Desired Properties of Traversable Wormhole

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Observable effects of wormholes.

Wormholes inhabiting the universe may produce visible effects. Cramer et al [23] suggested that a wormhole mouth embedded in high mass density may cause gravitational lensing effect and the resulting light enhancement can be observed. Torres et al, examined the scenario of extragalactic microlensing for natural wormholes in [102] and [103]. Anchordoqui et al [5] investigated a number of time profiles of gamma ray bursts searching for observable signatures produced by microlensing events related to natural wormholes. Safonova, Torres and Romero [104] provided computer simulations of exotic mass microlensing effects that can be produced by large wormholes and methods for finding wormholes in the observational microlensing experiments.

Metric and Einstein Field Equations

A solution of the equations describing some system is stationary if there exists a special coordinate system in which it is time independent, i.e. all metric terms are not explicit functions of time. Being static means even more: the solution cannot be evolutionary in any way in some coordinate system. In particular the static metric is invariant under a time reversal with respect to any origin of time. Thus all static metric terms of type gtα where t 6= α (t is a time coordinate, α is spacelike coordinate) must vanish in some coordinate system. Mathematically we can say that a space-time is stationary if and only if it admits a timelike Killing vector field. A space-time we call static if additionally this Killing vector field is hypersurface orthogonal.

METRIC AND EINSTEIN FIELD EQUATIONS 23

Spherical symmetry means that there exists a privileged point such that the system is invariant under spatial rotations around this point. Since the reflection, or a coordinate reversal is a special case of rotation, we cannot have terms of mixed type gαβ (α 6= β). Mathematically we would say that a space time is spherically symmetric only if it admits three linearly independent spacelike Killing vector fields Kα whose orbits are closed and which satisfy [26], [27] [Kα , Kβ ] = αβγKγ (2.1) where αβγ is an antisymmetric tensor such that 123 = 1. Since the unit line element of the 2-sphere is ds2 = dθ2 + sin2 θdφ2 we take our static and spherically symmetric metric to be in general form [79]

WORMHOLE EXAMPLES 31

where the index 0 indicates that we are operating in immediate throat surroundings. Above result is central to the wormhole analysis since it indicates that the tension has to be greater than the mass-energy density and this undermines the physical reasonability of stress-energy tensor. It is widely recognized that such a situation is impossible when we are dealing with classical matter only. The generalization of this statement is called “energy condition” and we are going to commit the whole next chapter to its description and analysis. We give name “exotic” to the matter that exhibits property (2.37).

VIOLATIONS 41

not hold when the curved background is applied [33]. For example Krasnikov [70] by creating an explicit example shows that QI break down when a conformal scalar field in the two-dimensional de Sitter space is considered. Other objections that can be applied to QI are listed in [77] and the strong defence of QI validity is presented in [92]. An interesting consequence of pointwise and averaged EC is the topological censorship theorem. It states that the topology of any physically reasonable isolated system (i.e. where EC and AEC hold) is shrouded. That is a spacetime may contain isolated topological structures such as wormholes, but there is no method by which this structure can be observed [93]. Let γ0 be a causal curve with past and future endpoints lying in an asymptotically flat region.

QUANTIFICATION OF EXOTIC MATTER 47

During inflation the second term is strongly dominant and quantum fluctuations will cause variation in V (φ). Thus H will oscillate, causing its derivative to change sign. This implies that Tabk ak b in (3.38) will occasionally become negative and thus NEC will not hold. There exist a number of other physical conditions causing EC violations. One could list Hawking evaporation [39], Hawking-Hartle vacuum [108], moving mirrors [13] and a couple of others [107]. The quantum-induced violations are usually very small, typically of order of ~, and it is questionable if they suffice to support traversable wormhole. To evaluate the worth of quantum effects for wormhole design one has to know how much of exotic matter is needed.

Contents :

• 1 Introduction
  • 1.1 Faster Than Light Mechanisms
  • 1.2 Historical Perspective
  • 1.3 Concept of a Wormhole
• 1.4 Current State of Research
• 2 Morris-Thorne Framework
  • 2.1 Desired Properties of Traversable Wormhole
  • 2.2 Metric and Einstein Field Equations
  • 2.3 Traversability Conditions
  • 2.4 Wormhole Examples
    • 2.4.1 Zero radial tides
    • 2.4.2 Zero density
• 3 Energy Conditions
  • 3.1 Pointwise and Averaged Energy Conditions
  • 3.2 Violations
  • 3.3 Quantification of exotic matter
• 4 Evolving Wormholes
  • 4.1 Types of Evolution
  • 4.2 Conformal Approach
  • 4.3 Inflated Wormhole
  • 4.4 Cosmological Wormhole
• 5 Rotating Wormholes
  • 5.1 Rotating Gravitational Fields
  • 5.2 Rotating Wormhole
• 6 Summary
• A Einstein Tensor for Morris-Thorne Wormhole
• B Einstein Tensor for Conformal Wormhole
• C Einstein Tensor for Inflating Wormhole
• D Einstein Tensor for Cosmological Wormhole
• E Cartan Calculus and Rotating Wormhole
• Bibliography

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STATIC AND DYNAMIC TRAVERSABLE WORMHOLES