Saturday, September 25, 2010

Time travel

Time travel is a concept used very often in science fiction. Basically it deals with questions like: what if we could travel to any point in time in either the past or the future? How could this be achieved? What would be the implications and consequences of such an endeavor?
As I've explained in the post dealing with sublight interstellar travel, Einstein's theory of relativity shows us that travelling close to the speed of light is time travel into the future. Time passes faster for everyone else except you. The theory of relativity has also shown us that an intense gravitational field produces the same effect. The closer you are to a gravity source (a massive body) the slower time will pass for you. For example, let's say we have a black hole with an event horizon at a distance R from it's center.

The event horizon is the surface beyond which light can no longer escape the intense gravity field of the black hole. If you go beyond it, there's no turning back, but as long as you keep your distance from the event horizon the black hole is just another massive object (it doesn't pull you in any more than any other object of the same mass). If you were at a distance of 2R from the black hole and stayed there for 7 years, you would notice that about 10 years have passed on Earth. As you can see, gravitational time dilation is a tiny effect.
So, we've solved the problem of time travel into the future. The effect has been measured, and current technologies like GPS satellites have to take time dilation into account so as to keep synchronized. So far we don't have the means to send a person into the future a meaningful length of time, but as technology advances and we develop faster propulsion techniques that moment will come some time in the near future.
Now, we deal with the really big question concerning time travel: is travel into the past possible?
For starters, let's tackle this from a purely logical manner. What would happen if you could go back in time? We can immediately see that many paradoxes can arise. For example, if you were to go back in time and kill your own grandfather before he met your grandmother, then you would cease to exist. But if that happens how can you go back in time and kill your grandfather?! This is known as the grandfather paradox. Another type of paradox that arises comes from conservation of mass/energy and information, the ontological paradox. If you go back in time to a point in which you already exist there will be two of you. And if both of you travel to a previous point, there will be three of you, and so on. Essentially you could fill the Earth with these "time clones" and it would appear that all of these versions of you are coming out of nowhere, a clear violation of conservation of mass/energy. There is also the problem of information: let's say that one day you find on your desk the plans for building a time-machine. So you build it and decide to send the plans back in time to your earlier self so that he can find them. Now comes the question: who invented the time-machine? You received the schematics from the future, built the time machine and then sent them back in time for you to discover. So no one actually wrote them. So where did they come from? These are examples of violations of causality, the basic notion that a cause produces an effect.
Another problem that arises is: if time travel into the past is possible, how come we haven't seen any time travelers from the future? Well, there are many ways to answer this question. Stephen Hawking postulated the chronology protection conjecture, which basically states that the laws of physics are such that they prevent time travel (well, except for submicroscopic systems which can't affect the timeline). Essentially, that would mean that travel to the past is impossible. Another answer would be that a time machine only allows you to go back in time to a point in which the device already exists, and since no one has built a time machine yet we don't have any visitors from the future. However, even if this is true, there are still the grandfather and ontological paradoxes to be solved. Fortunately, there are two solutions available both of which also solve the problem of why we haven't been visited by time travelers.
To begin with, we have the Novikov self-consistency principle which states that if time travel is possible then all time travel events are consistent with our current history. So basically, you could go back in time but you couldn't change anything from our current history. For example if you went back in time to kill your grandfather, you could be killed by some accident before you found him. It would also mean that you yourself are part of history, since your presence in the past is part of our current history. This solution to the paradoxes of time travel is somewhat unsatisfying as it implies a form of predetermination since your actions in the past seem to be restricted by history.
The second solution, which is my personal favorite, is the existence of parallel universes. The idea is that when you go back in time you are no longer in your own universe, but in another that is the product of your travel through time. So, in this case if you go back in time to kill your grandfather, it won't be your grandfather, it will be the one from that universe. In fact you can do whatever you want because, if let's say you are from the year 2050 and go back to the year 1950, in that universe the present is 1950, and the future of 2050 hasn't happened yet and that future will be determined by your actions in that universe. The existence of parallel universes is postulated in one of the interpretations of quantum mechanics, the many-worlds interpretation, but it makes the exact same predictions as the Copenhagen interpretation, which is currently used by the scientific community, making it difficult to assert the existence of such universes. At least it means it's plausible.
Now, let's tackle this from physics' perspective: how would a time machine work and how could we build one?
The simplest time machine can be constructed using a wormhole.

