Will The Theory of Relativity allow me to travel Backwards in Time?

PeterDonis

We agree that SR does not "know" which specific spacelike worldline an FTL rocket would travel on, assuming an FTL rocket were possible. We do *not* agree that SR can say nothing about what would happen if an FTL rocket could travel on a spacelike worldline in general. See below.

Wrong. SR *can* say things about the general consequences of allowing *any* spacelike curve to be causal, i.e., to be the worldline of an object like an FTL rocket. Those consequences are what I've been talking about. Nothing I've said requires any assumption about which specific spacelike worldline the FTL rocket travels on. In my post #29, I did make a particular assumption about that, but that was only for concreteness, to illustrate the kind of general consequences I've been talking about. I could have assumed *any* other spacelike worldline for the FTL rocket and the general consequences would have been the same; it just would have taken more words to describe them.

As I said, an FTL rocket would have to travel on *some* spacelike worldline. That means there has to be *some* frame in which the events of its "departure" and "arrival" are simultaneous. So it isn't a "coincidence" that you can find such a frame.

What you're basically saying is, normally we judge simultaneity indirectly, by measurements involving light signals; but if we allow FTL travel, we can judge some particular "simultaneity" directly, based on pairs of events linked by the FTL object's worldline. That's true, but it's not a "coincidence"; as I said, it's a logical consequence of allowing FTL travel at all.

They are ordered one way in frame Y, but they are ordered in the opposite way in some other frame, Z. There is no frame invariant order to the events. Frame X is just a frame in which this lack of invariant order is particularly obvious, since the events happen at exactly the same instant.

Yes; when I gave my own summary, I said exactly this.

I said that Lorentz invariance means you can't privilege either view, *if* FTL travel is possible; both views would have to be valid if FTL travel is possible and Lorentz invariance holds. But I also said that the two views were physically inconsistent; that's why I think FTL travel is impossible if Lorentz invariance holds.

See above. I wasn't contradicting myself; everything I've said has been consistent with the summary I gave several posts back and what I just said here.

Neither of these really captures what I meant. See above.

danmay

Well, check these out.
(1) http://en.wikipedia.org/wiki/Lorentz...ion#Derivation
(2) http://arxiv.org/pdf/gr-qc/0107091v2.pdf
Note especially the discussion on whether FTL violates causality, benign versus paradoxical tachyons, and the candidate definition of "cause and effect" using the "arrow of time" from thermodynamics.
(3) http://www2.physics.umd.edu/~yakoven...ng/Lorentz.pdf
(4) http://o.castera.free.fr/pdf/One_more_derivation.pdf
Note that in all of these derivations of SR/LorentzT, the case where the value of alpha or kappa violates of causality is dismissed before arriving at the conclusion that the other cases are Lorentz (<0, or in the limiting case, =0) and constitue what we usually know as SR. Alternatively, SR can be broadened to include transformations that are not Lorentz, or LorentzT can be re-defined to include all values of alpha or kappa, but I don't think either option is the concensus defition of SR/LorentzT.


Well, we can say if you could travel "backwards" in time as well as "forwards", then time would lose its "arrow"/asymmetry/causality, and be just like space. In which case you wouldn't be traveling "backwards" or "forwards" in time, but only along or opposite relative to the direction of other time travelers/or whatever reference you choose. In essence, if you were to travel both ways in time under SR, it would be just like traveling in space, and you would lose causality.

PeterDonis

Ah, I see; you are referring to alternate derivations of the Lorentz transform that don't assume the constancy of the speed of light; instead they make some alternate assumption that's related in some way to causality (link 2 calls it "pre-causality", for example). Yes, if you use one of the alternate derivations, then you're basically deriving the constancy of the speed of light from some kind of "causality", instead of deriving propositions about causality from the constancy of the speed of light. Since the constancy of the speed of light is an actual observable, I would tend to prefer using it as an axiom to trying to formulate "causality" as an axiom; but that's a judgment call.

