The United Federation of Paradox

In Star Trek, the Federation is a post-capitalist utopia where citizens act out of a desire to better themselves or civilization rather than attain monetary wealth. It's not entirely clear how this utopia came about, but we're often told humanity transcended its violent, greedy impulses through cultural evolution. A more cynical view is that the advent of replicators eliminated most scarcity, and with it any need to be violent or greedy.

I would like to offer an alternative hypothesis: Every episode, Starfleet ships employ technology that permits time travel, so the Federation should be able to Seven Days its way out of any mistake on the path to utopia. You see, the physics of the 20th century—special relativity—tell us that any method of FTL (whether warp drive, subspace communication, or galactic spore network) is also a method of time travel. FTL permits time travel because reality has no rigid, universal stage on which all events play out. Instead, space, time, and the events that occupy space and time are all linked together by a consistent set of interrelationships.

Galileo made this argument while trying to convince others that a spinning, moving Earth wouldn't throw everything out of whack. What we now call Galilean relativity says the laws of motion don't depend on your (inertial) frame of reference. A frame of reference is just a perspective from which to observe the universe. If you're sitting in a chair reading this, you and the chair constitute a frame; if you're hurtling through interstellar space (at a constant speed) in a starship, that's another frame. Also, go you.

Galilean relativity means that as long as you occupy an inertial frame, you never notice anything funny that doesn't accord with the laws of motion. Whether you're in a turbolift or a shuttlecraft, if your velocity is constant, a tossed ball will land where you expect it and you won't feel any mysterious forces pushing on you. The upshot is that no frame of reference is privileged or what's "really" happening. All are equally valid.

The tricky part is translating one reference frame to another. Walking down the aisle of a plane, everyone on the plane can treat you as moving only a few miles per hour. Everyone on the ground, however, needs a way to combine your velocity and the plane's. This feat is accomplished via a transformation, which is just a mathematical tool for moving between reference frames. In Galilean relativity, that transformation is easy and basically commonsense: to an observer on the ground, your speed = plane speed + walking speed.

It is these transformations—which spell out equally valid and consistent ways of interpreting reality from different frames of reference—that allow for time travel. To see how, we have to move from Galilean relativity to Einstein's special relativity.

Special relativity is a generalization of the Galilean variety. There are two postulates that end up having deep consequences:

(1) The laws of physics don't depend on your frame of reference.

This is an expansion of Galileo's rules to include electromagnetism.

(2) The speed of light (c) is a law of physics.

This postulate is implicitly included in the first one, because Maxwell's equations for electromagnetism predict a speed of light. It's the revolutionary part in all this, though, so Einstein spelled it out explicitly.

By itself, a law that dictates a speed is not terribly noteworthy. Any wave equation specifies the speed at which the wave travels. We usually think of waves as traveling through a medium, in which case Galilean relativity might apply. To an outside observer, the total wave speed = medium speed + wave equation speed. Physicists assumed this applied to light as well and proposed a luminiferous aether to serve as a reference frame and medium.

The trouble was, the properties required by a luminiferous aether (given how light behaved) seemed ludicrous and unphysical, and when measured, c always seemed to be the same. Additionally, and famously, the Michelson-Morley experiment failed to detect any sign of the aether. The alternative, according to Einstein, is that c is not defined relative to a frame of reference; instead, the speed of light is a law of physics and the same for all inertial observers.

But this violates the rules of the Galilean transformation, because it means you can't add velocities when light is involved. If a Klingon runs at you firing a laser pistol (canon in some of the TOS era), Galileo says the laser's speed = Klingon running speed + c. Einstein says the speed is always only c, for both you and the Klingon. And that means we need a new transformation that is, as before, equally valid and consistent for all inertial frames of reference. For special relativity, that's called the Lorentz transformation.

Rather than just show you the Lorentz transformation (it involves c and some square roots and reduces to the Galilean transformation at everyday speeds), I want to provide a visual explanation for how all observers can measure the same c. Memory Alpha says Vulcan is 16 light years from Earth. So let's imagine there's a starbase between the two planets, 8 light years from each. If the starbase emits a radio signal traveling at c, it reaches both Earth and Vulcan 8 years later. How do we represent this graphically?

Credit: Paramount/CBS for the Trek stuff and NASA for the Earth stuff.

The x-axis (horizontal) is distance in light years and the t-axis (vertical) is time in years. If our reference frame is the starbase and the planets are not moving relative to it, then they move upward in time without moving left or right through space. The radio signals, on the other hand, move 1 light year per year, so they travel 45 degrees out from the starbase. Where the radio signal and the world line of a planet intersect is the location in spacetime (at the planet, 8 years in the planet's future) where the signal reaches the planet.

Now let's say the Enterprise is at the starbase and starts heading toward Vulcan at sublight impulse speeds. What does that look like?

Credit: Paramount/CBS for the Trek stuff and NASA for the Earth stuff.

