On the cylindrically symmetric wormholes : The motion of test particles.
In this article we partially implement the program outlined in the previous paper of the authors AmChum2013 (). The program owes its origins to the following comment in paper CvKK (), where a class of static spherically symmetric solutions in -dimensional Kaluza–Klein theory was studied: “…We suspect that the same thing [as for spherical symmetry] will happen for axially symmetric stationary configurations, but it remains to be proven”. We study the radial and non-radial motion of test particles in the cylindrically symmetric wormholes found in AmChum2013 () of type in 6-dimensional reduced Kaluza–Klein theory with Abelian gauge field and two dilaton fields, with particular attention to the extent to which the wormhole is traversable. In the case of non-radial motion along a hypersurface (“planar orbits”) we show that, as in the Kerr and Schwarzschild geometries chan (), we should distinguish between orbits with impact parameters greater resp. less than a certain critical value , which corresponds to the unstable circular orbit of radius .111Note the difference in the notations: in chan () where is the radial spherical coordinate, here where is the radial cylindrical coordinate. For there are two kinds of orbits: orbits of the first kind arrive from infinity and have pericenter distances greater than , whereas orbits of the second kind have apocenter distances less than and terminate at the singularity at . For orbits of the first and second kinds merge and both orbits spiral an infinite number of times toward the unstable circular orbit . For we have only orbits of one kind: starting at infinity, they cross the wormhole throat and terminate at the singularity.
The general static cylindrically symmetric space-time metric can be written in the form
where is a cylindrical radial coordinate, is the longitudinal coordinate, and is the angular coordinate (see 1 ()).
The metric (1) has one timelike Killing vector and two spacelike Killing vectors , , which define the axial symmetry. It is invariant under the simultaneous interchange and ; one of the coordinates () was arbitrarily chosen to be periodic with period to represent a cylindrically symmetric (rather than plane symmetric) geometry. (Frequently the scale of is chosen uniquely to make the origin nonsingular, but this cannot be done for the metrics considered here.)
Radial null geodesics are described by , which will be non-zero for all . Therefore there is no horizon, and any singularity at the origin is visible from everywhere (“naked”).
By definition, metric (1) describes a cylindrically symmetric wormhole if the “circle radius” has an absolute minimum at some point and for all possible values of the metric functions , , and are smooth and finite 1 ().
A cylindrical hypersurface defined by the equation is called a throat of the wormhole , if can be presented as the union , where and are interpreted as the two universes at the ends of the wormhole.
The wormhole is called traversable if travel is possible from one universe ( or ) to another ( or , respectively), i.e., if a traversing timelike path through the wormhole’s throat is allowed in a finite time (see, for example, 3 ()).
In AmChum2013 () we studied cylindrically symmetric Abelian wormholes in -dimensional Kaluza-Klein theory. It was shown that static four-dimensional cylindrically symmetric solutions in -dimensional Kaluza- Klein theory with maximal Abelian isometry group of the internal space with diagonal internal metric can be obtained (as in the case of a supersymmetric static black hole CvYou ()) only if the isometry group of the internal space is broken down to the gauge group; the solutions correspond to dyonic configurations with one electric and one magnetic charge that are related either to the same or gauge field or to different factors of the gauge group of the effective six-dimensional Kaluza-Klein theory. We found new static cylindrically symmetric exact solutions of the six-dimensional Kaluza-Klein theory with two Abelian gauge fields , , a dilaton field , and a scalar field associated with the internal metric. We obtained new types of static cylindrically symmetric wormholes supported by radial and longitudinal electric and magnetic fields. In the case of radial gauge fields we found three types of static cylindrically symmetric wormholes: dyonic with nonzero electric and magnetic charges, with nonzero electric charge, and with nonzero magnetic charge. For longitudinal gauge fields we obtained nine types of dyonic wormholes , , the wormhole with nonzero electric charge, and the wormhole with nonzero magnetic charge. From the physical point of view these wormholes are interesting because they do not need exotic matter or phantom fields for support. However the universes and at the ends of the wormhole are not benign, for example contains a curvature singularity. We will leave exploration of these features to a later investigation and focus here primarily on the interesting, near-throat region.
