We investigate the traversable wormholes in (2 + 1) dimensions in the context of Gravity’s Rainbow which may be one of the approaches to quantum gravity. The cases in the presence of cosmological constant and Casimir energy are studied.

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Quantum gravity has been studied and investigated for a long time by lots of researchers. However we could not obtain even the twinkles of quantum gravity. Wormholes may give us some hints toward quantum gravity which may be different from black holes. Wormholes are the solutions to the Einstein field equations that have topological structures with a throat connecting two asymptotically regions of spacetime , called mouth.[1], [2], [3]. They can be interpreted, classically, as instantons describing a tunneling between two distant regions. The traversable wormhole solution was at first investigated by Morris and Thorne to verify the realistic possibilities of travelling through it [4]. A wormhole needs a negative energy in order to form and keep wormhole structure[4], though the standard energy conditions are violated. Without such a exotic matter which is defined to be violated energy conditions , a wormhole cannot keep the structure to collapses. Such a situation may suggest

that wormholes belong to the realm of a possible quantum gravity[14] . Casimir energy is one of the candidates as such a negative energy. A Casimir wormhole in D dimensions [6] can be constructed if D > 3. But a three dimensional traversable Casimir wormhole cannot be constructed since it has an event horizon [5][6]. Namely, three dimensional case is very different from D(> 3) dimensional case. A traversable wormhole has two ways, which situation is very different from a black hole which has one way through horizon. In addition the cosmological constant is also the candidate of the exotic matters.The negative energy is realized if Λ < 0.

In this report we will study wormholes in (2+1) dimensions , considering Gravity Rainbow. Traversable wormholes in (2+1) dimensions were studied in the past [5][9][10].

Note that the (2+1)case is very different from the case of (3+1) in the Morris Thorne wormhole It is generally believed that the geometry of spacetime should be fundamentally described by quantum gravity above and around the Planck energy [11]. Several candidates for that description are investigated with difficulties; string theory, non-commutative geometry, loop quantum gravity , Lorentzian dynamical triangulations , etc. A key points are to study how to transit between the fundamental quantum description and the effective low energy description in terms of classical general relativity. We explore the possibility that wormhole geometries are sustained by the Casimir energy and the Cosmological constant in three dimensional space-time in the context of Gravity’s Rainbow. Gravitational Rainbow consists of a distortion of the space-time metric around the Planck energy. A wormhole is a solution to the Einstein Field Equation, which is structured by two mouthes and a throat. A traversable wormhole has no space-time singularity and no horizon . Semi-classical theory of quantum gravity may play a crucial role since we have no full quantum gravity as yet The outline of the present paper is as follows: In Sec.2, we review the wormhole in (2+1) dimensional gravity with no Gravity Rainbow. In Sec.3 we obtain the metrics of the wormholes in (2+1) dimensions in the presence of Gravity Rainbow.

#### TraversableWormholes in (2+1) dimensions with no gravitational Rainbow

The general metric of a traversable circularly symmetric wormhole in three dimensions is represented as

We studied the traversable wormhole in the presence of the Casimir Energy and the cosmological
constant in three dimensions. Considering the deformation of the metric when
the energy tends to the Planck energy, we found the Rainbow metric for the traversable
wormhole in three dimensions in the presence of the Casimir energy and the cosmological
constant.

The author is grateful to Professor. Douglas Smith and the department of Mathematical
Sciences, Durham University for giving me an opportunity to stay there in the past .
Besides he thanks many researchers for giving me the useful informations and discussions
when he took part in the conferences (domestic and international). He also thanks coworkers
of medical hospital for their warm kindness

# Review Completed (Under Processing)

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##### Editorial Information

#### Article Type

Research Article

#### Publication history

Received date: February 02, 2024

Accepted date: March 06, 2024

Published date: ?, 2024

#### Copyright

©2024 Tatsuyuki Sugawa. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

#### Citation

Tatsuyuki Sugawa. (2024) Traversable Wormholes in (2+1) dimensions and Gravity’s Rainbow. *OSP Journal of Physics and Astronomy* 2: JPA-5-...?

##### Corresponding author

#### Tatsuyuki Sugawa

Internal Medicine Division, Shinseikai hospital Kitasakurazuka, Toyonaka, 560-0022, Japan and Osaka University Yamadagaoka, Suita, 565-0701 Japan.