Structural Engineering

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Worm Holes Worm Hole Resonance


The inflating wormhole: a MATHEMATICA animation

Author: Thomas A, Roman

Journal: Computers in Physics 8, 480 (1994); doi: 10.1063/1.168507

Publisher: American Institute of Physics


It has been recently speculated that the laws of physics might allow an arbitrarily advanced civilization to construct traversable wormholes for interstellar travel. If submicroscopic traversable wormholes existed in the very early universe, then it is conceivable that an inflationary phase might have enlarged some of them to macroscopic size. In a recent paper, the author explored this possibility, and in so doing found it useful to construct some visualizable (albeit simplified) models as an aid to intuition. The purpose of the current paper is to present the details of these computer simulations, and in the process introduce the reader to some techniques of simple graphing in MATHEMATICA. The models take the form of animated embedding diagrams of traversable wormholes in an inflationary universe.


Current research has explored the possibility that the laws of quantum field theory and general relativity might allow an arbitrarily advanced civilization to construct traversable Lorentian wormholes for interstellar travel (refs 1-3). A wormhole can be thought of as a "tube" which can connect two otherwise distant regions of space. Morris and Thorne (ref 1) (MT) have proposed specific models for the geometry of such wormholes and have analysed the properties of the matter and energy which would be required for their construction and maintenance. They showed that a necessary property is that, as seen by at least some observers, this material must have a negative energy density in order to hold the wormhole open against gravitational collapse.

Although classical matter and energy densities are always positive, negative energy densities and fluxes are allowed in quantum field theory. A well-known example is the Casimir effect (ref 4). However, quantum field theory also imposes some restrictions on negative energy, in the form of uncertainty principal-type inequalities on the magnitude and duration of negative energy fluxes (ref 5,6). Whether or not quantum field theory allows the existence of negative energy in the amounts required to support macroscopic traversable wormholes is presently unknown (ref 6,7). Morris, Thorne and Yurtsever (ref 8) later discovered that a "twins-paradox" motion of one wormhole mouth relative to the other leads to the conversion of the wormhole into a time machine for backward time travel. Frolov and Novikov (ref 9) showed that simply placing one wormhole mouth in a different gravitational potential than the other mouth generally results in transforming the wormhole into a time machine. It was subsequently discovered by Kim and Thorne (ref 10), and also by Frolov (ref 11) that the self-intersecting null geodesics that would arise in the course of time machine formation produce divergences in vacuum expectation values of the stress-energy tensor of test fields. These results led Hawking (ref 12,13) to propose his "Chronology Protection Conjecture:" {the laws of physics always prevent the formation of closed timelike curves}. It is unclear at present whether enforcement of chronology protection would necessarily also forbid the existence of traversable wormholes or only their conversion into time machines.

In classical general relativity there are restrictions on changes in the topology of space (ref 14,15), which might make it difficult for even an advanced civilization to create traversable wormholes. However it is possible that such wormholes, although perhaps forbidden classically, might be quantum-mechanically allowed on sub-microscopic scales (ref 16). If such wormholes existed in the very early universe, then it is conceivable that an inflationary phase (ref 17) might have enlarged some of them to macroscopic size. The author recently considered a metric of a traversable wormhole embedded in an inflating universe (ref 18). During the course of that investigation it proved useful to construct a visual model of the spatial hypersurfaces to illustrate the change in size of the wormhole with time. A series of embedding diagrams for different values of \(t=const\) were graphed and animated using MATHEMATICA (ref 19,20). The details of several such animations are presented here.

One motivation in writing this paper was to give some examples of the use of animated three dimensional graphics in MATHEMATICA to explore physically interesting problems. The standard reference (ref 20) provides only one short example. Although other books on MATHEMATICA treat the subject of animation to varying degrees, the half-dozen or so perused by this author had little or no discussion of how to animate 3-D graphics. In the course of the current treatment, the reader will be introduced to some simple graphing techniques in MATHEMATICA. Our sign and unit conventions are the same as those of Misner, Thorne and Wheeler (ref 21).