As explained in the previous post, a wormhole is a hypothetical tunnel through space-time which acts like a shortcut essentially uniting two separated regions. For example you could have one end of the wormhole on Earth and the other close to the Sun. Going from Earth to the Sun the classical route, takes a long time (even light takes about 8 minutes for this trip), but stepping through the wormhole will take you there instantly. Now, from what we've learned about time dilation we can take one end (A) of the wormhole and put it on Earth in the year 2010, and accelerate the other end (B) to near light speed or place it in a powerful gravitational field. Due to time dilation, time will pass slower at end B so that it can still be 2010 at that mouth and 2050 at mouth A. Now if we were to bring mouth B to Earth (where it is the year 2050), we have a wormhole that  takes you back to the year 2010. This is a type of time machine that can only send you back in time to a point in which it already exists. Since the wormhole didn't exist before 2010, you can't go to an earlier point in time. Unfortunately, as we know, creating a wormhole requires negative energy so this solution is impossible to implement for now. So what other options are there for time travel?
First of all, we need to find a proper definition for it. As we know from relativity, it is postulated that nothing that carries mass/energy or information can travel faster than the speed of light. Therefore we can construct the following space-time diagram (Minkowski diagram):

As we can see, points on the light cone are where x^2 = (ct)^2, which is equivalent to (x/t)^2 = c^2. Distance over time is velocity, so these are points that can be reached by travelling at the speed of light, hence the name light cone. Inside the cone are points where x^2 < (ct)^2 or (x/t)^2 < c^2 so these are points that can be reached by sublight travel and outside the cone we have the points which can only be reached at superluminal velocity. Keep in mind that these aren't points in space, they are points in space-time,  meaning that each point has an associated space and time coordinate and so travelling from one point to another is restricted by the lightspeed limit. For example, the origin can be (Earth, now); this is a space-time point, it defines the position on Earth and time as the present. A point which would be in the future light cone would be (Earth, now + 5 minutes), and a point in the past (Earth, now - 5 minutes). If we were to travel to the Sun, a point on our world line could be (Sun, now + 5 days), and the world line would be the curve between that point and (Earth, now). This would be inside the future light cone, as travelling from Earth to the Sun in 5 days can be achieved at sublight speed. What if I wanted to go to the point (Sun, now + 5 seconds)? That's not possible, since light takes 8 minutes to travel the distance so we can't go from Earth to the Sun in under that time, therefore the point (Sun, now + 5 seconds) is outside the light cone. Due to the  lightspeed restriction, observers are "trapped" inside the light cone, moving from past to future. The trajectory they follow is called a world line. For reasons beyond the scope of this post, points inside the cone are called "timelike" and points outside the cone are "spacelike". Now, if we could somehow make this world line loop back on itself it would essentially mean that the observer would be returning to a previous point in space-time and would have thus time traveled. Therefore time travel is defined by the existence of closed world lines, called closed timelike curves (CTC). They are called timelike, because they would be inside the light cone and so would not be in conflict with breaking the lightspeed barrier. An example of points which could be on a CTC are: (Earth, now), (Sun, now + 5 days), (Earth, now). Notice how we went from the present to 5 days in the future (which would have become the new present) and then 5 days back into the past.
How can we make CTCs?
The trick is to do something similar to FTL travel. If you recall, in that post, when I talked about the Alcubierre drive I explained how you couldn't pass the speed of light locally but you could globally. The situation is very similar. If the light cone only lets us go from past to future, then lets make a series of local light cones in which we're going from past to future but each light cone will be slightly tilted with respect to the previous so that in the global light cone we will have formed a closed curve like in the figure:

