Fate of the first traversible wormhole: blackhole collapse or inflationary expansion
Abstract
We study numerically the stability of Morris & Thorne’s first traversible wormhole, shown previously by Ellis to be a solution for a massless ghost KleinGordon field. Our code uses a dualnull formulation for spherically symmetric spacetime integration, and the numerical range covers both universes connected by the wormhole. We observe that the wormhole is unstable against Gaussian pulses in either exotic or normal massless KleinGordon fields. The wormhole throat suffers a bifurcation of horizons and either explodes to form an inflationary universe or collapses to a black hole, if the total input energy is respectively negative or positive. As the perturbations become small in total energy, there is evidence for critical solutions with a certain blackhole mass or Hubble constant. The collapse time is related to the initial energy with an apparently universal critical exponent. For normal matter, such as a traveller traversing the wormhole, collapse to a black hole always results. However, carefully balanced additional ghost radiation can maintain the wormhole for a limited time. The blackhole formation from a traversible wormhole confirms the recently proposed duality between them. The inflationary case provides a mechanism for inflating, to macroscopic size, a Plancksized wormhole formed in spacetime foam.
pacs:
04.70.Bw, 04.25.Dm, 04.40.Nr, 98.80.CqI Introduction
Wormholes are known as a kind of solution of the Einstein equations, and have become a popular research topic, raising theoretical possibilities of rapid interstellar travel, time machines and warp drives. These topics sound like science fiction, but after the influential study of traversible wormholes by Morris & Thorne MT , it became widely accepted as a scientific topic MVbook . The only physically nonstandard feature is that one has to assume negativeenergy matter to construct such a wormhole. However, such exotic matter occurs extensively in quantum field theory and in alternative gravitational theories such as scalartensor theories.
Until recently, wormholes have been studied mainly as static or cutandpaste models, or without an independently defined exotic matter model. Not long ago, one of the authors wh proposed a unified theory of black holes and traversible wormholes, arguing that the two are dynamically interconvertible, and that traversible wormholes are understandable as black holes under negative energy density. This opens a new viewpoint on the dynamical nature of both black holes and wormholes, including Hawking radiation. This synthesis has been examined using a lowdimensional model HKL , where the theory is affirmed.
The purpose of this article is to investigate wormhole dynamics in fourdimensional Einstein gravity, using numerical simulations. Our starting point is a static wormhole which is perhaps best known as Morris & Thorne’s opening example (their box 2). This metric is actually a solution for a massless KleinGordon field whose gravitational coupling takes the opposite sign to normal, as was shown earlier by several authors Cl , the earliest of which appears to be Ellis E , who called it a drainhole. To our knowledge, this is the earliest solution which would nowadays be called a traversible wormhole.
We study dynamical perturbations of this static wormhole, using the spherically symmetric Einstein system with the above exotic matter model, the massless ghost KleinGordon field. We developed a numerical code based on a dualnull coordinate system, in order to follow the horizon dynamics and radiation propagation clearly. Our main experiment is to add or subtract Gaussian pulses in the ghost field, i.e. respectively with negative or positive energy. We also consider Gaussian pulses in a normal KleinGordon field, to see the effect on the wormhole of normal matter, like a human being traversing the wormhole. We discover how the initially static wormhole will change its structure due to these dynamic perturbations. Although our model is specific to spherically symmetric spacetimes, we believe that it illustrates the dynamical nature of traversible wormholes. To our knowledge, this is the first numerical study of wormhole dynamics.