Interesting papers, btw; I've only briefly skimmed them and will have to read them in more detail when I get a chance.

Agreed.

Not necessarily; allowing timelike observers to go "backwards" along timelike worldlines is not the same as getting rid of timelike (and null) curves altogether and making all curves spatial. The latter changes the metric of spacetime to a positive definite one; the former keeps a non-positive definite metric. The two types of metric are physically different.

Austin0

OK We are the reverse observer approaching Mars< We have been observing the rocket on Earth. Suddenly the rocket appears on Mars because of course it has arrived there faster than the image of it's journey can reach us. So by the time we see it it has been on Mars the length of time it took for that image to reach us in space. At the same time we are still receiving images of the rocket on Earth because at the time it left Earth there was a complete string of images transmitting through space toward us.

As soon as we see the rocket appear on Mars we begin to receive the images of the trip in reverse order. So we are seeing the ship on Mars and still seeing it on Earth and at the same time receiving the reverse path images of the trip.

As soon as we receive the last image of the trip (I.e.The takeoff) the image of the ship completely disappears because this instant is coincident with the last image of the rocket on Earth that was strung out in space at ignition.

At the time we receive the first image of it's arrival on Mars we note the time. A simple calculation D/c returns the time of arrival on Mars At the time we observe the takeoff we note the time. Again a simple calculation based on distance from Earth gives us the time of takeoff.
It seems clear that this time would have to be earlier than the time calculated for arrival
So both visually and as a calculated chronology the temporal sequence is Earth to Mars.

Interestingly at small velocities above c the visual image of the reverse trip would appear to be much faster (take less time) than the actual trip. at very low c+ v's appearing almost instantaneous. The sonic boom analog I mentioned.
At greater velocities it could appear to take longer than the actual trip.

BTW i think you are mistaken in your description of the rocket exhaust appearing to be sucked back into the rocket.
It would still appear to flow backward from the engine , It would be the whole system appearing to move in reverse, No part of it would appear to move backward relative to the general motion toward Earth (as you described)
Not that is any less strange an image.

So let me know if this description is sufficient or if you see any flaws in this analysis.

PeterDonis

I see a number of flaws. But first of all, since you continue to refuse to give the sort of precise specifications I've been asking for, I'm going to give what I think are the precise specifications corresponding to the scenario you describe. If you think I've got the specs wrong, feel free to correct me, so I can be sure I'm analyzing the correct scenario.

Ok, so in the frame of what you are calling the "reverse observer", the rocket travels from Earth to Mars instantaneously? You don't say that explicitly but it looks like that from your description. So in this frame, with coordinates I will call (X, T), event E ("launch" from Earth) has coordinates (-XE, 0) and event M ("landing" on Mars) has coordinates (+XM, 0). I have to leave the X coordinates unspecified for now because you haven't said how far away we are from Earth or Mars at any specific event; but I'll specify them in a moment. The only thing we can say is that, since we are at the spatial origin, Earth's X coordinate must be negative (as shown), and Mars' must be positive (as shown).

(Note that this frame is *not* a frame in which Earth and Mars are at rest; they are both moving in the minus X direction, since you say this observer is "approaching Mars", meaning moving towards Mars and away from Earth. So this scenario is different from the one I described before, where events E and M were simultaneous in the frame in which Earth and Mars were at rest.)

Let's call the event where the first light signal from event M reaches us, at rest in the frame we're talking about, event S. You haven't specified how far away from Mars we are, in this frame, at time t = 0, when events E and M occur; so I'll assume that at that instant, we are exactly halfway between Earth and Mars, i.e., XE = XM = 0.5. So event E has coordinates (-0.5, 0) and event M has coordinates (0.5, 0). That means event S has coordinates (0, 0.5), since we are at rest in this frame and the light travel time equals the distance.

Actually, as should now be obvious from the above, event S is also the event at which the *last* light signal reaches us from Earth--i.e., the signal from event E, where the rocket "leaves" Earth. So event S is the event at which we stop receiving light signals from the rocket on Earth, and start receiving them from the rocket on Mars.