Pretty ingenious, but can it be done? Well, the good news is there is nothing in the laws of physics which prevents this from happening. The tilting of light cones can be achieved in curved space-time like the one we live in, and many theoretical models have been found. The bad news is none of those models can be physically constructed in our Universe. The very first such model was discovered by Kurt Gödel, one of the greatest mathematicians of the 20th century and a very close friend of Einstein. Gödel presented this model to Einstein as a gift for his birthday. However, his model only worked in a rotating universe, which isn't the case for the one we inhabit. After that, scientists began to search for alternatives in our own universe. One of them was the Tipler cylinder, a large rotating cylinder which would allow the existence of CTCs in its interior. Unfortunately, it was proven that this could only happen if the cylinder had infinite length, something which is physically impossible. All other models have similar problems, though in some cases it may only be a question of lack of technology, for example one model suggests that a rotating superconductor (electrical conductor with 0 resistance) would form CTCs provided the number of Cooper pairs (special groups of electrons) is several orders of magnitude greater than it is in current superconductors. This could be merely a question of developing better superconductors, but on the other hand it could be physically impossible.
Can we define time travel in some other way?
Yes, however these alternate definitions don't even have theoretical solutions. One way to define it is simply by "true" faster than light travel, that is, travelling locally faster than light. The theory of relativity predicts that this is equivalent to travelling backwards in time as seen by outside observers. There is even a funny limerick based on this definition:
There once was a girl named Blight,
Who could travel faster than light.
She took off one day,
In a relative way,
And came back the previous day.
The only true FTL travel method we know of is represented by tachyons, the theoretical particles which could only travel at superluminal speeds, mentioned in the previous post. But even if they do exist and we could detect them, it has been proven that they cannot be used to send information into the past and thus violate causality, since tachyons of the future are indistinguishable from tachyons of the present.
Another definition comes from quantum mechanics. I mentioned antimatter in a previous post, the opposite of matter which annihilates it. An interesting property of antimatter is that in quantum mechanics it is viewed as matter travelling backwards through time at the same rate that normal matter travels forward through time. So if you would reverse the time axis on matter you would get antimatter. It is unknown whether this could be used to engineer a time machine, it's debatable if this even qualifies as time travel. Since time passes at the same rate if you would travel through time using this method it could mean that to go back in time 5 years, you'd have to wait 5 years.
There is also an interesting insight from the Einstein-Cartan theory. This theory completes general relativity, by including spin into the theory. Spin is seen as dislocations in the fabric of space-time and one of the predictions made is that particles with spin, travelling on certain trajectories can translate into the past or the future by a tiny amount, but yet again there is no theoretical time machine based on this method.
Time travel and FTL travel are very similar, though it would seem that time travel has a better chance of being achieved. This is due to the fact that it only seems to be restricted by paradoxes not actual physical laws, while FTL travel faces a major obstacle: the lightspeed limit. Also, both concepts find ways to circumvent their restrictions but the solutions to time travel are somewhat less exotic, in my opinion, and many solutions to the problem of FTL travel would also result in time travel while the inverse is not always true. It may turn out that both are possible or neither. History has shown us that many challenges viewed as impossible by scientists of the time, were eventually proven to be possible and are currently all around us. Examples include: airplanes, supersonic flight, X-rays, space travel, atomic energy and others. Given enough time, man's understanding of physics advances, the number of possibilities increases, to quote the Russian writer Anton Chekhov from one of his plays: "The human race progresses, perfecting its powers. Everything that is unattainable now will some day be near at hand and comprehensible..."
If time travel is possible then it would be the ultimate irony, for the only obstacle we face in achieving it, is time itself.

Friday, September 10, 2010

Interstellar Travel (part 2)

