The time dilation equation: [tex]\Delta t' = \Delta t/\sqrt{1 - v^2/c^2}[/tex] Now lets consider a tachyon with v = 2c [tex]\Delta t' = \Delta t/\sqrt{1 - (2c)^2/c^2}[/tex] [tex]\Delta t' = \Delta t/\sqrt{1 - 4c^2/c^2}[/tex] [tex]\Delta t' = \Delta t/\sqrt{1 - 4}[/tex] [tex]\Delta t' = \Delta t/\sqrt{-3}[/tex] [tex]\Delta t' = \Delta ti/\sqrt{3}[/tex] I've always heard that tachyons, if they exist, move backward in time. I would've thought this would be expressed as a negative delta t, but from this it appears to be a complex number. Can somebody explain why a complex delta t describes backward time travel? Or am I going about this the wrong way?
As tachyons travel faster than light then you can always change to a different inertial coordinate frame so that they travel backwards in time (or instantaneously) according to that frame.
Is it not true that in a specific inertial frame timing the travel of a tachyon between an emitter and detector, the time can never be negative, but limits as instantaneous for infinite speed? It appears to me that the "traveling backwards in time" can only be apparent if you switch coordinates during the measurement.
I'm not sure I've understood your question. The identification of an "emitter" and a "detector" depends on the frame of reference. You can think of the tachyon's presence at the emitter as one space-time event and its presence at the detector as another. The key here is that the space-time interval between these events is space-like, not time-like. That means that while one observer sees the "emission" event occurring before the "detection" event, you can easily find other reference frames in which the "detection" event comes first (and so you could claim that the tachyon moved backwards in time in that frame, although that's arbitrary, since its direction in time is not absolute). There will also be a frame in which they are simultaneous. In that frame the tachyon actually exists at all points along its path simultaneously, which is kind of what you'd expect for events that have space-like separations.
Exactly. It is the way different observers in relative motion synchronize their frame's clocks. I still believe that no inertial frame that does an experiment with a hypothetical tachyon and synchronized clocks could find it to have "moved backwards in time" in that frame. If a tachyon moves at 2c in that frame, it will take half the time that light would take to travel from emitter to detector. This fact cannot depend upon who else is watching from some other inertial frame, or can it? Obviously, other frames timing "my tachyon" may get different answers from their respective synchronized clocks. BTW, how do we add super-luminal speeds relativistically?
Yes, but that's an example of what makes tachyons non-physical. They don't obey the basic notion of cause preceding effect.
Yes, that's true - no one observer would conclude that a tachyon moved backward in time (what would that look like, anyway?). What would be the case is that different observers would disagree whether it moved from pt. a to pt. b or from pt. b to pt. a. If one event were clearly defined as the "start", however, like a projectile being launched from a gun, then there would be some observers who would see it move "backwards" from its "destination" back to the gun. I wouldn't call that moving backwards in time; I'd just call it a non-physical sequence of events. The observer who see the object at all places at once is clearly going to have difficult time describing any kind of "motion" for this object.
Not exactly. If you try to interpret the field as the field of a particle, say an electron, then yes, it appears to move in reverse. That's part of the reason why it makes more sense to interpret it as an anti-particle - the positron in this case - that moves forward in time like anything else.
But if you accept that tachyons obey the principle of relativity--that they work the same way in all reference frames--then if it is possible for the receiving of a tachyon message to happen before the sending of that message in at least one frame, it must be possible in all frames. So if you are traveling away from me at sublight speeds, and I send you a tachyon signal which goes FTL in my frame but "backwards in time" in your frame (meaning just that in your frame you receive the signal before I sent it), then you immediately send a reply which travels FTL in your frame but backwards in time in mine, then it is possible for me to receive your reply before I sent the original message, a clear physical violation of causality. This is nicely illustrated with a spacetime diagram on this page: http://www.theculture.org/rich/sharpblue/archives/000089.html
Thanks for the diagrams, JesseM. By the way, it seems to me that this sort of thing would happen regardless of the method for superluminal velocity. Even if the object weren't traveling faster than light locally, as in the Alcubierre drive, it still looks like you could use it to violate causality. Is that correct? Wormholes might get around it, because if you construct one that violates causality vacuum noise causes it to collapse. But then if you can have wormholes why couldn't you have the Alcubierre drive? Sorry, slightly off topic. What does complex time mean, anyway?
