The Voyager 1 space probe is (as of 7:01 AM US Eastern time yesterday morning) 17 hours, 5 minutes, and 43 seconds of light travel time from Earth. That means sending a radio signal one way to the probe takes just over 17 hours, and any response from the probe will take an additional 17 hours. Voyager 1 was launched in 1977, so it took over 35 years to get that far—and it is just reaching the edge of the Solar System now. The nearest star system (the Alpha Centauri system, which includes our closest stellar neighbor, Proxima Centauri) is over 4 light *years* away, and so any significant interplanetary travel would take a long time using current technology. Admittedly, Voyager 1 wasn’t built to be an interstellar spaceship—it was designed to explore Jupiter and Saturn (especially Saturn’s moon Titan)—but even if we were aiming at reaching another star with a probe, we couldn’t go a lot faster without a significant advance in propulsion technology.

### Warp speed, Mister Sulu!

However, there are loopholes of sorts. One class of hypothetical particles known as *tachyons* always travel faster than the speed of light, which provides a *lower* speed limit: you can’t slow them down to speeds we think of as ordinary. Unfortunately, no current theory involving tachyons is consistent, either within itself or with physical reality. Besides violating causality (meaning: effects can happen before their causes when tachyons are involved), tachyons lead to problems with energy conservation and other important physical laws. Even if these problems could be sorted out, it’s uncertain how one could harness tachyons to build spacecraft.

Another loophole is far more promising, and involves Einstein’s *general* theory of relativity (henceforth GR for “general relativity”). Within GR, it’s possible for an object to move much faster than light relative to another object, without ever violating the special theory. This may sound exotic, but it’s not: our own Universe does this regularly! As we know, the Universe is expanding: spacetime is stretching, carrying galaxies farther away from each other. Because of this, distant galaxies appear to be moving away from us, and the farther they are, the faster they appear to be traveling—to the point where many can appear to be traveling several times faster than the speed of light. Are these galaxies *actually* moving that speed? No. This is another example of relative motion, complicated by the expansion of spacetime.

That’s obviously a large scale phenomenon: nearby galaxies don’t appear to be moving anywhere close to those speeds. However, inspired partly by *Star Trek*, some physicists have pondered whether the faster-than-light relative motion could be replicated on the small scale. The best known of these is due to Miguel Alcubierre, from a 1994 paper, with a new revival from a NASA researcher named Harold White. The basic idea runs as follows: a small region of spacetime forms a bubble, inside which any motion is slower than the speed of light relative to the bubble, but from the perspective of an outside observer, any motion appears *faster* than light. Special relativity is preserved, yet a body within the bubble can travel much faster than it could otherwise. The bubble isn’t a material object, so its “motion” is irrelevant: no *thing* is moving faster than light.

White in particular promotes the idea that we can build spacecraft that generate these bubbles, and use them as a *Star Trek*-style warp drive. (In fact, this is the usual “scientific” explanation I’ve heard for how the starship *Enterprise* can travel as quickly as it does.) Over the last few months, including a few stories this week alone, have presented White’s ideas uncritically (see these stories in io9 and Wired, for example), but as I hope to show, we’re still nowhere close to building warp drives.

In particular, its not certain such bubbles can exist, much less whether they can be generated by a spaceship. The problem lies not in the mathematics—that’s remarkably straightforward, by GR standards—but instead in the nature of the source of the bubble. I’ll elaborate on this in the next, more technical section, but think of it this way: the bubble is a manifestation of gravity, just as all distortions of spacetime are in GR. The source of all gravitational distortions is energy (or mass, which is just another form of energy), but to make a bubble like Alcubierre’s system, the density of energy needs to be *negative*, whereas all normal matter produces positive energy density. To put it another way, the bubble isn’t a vacuum, it’s kind of a expulsion, pushing everything out of its way so stuff inside it can travel faster than light. That’s not typical behavior for any kind of normal matter.

That leaves two possibilities: somehow generating the necessary negative energy density, or exploiting “exotic matter”. Negative energy density does show up in quantum field theory as an aspect of the quantum vacuum; I discussed that briefly in a previous post, in a parenthetical note about the Casimir effect. To summarize: if you place two metal plates in a vacuum, they will experience a very slight attraction to each other, simply because the quantum fields in the vacuum generate a negative energy density. The key here is *slightly*: the Casimir effect is very small, so even if it produces the kind of bubble needed for a warp drive, you’d have to somehow amplify it, or make a huge amount of quantum vacuum.