Ii Model and Numerical method
ii.1 Field equations
The field equations for a massless conventional and ghost KleinGordon field, and respectively, in Einstein gravity are, in standard notation,
(1)  
(2) 
where the coupling constants have been fixed by choice of units for and . We adopt a dualnull formulation developed by one of the authors dne . The spherically symmetric lineelement may be written in dualnull form
(3) 
where refers to the unit sphere. Then the configuration fields are as functions of . The Einstein equations in spherical symmetry are expressed in sph ; 1st , leading to the field equations
(4)  
(5)  
(6)  
(7)  
(8) 
To obtain a system accurate near , we can use the conformal factor instead of and, as firstorder variables, the conformally rescaled momenta
(9)  
(10)  
(11)  
(12) 
Here has the meaning of expansion, though it is rescaled from the usual definition, . Then are finite and generally nonzero at . The remaining firstorder equations follow from the above field equations and the spherically symmetric identity :
(13)  
(14)  
(15)  
(16)  
(17) 
These equations plus the inverted momentum definitions
(18)  
(19)  
(20)  
(21) 
constitute the firstorder dualnull form, suitable for numerical coding.
ii.2 Numerical method
We prepare our numerical integration range as drawn in Fig.1. The grid will cover both universes connected by the wormhole throat . The grid is assumed to be equally spaced in .
The basic idea of numerical integration is as follows. We give initial data on a surface and the two null hypersurfaces generated from it. Generally the initial data have to be given as
(22)  
(23) 
We then evolve the data on a constant slice to the next.
Due to the dualnull decomposition, the causal region of a grid is clear, and there are inbuilt accuracy checks: the integrability conditions or consistency conditions . In order to update a point N (north), we have two routes from the points E (east) and W (west). The set of equations, (13)(21), gives us direction (W to N) and direction (E to N) integrations together with the consistency conditions.
All the equations can be written as
(24) 
and the input variables, , in the righthandside term depend on the integration schemes. In more detail, we made the steps:

Integrate using the equations (update N from W): use .

Integrate using the equations (update N from E): use .

Update N from W again: use .

Update N from E again: use .

Check the consistency of at point N. If the convergence is not satisfactory, repeat back to the step 3. We set the tolerance as for .
As a virtue of the dualnull scheme, we can follow the wormhole throat or blackhole horizons easily. They are both trapping horizons, hypersurfaces where or 1st ; bhd . Another benefit is the singular point excision technique. As we described, the causal region of each grid point in the dualnull scheme is apparent. When a grid point is inside a blackhole horizon and near to the singularity, we can exclude that point and grid points in its future null cone from further numerical computation. Actually we excised grid points if one of exceeds . We found that this is useful for investigating inside a blackhole horizon, though there is naturally a limit on how closely a singularity can be approached.
ii.3 Static wormhole
The first MorrisThorne wormhole metric MT was given as
(25) 
where denotes the throat radius of the wormhole. Since it is an overall scale, we set for numerical purposes, but retain it in the text. Transforming the proper radial length and the static proper time to dualnull coordinates
(26) 
the analytic expressions for the Ellis wormhole solution E are
(27)  
(28)  
(29)  
(30)  
(31)  
(32) 
with the conventional field vanishing. The wormhole is symmetric in interchange of the two universes, , though this will not be so for the perturbations. The energy density is
(33) 
with units . The negative energy density is a maximum at the throat , tends to zero at infinity and is smaller for larger wormholes. Note that the wormhole throat is a double trapping horizon, . The behaviour of these generally different trapping horizons will be a key indicator of the evolution of the perturbed wormhole.
The dualnull initial data for the static wormhole is
(34)  
(35) 
ii.4 Gravitational massenergy
We find that the dynamical structure will be characterized by the total mass or energy of the system, the Bondi energy. Localizing, the local gravitational massenergy is given by the MisnerSharp energy , while the (localized Bondi) conformal flux vector components were defined in sph ; 1st :
(36)  
(37)  
They are related by the energy propagation equations or unified first law 1st
(38) 
so that one may integrate to
(39) 
where the integral is independent of path, by conservation of energy sph ; 1st . Here the initial condition corresponds to the static wormhole. The definitions are local, but is the Bondi energy and the Bondi flux for the righthand universe. For the static wormhole, the energy is everywhere positive, maximal at the throat and zero at infinity, , i.e. the Bondi energy is zero. Generally, the Bondi energyloss property, that it should be nonincreasing for matter satisfying the null energy condition sph , is reversed for the ghost field.
