Time Travel and Warp Drives. One type of non-traversable wormhole metric is the Schwarzschild solution see the first diagram:. Introduction History Mathematical formulation Tests. Such an interaction prevents the formation of a gravitational singularity.
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The bulk geometry is a solution of 5-dimensional second order Einstein Gauss Bonnet gravity and causally connects two asymptotically AdS space times. Based on the analysis, we propose a very simple geometric criterium to distinguish coupling from entanglement effects among the two set of degrees of freedom associated to each of the disconnected parts of the boundary.
There is a general consensus, based on several checks, for the dual interpretation of various asymptotically AdS geometries: a big black hole solution is supposed to describe a thermal QFT state , a bulk solution interpolating between an AdS horizon corresponding to an IR conformal field fixed point and an AdS geometry at infinity of different radii realizes the renormalization group flow between two conformal fixed points .
As a third possibility, certain regular solitonic charged AdS solutions are interpreted as excited QFT coherent states . We would revisit this statement and discuss the issue of whether the two dual theories are independent, decoupled or not. For completeness we quote that in the Euclidean context a theorem states that disconnected positive scalar curvature boundaries are also ruled for complete Einstein manifolds of negative curvature  see also .
The wormholes studied in  avoided the theorem in  since they were constructed as hyperbolic slicings of AdS and supported by extra supegravity fields. An interesting second Lorentzian example with two disconnected boundaries was constructed in  by performing a non-singular orbifold of AdS3.
The result of the construction led to two causally connected cylindrical boundaries with the dual field theory involving the DLCQ limit of the D1-D5 conformal field theory, but the coupling between the different boundaries degrees of freedom was not clarified. The main difference between these two examples is that in the last case causal contact exists between the conformal boundaries.
The no-go theorem  is bypassed in the second case because the performed quotient results in the presence of compact direction with the geometry effectively being a S 1 fibration over AdS2 where the aforementioned theorem does not apply.
The no-go Lorentzian wormholes theorem  is also bypassed when working with a higher order gravity theory, moreover, higher order curvature corrections to standard Einstein gravity are generically expected for any quantum theory of gravity. However, not much is known about the precise forms of the higher derivative corrections, other than for a few maximally supersymmetric cases.
Since from the pure gravity point of view the most general theory that leads to second-order field equations for the metric is of the Lovelock type , we will choose to work with the simplest among them known as Einstein-Gauss-Bonnet theory. The action for this theory only contains terms up to quadratic order in the curvature and our interest in the wormhole 1 solution, found in , is that its simplicity permits an analytic treatment. The resulting geometry is smooth, does not contain horizons anywhere, and the two asymptotic regions turn out to be causally connected.
A perturbative stability study for the 5-dimensional solution case of  was performed in . We will revisit in the present paper the Gubser-Klebanov-Polyakov-Witten GKPW prescription [2, 3] for extracting QFT correlators from gravity computations and discuss its application for the Lorentzian wormhole solution found in  mentioning along the way the similarities and differences with the AdS2 case see [15, 24] for other work on the wormhole background discussed here.
We recall that the GKPW prescription in Lorentzian signature involves not only boundary data at the conformal boundary of the spacetime but also the specification of initial and final states, we will show how these states make their appearance in the computations see [23, 28, 29, 30, 39, 40] for discussions on Lorentzian issues related to the GKWP prescription.
It is commonly accepted that the QFT dual to a wormhole geometry should correspond to two independent gauge theories living at each boundary and the wormhole geometry encodes an entangled state among them. On the other hand, the causal connection between the boundaries has been argued to give rise to a non-trivial coupling between the two dual theories .
The paper is organized as follows: in section 2 we review the GKPW prescription for computing QFT correlation functions from gravity computations mentioning the peculiarities of Lorentzian signature, in section 3 we extend the GKPW prescription for the case of two asymptotic independent boundary data. We apply it to AdS2 , reproducing the results appearing in the literature, and to the wormhole  showing their similarities. We summarize in section 5 the results of the paper.
To set out the notation we summarize the prescription for massive scalar fields highlighting the points important for our arguments. The incorporation of normalizable timelike modes induce an outgoing component from the horizon which is naturally interpreted as an excitation see [2, 28, 29, 30] and [17, 31, 32, 33] for related work. AdS global coordinates The recipe for obtaining QFT correlators from gravity computations involves evaluating bulk quantities at the conformal boundary, as might be suspected from 4 , 7 and 13 the evaluation leads to singularities and therefore requires a regularization.
