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Two New Kinds of Wormholes

by John G. Cramer

Alternate View Column AV-148
Keywords: cylindrical, cosmological, wormholes, energy, conditions, inter-universe, connections
Published in the July-August-2009 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted
3/5/2009 and is copyrighted ©2009 by John G. Cramer.
All rights reserved. No part may be reproduced in any form without
the explicit permission of the author.

Wormholes are shortcuts through space-time, constructs of general relativity (GR) that appear to offer a physics foundation for faster-than-light travel and even for travel back in time.  They first appeared in the physics literature in 1935, when Albert Einstein and his colleague Nathan Rosen discovered that implicit in general relativity is a tunnel-like structure in the topology of space-time connecting two separated regions.  Einstein and Rosen were actually trying to explain fundamental particles like electrons and proton.  They suggested that if lines of electric flux were threaded through such a structure, the flux would be trapped and one end would appear to be an isolated positive charge and the other end would appear to be a negative charge.  Later, however, general relativity was used to calculate the masses of such “particles” and it was realized that they would be have a mass of at least a few micrograms, far heavier than the mass of an electron or proton.

The motivation for the Einstein-Rosen work thus proved wrong, but the mathematics survived as a curiosity of general relativity that was for a time called an “Einstein-Rosen Bridge”.  Later, John Wheeler changed the name to “wormhole”, and that is the designation that has stuck.  The mathematical description (or "metric") of a wormhole portrays a curved-space object that is a shortcut through space-time itself, connecting two regions of space-time in the same universe or even connecting two separated universes.

Wheeler demonstrated that simple wormholes are so unstable that if one opened up spontaneously, it would close again before even a single photon of light could pass through it.   However, in 1988 Michael Morris and Kip Thorne of Cal Tech showed that stable wormholes are possible (see my AV column “Wormholes and Time Machines”, Analog, June-1989).  They described how a stable wormhole might be constructed by an "advanced civilization" (i.e., not us.)  by placing a region of negative mass-energy in the wormhole's "throat".   The requirement of negative mass-energy is something of a show-stopper, because at present we are able to produce negative energy only in very tiny amounts between the conducting plates of a capacitor using the Casimir effect.

Wormholes are, of course, of great interest for the underpinnings of science fiction, from hard SF to space operas, and over the years I have written many Alternate View columns in this magazine about them.  The wormhole solutions come from a non-standard way of using general relativity, an approach sometimes described as "metric engineering."  General relativity is normally done by considering a particular arrangement of mass and energy and asking what metric would result, how space-time would be warped, and what effects would be produced by such an arrangement.  In metric engineering, we do it backwards.  We specify how we want space to be warped in order to produce desired effects (e.g., a wormhole or a warp drive), and then ask what arrangement of mass and energy would be required to accomplish this. The usual outcome of this kind of GR solution, at least in the cases of wormholes and warp drives, is that a sizable quantity of negative mass-energy would be needed.

In this column, I want to present two new solutions to the equations of general relativity involving wormholes that do not require negative mass energy for stability, and that thereby avoid the objections that have been raised against the other wormhole solutions.

The first of these is the cylindrical wormhole.  “Standard” wormholes usually have spherical symmetry, and can be thought of a two spherical surfaces in separated regions of space which have been “stitched together”, so that an object passing through one surface emerges from the other.  However, the sphere is not the only possible geometry for a wormhole.

Cosmic strings (see my AV column #19, Analog April-1987) are strange massive objects that may (or may not) be present in our universe.  They would have formed shortly after the Big Bang when the energy saturated space of the early universe was being replaced by the more normal space in which we now live.  If they exist at all, cosmic strings would be infinitesimally small in cross section but very long, perhaps forming loops that encircle an entire galaxy. And they would be quite massive, producing strong and very odd gravity fields. Cosmic strings can be loosely described as "seams" or "cracks" in space, long closed-loop tangles in the fabric of space itself. In cosmology they are geometrical imperfections in the topology of space, produced as the universe was unfolded out of the Big Bang.

