What I do not understand is whether sending information faster than light would cause a paradox of any sort. It may be that it happens to be impossible. But is it necessarily impossible?
Yes, because of time dilation. In particular, suppose that, in some reference frame, you can send a message from location A to location B faster than light. In that reference frame, it arrives at B after it leaves A. Then, in some other reference frame (moving at less than the speed of light relative to the first), it arrives at B before it leaves A.
So, we can reverse that: Let's say I (in Boston) want to send you (in Toronto) a message that you'll receive before I send it. Then I simply figure out a reference frame where the message would travel forward in time but faster than light. Then I use the faster-than-light communication in that reference frame to send it to you.
If you now do the same -- choose a 3rd reference frame, where the message from Toronto to Boston travels forward in time but faster than light, but where on the Earth's surface it travels backward in time -- you can now send your reply before I send my message.
Then I can arrange a simple grandmother paradox. e.g. I plan to send a message to you at 2pm, but if I get your reply before 2pm, I don't actually send it.
EDIT:
In particular, if one observer measures an event as happening at position-time (x, y, z, t), and another observer is moving at speed u along the x axis, that observer will see:
x' = (x - ut) / sqrt(1 - u^2/c^2)
y' = y
z' = z
t' = (t - ux/c^2) / sqrt(1 - u^2/c^2)
(This is the Lorentz transformation.)
Suppose the first observer can send info faster than light. Suppose she sends it along the x axis at a speed of 2c. You can repeat the derivation with any kc where k > 1. If the sending event is at position-time (0, 0, 0, 0), and it takes one time unit to get there, then the receiving event in her frame is (2c, 0, 0, 1).
What does the other observer see? Let's suppose they're moving more than half the speed of light, say 3/4 the speed of light. The sending event is the origin, and the origin maps to the origin, so that's easy. For the receiving event:
x' = (x - ut) / sqrt(1 - u^2/c^2) = (2c - (3/4 c) * 1) / sqrt(1 - (3/4 c)^2/c^2)
y' = y = 0
z' = z = 0
t' = (t - ux/c^2) / sqrt(1 - u^2/c^2) = (1 - (3/4 c) * 2 * c/c^2) / sqrt(1 - (3/4 c)^2 / c^2) = (1 - 3/4 * 2) / sqrt(1 - (3/4)^2).
In the formula for t', note that the first term, 1 - 3/4 * 2, is negative.
So, if I can send messages at a speed of k * c, then an observer moving with a speed greater than c/k in the same direction sees the message arrive before it leaves. That description is very easy to reverse.
Granted, it's not impossible by that proof because we have faster means of communication (you can detect something before its sound reaches you) (or maybe because it's slower than light), but that's not a problem with faster-than-light communication either. By hypothetically accepting that there's a FTL communication method, didn't you just accept that there's something faster than light?
Wouldn't that imply there's something we don't know about time dilation (if FTL comm is achieved), given that it defines such a thing as impossible? It's a circular argument, it's impossible by this proof because this proof relies on it being impossible. So arguing a hypothesis which rejects that requirement with this proof is ignoring the hypothetical universe that was constructed, and it's not a proof at all.
http://en.wikipedia.org/wiki/Time_dilation
Ok, so let's take two twins, Raj and Ragan. Raj is orbiting a massive cosmological object, Ragan is on the surface. Ragan's time appears to travel slower than Raj's time.
Does FTL communication between Ragan and Raj ever create a situation where messages can travel backwards in time? Does bringing an observer into it help?
But as to FTL, let me horrifically simplify. Imagine space as a 2D grid of points. Vertical is time (up is the future), left-right is 1D of space. Take every point, and connect it to the one directly above it, the one above and right, and the one above and left, with a directed link. You are allowed to move through spacetime along those lines. You can see your past and future lightcones on this graph, if you look. Light always travels diagonally on this graph.
Normal travel in this horrifically degenerate spacetime is travel that can be expressed on those links. FTL is somehow figuring out how to bypass the graph structure of space, which the shape of that space doesn't permit. Space is literally shaped so that FTL isn't really possible.
Reality is of course substantially more complicated than this, and this model is useless for any purpose beyond trying to show what I mean by space being shaped such that FTL isn't possible.
This also sort of demonstrates another thing, which is that ultimately, any point that you can't reach is, from the point of view of where you are, really the same. They are all equally simply "unreachable". So you if you create an "unreachable" drive that lets you get there, you can basically just use it twice to jump far away from where you are now, then from there just jump directly into your past. Also, there is no compelling reason to think you would have a drive that would only let you go "one light year"; it would open the entire universe to you. In General Relativity, this still basically holds true, though it requires more explanations about how there isn't one true reference frame, etc, etc, but FTL really is time travel and it really does just shoot physics and causality full of holes.