We will discuss in what follows how this is done in the AdS global coordinate system since the wormhole case we will discuss later coincides in its asymptotic region with that coordinate system and will therefore be regularized in the same way. The regularization of 16 is performed imposing the boundary data at some finite distance in the bulk and taking the limit to the boundary at the end of the computations see  for a subtlety when taking the limit.
The observation is simple, any particular choice of contour is equivalent by deformation to choosing the Feynman contour plus contributions from encircling the poles Choosing the retarded contour as reference should be interpreted as giving rise to response functions instead of correlation functions.
AdS2 Lorentzian strip We will apply in this section the prescription developed above to Lorentzian AdS2 reobtaining previous results. The integrals 47 - 48 can be computed using the residue theorem once a contour in the complex plane is chosen. The observed periodicity in time relates to a peculiar property of AdS, this is the convergence of null geodesics when passing to the universal cover and can be also understood as a consequence of the eigenmodes 46 being equally spaced see -.
The geometry does not contain horizons anywhere, and the two asymptotic regions are causally connected. Coupling In this section we will review several thoughts regarding the interpretation of the results 51 and We would like to address the issue of whether the results 51 and 62 are the consequence of: i an interaction between the two dual QFT theories or ii due to the correlators being evaluated on an entangled state or iii both.
Entanglement Entropy The entanglement entropy SA is a non local quantity as opposed to correlation functions that measures how two subsystems A and B are correlated. In particular we argue that a coupling between the field theories will exist whenever the asymptotic regions are in causal contact. Consider the simplest situation corresponding to choosing C to be the Feynman contour, this is, we are computing the vacuum to vacuum transition amplitude on the field theory side.
The absence of singularities in the Euclidean continuation implies that the wormhole spacetime can be in equilibrium with a thermal reservoir of arbitrary temperature, or stated otherwise, the periodicity in Euclidean time is arbitrary. Let us see now that an interaction term Hint should be present in the Hamiltonian in order to avoid a contradiction. Note that in the Euclidean context the solution in the interior bulk is completely specified by the boundary data, no normalizable solutions exist and rotating the Feynman contour to the imaginary time axis is straightforward and leads to a non-singular solution.
Highlights and Applications The outcome of the above observations is that the wormhole geometry encodes the description of a dual field theory with two copies of fundamental fields in interaction.
Moreover, the quantum state described by the wormhole is entangled. In particular, the Euclidean continuation can be seen as prescription to separate, in the dual field theory, entanglement effects from possible interaction terms Hint. A non vanishing Euclidean two point function between operators located at different boundaries must be interpreted as originated from an interaction term Hint , rather than an entanglement effect.
The Euclidean section of the global metric 39 has two boundaries upon compactifying the time direction, and the Euclidean correlation functions can be explicitly obtained from the formulas To compute Lorentzian quantities one needs to fix a refer- ence contour Cref and the two sensible choice are retarded or Feynman. These choices relate on the QFT side to being computing either response or correlation functions. When choosing the Feynman path as reference contour, any given contour C differs from C F by contributions from encircling poles, these encircled poles fix the initial and final states that appear in the correlation functions see In section 3 we extended the GKPW prescription to spacetimes with more than one asymptotic timelike boundaries, in particular we studied the simplest two boundaries case.
In section 4 we applied the ideas on holographic entanglement entropy developed in  to the wormhole geometry. On the other hand, the causal connection between the boundaries suggest that a coupling might exist as 16 well. Summarizing, the number of disconnected boundaries of the Euclidean section determines the amount of physical degrees of freedom. We would like to emphasize finally the implications of this approach on quantum gravity, which could be seen as one of our main motivations.
This subject has been discussed in different contexts in the last years and referred to as Emergent Spacetime . In this sense, we showed how important topological and causal properties connectivity of the space time geometry are encoded in the QFT action, and that part of this information is not only in the ground state but in its interacting structure.
We hope this conclusion might contribute to the construction of rules towards a geometry engineering. We should mention that in the presence of interactions one needs to address the way the points on opposite boundaries are identified. Acknowledgments We thank D. Correa, N. Grandi, A. Lugo and J. Maldacena for useful discussions and correspondence. References  J. Maldacena, Adv. Gubser, I. Klebanov and A. Polyakov, Phys. Witten, Adv. Freedman, S. Gubser, K. Pilch and N. Warner, Adv. Lin, O. Lunin and J.