Iii Dynamical behaviour of traversible wormholes
iii.1 Evolution of static wormhole
We start with the numerical evolution of the static wormhole. We find that numerical truncation error can quite easily destroy the stability. In Fig.2, we show our tracking of the location of the trapping horizon for static wormhole initial data, with different numbers of grid points: 801, 1601, 9601, for . The location of the throat, where , is supposed to be stationary at . The figure shows that at sufficient resolution, we can maintain a static wormhole for a given time. Vice versa, this shows that in our dualnull scheme and the current applied numerical integration scheme, the lack of resolution inevitably causes an unstable evolution eventually. We hereafter discuss the dynamical structure of perturbations of the wormhole only in the reliable evolution range, with the resolution kept good enough (9601) to preserve the static case.
In Fig.3, we demonstrate the numerical evolution of the static wormhole, giving the expansion and the local gravitational massenergy as functions of .
iii.2 Gaussian perturbations in ghost field
We first put perturbations of the static wormhole in the form of Gaussian pulses, input from the righthand universe . The initial data on is
(40) 
with all other initial data as for the static wormhole. Here are parameters, with runs being performed for increasingly small values of the amplitude , and . That is, the pulses will hit the wormhole throat at . For positive or negative, this corresponds respectively to enhancing or reducing the supporting ghost field. Note that this pulsetype perturbation starts only from the righthand universe. This situation is available because our numerical region covers both sides of the wormhole, though we assume that the spacetime is locally spherically symmetric.
Fig.4 shows the horizon locations for both and , for . We plotted three cases, and , and it can be seen that in each case, the double trapping horizon comprising the initial wormhole throat bifurcates when the pulse hits it. The two horizons then accelerate away from each other, rapidly approaching the speed of light and appearing to be asymptotically null. Due to the grid excision technique for strong curvature regions, the final horizon coordinate , where the righthand trapping horizon becomes null, is often excised from the numerical computation. In such a case, we determine by extrapolating the horizon trajectory with a function , where and are unknown constants. A partial Penrose diagram of the evolved spacetime is given in Fig.5, which is like the prediction in wh . However, the two horizons move in opposite directions depending on the sign of . In the following, one should bear in mind that initially in the righthand universe , with signs reversing for the lefthand universe.
When we supplement the ghost field, , we see that the horizons move respectively in the directions, meaning that in the region between them. This defines past trapped surfaces, which occur in expanding universes, such as the de Sitter solution. In Fig.6 we plot the areal radius of the ‘‘throat’’ ^{1}^{1}1Once the wormhole is perturbed, this hypersurface loses its specific geometrical meaning as the wormhole throat . However, one can say that it is composed of minimal surfaces in some family of spatial hypersurfaces, which are locally close to the constant family for small perturbations. , as a function of proper time
(41) 
It can be seen to rapidly increase, actually exponentially: the lines in the figure can be expressed with a function where and are constants, and we show the Hubble constant in Table 1. This exponential expansion, combined with the horizon structure, indicates that the wormhole has exploded to an inflationary universe. The two trapping horizons become the cosmological horizons. This connection between wormholes and inflation was unanticipated, but is consistent with the general theory wh . The two universes connected by the wormhole have essentially been combined into one universe.
On the other hand, when we reduce the ghost field, , the horizons move respectively in the directions, meaning that in the region between them. This defines future trapped surfaces, which occur inside black holes, such as the Schwarzschild solution. Fig.6 shows that the areal radius is heading towards zero in finite proper time, though the code cannot follow it into the apparent central singularity. This and the horizon structure indicates that the wormhole has collapsed to a black hole. The trapping horizons appear to become asymptotically null, where they would coincide with the event horizons of the black hole. This occurs around for the righthand universe.