Half right. The "trip" worldline, at least on the assumption that it's a straight line, must be the line from event E to event M; i.e., the line (-0.5, 0) to (0.5, 0). Which immediately tells us the following:

* We receive our first information about the trip at event O, coordinates (0, 0), which is when the rocket instantaneously flies past us (since that event is on the straight line I just described, and in fact is its midpoint). The light signal from the rocket at event O reaches us instantly, since we are right there.

* During the period from event O to event S, (0, 0) to (0.5, 0), we receive light signals from both halves of the rocket's worldline, from points gradually spreading outward from event O to events E and M. That is, we see signals from the rocket flying towards Mars, *and* from the rocket flying towards Earth; we see it appear to fly in both directions--towards the Earth in "reverse" order, towards Mars in "forward" order.

Half right again. We are seeing images from Earth, but *not* from Mars; event S is the first event where we get a signal from Mars, but the last where we get a signal from Earth. So between event O and event S, we get signals from Earth "overlapping" with signals from the Earthbound half of the trip (in reverse). We also get signals from the Mars-bound half of the trip (in forward order), as above, but not yet from Mars itself.

For the Earth images, yes, this is correct. But only for the Earth images.

Wrong. We see the images from the takeoff and the arrival at the same instant, and they are equidistant from us, so we conclude that the two events are simultaneous.

Note, however, that Earth and Mars are not at rest in this scenario, so we have to use previous light signals from both planets to conclude that we are exactly halfway between them at event O, the instant we receive the takeoff and arrival light signals from events E and M.

Wrong. We conclude that the travel was instantaneous, so *all* events between E and M happen simulataneously. We cannot assign a "temporal sequence" to *any* portion of this line.

I won't try to calculate whether this is correct or not, since I've already shown that you got some key things wrong above.

Not sure how you're coming up with that. In the "forward" temporal sequence, the rocket exhaust moves backward from the nozzle while the rocket moves forward. In the "reversed" sequence, the exhaust would therefore move forward into the nozzle, while the rocket would move backward to "suck up" the exhaust.

PeterDonis

I'll post separately on these two statements in particular, for reasons that should soon become evident. When you say the rocket "appears" on Mars, do you mean that's when it arrives there, or that's when the light from its arrival reaches us? It seems to be the former because "it has arrived there faster than the image...can reach us". But that leaves three possibilities, none of which appear to disagree with anything I've been saying.

The first possibility is the one I analyzed in my last post: you intended for events E and M to be simultaneous, in the frame of the "reverse observer". My last post shows that, if that is the case, what that observer will actually observe, by means of light signals, is not what you thought.

The second possibility is that you intended for event E to be earlier than event M, but for the travel to still be FTL. That would be consistent with the second statement I quoted above, that "both visually and as a calculated chronology" the trip is from Earth to Mars. I assume that by "calculated chronology" you mean "the actual coordinates assigned to events in the reverse observer's frame". But this possibility is precisely the one that we have never disagreed on to begin with! I have never said that an observer who sees event E before event M will observe anything physically unreasonable. I have only said that, if the curve from event E to event M is spacelike, there will be *some* observer who sees event M before event E, and *that* observer will see things that are physically unreasonable. Everything in your post talks only about what one particular observer will see, the "reverse observer"--but if this second possibility is correct, that is a misnomer, because this so-called "reverse" observer sees the "forward" version of the trip. So if this possibility is correct, you were looking at the wrong observer; you should have looked at some other observer, moving relative to the "reverse" observer, who sees event M before event E. Such an observer must always exist, so if you don't analyze what he observes, you're missing the whole point. (The analysis I'll give in a moment of the third possibility is what you would come up with if you analyzed what that other observer observes.)