Model Geometry

The metric for a spherically symmetric, static, MT traversable wormhole is given by (ref 1):

$$ds^2=-e^{2\phi(r)}dt^2+\frac{dr^2}{1-\frac{b(r)}{r}}+r^2(d\theta^2+sin^2(\theta) d\phi^2).............Eq. (1)$$

The two adjustable functions \(b(r)\) and \(\phi(r)\) are referred to as the "shape function" and the "redshift function", respectively. The shape function \(b(r)\) controls the shape of the wormhole as viewed, for example, in an embedding diagram. The radial coordinate \(r\) is chosen so that the circumference of a circle centered on the throat of the wormhole is given by \(2\pi r\). Note that the coordinate \(r\) is nonmonotonic in that it decreases from \(+\infty\) to a minimum value \(b_0\), representing the location of the throat of the wormhole, and then it increases from \(b_0\) to \(+\infty\). This behaviour of the radial coordinate reflects the fact that the wormhole connects two separate external "universes" (or two regions of the same universe).

There is a coordinate singularity at the throat, \(r=b=b_0\), where the metric coefficient \(g_{rr}\) becomes divergent. However, the radial proper distance \begin{equation}\label{eq2} l(r)=\pm \int _{b_0}^r \frac{dr}{[1-\frac{b(r)}{r}]^{\frac{1}{2}}}.............Eq.(2) \end{equation} must be required to be finite everywhere. At the throat \(l=0\), while \(l <0\) on the "left" side of the throat and \(l>0\) on the "right" side. For the wormhole to be traversable it must have no horizons, which implies the constraint that \(\phi(r)\) must be everywhere finite.

The technique of MT was to first choose \(b(r)\) and \(\phi(r)\) so as to obtain a wormhole with the desired characteristics. They then plugged the metric Eq.(2) into the Einstein equations in order to determine the properties of the stress-energy tensor required for wormhole maintenance. MT discovered that the weak energy condition (WEC) must be violated near the throat of the wormhole. The WEC requires (ref 22) that \(T_{\mu \nu}W^{\mu}W^{\nu} \geq 0\), for all timelike and null vectors \(W^{\mu}\). They called this WEC-violating material: "exotic matter".

A simple time-dependent generalization of the MT wormholes can be obtained (ref 18) by multiplying the spatial part of the metric Eq.(1) by a deSitter scale factor \(e^{2 \chi t}\), where \(\chi=\sqrt{\frac{A}{3}}\) and \(A\) is the cosmological constant. The resulting metric is given by \begin{equation}\label{eq3} ds^2=-e^{2\phi(r)}dt^2+e^{2 \chi t}\left(\frac{dr^2}{1-\frac{b(r)}{r}}+r^2(d\theta^2+sin^2(\theta) d\phi^2)\right).............Eq.(3) \end{equation}. Our coordinate system is chosen to be "co-moving" with the wormhole geometry, so that the coordinates \(r\), \(\theta\), \(\phi\) have the same geometrical interpretation as before. In particular, circles of constant \(r\) are centered on the throat of the wormhole, and the throat is always located at \(r=b=b_0\) for all \(t\). For \(\chi=0\), Eq. (3) becomes the static wormhole metric Eq. (1), while for \(\phi(r) = b(r) = 0\), it reduces to a flat deSitter metric. One may let \(\phi_{(r)}\rightarrow 0, \frac{b}{r} \rightarrow 0\) as \(r \rightarrow \infty\), so that the spacetime is assymptotically deSitter, or choose \(\phi(r)\), \(b(r)\) so that they go to zero at some finite value of \(r\), outside of which the metric is deSitter. (The properties of the stress-energy tensor associated with the metric given by Eq.(3) are discussed in detail in Ref. 18).

The onset of the inflationary phase is taken to be at \(t=0\). The functions \(\phi_{(r)}\) and \(b(r)\) can be suitably chosen so as to give a reasonable wormhole metric at \(t=0\). The inflation of the surrounding space subsequently enlarges the size of the wormhole. The proper circumference \(c\) of the wormhole throat, \(r=b=b_0\), for \(\theta=\frac{\pi}{2}\), at any time \(t=const\) is given by: \begin{equation}\label{eq4} c=\int_0^{2\pi} e^{\chi t} b_0 d\phi = e^{\chi t} (2 \pi b_0).............Eq.(4) \end{equation}. This is simply \(e^{\chi t}\) times the initial circumference. At any \(t=const\), the radial proper length through the wormhole between any two points A and B is \begin{equation}\label{eq5} l(t)=\pm e^{\chi t} \int_{r_A}^{r_B} \frac{dr}{[1-\frac{b(r)}{r}]^{0.5}}.............Eq.(5) \end{equation} which is just \(e^{\chi t}\) times the initial radial proper separation.