If the amplitude of the pulse is reduced, the black hole forms later, as shown in Fig.4(b2) for . As a physical measure of the size of the perturbation, we take the initial Bondi energy
(42) 
scaled by the initial throat radius . In practice we take and subtract the corresponding value for the static wormhole. Some data are presented in Table 1. We find that positive will cause collapse to a black hole, while negative will cause explosion to an inflationary universe. Also, the speed of the horizon bifurcation increases with the energy. One could also scale the total energy by the maximal initial energy , but this occurs at the throat and is . In either case, the perturbations are small, down to 1% in energy, yet the final structure is dramatically different. Thus we conclude that the static wormhole is unstable.
Accepting that the static wormhole is stable to linear perturbations BCCF ; AP , we seem to have discovered a nonlinear instability, something of interest in itself. Circumstantial evidence for a quadratic instability is that it is first signalled by nonzero values of , and the propagation equations (15) for have quadratic terms on the righthand side, which should cancel to zero in the static solution. This could be addressed by secondorder perturbation theory, but we know of no such studies for wormholes.
In Fig.7, we plot the energy as a function of , for for the three cases in Fig.4. In each case, the mass increases rapidly in as the horizon forms, but rapidly approaches a constant value in at the horizon. This value
(43) 
is the final mass of the black hole. Obtaining this easily is another virtue of the dual null scheme. The values are shown in Table 1. The graphs indicate that an observer at infinity will see a burst of radiation as the wormhole collapses or explodes. For collapse, a certain fraction of the field energy radiates away, the rest being captured by the black hole, constituting its mass. For explosion, the radiated energy continues to rise in an apparently selfsupporting way as the universe inflates.
Fig.8 gives a survey of various parameters of and , fixing . Fig.8(a) shows how quickly the blackhole horizon, the outer trapping horizon , develops after the pulse hits the throat. We see that the dots are close to one line, indicating a logarithmic relation between the collapse time and the initial energy of the perturbation,
(44) 
Fig.8(b) is the final black hole mass as a function of the initial energy of the perturbation. Interestingly, as the perturbations become small, the blackhole mass seems to approach a nonzero constant,
(45) 
In the inflationary case, the Hubble parameter (or horizon mass ) also seems to have a limiting value for small perturbations, as seen in Table 1:
(46) 
Since these masses are the same order as the wormhole scale , we guess that the static wormhole is something like an instanton with two different attractors nearby. This scenario requires more detailed study with different models, and we plan to report it subsequently.
final structure  
3.0  inflation  1.56  (Fig.4a)  
3.0  inflation  1.19  
3.0  inflation  1.13  
3.0  inflation  1.12  
3.0  0.0  static  (Fig.3)  
3.0  black hole  
3.0  black hole  
3.0  black hole  (Fig.4b2)  
3.0  black hole  (Fig.4b1)  
6.0  black hole  
9.0  black hole  
iii.3 Gaussian pulses in conventional field
Similarly, we next consider adding a small amount of conventional scalar field to the static wormhole solution. This is a simplified model of a situation that a positiveenergy creature jumps into a traversible wormhole. Again we add Gaussian pulses to the initial data on as
(47) 
with all other initial data as for the static wormhole. We set , so that the pulse will again hit the wormhole throat at .
We show the horizon structure in Fig.9 and initial total energy in Table 2. The additional normal field slightly enhances the total energy, with either sign of amplitude, and the wormhole collapses to a black hole. The fundamental dynamical behaviour is the same as for the previous ghost perturbations with positive energy. We see again a logarithmic relation between the time and the initial energy of the perturbation, , Fig.10(a). The critical exponent is consistent with that found for perturbations in the ghost field, indicating a new kind of critical behaviour reminiscent of that found by Choptuik Ch in blackhole collapse. In the limit of small perturbations, the blackhole mass also appears to approach the same nonzero constant, , Fig.10(b). Although the current code has a limit on how small perturbations can be, since the blackhole collapse should occur within the reliable evolution range, the data suggest the existence of a critical solution with a certain blackhole mass.
For these perturbations, the energy perturbation is as small as about 0.01%, a quite convincing indication of instability. From this and general principles wh , we expect that any normal matter traversing the wormhole will cause its collapse to a black hole.
iii.4 Wormhole maintenance
Supposing the normal field pulse represents an actual traveller of the traversible wormhole, then he, she or it may go through the wormhole and exit safely into the other universe if the speed is high enough, as can be seen from the Penrose diagram obtained from Fig.9. However the wormhole itself will collapse to a black hole, ending its usefulness for travellers.
