists of many de-Sitter branes born with same creation of time of all the de-Sitter branes.
3.3 Braneworld Consists of Anti de-Sitter and de-Sitter Spacetimes
In this subsection, we have discussed the embedding of different types of Karch-Randall branes in the different bulks which are disconnected from each other. Authors in [55] have discussed
the various possibilities of embedding of different types of branes, e.g., Minkowski, de-Sitter and anti de-Sitter branes in the same bulk. Existence of various branes are characterized by creation time τ∗. There is finite amount of time for which Minkowski and de-Sitter branes born and there is no cration time for anti de-Sitter branes. Out of various possibilities discussed in [55], it was pointed by authors that one can see Minkowski, de-Sitter and anti de-Sitter brane at the same time with creation time τ∗ = −π/2 in a specific bulk. In this case, branes have time dependent position. First we will summarise this result [10] and then comment on the realization of the same from wedge holography.
Bulk AdS5 metric has the following form:
Comment on the Wedge Holographic Realization of Mismatched Branes: One can construct doubly holographic setup from (19) using the idea of AdS/BCFT. Let us state the three possible descriptions of doubly holographic setup constructed from (19).
• Boundary description: 4D quantum field theory (QFT) at conformal boundary of (19).
• Intermediate description: Dynamical gravity localized on 4D end-of-the-world brane coupled to 4D boundary QFT.
• Bulk description: 4D QFT defined in the first description has 5D gravity dual whose metric is (19).
Due to covariant nature of AdS/CFT duality it remains the same if one works with the changed coordinates in the bulk i.e. different parametrisations of AdS does not imply different dualities [11] and therefore in the above doubly holographic setup, we expect defect to be 3-dimensional conformal field theory because 4-dimensional gravity is just FRW parametrization of AdS4 spacetime (20). Relationship between boundary and bulk description is due to AdS/CFT correspondence, in particular, this kind of duality was studied in [56] where bulk is de-Sitter parametrization of AdS4 and conformal field theory is QFT on dS3. As discussed in detail in appendix A of [55] and summarised in this subsection that one can also have de-Sitter and Minkowski branes in this particular coordinate system (19). If one works with de-Sitter metric (21) on end-of-the-world brane then we expect defect CFT to be non-unitary. Due to dynamical nature of gravity on Karch-Randall brane, holographic dictionary is not well understood in the braneworld scenario.
Now let us discuss what is the issue in describing wedge holography with “mismatched branes”. Wedge holography has “defect CFT” which comes due to dynamical gravity on Karch-Randall branes. Suppose we have two Karch-Randall branes with different geometry, one of them is AdS brane and the other one is de-Sitter brane. Then due to AdS brane, defect CFT should be unitary and due to de-Sitter brane, defect CFT should be non-unitary. It seems that we have two different CFTs at the same defect. This situation will not change even one considers four branes or in general 2n branes. Hence, one may not be able to describe “multiverse” with mismatched branes from wedge holography. That was just an assumption. Common boundary of multiverses M1 and M2 (described in Fig. 5) can’t be the same even when geometry is (19) due to “time-dependent” position of branes. All the AdS branes in M1 can communicate with each other via transparent boundary conditions at the defect and similarly all the de-Sitter branes in M2 are able to communicate with each other. But there is no communication between M1 and M2 even in (19).
Therefore we conclude that we can create multiverse of same branes(AdS or de-Sitter) but not the mixture of two. Hence issue of mismatched branes do not alter from wedge holography perspective too. Multiverse of AdS branes exists forever whereas multiverse of de-Sitter branes has finite lifetime [12] .
[3] It seems that some of branes have negative tension. Let us discuss the case when branes are located at −nρ1 and nρ2 with ρ1 6= ρ2. In this case tensions of branes are (d − 1) tanh(−nρ1) and (d − 1) tanh(nρ2). Negative tension issue can be resolved when we consider ρ1 < 0 and ρ2 > 0 similar to [48]. Therefore this fixes the brains stability issue in our setup. This discussion is also applicable to the case when ρ1 = ρ2.
[4] When we discuss multiverse then α and β will take 2n values whereas when we discuss wedge holography then α, β = 1, 2
[5] Explicit derivation of (14) was done in [42] for two Karch-Randall branes. One can generalize the same for 2n Karch-Randall branes. In this setup upper limit of integration will be different for different locations of Karch-Randall branes.
[6] See [42] for the explicit derivation. Only difference is that, in our setup, we have β = 1, 2, ..., n.
[7] Creation time is defined as the “time” when any universe born [55].
[8] In this case, warp factor will be different in the bulk metric. Exact metric is given in (45).
[9] We thank J. Maldacena for comment on this.
[10] For more details, see [55]
[11] We thank K. Skenderis to clarify this to us and pointing out his interesting paper [56]
[12] We thank A. Karch for very helpful discussions on the existence of de-Sitter branes and issue of mismatched branes in wedge holography.
This paper is available on arxiv under CC 4.0 license.