People assume that time is a strict progression of cause to effect, but actually, from a non-linear non-subjective viewpoint,it’s more like a big ball of wibbly-wobbly, timey-wimey… stuff.
~The Tenth Doctor (David Tennant)

This is part three of a multipart series on faster-than-light travel. In the first part of the series, I explained why the speed of light is constant, no matter the observer. In part two, I explained why this invariance prevents us from going faster than light. This time, I’ll explain how we might use general relativity to get around this restriction. Fair warning: although general relativity is generally accepted, using it to travel faster than light is very much in the realm of science fiction.

Before we can understand how to travel faster than the speed of light, we need to understand a little about general relativity.

Gravity and Acceleration

In the world of Sir Isaac Newton, the forces of nature–electromagnetism, gravity, etc.–cause objects to accelerate. Objects “resist” this acceleration with their “inertial mass,” . This is the classic equation from high school physics:

   

Newton’s theory of gravity predicted that the force exerted on an object by the Earth’s gravity was proportional to the object’s “gravitational mass.”

   

So we can calculate the acceleration on an object due to gravity by letting :

   

You may be wondering why I’ve written two different masses: “gravitational mass” and “inertial mass.” Usually, we only talk about one type of mass; I wrote them as separate objects to emphasize how surprising this is. Every  force other than gravity has a “mass” different from the inertial mass associated with it. Electromagnetism has “charge” and the strong force has “color.” The equation for the acceleration due to an electric field is

   

or

   

The “inertial mass” is still , but the “electric mass” is the charge , which is acted upon by an electric field .

In truth, as far as we can tell from experiment, gravitational mass and inertial mass are the same, . The equation I wrote becomes

   

and we can cancel the mass term to find

   

This implies a special relationship between gravity and acceleration, one that is absent for the other fundamental forces. Albert Einstein picked up on this relationship and took it one step further. In Einstein’s view, gravity doesn’t cause acceleration–gravity is acceleration.

Einstein’s Elevator

Einstein postulated the following thought experiment. Imagine that Emmy Noether is inside a completely sealed elevator, such that she can’t observe the world outside. She recognizes that she is being held to the floor by the same strength of force as standard Earth gravity. However, Noether can’t tell whether this is because the elevator is sitting stationary on the Earth or because the elevator is accelerating through space at a rate of 9.8 meters per second squared.

Einstein argued that Noether can’t tell the difference between acceleration and gravity because there is no difference. Explaining this was Einstein’s primary motivation for inventing general relativity. To make it work, Einstein had to introduce some radical ideas.

Space and Time as One

As we discussed last time, special relativity predicts that space and time both change to respect the constancy of the speed of light. This suggests that space and time are very similar. In general relativity, space and time are considered to be two parts of the same, unified object, called spacetime. We thus live in a four-dimensional world–three dimensions of space and one of time, all of which can mix together. (This “mixing,” by the way, can explain special relativity effects much more elegantly than the method I described last time.)

In general relativity, spacetime can curve. What I mean by “curve” needs some explanation, however. Imagine that you’re driving from your hometown of City to the capital, Metropolis, but there’s a mountain in the way. Travelling over the mountain takes more time than travelling around, both because the mountain is tall and because the vertical climb is more difficult.

A three-dimensional picture of what’s going on would show that the ground is curved upward into the shape of a mountain, forcing you to go around. However, it’s possible to encode the same information in two dimensions. If we draw the two paths on a map, the path over the mountain looks straight and the path around it looks curved. However, we define the straight path to be longer than the curved one, even though our Euclidean eyes tell us otherwise.

Incidentally, if the path around the mountain is the shortest possible path between two points (in our case, City and Metropolis), that path is called a geodesic. All objects in free-fall (including light rays) travel along geodesics.