We here demonstrate a kind of temporary maintenance of the wormhole, by sending in an additional ghost pulse just after the passing of the traveller. In Fig.11 we show the results for pulse parameters for the normal field representing the traveller, combined with a balancing pulse , case B, or , case C. If we do not send the balancing pulse, the wormhole collapses to a black hole with horizons given by case A in the plot. The case B ends up with an inflationary expansion, while the case C keeps the wormhole structure at least until . Since the final fate of the wormhole is either a black hole or an inflationary expansion, to keep the throat as it was requires a finetuning of the parameters, and may not be realistic. However, it shows how the wormhole life may be extended. This indicates that the wormhole might be maintained by continual adjustments to the radiation level, though it would be a neverending project.
Iv Discussion
We have numerically studied the dynamic stability of the apparently earliest E and most frequently rediscovered wh ; Cl traversible wormhole, most famous as the first MorrisThorne wormhole MT . We observe that the wormhole is unstable against Gaussian pulses in both exotic and normal massless KleinGordon fields. The wormhole throat suffers a bifurcation of horizons and will either explode to an inflationary universe or collapse to a black hole, if the total energy is respectively negative or positive. For the normal matter, the total energy is necessarily positive and a black hole always results. This blackhole formation from a traversible wormhole confirms the unified theory of both wh .
The inflationary expansion provides a mechanism for enlarging a quantum wormhole to macroscopic size. Wheeler introduced wormholes to gravitational physics as part of his stillpopular idea of spacetime foam W , which envisages transient wormholes at the Planck scale, due to quantum fluctuations in spacetime topology. Morris & Thorne imagined pulling a wormhole out of this foam and enlarging it somehow. Actual mechanisms are scarce: Roman R suggested inflation, though without an independently defined exotic matter model, and Redmount & Suen RS found a quantummechanical instability of certain cutandpaste wormholes in a spacetime foam model, also leading to unbounded expansion. We have discovered that for one of the simplest classical exotic matter fields, wormholes can naturally inflate. The problem is only to stop the inflation proceeding indefinitely.
The results also provide an entertaining answer to the question of what happens if someone attempts to traverse the wormhole. At best, with sufficient alacrity, our hero could exit into the other universe, only to look behind and see that the passage has caused the wormhole to collapse to a black hole, thereby sealing off the causal connection to the home universe. Perhaps there was also an unlucky companion who did not make it through, suffering the wellknown grisly fate of being trapped in a black hole. The survivor would be left to ruminate on the ironic fate of those who would tamper with the fabric of spacetime: stranded forever in the twilight zone. As we showed, such a tragedy could be avoided by a carefully calculated burst of ghost radiation, just before or after the traveller passes, which maintains the static wormhole for a certain time. The traveller then has a limited time to explore the other universe before the wormhole collapses.
For our colleagues who are understandably sceptical about wormholes and exotic matter, we hope that our numerical study has at least demonstrated that they can be studied and understood by the same local, dynamical methods as used for black holes with normal matter. Indeed, the results lead straight back to mainstream physics of black holes and inflation, with new discoveries about both, particularly the apparent critical behaviour. For a general audience, perhaps the message is that (albeit theoretical) science can be stranger than science fiction.
Acknowledgements.
HS would like to thank Y. Eriguchi, Y. Kojima, T. Harada and M. Shibata for comments, and SW. Kim for hospitality in visiting Ewha Womans University. SAH would like to thank SW. Kim, S. Sushkov, K. Bronnikov and C. ArmendarizPicon for discussions. HS is supported by the special postdoctoral researchers program at RIKEN, and this work was supported partially by the GrantinAid for Scientific Research Fund of Japan Society of the Promotion of Science, No. 14740179. SAH is supported by Korea Research Foundation grant KRF2001015DP0095.References
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