Recently, Bronnikov and Lemos have considered the possibility of a wormhole that surrounds a cosmic string, with the geometry of a very long cylinder instead of a sphere.  What they find is that wormholes are much better behaved in this geometry.  They do not require negative mass-energy for stabilization and do not violate the weak or null energy conditions (see my AV column “Outlawing Wormholes and Warp Drives”, Analog May, 2005) , violations of which have been used label solutions of general relativity as “unphysical”.

The problem with Bronnikov-Lemos wormholes is that they should be infinite in length, and that is difficult in a universe that may not itself be of infinite extent.   This difficulty can be avoided by “putting the snake’s tail in its mouth”, in other words, bending the cosmic string in a circle, so that the wormhole becomes a torus (doughnut-shape).  Bronnikov and Lemos have investigated this possibility, but the results are inconclusive.  It is not at the moment clear whether a toroidal wormhole needs negative mass-energy for stabilization or violates the weak and/or null energy conditions.  But there are indications that the desirable properties of the cylindrical geometry may be retained when the infinite string becomes a circle and the infinite cylinder becomes a torus.  Therefore, perhaps we should be searching for indications of wormhole leftovers from the Big Bang in the form of doughnut shaped objects.

            A second recent advance in our understanding of wormhole physics came from the work of Maeda, Harada and Carr.  Their work was motivated by their investigation of numerical relativity, in which the equations of general relativity are solved in a dynamic situation where conditions are changing (for example, the universe is expanding) using numerical approximations on a large computer.  That numerical work pointed to dynamic wormhole solutions that had unusual properties, and caused the authors to look for corresponding algebraic GR solutions.  The result is what the authors describe as cosmological wormholes.  These are dynamic wormholes that cannot connect within a single universe, but instead must connect one Friedmann universe to another.

            Here, a Friedmann universe is the present Standard Model of cosmology.  It uses general relativity to describe a simplified version of the universe in which we live.  It is a universe that is expanding at a regular rate and that contains matter that is uniformly distributed and that acts as a fluid characterized by pressure and density.  The lumpy stars and galaxies of our universe are averaged out in the Friedmann model and characterized by a universal fluid, a good approximation if one takes a very large scale view of the universe.

            The cosmological wormholes of Maeda, Harada and Carr connect two Friedmann universes, (presumably ours and another one).  They are dynamic, changing with time.  They satisfy all of the energy conditions, and they do not require any negative mass-energy for stability.  At least in isolation, they cannot lead to time-like loops and time travel paradoxes because they lead to another universe with its own time structure (and perhaps its own laws of physics).  The authors suggest that such wormholes may have formed naturally in the early phases of the Big Bang and may have influenced the behavior of the universe during its initial expansion phase.  

            The Maeda, Harada and Car paper does not address the issue of whether two independent cosmological wormholes might connect the same pair of universes.  Their calculations require certain symmetries and probably not tolerate the consideration of two wormhole connections in the same universes.  However, the presence of two wormhole paths connecting arbitrary space-time points in a pair of universes would make possible time-like loops that threaded both wormholes and lead to time paradoxes.  I suspect that there is an underlying exclusion principle implicit in the wormhole mathematics that prevents such dual connections.

            The implication of this work is that we now know of two wormhole types that do not require negative mass-energy for stability.  These provide an “existence theorem” that negative mass-energy is not always required for stable wormholes (and perhaps also for warp drives as well.)  Further, the cosmological wormholes of Maeda, Harada and Carr could have been produced during the Big Bang, perhaps in great numbers.   They could still be around and could provide gateways to many other universes.   Writers of inter-universe science fiction should take note.


“Cylindrical wormholes”, K. A. Bronnikov and J. P. S. Lemos, arXiv preprint 0902.2360v3 [gr-qc], February 24, 2009.

“Cosmological Wormholes”,  H. Maeda, T. Harada and B. J. Carr, arXiv preprint 0901.1153v3 [gr-qc], March 3, 2009.

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