In this article we consider the geodesic structure of the wormhole solutions of type with the space-time metric in coordinates (AmChum2013 (), Eqs. (35)-(38))
The throat radius of the is
and the wormhole is generated by the following Abelian gauge field , dilaton , and scalar field :
Further in this paper we put and consider a 3-parameter family of the wormholes with the line interval
and the throat radius
defined in coordinates by the formulas:
Further, we put , whence , , and , since the opposite choice is reduced to the previous one by a change . Metric (2) and its geodesics as well as dilaton field and scalar field are invariant under this change, while the gauge field changes sign.
is the throat of wormhole (3), which is generated by the following vector, dilaton, and scalar fields:
Note that metric (3) is not asymptotically flat, which confirms the “no-go” statement about the nonflat asymptotic behavior of a cylindrically symmetric wormhole in the absence of ghost fields, i.e., fields having negative kinetic energy 1 ().
As the main source of information about the structure of any physical field is the behavior of test bodies, we will focus on the motion of test particles in the wormhole whose trajectories are geodesics.
In 3 () a detailed derivation of solutions of the Einstein field equations was presented, which describe traversable wormholes that, in principle, could be traversed by human beings.
The creation of wormholes and their conversion into time machines as well as quantum-field-theoretic stress-energy tensors that are required to maintain a two-way traversable wormhole were discussed in ttravel ().
In tog1 () the motion of massive and massless test particles in a space-time of a slowly rotating spherically symmetric wormhole with a ghost scalar field as a source was considered, and it was shown that after crossing the wormhole throat the particles (massive or massless) moving initially radially will move in a spiral away from the throat.
The study of dynamics of null and timelike geodesics for traversable static spherically symmetric Schwarzschild and Kerr thin–shell wormholes constructed by cut-and-paste method was presented in kag ().
The radial geodesic motion of a massive particle into a version of an Einstein–Rosen bridge was considered in Poplawski (). This wormhole was constructed by gluing regions I and III of the Kruskal space-time along the future resp. past horizons, which requires a delta-function matter source violating the energy conditions. This wormhole is traversable by complete timelike geodesics. The author suggests that observed astrophysical black holes may be Einstein–Rosen bridges, each with a new universe inside.
In Hackmann () the geodesic motion of charged test particles in the gravitational field of a rotating and electromagnetically charged Kerr–Newman black hole was studied; the colatitudinal and radial motions of particles moving along timelike world lines were classified.
A class of axially symmetric stationary exact solutions of the phantom scalar field in general relativity describing rotating and magnetised wormholes was found in Matos2 ().
In Matos () properties of a Kerr-like wormhole supported by phantom matter were studied. The geodesics of the Kerr-like phantom wormhole were analysed in Miranda (), where it was shown that the wormhole can be traversable for an observer like a human being.
This article is organized as follows. In section II we discuss the general properties of geodesic motion in space-time (3). We compute Kretschmann’s invariant and show that the space–time (3) becomes singular as and has a physical (naked) singularity at (for ). In section III we describe radial motion of massive test particles. We find the turning points of the particle motion and the regions accessible by a particle with energy . Equating and , we find , We prove that the particle located at will be at the point of unstable equilibrium and study in detail the character of motion of particles with energies greater and/or less than . We find a (lower) energy threshold of traversability of the wormhole throat for a radially moving massive test particle. In section IV we consider radial trajectories of photons and prove that each of such trajectories crosses the throat. Non-radial motion along a hypersurface (“planar orbits”) is studied in section V where we draw parallels with the Kerr and Schwarzschild geometries. In section VI we investigate non-radial motion in the plane . We prove that all null orbits have a pericenter distance and terminate at radial infinity and identify four types of massive particle behavior. Conclusions are given in section VII.
Ii The general properties of geodesic motion in the space-time (1).
