While wormholes are as good a prediction of Einstein’s theory as black holes, they are subject to severe restrictions from quantum field theory. In particular, holding a wormhole open requires a violation of the null energy condition, calling for the existence of exotic matter. The Casimir effect has shown that this physical requirement can be met on a small scale, thereby solving a key conceptual problem. The Casimir effect does not, however, guarantee that the small-scale violation is sufficient for supporting a macroscopic wormhole. The purpose of this paper is to connect the Casimir effect to noncommutative geometry, which also aims to accommodate small-scale effects, the difference being that these can now be viewed as intrinsic properties of spacetime. As a result, the noncommutative effects can be implemented by modifying only the energy momentum tensor in the Einstein field equations, while leaving the Einstein tensor unchanged. The wormhole can therefore be macroscopic in spite of the small Casimir effect.
References
[1]
Morris, M.S. and Thorne, K.S. (1988) Wormholes in Spacetime and Their Use for Interstellar Travel: A Tool for Teaching General Relativity. American Journal of Physics, 56, 395-412. https://doi.org/10.1119/1.15620
[2]
Casimir, H.G.B. (1948) On the Attraction Between Two Perfectly Conducting Plates. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, 51, 793-795.
[3]
Garattini, R. (2019) Casimir Wormholes. European Physical Journal C, 79, Article ID: 951. https://doi.org/10.1140/epjc/s10052-019-7468-y
[4]
Witten, E. (1996) Bound States of Strings and p-Branes. Nuclear Physics B, 460, 335-350. https://doi.org/10.1016/0550-3213(95)00610-9
[5]
Seiberg, N. and Witten, E. (1999) String Theory and Noncommutative Gometry. Journal of High Energy Physics, 9909, Article ID: 032. https://doi.org/10.1088/1126-6708/1999/09/032
[6]
Smailagic, A. and Spallucci, E. (2003) Feynman Path Integral on a Non-Commutative Plane. Journal of Physics A, 36, L-467-L-471. https://doi.org/10.1088/0305-4470/36/33/101
[7]
Nicolini, P., Smailagic, A. and Spallucci, E. (2006) Noncommutative Geometry Inspired Schwarzschild Black Hole. Physics Letters B, 632, 547-551. https://doi.org/10.1016/j.physletb.2005.11.004
[8]
Nicolini, P. and Spallucci, E. (2010) Noncommutative Geometry-Inspired Dirty Black Holes. Classical and Quantum Gravity, 27, Article ID: 015010. https://doi.org/10.1088/0264-9381/27/1/015010
[9]
Rinaldi, M. (2011) A New Approach to Non-Commutative Inflation. Classical and Quantum Gravity, 28, Article ID: 105022. https://doi.org/10.1088/0264-9381/28/10/105022
[10]
Rahaman, F., Kuhfittig, P.K.F., Ray, S. and Islam, S. (2012) Searching for Higher Dimensional Wormholes with Noncommutative Geometry. Physical Review D, 86, Article ID: 106101. https://doi.org/10.1103/PhysRevD.86.106010
[11]
Kuhfittig, P.K.F. (2013) Macroscopic Wormholes in Noncommutative Geometry. International Journal of Pure and Applied Mathematics, 89, 401-408. https://doi.org/10.12732/ijpam.v89i3.11
[12]
Nozari, K. and Mehdipour, S.H. (2008) Hawking Radiation as Quantum Tunneling from a Noncommutative Schwarzschild Black Hole. Classical and Quantum Gravity, 25, Article ID: 175015. https://doi.org/10.1088/0264-9381/25/17/175015
[13]
Liang, J. and Liu, B. (2012) Thermodynamics of Noncommutative Geometry Inspired BTZ Black Hole Based on Lorentzian Smeared Mass Distribution. Europhysics Letters, 100, Article ID: 30001. https://doi.org/10.1209/0295-5075/100/30001
[14]
Kuhfittig, P.K.F. (2020) Accounting for the Large Radial Tension in Morris-Thorne Wormholes. European Physical Journal Plus, 135, Article ID: 50. https://doi.org/10.1140/epjp/s13360-020-00511-8