The third possibility is that you intended for event M to be earlier than event E, in the frame of the "reverse observer". This would make that observer correctly named, but it would be inconsistent with the second quote above, since by hypothesis the "calculated chronology" would be that the trip was from Mars to Earth. However, I'll go ahead and briefly analyze this scenario as well. It works much the same as the one where the events are simultaneous; the only difference is that now there are two events, SM and SE, instead of a single event S. Event SM is the first event where signals from the "landing" on Mars reach the observer; event SE is the last event where signals from the "takeoff" from Earth reach the observer. Event SM comes first (because event M is earlier), so there is a period between events SM and SE where the observer is receiving signals from the ship on Mars (after "landing"), from the ship on Earth (before "takeoff"), *and* from the Earthbound half of the trip itself (in "reverse" order).

What would the observer conclude from this? He would conclude that there were two ships and one "antiship"! He would conclude that there was a ship on Earth; then, at some instant, a ship and an "antiship" appeared on Mars out of the vacuum; the ship stayed on Mars, while the "antiship" flew to Earth and annihilated the Earth ship, vanishing with it into the vacuum.

You may say that that doesn't make sense--wouldn't it make more sense to say that a single ship started on Earth, went backwards in time to Mars, and then stayed on Mars? The problem with that is that it isn't physically reasonable for at least two reasons:

(1) The portion of the ship's worldline between events E and M violates the second law of thermodynamics; the ship's entropy decreases from event M to event E, which is the forward direction of time.

(2) The whole portion of the spacetime between the time of event M and the time of event E violates energy conservation: there are three ships there, whereas at other times there is only one.

The "antiship" version I gave above avoids these problems because the "antiship" would have negative energy, canceling the energy of one of the ships to keep total energy conserved; and its entropy would run in reverse (because of the negative energy), so there would be no violation of the second law. However, that doesn't (IMO) make an "antiship" physically reasonable either; it just illustrates why FTL travel is disallowed if you want to keep Lorentz invariance.

Austin0

Since the scenario under discussion was for two observers. One seeing the normal forward motion of the ship. This observer naturally being in a frame traveling toward Earth on a trajectory parallel to the Earth Mars line (but outside the Earth-Mars interval), traveling in the positiv3e x direction. But able to see the takeoff,,, so as this wasn't specified I assumed take off was from an equatorial position orthogonal to earth -Mars line and thus visible to such on observer.
The observer seeing the reverse translation being on a parallel course in the -x direction approaching Mars from outside the Earth-Mars interval with the landing likewise being visible on the equator.
Otherwise the scenario made no sense.
There was no mention of being in the middle or both observers traveling in the same direction which makes no sense in any such scenario.
Does this clear it up?

i made no mention of instantaneous although I did specifically mention various velocities greater than c. SO when I said appeared I meant as said that the light from the intermediate trip had not arrived not that the trip was instantaneous..

If this is not enough information I will get back to you

danmay

Actually, they had all assumed constancy of c before even getting into the point on causality.


I only meant in terms of freedom of travel. Space is not actually turning into time or vice versa.

Austin0

It sounds like you are starting to get the picture. But your anti-ship ,and entropy violation etc.,etc. don't make any sense whatsoever. There is no 2nd and 3rd ship.Merely optical illusions as I said much earlier.
There is no time travel unless you adopt a nonsensical interpretation of visual perceptions.

so do you understand my description?

Or do you require more explication?

Would you now say it was inaccurate in any way??

PeterDonis

Ok, so we have two observers, and they are located somewhat differently from what I assumed. Fair enough; see revised analysis below. None of the key conclusions change.

Observer A is moving in the positive X direction and is further in the negative X direction than Earth is; observer B is moving in the negative X direction and is further in the positive X direction than Mars is. Event E is still the event of the ship's "takeoff" from Earth; event M is still the event of the ship's "landing" on Mars. I'll define other important events in the new scheme below.