Maoz and A. Galloway, K. Schleich, D. Witt and E. Woolgar, Phys. Skenderis and B. Maldacena and L. Arkani-Hamed, J. Orgera and J.
Polchinski, JHEP , , Witten and S. Yau, Adv. Cai and G. Galloway, Adv. Buchel, Phys. Correa, J. Oliva and R. Troncoso, JHEP , Barnes, D. Vaman, C. Wu and P. Arnold,
Schwarzschild wormholes[ edit ] The first type of wormhole solution discovered was the Schwarzschild wormhole, which would be present in the Schwarzschild metric describing an eternal black hole, but it was found that it would collapse too quickly for anything to cross from one end to the other. Wormholes that could be crossed in both directions, known as traversable wormholes , would be possible only if exotic matter with negative energy density could be used to stabilize them. In order to satisfy this requirement, it turns out that in addition to the black hole interior region that particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region that allows us to extrapolate the trajectories of particles that an outside observer sees rising up away from the event horizon. And just as there are two separate interior regions of the maximally extended spacetime, there are also two separate exterior regions, sometimes called two different "universes", with the second universe allowing us to extrapolate some possible particle trajectories in the two interior regions. This means that the interior black hole region can contain a mix of particles that fell in from either universe and thus an observer who fell in from one universe might be able to see light that fell in from the other one , and likewise particles from the interior white hole region can escape into either universe. All four regions can be seen in a spacetime diagram that uses Kruskal—Szekeres coordinates. Note that the Schwarzschild metric describes an idealized black hole that exists eternally from the perspective of external observers; a more realistic black hole that forms at some particular time from a collapsing star would require a different metric.
Wormhole Theory (Lorentzian, Schwarzschild, Euclidean)
A good popular level description of Lorentzian wormholes can be found in the book Black Holes and Time Warps: After a brief overview of general relativity and quantum field theory, the author devotes the first part of the book to the history of wormhole physics. Other non-traversable wormholes include Lorentzian wormholes first proposed by John Archibald Wheeler inwormholes creating a spacetime foam in a general relativistic spacetime manifold depicted wormnoles a Lorentzian manifold and Euclidean wormholes sormholes after Euclidean manifolda structure of Riemannian manifold. Thermodynamics Schwarzschild radius M—sigma relation Event horizon Quasi-periodic oscillation Wormhlles sphere Innermost stable circular orbit Ergosphere Hawking radiation Penrose process Blandford—Znajek process Bondi accretion Spaghettification Gravitational lens. Study of such strange geometries can help better distinguish the boundaries of behavior permitted in the theory of general relativity, and also possibly provide wormhokes into effects related to quantum gravity. However, a light beam traveling through the same wormhole would of course beat the traveler. In addition, the black holes formed by a collapsing star have no associated wormhole at all.
The bulk geometry is a solution of 5-dimensional second order Einstein Gauss Bonnet gravity and causally connects two asymptotically AdS space times. Based on the analysis, we propose a very simple geometric criterium to distinguish coupling from entanglement effects among the two set of degrees of freedom associated to each of the disconnected parts of the boundary. There is a general consensus, based on several checks, for the dual interpretation of various asymptotically AdS geometries: a big black hole solution is supposed to describe a thermal QFT state , a bulk solution interpolating between an AdS horizon corresponding to an IR conformal field fixed point and an AdS geometry at infinity of different radii realizes the renormalization group flow between two conformal fixed points . As a third possibility, certain regular solitonic charged AdS solutions are interpreted as excited QFT coherent states .
LORENTZIAN WORMHOLES VISSER PDF
Another way to imagine wormholes is to take a sheet of paper and draw two somewhat distant lorfntzian on one side of the paper. The sheet of paper represents a plane in the spacetime continuumand the two points represent a distance to be traveled, however theoretically a wormhole could connect these two points by folding that plane so the points are touching. Kip Thorne and his graduate student Mike Morrisunaware of the papers by Ellis and Bronnikov, manufactured, and in published, a duplicate of the Ellis wormhole for use as a tool for teaching general relativity. For a wormholea notion of a wormhole, space can be visualized as a two-dimensional 2D surface.
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