Because impulse is slower than light, its path is tilted more toward the vertical than the radio signal; more time is required to go the same distance. Since we’re dealing with special relativity, there is an inertial reference frame following along with the Enterprise, and from that frame we have to measure the same c. According to the graph, this doesn't seem possible. It sure looks like the radio signal hasn’t gotten as far away from the Enterprise as it has the starbase (horizontal distance) in the same amount of time (vertical distance).

So here's where we need to perform a coordinate transformation that takes us from the reference frame of the starbase to the reference frame of the Enterprise. For a frame centered on one inertial object, the object's position doesn't change in time. For the starbase, that means its path through spacetime follows the vertical—or time—axis. So then let's define a new time axis (t') for the Enterprise which follows its diagonal path. If c is the same in all references frames, that means we also need a new space axis (x'), which has the same angular separation from the radio signal as t’.

Credit: Paramount/CBS for the Trek stuff and NASA for the Earth stuff.

Because x' and t' are tilted toward the radio signal by the same amount, the signal still moves 1 light year per year in this new reference frame; the ratio doesn't change. This has weird consequences, though. For starters, reconciling a constant c seems to have involved squishing space and time together. But it gets worse.

In the starbase reference frame, lines parallel to the x-axis are single moments in time. Any event on such a parallel line happens simultaneously for all observers sharing that frame. For the Enterprise frame, simultaneous events happen on lines parallel to the x' axis, which is a diagonal line that cuts through time in the starbase frame. This means events that are simultaneous in the Enterprise frame happen at different times for observers in the starbase frame, and vice versa.

For example, if you draw a line parallel to the x'-axis through the moment when the radio signal reaches Vulcan, you see that the event of the signal reaching Earth is ahead of that line; it happens later in the Enterprise's frame, despite the two planets being equidistant from the starbase. This is (a) the relativity of simultaneity, (b) patently ridiculous, (c) absolutely true, and (d) the feature we want to exploit to travel through time and create a problem-free utopia.

Normally (in special relativity), observers disagreeing on the order of events doesn't matter. If observers are limited to light speed or less, by the time they're able to meet up and discuss the discrepancies, all the events they disagree about are in everybody's past. FTL lets you circumvent this restriction.

So here's how to resolve every 42-minute Star Trek plot in 3 easy steps. The scenario presented here is set up for graphical simplicity; it smooths over a few wrinkles and might not perfectly align with Star Trek technology. (Then again, neither does Star Trek technology.)

Step 1: A space-ooze-energy monster attacks the Defiant, but it turns out the creature is just misunderstood. To restock on redshirts, Worf activates the Lorentz Protocol! Via subspace, the Defiant sends a message to Deep Space Nine.

Credit: Paramount/CBS

If subspace communication is instantaneous (which it looks close enough to being in most episodes), then Worf just finds the Bajoran system along the x-axis and puts the message there. Because no time passes, the message arrives along the x-axis.

Step 2: On DS9, Sisko gives the message to O'Brien, who hops into a runabout and flies away from the Defiant at impulse (some speed close to c).

Credit: Paramount/CBS

In our diagram, we're now switching to the runabout's moving reference frame. Its speed relative to the Defiant establishes a new frame of reference.

Step 3: The runabout sends a warning about the interdimensional slug to the Defiant's location in space via subspace.

Credit: Paramount/CBS

Because we are in a new reference frame moving relative to the Defiant, an "instantaneous" subspace message no longer appears somewhere on the horizontal line but along the runabout's x'-axis, which intersects the Defiant's spacetime location in its past.

Ultimately, the speed of the runabout and its distance from the Defiant determine, via a pretty simple triangle, how far into the Defiant's past the subspace warning goes. Arrange things correctly and Worf gets the warning before ever running into the crystalline spider-snake.

But of course, now Worf's gone and killed his own grandfather (who he may have been?—time travel!). That is, if he receives the warning before sending out the message to request a warning, then he avoids the cybernetic mind worm attack and never needs to send out a message in the first place. Paradox!

This is the central reason why physicists think FTL communication or travel is a non-starter. Other aspects of special relativity prohibit reaching c, but there’s nothing about naturally faster-than-light processes. They do, however, invariably lead to issues with causality.

There's a saying about this. Pick two: special relativity, FTL, or causality.

As we've just seen, special relativity + FTL means you lose a coherent narrative leading from the past to the future. You can preserve causality with FTL but only if you abandon the rules of special relativity. Or you can live in the universe we seem to inhabit, which has relativity and causality but loses all that FTL fun.

Of course, when asked to pick two, Star Trek usually just picks one: FTL. Most time travel stories in Trek are rife with causality issues that are usually intentionally ignored, except by having characters say things like, "Oh yeah I totally flunked temporal mechanics at Starfleet Academy, haha!" And relativity is almost entirely absent, because there's rarely any mention of time dilation or length contraction or all the other whacky things that happen when you get close to c.

Nevertheless, the United Federation of Planets is a utopia, and it must have gotten there somehow... or will get there... or will have already gotten there. (Oh boy. Consult Dr. Streetmentioner's book for tense corrections.) Or maybe not—after all, utopia does mean no-place.