Last year I got talking to theoretical physicist Aron Wall about the thermodynamics of quantum gravity. Now that’s a deceptively beautiful phrase: in four words, you get three of the deepest areas in modern science. Their union promises answers to such mysteries as the arrow of time and what the heck time actually is. And in reading one of Wall’s papers, I learned that thermodynamics might finally settle whether time machines are possible. So I invited him to describe his ideas for us. Wall, a postdoc at U.C. Santa Barbara, has his own blog Undivided Looking, and the discussion of this post will continue there; he says he’s happy to answer any questions you may have.–George Musser
We’ve all seen those movies where someone goes back in time and tries to change something (the classic “Grandfather Paradox”: what happens if you go back in time and try to kill your grandfather?). Sometimes the attempt results in a new timeline, where everything becomes different and only the time traveler remembers what things used to be like. In more fragile universes, the paradox threatens to destroy the entire spacetime continuum unless the intrepid hero puts everything back the way it is supposed to be. But my favorite is when writers are clever enough to imagine a single consistent timeline in which everything makes sense.
Is this something we have to worry about in the real world? Suppose someone tried to build a time machine in the laboratory. Is there some law of nature which would prevent them from doing so? Theorists have debated this question for decades, but recently I have proposed an argument based on the Second Law of Thermodynamics which would rule time machines out.
In modern physics it’s not actually obvious that you can’t build a time machine. The reason comes from our best theory of gravity, general relativity. Einstein showed that the matter can cause distortions in space and time. For example, near the surface of the Earth, time goes slower than in outer space, by about one part in a billion. (This is actually what causes things to fall down.) One part in a billion is not very much; in other words, the gravitational field of the Earth is fairly weak. But there are stronger gravitational fields in the vicinities of neutron stars and other really massive objects. The most extreme examples involve black holes, where the gravitational fields are so strong that not even light cannot escape–if, that is, the light gets too close and falls in past the event horizon, the point of no return.
Is it possible for a really strong gravitational field to distort time so much that, by guiding your spaceship through just the right trajectory, you end up meeting a past version of yourself? Physicists call a situation like that a closed timelike curve (CTC). A “curve” is a trajectory through spacetime. The word “timelike” is code for saying that the curve doesn’t go faster than light, so that a physical object (like a spaceship) could in principle follow it. And “closed” just means that the curve meets itself again at an earlier point.
One possible strategy for building a time machine would start by finding or constructing a traversable wormhole. A “wormhole” (you may recall from various science-fiction stories) is a tunnel through space which connects two very distant regions (see artist’s conception above). It has two ends (you could imagine these as being roughly shaped like spheres) that are connected to each other by a tube of space called the “throat.” It is “traversable”–and the ones in science fiction nearly always are–if you can drive a spaceship through it to get to the other side.
It’s actually pretty easy in general relativity to construct mathematical solutions that look like wormholes. In fact, if you take the equations that describe the gravitational field outside an eternal, unchanging black hole and extrapolate them to inside the event horizon, you find that the black hole is actually a gateway to another universe. The hard part is keeping this gateway open long enough to get through it. In the case of a regular black hole, the wormhole throat collapses so quickly that you would instead hit the singularity in the middle and die.
But let’s suppose you overcame this problem and figured out how to build a traversable wormhole between two parts of our universe. You might then bring the two ends of the wormhole closer together. Let’s say that one end is on Earth and the other is in orbit. Astronauts use it to commute to the space station to do their day job and zip back home again in time for dinner. Remember that time is going slower on the Earth. That means that one wormhole end is traveling through time faster than the other. The time difference would accumulate. The astronauts would start to notice that someone traveling from Earth to space would arrive ever so slightly earlier than they left. A year or so after the wormhole had been put in place, there would start to be a closed timelike curve. Eventually, an astronaut would be able to hop through the wormhole and send a radio signal back in time to an earlier version of themselves! An astronaut might go through the Earth end of the wormhole at 9:00 a.m. and arrive at 8:59 a.m. in the space station. She could then send a radio signal to herself back on Earth, telling herself not to bother to go to work that day, creating a potentially troublesome situation like the grandfather paradox.
Fortunately for grandfathers everywhere, it turns out there’s a theorem in general relativity which says wormholes are never traversable unless you have hold them open using negative energies. To see why, imagine sending a bunch of light from different angles into one end of the wormhole. The light is initially contracting; the rays are getting closer together. But when the light rays pop out the other end and radiate away from one another, they are expanding. This means that the gravitational fields in the wormhole caused the light rays to defocus, or bend away from one another. But normal, positive-energy matter, which has attractive gravity, always causes light to focus. To hold a wormhole together, you’d need some sort of anti-gravitating matter with negative energy. That would be quite weird, because normal objects always have positive mass and therefore (by E=mc2) positive energy.
A theorem due to Stephen Hawking says that if you have a spacetime that starts out without any time machines, then you can never build one unless you have negative energies. So the mystery is solved, it seems. So long as all objects in nature have positive energies, time machines are impossible. There’s just one little problem, which is that the premise isn’t true.