According to the general theory of relativity, trajectories of test particles in the absence of force fields are geodesics, which are integral curves of the equations
where denotes the absolute derivative with respect to an affine parameter , is the 4-velocity vector of the test particle, are the Christoffel symbols, and overdots denote derivatives with respect to . For metric (1) the non-vanishing Christoffel symbols are
and Eq. (6) gives
(, and the prime denotes the derivative with respect to the radial coordinate ).
The equations of motion can also be obtained from the Lagrangian
The canonical momenta are
and the Hamiltonian function is From the Hamilton’s equations we find
They are generated by the three Killing vector fields , and . The constants of motion and are interpreted as the energy and angular momenta per unit mass of the particle with non-zero rest mass.
From Eq. (12) we have
here denotes the initial value of the parameter and the integration constants , , and are the initial values of the coordinates , , and , respectively.
From Eq. (12) and the equality , where for time-like geodesics and for null geodesics, we get
or, in terms of an effective potential,
where is the conserved energy and
is the effective potential for the geodesic motion.
In the following sections we consider three possible cases: 1) , 2) , , and 3) ,
Non-zero components of Riemann curvature tensor of metric (3) are
From here we can calculate the Ricci tensor and scalar curvature :
and from these we can compute the Kretschmann invariant :
where , and .
It can be easily verified that (for ). From here it follows that space–time (3) becomes singular as and has a physical singularity at (for ).
Iii Radial motion of massive test particles.
We consider the first case: . From (12) we have and . Hence, only and can depend on . The motion in this case is called radial.
where we have denoted
It follows from Eq. (17) that the function must be nonnegative
This condition is satisfied for all real when and for with
The turning points of the particle motion are defined by equation or, by virtue of (17), by equation Solving this equation, we find or, in terms of the coordinate ,
The circle radii of the turning points are
The regions accessible by a particle with energy are () or (). For a given there is a potential barrier at (), whereas for the potential barrier vanishes.
A value of may be chosen such that and coincide, and the potential barrier vanishes. Equating and , we find (), , and from (17), (18) it follows that , . The particle located at () will be at a point of unstable equilibrium. Actually, the only equilibrium point () along radial timelike geodesics is found by solving the equation . The equilibrium is unstable, since , and the effective potential has a maximum at .
We consider trajectories of particles with moving from the turning points or . It follows from (18) that initial accelerations at the turning points are
The initial acceleration is positive for and negative for . A particle starting from rest at moves away from the singularity at (), and a particle that starts from rest at moves in the opposite direction and falls into this singularity in a finite proper time. Namely, for the latter particle we have from (17)
We also assumed that at the starting point and took into account the series expansion
Similarly, for the particle that starts from rest at we have
The particle starting from rest at is separated by the potential barrier from the singularity at . It moves away from the singularity and reaches infinity in infinite proper time .
From here we find a (lower) energy threshold of traversability of the wormhole throat for a radially moving massive test particle:
Radially moving particles starting from rest at with energy satisfying pass through the throat and fall to the singularity in a finite proper time. Radial trajectories of massive particles starting from rest at with energy do not pass through the wormhole throat.
Massive particles with energy start at the turning point on the wormhole throat and fall to the singularity in a finite proper time.
Radially moving particles with energy that start from rest at do not traverse the wormhole throat; they move away from the throat and reach infinity in infinite proper time.
In the case , i. e., when , the right-hand side of (17) is strictly positive and for all real . A particle with an initial value of the radial coordinate and positive initial radial velocity moves away from the singularity and reaches infinity in infinite proper time. If , the particle crosses the wormhole throat in a finite proper time . In the case of negative initial radial velocity the particle moves toward the singularity, crosses the wormhole throat in a finite proper time (only if ), and falls to the singularity in a finite proper time (see Eq. (21)). Thus, for the wormhole is traversable for radially moving test particles with non-zero rest masses.
and study the nature and stability of the corresponding fixed points in the phase plane . They are given by the equations whose solution is the only fixed point . The corresponding Jacobian matrix about the fixed point has the form
A particle with energy and initial values moves toward the singularity, crosses the wormhole throat (only if ) and falls to the singularity in a finite proper time