(Btw: when you use the word "see" in the above, I believe you mean "receive light rays from", because otherwise you have the observers backwards. As we will see, Observer A will assign an earlier time to event M than event E; his "calculated chonology" has the trip going from Mars to Earth, even though he receives light rays from the Earth end of the trip first. Observer B will assign an earlier time to event E than to event M; his "calculated chronology" has the trip going from Earth to Mars, even though he receives light rays from the Mars end of the trip first. This doesn't really affect my conclusions below, but I wanted to make clear how I'm interpreting what I quoted.)

It will be easiest here to adopt a reference frame in which events E and M are simultaneous, and to assume, as I did in my original elaboration on Saw's scenario, that this is the frame in which the Earth and Mars are mutually at rest. You didn't specify any of this in your statement, but nothing you specified contradicts it, and it makes the analysis simpler without affecting any of the key conclusions (which, as I've said several times now, apply regardless of which particular spacelike curve you choose for the trip). In this frame, we set the origin halfway between Earth and Mars at the instant the trip occurs (since the trip is instantaneous in this frame), so event E, once again, has coordinates (-0.5, 0), and event M has coordinates (0.5, 0).

At time t = 0 in this frame, I'll also assume that Observer A is at coordinates (-1, 0), moving in the positive X direction. You didn't specify a speed, so I'll assume it's 0.5. Similarly, I'll assume that Observer B at time t = 0 is at coordinates (1, 0), moving in the negative X direction, with speed -0.5.

Observer A will therefore see the light signal from event E at event EA = (-.833, .333), and from event M at event MA = (-0.5, 1); in between those two events, he sees light signals from intermediate points on the trip, in "forward" order. Since he is further away from Mars than Earth, he sees *no* "overlapping" light signals from the rocket being on Earth or Mars during this period; he sees light signals only from one "segment" of the rocket's total worldline at a time.

However, Observer A's "calculated chronology" concludes that event M happened *before* event E. This is easiest to see if we keep the origins of all the frames the same (meaning that observer A is *not* at the origin of the "moving frame" in his direction); then we can run a simple Lorentz transformation with v = 0.5 to find (X', T') for event E = 1.15 (-0.5, 0.25), and for event M = 1.15 (0.5, -0.25).

Observer B sees the light signal from event M at event MB = (.833, .333), and from event E at event EB = (0.5, 1). In between those two events, he sees the light signals from intermediate points on the trip, in "reverse" order. He also sees "overlapping" light signals from the rocket being on Earth before the "launch", in "forward" order, during this entire period (and, of course, he sees the rocket on Earth before that period as well).

However, Observer B's "calculated chronology" concludes that event E happened before event M; a similar procedure to the above but with v = -0.5 gives (X'', T'') for event E = 1.15 (-0.5, -0.25), and for event M = 1.15 (0.5, 0.25).

So after all this, I'm still not sure what you think it's all supposed to prove. There is an observer who sees (receives light signals from) the rocket's entire history in "forward" order--but that observer concludes, after allowing for light speed time delay, that the trip actually occurred in "reverse" order, from Mars to Earth, which raises all the issues I raised in my previous posts (either he concludes that there are two ships and an "antiship", or he concludes that the trip portion of the ship's history violates energy conservation and the second law of thermodynamics).

There is also an observer who sees "weird" stuff like light signals from the trip "overlapping" with light signals from the rocket's period on Earth before "launch", and the trip light signals arriving in "reverse" order. But *this* observer is the one who concludes, after allowing for light speed time delay, that the trip occurred FTL, but the rocket's entire history happened in ordinary "forward" order in time--no violations of anything, once he allows the possibility of FTL travel. So the observer who sees (receives light signals from) the "weird" stuff is *not* the same one who has to conclude that physical laws have been violated.

And in any case, there *is* an observer who has to conclude that physical laws have been violated; and it shouldn't be too hard to see, after the analyses I've already posted, that there will be some such observer regardless of which particular spacelike trajectory the trip is assumed to follow. So I'm still seeing this scenario as a good reason to disallow FTL travel in the first place.

PeterDonis

No, they're not. They are conclusions those observers are *forced* to, after allowing for light speed time delay and all other optical effects.

