We develop an iterative approach to span the whole set of exotic matter models able to drive a traversable wormhole. The method reduces the Einstein equation to an infinite set of algebraic conditions and easily allows the implementation of further conditions linking the stress-energy tensor components among each other, like symmetry conditions or equations of state

Manfredonia, Mattia

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DOI 10.1393/ncc/i2018-18061-4Colloquia: IFAE 2017IL NUOVO CIMENTO 41 C (2018) 61Exotic matter driving wormholesMattia Manfredonia(1)(2)(1) Dipartimento di Fisica Ettore Pancini, Università di Napoli Federico IIComplesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy(2) Istituto Nazionale di Fisica Nucleare, Sezione di NapoliComplesso Universitario di Monte S. Angelo, Via Cintia Edificio 6, 80126 Napoli, Italyreceived 21 April 2018Summary. — We develop an iterative approach to span the whole set of exotic mat-ter models able to drive a traversable wormhole. The method reduces the Einsteinequation to an infinite set of algebraic conditions and easily allows the implementa-tion of further conditions linking the stress-energy tensor components among eachother, like symmetry conditions or equations of state.1. – IntroductionWormhole solutions to Einstein equations can be seen as handles connecting twoasymptotically flat regions of the space-time manifold, i.e. a short-cut or a bridge link-ing together two distant portions of the universe. An example of wormhole is givenby the Einstein-Rosen bridge [1]. However, such a bridge is not traversable because itis part of the Schwartzschild vacuum solution [2]. In 1988, Thorne and Morris [3] in-troduced the concept of traversable wormhole and found the condition the bridge mustsatisfy in order to be safely crossed by hypothetical travellers. In ref. [4] we consid-ered a static spherically symmetric traversable wormhole. Following refs. [2, 3, 5-7] themost general metric without horizons in the Schwarzschild coordinate system (t, r, θ, φ) isds2 = gμνdxμdxν = −e2Φ(r)dt2 +(1 − b(r)r)−1dr2 + r2 (dθ2 +sin2 θdφ2), where b(r) andΦ(r) are the so-called shape and red-shift functions, respectively. A traversable worm-hole has to satisfy the flaring-out condition b′(r) < b(r)r , near the wormhole “throat” (theregion where the radial coordinate takes its minimum value r0 [2, 3]). Furthermore, rand b(r) must have the same value at the throat, namely b(r0) = r0 [3]. Hence at r0 onegets b′(r0) < 1. According to ref. [3], Birkhoff’s theorem forbids a traversable wormholewith the desired symmetries to occur in vacuum. Thus, we consider Einstein equationsin presence of some matter with non-trivial stress-energy tensor [2, 3]. We adopt theCreative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0) 12 MATTIA MANFREDONIAproper reference frame (t̂, r̂, θ̂, φ̂) introduced in refs. [3,8]. The metric gμν locally reducesto the Minkowski’s one and the Einstein tensor Gμν is diagonal. The Einstein equa-tions imply that the stress-energy tensor Tμν is diagonal as well and its non-vanishingcomponents are Tt̂t̂ ≡ ρ(r), Tr̂r̂ ≡ −τ(r), Tθ̂θ̂ = Tφ̂φ̂ ≡ p(r). The flaring-out conditionimplies that τ(r) > ρ(r) so the considered material violates the Null Energy Condition(NEC) [2,3, 7, 9].2. – Exotic matter in a neighbourhood of the wormhole throatIn the chosen frame Gμ̂ν̂ and Tμ̂ν̂ are both diagonal and Einstein equations readGμ̂ν̂ = 1m2PTμ̂ν̂ , where mP is the Planck mass. We recast all quantities involved in termsof dimensionless ones, namely r̄ ≡ mP r, b̄ ≡ mP b, Φ̄ = Φ, ρ̄ ≡ ρ/m4P , τ̄ ≡ τ/m4Pand p̄ ≡ p/m4P . We aim to characterise the whole set of exotic matter models ableto drive a traversable wormhole. To achieve such a goal we expand around the throatall quantities entering in equations. A set of algebraic conditions is obtained order byorder in the relative distance