Besides general relativity, our other best theory of the universe is quantum mechanics. And it turns out that if you study the quantum-mechanical properties of, say, the electromagnetic field, you can make negative energies. A classic example is the Casimir effect, which has been experimentally measured in the laboratory. If you take two electrically conducting plates and put them very close together, the plates affect the empty space in between them in such a way as to create a negative energy density. Granted, the negative energy in the gap between the plates is much smaller than the positive energy density in the plates themselves. But the presence of any negative energy invalidates theorems such as Hawking’s which rely on energy being positive everywhere.
The question is whether some replacement principle still holds even in quantum situations. In my own work, I’ve proposed that there is such a principle, related to the Second Law of thermodynamics. This, you might recall, is the law of nature which says that you can’t build a perpetual motion machine; there exist processes in nature which are impossible to reverse.Technically, physicists define a number called the entropy to measure how scrambled-up the universe is at a given moment of time. The Second Law says that this number always increases as time passes.
Even black holes obey a version of the Second Law. Their entropy turns out to be proportional to the area of their event horizon. In other words, the area of a black hole, plus the entropy of any matter outside of it, always increases with time. This formulation of the Second Law, discovered by Jacob Bekenstein and Stephen Hawking, is called the Generalized Second Law (GSL). In my dissertation research, I proved mathematically that the GSL is true in a broad range of circumstances.
But here’s the important point: you don’t need a black hole to talk about the thermodynamics of horizons. The same principle applies to any observer at all. If some regions remain out of that person’s sight forever, those regions are behind the observer’s own personal horizon. No black hole is necessary. For example, if you get in a spaceship and accelerate, then as long as you have a sufficient head start, some light rays will never catch up to you. They are behind your own personal horizon. This type of horizon is called a Rindler horizon.
A similar thing happens in cosmology. The expansion of the universe is accelerating as time passes. This means that if another galaxy is sufficiently far away, we will never be able to see it, no matter how long we wait. This is called a cosmological or de Sitter horizon. Every instant of your life, you are falling across a horizon as seen by aliens on some sufficiently distant planet!
But here’s the amazing thing. All of these types of horizons also obey the Second Law, just as black holes do. It doesn’t matter that the horizons are defined in a more subjective way. It is still the case that their area, plus the entropy of anything that can be seen by the observer, is increasing as time passes. The Generalized Second Law still applies.
And this is sufficient to rule out time machines. Suppose you could manufacture a spacetime with a CTC. The closed timelike curve would itself have a horizon. To see why, consider what you would see if you followed the path of the CTC. You strap yourself into your space pod, travel through space to the wormhole mouth, and pass through the wormhole to return to your original place and time. You get on the ride again and keep going around forever and ever. (This scenario has many complications, but let’s ignore them, since we’re just using this concept of “observer” to illustrate the behavior of the CTC trajectory.)
Now suppose someone else–say, the Intergalactic Advertising Agency–tries to send a radio signal to you from a distant planet. As their signal moves towards you, it advances to later and later times (just as most things do). The farther away it starts, the later it will be before the advertisement arrives. You, on the other hand, keep experiencing the same moments of time over and over again. So if the IAA starts sufficiently far away, its ad will never be able to reach you. That means that there is a horizon separating the places that can reach you in time from those which cannot.
I’ve drawn a picture of this scenario on the left. It shows a spacetime diagram of a universe containing a CTC. Light travels at 45 degrees. Imagine that the two blue slits are two ends of a traversable wormhole. The CTC is the red curve. The wiggly part is your journey through spacetime to the wormhole mouth; the dotted line represents your passage through the wormhole back to your original location. The black cone represents the horizon, separating points that can send signals to the CTC from those which cannot. The earlier green signal is capable of reaching the CTC, while the later one is not. At early times, the horizon has a large and shrinking area.
The horizon (which is just the boundary of what can be seen by the CTC) exists even before the CTC forms. At very early times, the horizon would be a gigantic sphere contracting at the speed of light. Because it is shrinking, so is its entropy. To preserve the GSL, the entropy of some other matter system must increase. The trouble is that the total decrease in entropy is actually infinite, and nothing could compensate for that. Therefore the time machine would violate the GSL.
So we learn that the GSL forbids time machines. Actually, it’s so strict that it forbids lots of other things, too. You can’t have traversable wormholes (even ones that aren’t time machines) because trajectories going through the wormhole have horizons with decreasing area. You can’t make a warp drive for similar reasons. To science-fiction fans, I’m terribly sorry for pouring cold water on your dreams, but this seems to be how things are.
You can also use the GSL to show that time has to end at a singularity inside of black holes and had to have a beginning at the big bang. Otherwise, there’d be observers whose horizons wouldn’t satisfy the GSL. Admittedly, no one really understands the laws of physics near singularities, so this part of my research is speculative and depends on what kinds of new physics might be relevant. It is conceivable that, even though the GSL is valid in all the situations we’ve been able to check so far, it is false near singularities. But I wouldn’t bet on it. Something has to make it impossible to kill your grandfather, after all.
Wormhole figure by George Musser; spacetime diagram by Aron Wall