Why Wormholes Don't Exist
From Robert Heinlein's novel "Tunnel in the Sky" to the television series "Star Trek: Deep Space Nine" teleportals have been (and still can be) an enjoyable fantasy. But we must regard them, as I shall demonstrate, emphatically as fantasy, not as science fiction; that is, they violate known laws of physics in ways that we cannot mitigate by the discovery of new laws of physics, so they cannot exist in what we call Reality.
I have taken as my ultimate goal in these essays the confrontation of the question of whether Existence has so structured Reality that we can eventually build a working hyperdrive. In the pursuit of that question I find that I must address certain other concepts from the genre of science fiction.
Whether we call them spacebridges, stargates, teleportals, or syntopes, wormholes have become a minor theme in the grand symphony of science fiction. The wormhole gives us a variation on the theme of teleportation by way of the theme of hyperspace. We have this basic idea:
Two portals give us a visible manifestation of the wormhole, material openings through which we may enter the otherwise invisible shortcut through space that the wormhole provides. Between those portals we postulate two distances; the distance through normal space and the presumably much shorter distance through the wormhole. In many instances authors take the internal distance between the portals as effectively zero, even though many lightyears may separate the portals in normal space. This shortcut through space manifests the defining and best-known feature of wormholes. In mathematical terms, a wormhole makes the distance between two points in an inertial frame of reference a double-valued function.
If you want to cross interstellar distances in reasonable elapses of time, you would do well to have such a phenomenon available. But just as Special Relativity prevents us from simply accelerating rocketships to speeds faster than that of light, so it also prevents wormholes from existing in this Reality.
Imagine that two portals of a wormhole lie stationary relative to each other on some arbitrarily defined x-axis. Think of a railroad tunnel parallel to an east-west double track. One track goes through the tunnel and the other track goes alongside it (if such a thing makes sense). A surveyor measures the distances between the east and west portals of the tunnel as zero through the tunnel and X alongside it. He has mounted a clock at the entrance to each portal and has synchronized both clocks in his inertial frame of reference, the frame in which the portals do not move. After the surveyor leaves a Timekeeper sets up camp by the east portal. She sets up an apparatus that uses a pair of parallel telescopes to look through the tunnel and alongside it to project onto a screen images of the west clock as seen through the tunnel and through normal space. She then adds a lens that projects an image of the east clock onto the screen as well. On the screen, then, she sees the image of the west clock seen through the tunnel showing the same time as the east clock and the image of the west clock seen through normal space showing a time retarded relative to that shown on the east clock by an interval T = X/c.
Now a westbound train approaches the Timekeeper's position, moving down the track at speed V relative to the Timekeeper's frame. The conductor aims a camera at the screen and takes a picture of the three images of the clocks as the train passes the Timekeeper. When he develops the photograph, will it display a triple image of the clocks showing the same times that they showed the Timekeeper when the conductor snapped the shutter?
Consider first the image of the west clock that came to the screen through normal space. First, we know that the light carrying that image had to travel a distance
We also know that in the time T' that the light needed to cross that distance the train had to cross a distance VT' to reach the Timekeeper when the light from the west clock did. In the train's frame the Timekeeper appeared to come to the train at the speed V and the light from the clock came to the train at the speed of light, so the conductor can calculate the time T' by thinking as if the light from the clock overtook the Timekeeper at a relative speed of c-V, so he has T' = X'/(c-V). And we also know that the west clock emitted the light captured by the conductor's camera when the train had a distance of VT' between it and the Timekeeper, so the temporal offset on that clock must have had the value
From that number we must subtract the amount of time that elapses on the east clock as the light bearing the image of the west clock crosses the distance between the clocks.
We must also take into account the fact that the conductor synchronized his clocks with those in the Timekeeper's frame when the train was a distance VT' from the Timekeeper. That means that he must also take into account the temporal offset that the east clock will display at that instant, so he must also subtract the time
Thus the conductor must calculate the time by which the image of the west clock passing through normal space lags the time on the image of the east clock as
But that result expresses the time lag in units elapsed on the train's clocks. In order to express the lag in the dilated units elapsed on the clocks by the tunnel the conductor must multiply that expression by the inverse of the Lorentz factor between his frame and the Timekeeper's. He thus obtains for the lag T''' = X/c, which agrees perfectly with what the Timekeeper sees.
Now the conductor must examine the image of the west clock that came through the tunnel. Because light passes through the tunnel in effectively zero time, the image shows the time on the west clock at the instant the train passes the Timekeeper. But the train's motion relative to the clocks has offset the west clock's time into the future relative to the time on the east clock by an amount which emphatically does not agree with what the Timekeeper sees.