The other option is exotic matter, and we’re not talking about the ordinary exotic matter like the Higgs boson or dark matter, because these are associated with positive energy density. The kind of exotic matter needed for warp drives is stuff like…yes, the tachyons I mentioned earlier, which may not exist at all, and may not be exploitable even if they do exist. If your grand solution for sailing to the stars depends on matter for which we don’t even have a coherent theory, much less experimental evidence, the idea isn’t promising. I give White props for his efforts to try to test his ideas in the lab—for all the problems I see with his scheme, he’s no crackpot—but I’m not placing any odds on his success.

At this point, I want to delve into the structure of general relativity a little bit. There will be a few equations, but don’t let that stop you from reading! I think it may help you understand how the Alcubierre-White scheme can simultaneously be correct mathematically, but unlikely to be physically realized.

### General relativity in ~~boxers~~ brief

The central tenet of general relativity is best summed up by physicist John Archibald Wheeler: “Matter tells space how to warp. And warped space tells matter how to move.” (This is often misquoted, replacing “warp(ed)” with “curve(d)”. I’ve done it myself.) That concept is simple, but deceptively so: matter shapes the environment in which it exists, a nonlinear process that makes many calculations in GR very complicated.

The first bit of Wheeler’s description takes the mathematical form of *Einstein’s equations*, which despite the plural, are written as follows:The **G** on the left side is the geometry of spacetime (technical name: the Einstein tensor), while the **T** on the right contains the contribution of matter and energy (the energy-momentum tensor); *κ* is simply a number that sets the strength of gravity, and is determined by experiment. The *a* with an arrow is a “vector”, a number that describes the direction of motion along a path in spacetime. We can also include a *cosmological constant* in this equation, which is one possible way to describe dark energy—the acceleration of the cosmos. However, the cosmological constant is a relatively (ha!) small contribution for most applications of GR, including studies of the Solar System, black holes, or warp drives.

(Experts may or may not recognize this way of writing Einstein’s equations. Often you’ll see it written in “index notation” instead, but the way I’ve written it means the same thing. I prefer this version for most applications, which is the “machine” or “coordinate-free” notation. Instead of thinking of a tensor as a complicated matrix, you think of it as a function that takes a vector as an argument and (in this case at least) returns a vector as output.)

Since spacetime is four-dimensional (three space dimensions, plus one time dimension), there are four distinct vectors you can choose to plug in, which is why we say that this is a set of four equations rather than one single equation. Basically, you pick a path—say, the trajectory of a particle like an electron or a politician—and insert the vector that describes the path into the equation. **G** then tells you the landscape of gravitation: which way “downhill” is, no matter what path the electron or politician happens to be following.

Ideally, Einstein’s equations work like this: you have a distribution of matter and energy (a star or galaxy or black hole or whatever), which gives you **T**. Then you “solve” the equations to get **G**, the landscape of gravitation. The solutions are themselves a set of equations, which are often complicated. In practice, it isn’t always easy to get the solutions from the equations, though: they are not the kinds of equations we all solved in high school algebra. As a result, many solutions have been found by educated guesswork: starting from an idea of what the solution should look like, and working backward. Many of the most important, powerful solutions to Einstein’s equations have been found this way, including those describing black holes: the *Schwarzschild* and *Kerr* solutions, for non-rotating and rotating black holes, respectively. Then you use that solution in another set of equations (called the *geodesic equations*), which is the second bit of Wheeler’s description: how the landscape of gravitation dictates the motion of matter. That’s GR in one paragraph, kind of.

Alcubierre’s solution is one of these solutions, so it’s not in the same category as crackpot theories I’ve discussed here before. However, note that in the previous paragraph, I never said every solution to Einstein’s equation actually corresponds to a physically real situation. In fact, *most* solutions in GR aren’t physically realized in our Universe! They describe rotating universes, as in Kurt Gödel’s solution, which he presented to his friend Einstein as a birthday present, in which time travel is an immediate consequence. They describe infinitely long, infinitely dense strings of matter running perfectly straight across the cosmos. (These *cosmic strings* may be mathematical idealizations of real objects (which would *not* be infinitely long), though astronomers have yet to see one in the wild.) And they may describe a bubble that allows faster-than-light travel, but which requires as its cost a **T** that cannot be made by any matter or energy distribution known to science.

You see, Einstein’s equations are mathematical objects. Like computer programs, the quality of the output depends on the quality of the input. You can produce some beautiful fantasies in GR, and I have spend a fair amount of time doing that, but in the end the value of the solution lies in whether **T** corresponds to a distribution of matter and energy that exists in nature, not in whether **G**—the landscape of gravitation—is beautiful.