Who says that makes more sense? So in other words all anti-matter could simply be explained by normal matter moving backwards in time? That sounds like a much simpler explanation than doubling the number of fundamental particles just to preserve a forward-only time dimension (for which no one knows _why_ it must always be forward).
My understanding is that "moving forwards in time" vs. "moving backwards in time" doesn't really have any clear physical meaning (if you draw a vertical line on a piece of paper, is it 'moving up the page' or 'moving down the page'? Why should a timelike worldline in relativity be any more directional than a line on paper?), and that the idea of antiparticles being like regular particles "moving backwards in time" is more like a mathematical trick that can simplify things when summing feynman diagrams, something like the difference between doing an integral from [tex]\int_{t_0}^{t_1}[/tex] vs. an integral from [tex]\int_{t_1}^{t_0}[/tex]. I could be wrong about this, but this statement from an FAQ on virtual particles seems to be saying something along these lines:
That is a very interesting point. Since there is only one time dimension (dimension with opposite sign in spacetime interval) timelike lines have a definite before-end and after-end which cannot be swapped. On the other hand with three spatial dimensions spacelike lines don't have a definite left-end and right-end since you can always rotate things around. I have no idea what the implications are for something like that, but it makes a clear difference between a tachyon and a positron-is-an-electron-running-backwards-in-time. In the case of the positron the worldline is still timelike, just running in reverse. That seems to be a much more clear "backwards in time" than the spacelike worldline of a tachyon.
What do you mean "cannot be swapped"? The laws of physics are time-symmetric (or in quantum field theory, charge-parity-time-reversal invariant) so you're free to reverse the labels of which direction in time you call the "future" and which you call "the past", and for any physical process you can find another physically allowable process which looks like a backwards version of the first (though you may need to flip the charges and parity of the system of particles in the second as well). What do you mean "running in reverse"? What physical statement are you making here? Again, as far as I know the concept of treating a positron like an electron "going back in time" is just a mathematical trick for summing feynman diagrams.
I mean that you cannot rotate in a single dimension. I am not talking physics here, just geometry. In a single dimension you cannot rotate an object so there is a clear directionality to the two ends of a line. In two or more dimensions there is no longer a clear directionality to the two ends of a line. You can always approach the line from the other side and then the sense of the direction of the line is reversed. I thought that was the point you were making with your "Why should a timelike worldline in relativity be any more directional than a line on paper?" comment.
I've just never heard an explanation as to why a "normal particle backwards in time" model is any less "real" and more of simply a "trick" than a "anti-particle forwards in time" model. Seems to me like people use the latter view just to comfort themselves into believing in the absolute nature of linear time. I also find it incredibly interesting that a 4-dimensional Euclidean view of Relativity has particles moving faster than the speed of light as moving backwards in time as well. These particles would be indistinguishable from anti-particles. But, again, we don't use that model, and instead use the Minkowski model, simply to keep our minds at ease that time always moves forward. [Edit] Just wanted to add, in that model,whether the particle is a "real" particle or "anti" particle is frame-dependent. In other words, _all_ of the quantum numbers, not just mass, become frame dependent. Doesn't this make for a simpler unified theory?
But what do you mean by "forwards in time" or "backwards in time"? As far as I know particles don't "move" in time in either direction in any meaningful physical sense, they just have worldlines in spacetime. Like I said, maybe in the course of certain mathematical procedures you would integrate along the worldline from one end to the other or something like that, but I don't think there's any reason to take this too literally as some sort of physical reality (and note that even as a mathematical procedure, I don't think you're forced to treat antiparticles as normal particles moving backwards, it just simplifies the calculation, and I imagine you could equally well say antiparticles are moving forwards and normal particles are antiparticles moving backwards.) The minkowski model doesn't say time moves forward or backward, any more than it says space moves left or right, at least not as far as I can tell. And what do you mean by "4-dimensional Euclidean view"?