What other interpretation would you give? Would you have the observer say that the events from which "optical illusion" light rays are received weren't real? Then you have the problem that the rocket's worldline has "gaps"--it disappears from one event in spacetime and appears at another, without traversing any intervening events. That also violates energy conservation.

Austin0

As I said so far I was only interested in the visual perceptions. I had only examined the question through coordinate events and charts and never thought FTL objects would be visible so when I read your description , (it was your scenario not saw's)
talking about seeing exhaust in reverse being sucked up etc. it seemed to be refering to visual perceptions , not coordinate calculations and it got me thinking.
Since the case of the observer that saw forward motion was uninteresting i restricted my thought to observer B and the optical effects



it seemed so straight forward I assumed it would be obvious simply from a verbal description. I wasn't trying to be difficult.

I never imagined you would go to the trouble you did defining possible scenarios and I apologize for that . I should know better and not assume anything is obvious when dealing with these questions. I will remember in the future.



.I call this Case 2) later in this post

.I call this Case 1) later in this post
.



OBSERVER A

OBSERVER B

So here you are now applying the anti-ship interpretation to observer A
You are quite right. It works in many interesting ways

Case 1) There is forward spatial motion wrt both frames A, sees forward motion in space but calculates that it is translating backward in time.
B sees backward spatial motion through optical effect but calculates the motion is forward in both dimensions. A single ship.

Case 2) The ship jumps directly to the destination at an earlier time without passing through the intermediate space and then translates in reverse motion back to the origin and the moment of launch. In this instance the A observer does not see the motion until the ship reaches the launch and then it only appears to move towards the destination through the same optical effect that takes place in B in the reverse direction. Amazing symmetry.
B sees an effect and it appears strange being in reverse. A sees the same effect but as it conforms to expectations it is deemed ordinary.
Here there are multiple ships.

If we place intermediate observers throughout both of the frames the observations will be equivalent:
the ship appearing out of thin air and racing off in both directions With no clue to determine direction.
In case 1) in both frames the optical part would be directed back to Earth and the normal visual towards Mars.

In Case 2) B's optical effect would point to Earth while A's would point to Mars

this all appears symmetrical and consistent but there is a point of contradiction.

If we have all the observers in both frames set off a flare at the instant the ship is collocated:

In case 1) we can assume that the flares would progress from Earth to Mars in both frames

case 2) it would seem that the flares in B would progress E--->M but the flares in A would have to progress in the opposite direction.

A clear lack of frame agreement on proximate events. I would think that this alone would be enough to eliminate case 2) as a possible option. Would you agree??

In addition case 2) has the problem you mentioned of having two or three ships existing at once , not as optical effects, as in case 1), but as actualities.

.

I am confused here. You say a single ship ((case 1) doesn't make sense but them proceed to give reasons that only seem apply to case 2) Am i missing something here?
Why you think the single ship is less reasonable that case 2) ?

Again this seems to apply to case 2)

So i hope we are on the same track now and can go on to the interoretation. ;-)

PeterDonis

Why wouldn't they be? They're still there; they're just moving FTL. Wouldn't the simplest assumption be that they act like ordinary objects in all other respects?

I am saying that observer A's "calculated chronology" has event M before event E--the ship is on Mars before it is on Earth, yes. The "anti-ship" interpretation is the only way I can see for observer A to interpret that chronology without violating energy conservation and the second law of thermodynamics; but that doesn't mean I think the "anti-ship" interpretation is physically reasonable.

I'm not sure I follow these alternate "cases"; some of what you're saying doesn't appear to make sense, other things just seem to be restatements of things we've already said and (I think) agreed on. Rather than try to comment in detail, let me just make a few general remarks:

(1) Once you have specified the worldline of the FTL rocket and of observers A and B, the scenario is entirely fixed; there's no further room for anything to vary as far as the spacetime coordinates of events are concerned. Once the worldlines are defined, all of the events of interest are fixed and their coordinates in all of the frames of interest are also fixed.