This is not necessarily a problem - in a quantum mechanical formulation, it's simple enough to resolve (for instance) a grandfather paradox by looking at quantum interference around causal loops (almost 100% analogous to the way interference on a violin string will restrict stable modes to certain frequencies, interference around a causal loop will only allow stable self-sustaining situations to survive, so any grandfather paradox would self-interfere and not be allowed). The main difference is that with closed timelike curves the quantum effects can get a lot stronger than without them, even affecting very large objects.
(I can't vouch for its accuracy. It seems that the grandfather paradox is the biggest problem with FTL communication)
j0.
Indeed it does seem to allow that, and this is right at the heart of the quantum measurement problem. The three popular ways out of this are:
1. To ignore it entirely, stating that we don't really care about the implications of quantum mechanics and are just happy that it's useful for calculating things. LA LA LA!
2. To declare that yes, well, maybe it sort-of does, but it doesn't matter because there's no actual information being transmitted. I can't use quantum entanglement to send a message to Alpha Centauri, since I have no control over the sequence of bits which get transmitted; the most I can do is ensure that both they and I wind up with the same string of completely random bits.
3. Go to a no-collapse ("many-worlds") theory of quantum mechanics, in which there isn't any superluminal communication required (why? It's nonobvious and hard to explain but if you're willing to trust me then you can just go along with it for now...)
My personal preference is for (3) but the implications of many-worlds theories are so horrifying that it remains quite controversial.
Here is how I remember it (and I am just a comp sci nerd, not a physicist so please correct me):
* you have 2 qubits Q1 and Q2. You entangle them. Now you have an entangled system Q1Q2.
* You separate Q1 and Q2 in space. Let's say you put Q2 on a spaceship and blast it into space, but leave Q1 in the lab on earth.
* After some time you measure Q1 to get its value. At that point the entanglement of Q1Q2 collapses. You now know the value of Q1 that you just measured and at the same time Q2 is forced to a known value too.
* At _appears_ as if you could send information this way but you cannot. Think about it. You don't know what value you'll measure in the lab for Q1. So you can't force Q2 to be a certain value either.
* Let's think about it another way -- suppose you tell the spaceship operator that if Q2 collapses to |0> then they should turn the spaceship immediately around and head back home and if it collapses to |1> they should arm their weapons and prepare for an alien attack. Now you are on earth in control of Q1, and you want to force Q2 to be measured to |1> because you know the aliens are coming. There is nothing you can do to Q1 to force Q2 to be measured as |1>.
http://en.wikipedia.org/wiki/Bells_theorem
See also:
As far as I know, general relativity relies on the existence of a finite speed upper bound. It appears that this bound is equal to the speed of light, but any higher speed would do.
(I am not sure if this analogy works - if it is flawed, please help me fix it until it is correct)
Lets assume that you are on a computer connected to another machine that is at the exact furthest opposite possible on the globe, via fiber.
The remote machine will display a question, which you have never seen before and do not know the answer to, but you can read the question and then the answer to the question will be displayed.
Is it impossible that you could not derive an answer to the question fast than the answer could be displayed?
Thus, information (your derived answer) arriving in your head faster than the information arriving on screen as delivered at the speed of light?
My assumption is that information can be derived/created/transferred/understood faster than linear light travel as the processing of thoughts in our minds is semaphoric/symbolic and parrallel - light transmission is serial.
btw, a great book about time (and time loops) is Igor Novikov's "River of time"
In particular, the tying together of speed and time.
"By 'world line' we mean a curve traced out in the four dimensions of space-time". Time is not a dimension. Time is a human concept. Clocks don't measure a physical dimension or force called time, they repeat mechanical motions at regular intervals that we label as time passing. Same for atomic clocks. We measure the spin rates of cesium atoms but that isn't any different than mechanical clocks. Time is not a force that figures into physics equations.
Further, time doesn't exist. Not in the classical sense. What we experience as time is the entropy of energy in the universe winding out. We remember things that happened before and imagine things in the future, but since there is no physical "time" dimension there is no traveling forward and back.
If you are on a ship traveling close to c, the rate of entropy in the matter and energy on your ship is lower than the outside universe. That's why time seems slower.
There's a classic thought experiment that you can build a time machine by putting one end of a wormhole on a ship, send it out for 20 years near c, then bring it back. At this point you would have a wormhole with ends in two different times. It doesn't work like that though. Passing through it will just put you wherever it opens to, and you'll just end up in whatever local entropy rate is going on.
In physics, time is a dimension in all but the most obscure and fringe formulations - whether you're talking about classical theory, special relativity, quantum mechanics, general relativity, quantum field theory, string theory, M-theory, there's always a timelike dimension that shows up somewhere in the formulation.
Are you actually arguing that all of standard physics is in some way measurably wrong, or are you merely trying to make a philosophical point about the definition of the word "time"?