Having thus inferred two mutually exclusive descriptions of the same image, we must now falsify at least one of the premises that led us to that inference. We won't dismiss Relativity and the Lorentz Transformation: we have too much evidence in their favor. Rather, we must declare our postulate that wormholes exist false to Reality.
Not convinced? Well, the conservation laws add their own weight to the proposition that wormholes don't exist. Since we deduced the postulates of Relativity from the conservation laws this may seem a little tautological, but I believe the results will justify this little detour.
Let's imagine that we have mounted one portal of a wormhole on the platform of a station and the other portal on a flatcar coupled to a westbound train moving at the speed V. If the station-master were to pass a package of mass M through the wormhole, that package would go from having zero linear momentum relative to the station to having momentum MV relative to the station due to the motion it shares with the train. That clearly violates the conservation law, but before we dismiss the wormhole out of hand let's look at one way in which Reality might fix that situation. If the station-master must push the package, must apply a force to get the package through the portal, then the wormhole will satisfy Newton's third law: the equal and oppositely directed reaction in this case causes the station-master to push eastward against the platform as he pushes the package west onto the train. Thus the wormhole can satisfy the conservation law if it generates a forcefield that resists the passage of objects through it.
Oops! The station-master forgot to stamp the package. He must get it back. Now, though, if he is to satisfy Newton's third law, he must exert a force to diminish the package's westward linear momentum; that is, he must exert an eastward force, the reverse of the force he had to exert to get the package onto the train. But now we can see that the station-master had to do work upon the package both to get it onto the train and then again the get it back, to bring it back to its original state. In net result he has annihilated the energy that his work represents and thereby violated the conservation law pertaining to energy.
That fact does not give us sufficient reason to dismiss the wormhole's existence. We know that the conservation of energy theorem admits exceptions and in the wormhole we may have found one. We need to look further to see how this double-sided forcefield necessitates the non-existence of wormholes.
So far we have seen the station-master putting a package into the east side of the station portal and the package coming out of the west side of the train portal. Now suppose the station-master were to put the package into the west side of the portal.
In that case the station-master looks through the portal toward the rear of the train. He must know that in the train's frame the package in his hand has an eastward momentum of MV. It comes as no real surprise to him, then, when he reaches the package toward the portal only to have it yanked from his hand and catapulted toward the caboose. Fortunately, the train's crew have set up a net behind the portal, lest objects pelt the cars behind the portal. Unfortunately, the net has just enough elasticity to toss the package back at the portal. Yanked back through the wormhole, the package slams into the station-master's belly and knocks the wind out of him. In this example the package's passages through the wormhole have created energy ex nihilo.
Again, we don't have sufficient reason to dismiss the existence of wormholes. What does give us sufficient reason comes from the fact that the forcefields that we have described produce a weird kind of time travel. And on that subject I must first digress slightly off our current path of reasoning.
Many physicists believe that a lesser feature of the wormhole gives us a restricted form of time travel (See Cramer, John G., "Wormholes and Time Machines" in The Alternate View, Analog Science Fact and Science Fiction, June 1989). If one portal moves relative to the other, then a clock attached to the moving portal, as viewed from a point outside the wormhole, will lapse time that the relative motion dilates relative to the time elapsed on a clock attached to the stationary portal. The presumption that leads to the time travel hypothesis says that, viewed through the wormhole, the moving clock lapses time at the same rate as does the stationary clock. Thus, when the portals come back together again, one can travel through the wormhole to the recent past or the near future, depending upon the direction in which one traverses the wormhole.
Actually, the requirement that Reality conserve that total linear momentum whenever an object passes through the wormhole invalidates that presumption. The conservation law requires Existence to so structure Reality that a pseudo-gravitational field appears in any wormhole whose portals move, one relative to the other: that field obliges any object going through the wormhole to be subjected to a force that confers upon it the momentum that it would have to be given if it were to be made to pace the moving portal on a path outside the wormhole.
That fact applies to anything that carries linear momentum. Light carries linear momentum, so we must infer that the forcefields in the portals impose either a redshift or a blueshift upon light passing through the wormhole. But we also know that light serves Reality as the touchstone for the relationship between space and time; specifically, we know that if some force causes light to suffer a redshift, then that force also dilates the time elapsed on clocks at the light's source. Likewise, we know that if some force causes light to suffer a blueshift, then that force also contracts the time elapsed on clocks at the light's source.