(2) Once you assume that the rocket can emit light rays, so it can be seen, and that observers A and B receive those light rays, you can map out the paths of all those light rays in spacetime and tell exactly at what events on A's and B's worldlines they will receive light rays from what events on the rocket's worldline. That's what I did in putting together my descriptions. There is no way for A or B to label some of those light rays as "just optical effects" and others as "normal visual". They're all bona fide light rays and they all have the same status when the observers see them.

(3) A and B can each construct a "calculated chronology" for their frames, from the optical data they receive and knowledge of the worldlines of Earth and Mars. Each of their chronologies is also fixed by the scenario; there's no room for either one to vary once the above items are fixed. Also, each calculated chronology is a chronology of "real" events; the observer is committed to the belief that those events actually happened, at the time and space coordinates, in his frame, that the chronology assigns to them. We are ruling out hallucinations and false data for this scenario.

(4) The above things are *prior* to any "interpretation" of what's going on by A or B; they are fixed data that any "interpretation" must be consistent with. That means, again, that there is no room for an interpretation to somehow declare that "well, my chronology makes it *seem* like one portion of the rocket's worldline is physically unreasonable, but that's just an optical illusion". If an observer's calculated chronology says that something physically unreasonable has to have happened, then that's the inescapable consequence of the scenario.

In this scenario, as I've said, there *must* be some observer--observer A, in the specific case we've been discussing--who will construct a calculated chronology that includes physically unreasonable events, *if* we allow FTL travel and also retain all the rest of standard SR, including Lorentz invariance. The only residual discussion, it seems to me, is over how to describe the physical unreasonableness. Do we say that observer A would have to believe that an "antiship" existed, and since "antiships" are not physically reasonable, that rules out this scenario? Or do we say that observer A would have to conclude that conservation of energy and the second law of thermodynamics were violated, and that rules out this scenario?

Some of what you say in your cases 1 and 2 could be seen as trying to pin down how to describe the unreasonableness; but to me that's a side issue. The question is, is there *any* way observer A could interpret his calculated chronology which *isn't* physically unreasonable? I don't see how there can be, and that's the critical point.

Saw

Well, the subject of our discussion is precisely the (unlikely) hypothesis that FTL is possible. So for simplicity we can leave out that IF and thus your statement boils down to the following:
Hmm. This looks quite similar to the way I had tried to express your opinion on this occasion:
Never mind. Anyhow, if you substituted “consistence” for invariance, the sentence could be rephrased as follows:
Obviously, I assume you would reject this rephrasing on the grounds that “invariance” and “consistence” have in this context different meanings...

Since this is not clear to me, I will try to delve into the definition of Lorentz invariance (or covariance?). I think we can equate it with the principle of relativity = laws of physics are the same in all inertial frames, so that:

(a) If in two different inertial frames the same TWO experiments (same initial conditions) are carried out, then you get in both frames the same results (same final conditions).

Once that you describe such results in terms of physical laws, this means that in both frames those results could have been predicted by application of the same equations.

(b) If only a SINGLE experiment is carried out and it is analyzed in two different inertial frames, then the initial conditions may be equal but forcefully the final conditions will differ.

But these final conditions can be related through transformations, in particular Lorentz transformations. Thus if you know the final coordinates in frame X and its relative velocity wrt frame Y, then you can guess the final coordinates in frame Y.

In any case, both frames, based on their respective sets of coordinates, should be able to reach a common conclusion about what happens, by feeding their coordinate and frame-dependent values into equations that give out invariant or frame-independent conclusions. For example, if you send a moun from event A to B, you should be able to predict in any frame, by application of the same formula, the length of its timelike spacetime interval, i.e. its proper time and hence whether it reaches event B before disintegration.

In our case, we have situation (b), a SINGLE experiment = something that is on the Earth at event A and travels to Mars, arriving at event B.

Unfortunately here events A and B are simultaneous in the frame Earth-Mars (frame X), so we talk about a spacelike trajectory, requiring FTL travel.