Faster than light travel

Last post I mentioned one of the postulates of relativity: the speed of light in vacuum is constant for all observers and nothing that carries mass, energy or information can travel faster locally. You may be wondering how we could circumvent this. There are many ways to "attack" this postulate. For starters there are many phenomena in which the speed of light is exceeded, and yet all of them respect the postulate. Let's take some examples: if you were to point a laser at the Moon and then move the laser spot on the Moon very fast you could make the spot move faster than light-speed since the distance in which you move the laser is small, but the distance the spot travels is large. So how does this help you? It doesn't. This method doesn't allow you to transmit neither energy or information faster than light. The laser spot will reach the Moon at the speed of light, and assuming it travels between two Lunar bases faster than light, none of those bases can control how it moves, since the source is on Earth and communicating with Earth requires sending a signal which can only travel at the speed of light. Therefore, none of the bases can control the information the laser spot sends to one another. Another example is a wave's phase velocity. Phase velocity is the rate at which a wave's phase propagates through space, and mathematically it is the product between wavelength and frequency. An interesting property of this velocity is that for electromagnetic waves travelling in certain media, presenting anomalous dispersion, (like the ionosphere for example) it can exceed the speed of light. This however, again does not mean that you can transmit information, energy or matter faster than light. Information and energy are associated with a wave's group velocity, the speed of a wave packet. This velocity cannot exceed lightspeed. In quantum mechanics we have another intriguing phenomenon: quantum entanglement. It is a kind of connection between two objects which form a quantum system, in which by measuring a certain property of one object you instantaneously gain some information about the second. This link is independent of the space between the two bodies. For example if you have two entangled electrons and you separate them by whatever distance you want and you measure the spin of one of them, you will immediately know the spin of the other one, regardless of where it is. Thus, it would seem that the information regarding the second electron's spin, an intrinsic property of that particle, had traveled instantly. Einstein called this phenomenon "spooky action at a distance" and contributed to the formulation of the Einstein-Podolsky-Rosen paradox which deemed quantum mechanics an incomplete or incorrect theory as it seemed to allow instantaneous communication. The apparent paradox was solved by the no-communication theorem which states that information cannot be exchanged instantly, however this does not prohibit FTL (faster-than-light) communication. But how could it be achieved?
Let's tackle another aspect of the mentioned postulate of relativity: locality. What does it mean to travel faster than light locally? It's difficult to give a precise, non-mathematical definition, but locality essentially means with respect to the immediate surroundings. It's easier to define non-local faster than light travel, which isn't prohibited by relativity, with an interesting example: the Alcubierre drive (very similar to the Star Trek warp drive). You might be familiar with the concept of space-time. Basically it's the joining of our 3-dimensional space with time thus forming the 4-dimensional universe in which we live in. Einstein's general theory of relativity says that gravity is not a force but the geometry of space-time itself. Therefore anything that produces gravity will affect space-time. What produces gravity? Mass and/or energy. Thus, a certain configuration of mass-energy will create a certain configuration of space-time. A useful analogy is to think of empty space as a stretched-out blanket. When you place an object on the blanket, say a heavy ball, it curves and other objects placed near it tend to fall towards the ball. If you throw a smaller ball on the blanket, due to it's initial velocity, it will circle the large ball, thus orbiting it. This is reminiscent of our own solar-system in which the planets orbit the Sun. The relationship between mass-energy and the geometry of space-time is neatly expressed in Einstein's field equation, an elegant tensor equation that is the core of general relativity. All geometrizations of space time (so called metric tensors) must satisfy Einstein's field equation if they are to exist in our own Universe. One such solution is the metric for the Alcubierre drive. Going back to the blanket analogy, the Alcubierre drive works something like this: if I'm on the blanket and I want to move forward I contract the blanket in front of me, and expand it at the back. Essentially this is warping space-time around me. Assuming I could to this, there is no limit to how fast I could go because I'm not locally going faster than light. In my local space-time, which is the one inside my "warp bubble" I still can't go faster than light. But to an outside observer, that is non-local space, this ship is travelling faster than light. Voila! We have achieved FTL travel without breaking the laws of physics. So what's the catch? When you think about the blanket analogy you realize that all objects (objects with mass/energy) bend the blanket downwards (negative curvature), but in the Alcubierre drive when you contract the space in front of your ship, you bend it upwards (positive curvature). This is a problem, since creating positive curvatures would require negative mass or negative energy. There is no law of physics which prevents the existence of this but we've never encountered it and we don't know if it exists. Another problem is the enormous amount of normal mass/energy required (comparable to the Sun).