Not trying to be facetious, this is a genuine question, but rate of change in entropy with respect to what?
So really what you are really measuring is the difference in change of entropy from your local frame of reference to the frame of reference of the ship. There is no such thing as absolute rate of change. All is relative.
With respect to time, if you could somehow reverse all the motions of every single subatomic particles in a particular frame of reference then you would essentially be moving back in time in that specific frame of reference. To reverse time you have to reverse the motion of every single particle, sub-particle. Is almost the same thing as simply playing a movie in reverse. Now the real issue with this methodology is that the particle movements are not being recorded anywhere as far as we can tell. So we need to first find a way to record the movement of all the particles in a frame of reference and then find another way to run the entire recording in reverse. I wonder if exceeding the speed of light would actually reverse the motion of particles. That would imply that the motion is somehow being recorded? Who knows, just thinking out-loud.
Another really good question that I've been wondering about is why does entropy decreases when you move faster? What is it that causes entropy to decrease? Is it some sort of "friction" with space-time? Anybody have any good suggestions? It may have to do something with conservation of energy. Or conservation of something. The faster it moves the slower the particles move. Something is being compensated for.
If I am standing on the Earth with my laser and I place two observers on either edge the moon, we may say colloquially that a spot has moved from one observer to the other faster than light can travel, but in actual fact what has really happened is that light has travelled from me to the first observer and then to the second observer at the speed of light, and it is just that my signal to the second observer arrived extremely soon after my signal to the first observer.
Imagine you have a device with two lasers (Laser A and Laser B) that each point down to a white surface. The lasers are one meter from one another, and are calibrated such that Laser B fires exactly 1/(600,000,000)th of a second after Laser A. It might look that the dot is moving faster than light, since light travels at 300,000,000 m/s; however, as we know, nothing went from Point A to Point B.
You can also think about the beacon that a pulsar gives off. We can detect the pulses happening every few seconds-- so what happens when our friend on Alpha Centauri also detects the pulses? The pulsar 'spotlight' is cycling between Alpha Centauri & Earth in a fraction of a second, even though we're 4 lightyears away from eachother.
It's not really FTL, only from a certain point of view. As the author says: "These are all examples of things that can go faster than light, but which are not physical objects. It is not possible to send information faster than light on a shadow or light spot, so FTL communication is not possible in this way. This is not what we mean by faster than light travel, although it shows how difficult it is to define what we really do mean by faster than light travel."
you have a point lightsource in the center of a circle. The radius of the circle is R.
You have a point on a concentric circle with radius R/2. This point is moving on the circumference of this small circle with the speed of light.
If my calculations are right the shadow of the point on the big circle runs with 2*c.
Physicists choose Relativity since it has been tested to many decimal places and always comes through with flying colors. Relativity means that FTL travel is the exact same thing as a time machine. Time machines can cause temporal paradoxes like the Grandfather paradox. Temporal paradoxes render Causality impossible. And without Causality, the entire structure of science crumbles.
The only way to have all three is if there is some weird magic law that makes it impossible to use time machines to create temporal paradoxes.
And why, exactly, would we assume a thing like that?
- quantum entanglement (we don't know the machinery behind it, very-very possible that the SR's smoothness of space is also only a rough approximation of the real world.) Obviously any attempts to find the "signal" between the entangled particles in the conventional sense (something traveling through smooth static spacetime) have so far been and will continue to be futile.
- relative motion of the parts of the (our expanding) Universe that are far enough from each other (observed, space expansion, described by General Relativity, at this scales of spacetime the SR's condition of static metric just isn't valid anymore )
>Now consider the description of an EPR-entangled pair of photons:
(|↑↓> + |↓↑>)/√2
At first glance this looks very much like the single-photon case, except that where before we had ΨU and ΨL we now have |↑↓> and |↓↑>, representing respectively photon 1 being in the upper slit and photon 2 being in the lower slit and viceversa. But this distinction is crucial because it turns out that there is some notational sleight-of-hand going on here. First, |↑↓> is shorthand for |↑>|↓>. Second, the arrow symbols have no semantic significance; they are just compact mnemonic identifiers. We could just as well have written |UL> and |LU> (which of course is shorthand for |U>|L> and |L>|U>) as |↑↓> and |↓↑>. Finally, ΨU is just another way of writing |U>.
So if we employ alternative notation we get the following description of two entangled photons:
(ΨU |U> + ΨL |L>)/√2
As I have probably demonstrated other places, IANAQM. But I don't follow that last transformation. If |↑↓> == |↑>|↓> == |U>|L> == |UL> and ΨU == |U>, how does (|↑↓> + |↓↑>)/√2 == (ΨU |U> + ΨL |L>)/√2 and not (ΨU |L> + ΨL |U>)/√2 ? Shouldn't the |U> and |L> be reversed?