Now we know that if the station-master looks through the east side of the station portal, he will see the train's clock, reflected in a mirror, falling progressively farther behind the station's clock. But if he looks through the west side of the portal, he will see the train's clock, reflected in another mirror, gaining time over that shown on the station's clock.
When no one is looking, a certain young Mister Coyote, filled with the Trickster spirit, comes to play a prank. With barely controlled glee, he sets up an array of mirrors, a shutter driven by a photocell, and a flashbulb with a collimating lens around the station portal. When the station clock shows noon the flashbulb pops, the collimating lens shapes the pulse into parallel rays, which then go through the east side of the portal, bounce off several mirrors that bring it through the east side of the train's portal, out the west side of the station portal, and onto the photocell, which sends out a pulse of electricity that closes the shutter, which Mister Coyote has placed directly in front of the collimating lens. But when the station clock shows noon, the train's clock, seen through the east side of the station portal shows five seconds to noon. Seen through the west side of the station portal, the train's clock showed five seconds to noon ten seconds before the station clock did, so at fifteen seconds before noon on the station clock the shutter closes and prevents the pulse from going through the wormhole.
That fact proclaims loudly and clearly, "This absolutely cannot happen." Existence absolutely will not admit a situation in which a pulse of light passing through some region of space prevents itself from passing through that region of space. No paradox confronts us here. We know exactly what our description of Reality requires of us - a firm statement that wormholes absolutely do not exist.
However, we have already mooted that analysis by the fact that, General Relativistic considerations notwithstanding, Special Relativity contradicts the existence of wormholes. Bearing in mind all of the evidence weighing against the existence of wormholes, we may well wonder how these things ever became part of our scientific culture in the first place.
The theory of wormholes evolved from a paper written by Albert Einstein and Nathan Rosen to resolve what they saw as a problem in General Relativity (Einstein, A. and N. Rosen, "Particle Problem in the General Theory of Relativity", Physical Review, Vol. 48, Pg 73-77, 1935 Jul 01). In the case of a body so compact that its mass lies within its Schwartzschild radius ( R = 2MG/c2) they found that the radial spacial coefficient of the metric equation "blows up"; that is, it becomes infinite. Likewise the coefficient of the temporal component goes to zero. Those facts represent what physicists call an event horizon and create a serious problem in our understanding of Reality. Einstein and Rosen sought to solve that problem by devising a coordinate transformation that would eliminate the infinities from the Schwartzschild and Reissner metrics. For the Schwartzschild metric
they made the substitution
thereby converting the metric equation into
In the authors' analysis the case U = 0 produces a hypersurface that joins two congruent sheets (U<0 and U>0) as a "bridge" (the authors' own term). This Einstein-Rosen bridge is the concept that evolved into the wormhole.
But, as I will show you in the section on General Relativity, the Einstein-Rosen bridge was based on a misunderstanding of the Schwartzschild metric. The equation that Einstein and Rosen modified is semi-classical, in the sense that the 2MG/r factor in the coefficients comes from representing the square of relative velocity in terms of the equivalent gravitational potential via the Newtonian expression for the kinetic energy. But when we use a fully relativistic version of the Schwartzschild metric, we don't get an event horizon in our description and, thus, have neither the need nor the opportunity to build an Einstein-Rosen bridge.
This analysis may offer us a cautionary tale in our pursuit of a Rationalist physics. By making a slight foray into the territory of Rationalism, Einstein and Rosen went wrong and led physicists on a wild-goose chase because they did not understand clearly what the Schwartzschild metric represented. But we must make such forays or we will get nowhere with our efforts to understand Reality. We cannot rely on pure Empiricism and a faith in serendipity to advance our cause.
We cannot rely on pure Empiricism because ultimately Empiricism lacks vision. That's why Hans Christian ěrsted failed to discover Faraday's law of electromagnetic induction, though he could have done it easily (as I will show in another essay).
On the other hand Rationalism promotes too much vision. Wishful thinking can intrude too easily into our efforts to develop knowledge through the application of pure Reason. Insufficient understanding of the phenomena that we seek to deduce may also lead us astray.
We can, of course, balance Empiricism and Rationalism against each other, using the empirical-inductive Scientific Method to act as a Reality check on the axiomatic-deductive method of Rationalism. But we might also correct Rationalism by developing a pseudo-empiricist test for Rationalist hypotheses - that is, a protocol for imaginary experiments. But that, in itself, may lead us in circles since the protocol must come from a Rationalist analysis of what we have accomplished. Quis custodiet custodiens? How can we test the test?
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