Other solutions to Einstein's equation exist in the form of wormholes, shortcuts through space-time. The name originates from an analogy: to get to the other side of an apple a worm doesn't have to travel on the surface, but can dig a hole through it. Wormholes are of many types and would theoretically allow travel between two points in space, two points in time, even between universes. Unfortunately, as with the Alcubierre drive, problems arise such as the necessity for negative energy density or exotic matter. It is theorized that you don't need any of that to create a wormhole but you do in order to stabilize it, more precisely the wormhole would collapse before anything could pass through it. It could be possible that natural wormholes exist somewhere in the Universe, for example it is believed that ring singularities (rotating black holes) could form wormholes. Even if wormholes exist somewhere, we can say with a fair amount of certainty that there aren't any in our solar system and it doesn't look like we'll be creating any in the near future.
The problem of negative energy density could be solved by a quantum physics phenomenon known as the Casimir effect. The experiment which led to its discovery is this: if you take two uncharged parallel plates and bring them extremely close to each other, a force will act upon them pushing them even closer. This can be explained through vacuum fluctuations. What we think of as vacuum is empty space, nothingness, but this isn't the physical vacuum which exists in the real world. Vacuum has an associated energy and is thought to be made of virtual particles. Why? It's a result of quantum mechanics which has shown us that a system can only occupy discrete energy levels (i.e. quantification) and that it is impossible to measure certain physical properties with absolute precision (Heisenberg's uncertainty principle). Therefore, the lowest possible energy level cannot be 0, as this would mean you could know energy with absolute precision, but is slightly above 0. This also explains why no physical system could ever reach absolute 0 temperature. Temperature is a measure of particle movement. The uncertainty principle tells us that you can't simultaneously know the position and momentum of a particle with infinite precision and if a system were at absolute 0, the particles's positions would be fixed and their momenta would be 0. Going back to the Casimir effect, vacuum has energy in the form of virtual particles. When you bring the two plates close to each other, the number of particles outside of the plates is greater than the number inside and therefore creates pressure, pushing the plates closer to each other. This system has an associated negative energy density, though it is unknown how it could be harnessed for stabilizing wormholes or constructing an Alcubierre drive. It is a problem of incorporating gravity into quantum mechanics (which for the moment are separated), something which can only be solved by a quantum theory of gravity.
The Casimir effect offers insight into another FTL phenomenon: considering that vacuum has an associated energy, this energy is thought to be responsible for the values of electric permittivity and magnetic permeability in vacuum. These are two very important constants. They are measures of the resistance of forming electric and magnetic fields in vacuum. Light is an electromagnetic wave and from Maxwell's equations it can be shown that the speed of light is inversely proportional to the square root of the product of these two constants. But what if the constants aren't constant? A vacuum with a lower associated energy would have lower constants and therefore a larger value for the speed of light. It is believed that due to the smaller density of virtual particles between the plates described in the Casimir effect experiment, there would be a lower vacuum energy and therefore light would travel faster. The difference would be very small, and this has made it difficult to measure such an effect. On a side note, it is also possible that we live in a false vacuum: a local area of space in which the associated vacuum energy is larger than everywhere else where we have the "true" vacuum. If something were to happen to cause even a tiny region of our space to tunnel to that lower energy level of true vacuum, it would cause a so called "vacuum metastability event", a doomsday scenario in which a bubble of true vacuum would expand at near light speed changing the very fabric of our space-time.
Another concept related to FTL travel is the tachyon. These are theoretical particles which can travel only above the speed of light and can never slow below it. They are predicted in string theory, though it is believed that even if they do exist they cannot be used to transmit information faster than light.
FTL travel is often associated with time travel as almost all FTL solutions would also permit time-travel. I will talk about time-travel in another post so I'm not going to go into details here, suffice to say that the paradoxes associated with time-travel would present a problem for developing FTL travel.
Other methods used in science fiction involve hyperspace. The idea is for the ship to go into another dimension where fundamental physical constants, like the speed of light, don't exist and a ship could travel infinitely fast. While the possibility of other dimensions is explored by string theory (which postulates the existence of between 10 and 26 dimensions) these other dimensions are theorized to be curled up at extremely small distances and could only be accessible to high-energy subatomic particles. There are many other science fiction FTL drives like: jump drives, slipstream drives etc but all of these achieve FTL travel by assuming that our current understanding of physics is either fundamentally wrong or largely incomplete. It is true that our current understanding of physics is incomplete as we have yet to find a theory of everything which will ultimately answer the question regarding the possibility of FTL travel.
Personally, I am unsure if FTL travel is possible but I am confident that a viable means of interstellar travel will be discovered some time in the future. It is something which I find inevitable, motivated not just by curiosity and ambition but by our need for survival. The method which will be used could be a variation of the ones discussed here or it could be something completely new. As someone once told me science fiction of today will shape the science of tomorrow.