This is what I mean when I say that spacetime is curved. We can measure (using light as a meter-stick) that distance itself changes depending on our position in spacetime. We call this change in distance the curvature of spacetime. It is a curvature in spacetime, not just space, because we can measure the “temporal distance” to change (a form of time dilation) as well as the spatial distance. Note that the curvature of spacetime affects spatial and temporal distance in ways not predicted by special relativity. There’s more going on here than the constant speed of light.

So how does spacetime become curved? This is the second part of Einstein’s theory. Mass and energy generate curvature. A massive object curves spacetime such that the distance around that object shrinks–the more massive, the bigger the curvature. If a light ray passes by the object, the light ray will be deflected towards the object, because the path to wherever the light is going will be shorter if it goes through the contracted space by the object. This is the cause of gravitational lensing.

Matter is similarly deflected by curvature. John Wheeler famously said, “Matter tells spacetime how to curve, and curved spacetime tells matter how to move.” That’s important enough to repeat.

Matter tells spacetime how to curve, and curved spacetime tells matter how to move.
~John Archibald Wheeler

Note that matter pulls on time, too, not just space.

Shortcuts

The ability to bend spacetime using mass means that we can change the shape of the universe; in turn, we could theoretically manipulate distance so that distant points become very close, allowing us to effectively go faster than light. The most obvious way to do this is with a wormhole, which is literally a shortcut through spacetime.

Wormholes are created by a black hole–white hole pair and would allow instant travel from the black hole to the white hole. Unfortunately, it is believed that a wormhole would collapse as soon as even a single photon travelled through it. Exotic matter might be able to solve this problem, however.

We could also create “warp paths” through spacetime by making a “daisy chain” of extremely massive objects, creating a path between points that is much shorter than it would otherwise be. The problem with this approach is that we have to get somewhere the slow way before we can make a warp path to it. Furthermore, our “daisy-chained” massive objects would likely have to be Jupiter-sized or larger to have the desired effect. Our civilization would need a level of technological sophistication capable of tearing apart multiple solar systems for raw materials, then reassembling them however we wanted. So your great-great-(x150)-grandchildren would be lucky to see a warp path begin construction, let alone enter operation.

Of course, why make a permanent warp path when you could take one with you? This is the idea behind the Alcubierre Drive. An Alcubierre drive shrinks space ahead of it and grows space behind it, so that a spaceship is pulled along by the very fabric of spacetime. The Alcubierre drive relies on the fact that, although no particle can travel faster than light, spacetime can stretch or shrink at any speed. However, there are major problems with the Alcubierre drive. It needs massive amounts of exotic matter–not as consumable fuel, exactly, but as a facilitator to move spacetime around the spaceship–and it is possible that an Alcubierre field could only exist if it existed since the beginning of the universe.

Of course, all of these ideas are very much in the realm of science fiction. Although they’re all theoretically possible, it is unlikely we’ll see any of them in the near future.

Further Reading

• The Alcubierre Drive has gotten some press recently. Dr. Harold White at NASA is currently working on Alcubierre theory. He plans to test the effect at a very small scale in the lab.
• Odd Factor has a nice compilation of quotes on Einstein’s gravity.
• The website for the Gravity Probe B experiment has a very thorough description of general relativity and related phenomena.
• The Wolfram Site has a decent demonstration of the difference between Newton’s and Einstein’s gravities.
• The Physics of the Universe, an excellent layman’ website, has a description.
• The Encyclopedia of Science has a nice description of wormholes.
• Robert Johnston’s site has a summary of general relativity.
• How Stuff Works has a very accessible article.
• For the truly brave, the canonical textbook on general relativity is this behemoth by the physics giants Charles Misner, Kip Thorne, and John Wheeler.
• Michio Kaku‘s famous book (and any number of popular-science books).
• Ask me for more! General relativity is one of my favorite topics in physics, and I’d love an excuse to describe it in more depth.

Questions? Comments? Hatemail?

As always, if you have any questions, corrections, comments, or complaints, please let me know in the comments!