Let us now consider embedding a \(t=const\), \(\theta=\frac{\pi}{2}\) slice of the spacetime given by Eq. (3) into a flat three-dimensional Euclidean space with metric: \begin{equation}\label{eq6} ds^2=d\overline{z}^2+d\overline{r}^2+\overline{r}^2 d\phi^2.............Eq.(6) \end{equation} (The standard methods for constructing embedding diagrams are discussed in detail in Ref.1, Ref.21 pp612-615, pp836-840, and Ref.18). The metric on this two-dimensional slice is given by \begin{equation}\label{eq7} ds^2=\frac{e^{2 \chi t} dr^2}{1-\frac{b(r)}{r}} + e^{2 \chi t} r^2 d\phi^2.............Eq.(7) \end{equation} A comparison of the coefficients of \(d\phi^2\) yields \begin{equation}\label{eq8} \overline{r}=e^{\chi t}r|_{t=const}.............Eq.(8) \end{equation} \begin{equation}\label{eq9} d\overline{r}^2=e^{2\chi t} dr^2|_{t=const}.............Eq.(9) \end{equation} Equations (8) and (9) correspond to a "rescaling" of the \(r\) coordinate on each \(t=const\) slice. By using Eqs (8) and (9) and \begin{equation}\label{eq10} \overline{b}(\overline{r})=e^{\chi t}b(r).............Eq.(10) \end{equation} the metric on the embedded slice can be rewritten as \begin{equation}\label{eq11} ds^2=\frac{d\overline{r}^2}{1-\frac{\overline{b}(\overline{r})}{\overline{r}}}+\overline{r}^2 d\phi^2.............Eq.(11) \end{equation} Here, \(\overline{b}(\overline{r})\) has a minimum at \(\overline{b}(\overline{r}_0=\overline{b}_0=\overline{r}_0\). It can also be easily shown that (Ref 18) \begin{equation}\label{eq12} \overline{z}(\overline{r})=e^{\chi t} z(r).............Eq.(12) \end{equation} where \(z=z(r)\) is the usual "lift" function (Ref 1,21) At \(t=0\), Eq.(6) reduces to \begin{equation}\label{eq13} ds^2=dz^2+dr^2+r^2d\phi^2.............Eq.(13) \end{equation} The metric on the embedded surface can be expressed as \begin{equation}\label{eq14} ds^2=\left(1+\left(\frac{dz}{dr}\right)^2\right)dr^2+r^2 d\phi^2.............Eq.(14) \end{equation} Equation (14) will be the same as Equation (13) if we identify the \(r, \phi\) coordinates of the embedding space with those of the wormhole spacetime, and also require \begin{equation}\label{eq15} \frac{dz}{dr}=\pm\left(\frac{r}{b(r)}-1\right)^{-\frac{1}{2}}.............Eq.(15) \end{equation} The relation between our embedding space at any time \(t\), given by Eq. (6) and the initial embedding space at \(t=0\) given by Eq. (13) is: \begin{equation}\label{eq16} ds^2=d\overline{z}^2+d\overline{r}^2+\overline{r}^2 d\phi^2=e^{2 \chi t}(dz^2+dr^2+r^2 d\phi^2).............Eq.(16) \end{equation} Note that the embedding space coordinates (\(\overline{z}\), \(\overline{r}\)) "scale" exponentially with time. This scaling of the embedding space compensates for the expansion of the wormhole so that, relative to the \(\overline{z}\), \(\overline{r}\), \(\phi\) coordinate system, the wormhole will always remain the same size. However, the wormhole will change size relative to the initial \(t=0\) embedding space, and it is this feature of the model which is examined in the following section.

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Quaerendo Invenietis

By seeking, you will discover ( JS Bach )