from the throat (r̄ − r̄0)/r̄0. Due to flaring-out, one hasto fix b̄0 = 1 and b̄1 < 1. The zero-th order equations are {b̄1 = ρ̄0 < 1, τ̄0 = 1,Φ̄1 = 2p̄01−ρ̄0 − 1}, and first order conditions are {b̄2 = 2ρ̄0 + ρ̄1, τ̄1 = −(4p̄0 + ρ̄0 + 1),Φ̄2 = 6p̄0+2p̄13(1−ρ̄0) +2p̄0(2ρ̄0+ρ̄1)3(1−ρ̄0)2 −13 (2p̄01−ρ̄0 − 1)(4p̄01−ρ̄0 + 1)}. For a more detailed list see [4].All the above equations hold for an arbitrary throat size r0. For each order five newcoefficients enter into the game, whereas three new conditions must be satisfied, henceone can fix two of such quantities to obtain all the others. The set of equations furthersimplifies if one considers additional constraints F (ρ̄, p̄, τ̄) = 0 linking the componentsof the stress-energy among each other (for example symmetry conditions or equationsof state). By expanding F we get additional conditions {F0 = F (ρ̄0, p̄0, τ̄0) = 0, F1 =1r̄20[∂F∂ρ̄ |r̄0 ρ̄1 +∂F∂p̄ |r̄0 p̄1 +∂F∂τ̄ |r̄0 τ̄1] = 0, . . . }. Far away from the wormhole throat, themetric must approach the Schwartzschild vacuum. We bound the exotic matter into afinite spherical region of radius R̄ surrounded by a shell of different material [3, 2, 8].However, the size of this finite region is not completely arbitrary. Indeed, we requirethe shape function b̄(r̄) to have only one point where b̄(r̄) = r̄, which by construction isalready verified for r̄ = r̄0. If a radius r̄∗ > r̄0 such that b̄(r̄∗) = r̄∗ exists then the exoticmaterial must be confined in a smaller region (this provides the upper bound R̄ < r̄∗).Outside the shell b̄(r̄) is constant. We related this value with the dimensionless mass M̄featuring in the outer Schwartzchild solution [4]. Such a mass M̄ =∫ R̄r̄0ρ̄(r̄)r̄2dr̄ + r̄0 isa function of the stress-energy and of the radii r̄0 and R̄ .3. – ConclusionsBy performing series expansion of functions b(r), φ(r), ρ(r), p(r), τ(r) we reduce theEinstein equations to an infinite set of algebraic conditions. For each order we havefive new coefficients and three new conditions to be satisfied. This allows one to fixarbitrarily two quantities out of five to get all the others by iterating the procedure. Inorder to reduce the d.o.f. of the solution, it is possible to implement an extra constraintsF (ρ̄, p̄, τ̄) = 0. In ref. [4], we applied the method to some relevant examples characterisedby constant mass-energy density and isotropy, or described by quintessence-like equationsof state [10]. The approach provides up to a certain order in the relative distance fromthe throat the radial profile of all quantities of the described wormhole source.EXOTIC MATTER DRIVING WORMHOLES 3∗ ∗ ∗The author acknowledges support by the Istituto Nazionale di Fisica Nucleare I.S.TAsP and the PRIN 2012 “Theoretical Astroparticle Physics” of the Italian Ministerodell’Istruzione, Università e Ricerca.REFERENCES[1] Einstein A. and Rosen N., Phys. Rev., 48 (1935) 73.[2] Visser M., Lorentzian Wormholes. From Einstein to Hawking (American Institute ofPhysics) 1996.[3] Morris M. S. and Thorne K. S., Am. J. Phys., 56 (1988) 395.[4] Chianese M., Di Grezia E., Manfredonia M. and Miele G., Eur. Phys. J. Plus, 132(2017) 164.[5] Morris M. S., Thorne K. S. and Yurtsever U., Phys. Rev. Lett., 61 (1988) 1446.[6] Visser M., Phys. Rev. D, 46 (1992) 2445.[7] Lobo F. S. N., Int. J. Mod. Phys. D, 25 (2016) 1630017.[8] Misner C., Thorne K. and Wheeler J., Gravitation (W. H. Freeman) 1973.[9] Visser M. and Barcelo C., Energy conditions and their cosmological implications, inProceedings, 3rd International Conference on Particle Physics and the Early Universe(COSMO 1999): Trieste, Italy, September 27-October 3, 1999 (World Scientific) 2000,pp. 98–112.[10] Lobo F. S. N., Phys. Rev. D, 71 (2005) 084011.

Exotic matter driving wormholes

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Exotic matter driving wormholes

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