We assume that in spite of that, the final coordinates in frame X and Y or any other one ARE related by the Lorentz transformations.

However, we do not have the other advantage. We have assumed for the sake of discussion that the spacetime spacelike trajectory in question ends in event B, but we could not predict it. Furthermore, if we place a moun on the FTL rocket, we cannot guess whether it will “survive” to reach the target: the length of the spacetime interval (which is its proper time in timelike trajectories) is now an imaginary number, ie no solution to the question.

Conclusion: our life is more complicated now. We have two sets of coordinates, frame X’ and frame Y’s, describing the same events. They are Lorentz-invariant in that they meet the first requirement: they can be mutually related by Lorentz transformations. But they fail to meet the second, which is also part of the usual meaning of Lorentz invariance: they have no predictive value.

Given this, my reaction is quite straight-forward:

- Those space and time coordinates differ because they have to. They are measured from different states of motion, which has a bearing on the result, so it is no surprise they are frame-variant.
- However, they are good as clues for guessing what happens because when you mix them into spacetime formulae, they all lead to the same predictions. That is Lorentz invariance. “Consistence” in predictions.
- If we now refer to FTL travel, we could talk about building clocks and rods based on the FTL mechanism, if that were possible. Then we should see how the diagram is re-constructed and whether the same equations apply. In the (impossible) limit, if we had a really instantaneous agent travelling at infinite velocity from two places, we could synch clocks thereby and would thus have absolute simultaneity and Galilean invariance, at least for an instant. (I cannot even think of how to build an absolute system for registering durations, since an agent of infinite velocity enclosed in a box would not “change”.)
- That is not our assumption, however. We assume that the diagram is the same as before FTL came into play. But then we must also acknowledge that it is not apt for the analysis of the FTL (theoretical) challenge. Remember: the coordinates are not, strictly speaking, “views” about what happens with the rocket or the muon, they are primarily measurements or views about what happens with the clocks and rods. Normally speaking, they are also good indirect clues about what happens with the rocket and its passengers, but not when faced with FTL travel. They are not “valid” for this more ambitious purpose, because by definition non-FTL instruments do not “mirror” FTL agents.

Instead, I do not understand your statement. The two “views” (actually, to be accurate, “measurements”) cannot be both valid in some mysterious sense. They are valid or not in the sense that they meet together at the goal, they are consistent in providing the same predictions. If they do not, then the statement that they are still valid does not make any sense. So you cannot even say that that they are “physically inconsistent”.

PS: A different thing is whether *one* frame could hold that only its (non-FTL) measurements are valid for predicting what happens FTL. That would be for example an aether frame, which would thus be “preferable” only in this specific domain (FTL maters), relativity still holding for the rest of problems (non-FTL ones). Such frame would be undetectable in the non-FTL world. I suppose that in a FTL thought experiment you can argue that such frame is the one where the instantaneous trajectories, regardless direction, always coincide with that frame’s line of simultaneity. I would not find all that abnormal, though. SR (IMO) holds that all frames are equal (in their predicting capabilities) assuming that FTL is impossible, it has no problem with admitting a theoretical prefered frame in a theoretical world.

PeterDonis

I would say this: Lorentz invariance requires that the "views" of a given scenario from all inertial frames must be valid--i.e., they must be physically reasonable and consistent with the laws of physics. In this case, we have at least one inertial frame whose "view" of the scenario is *not* physically reasonable and/or not consistent with the laws of physics. So Lorentz invariance says that this scenario is not possible. This seems pretty straightforward to me.

You could view Lorentz invariance as a more specific statement of the POR, yes; Lorentz invariance tells you more about what "laws of physics are the same" means. But you still have to be careful about applying the principle.

By "same initial conditions" I assume you mean "same conditions relative to the two different inertial frames". For example, if we start the experiment in frame A with the apparatus at rest in frame A, then we must start the experiment in frame B with the apparatus at rest in frame B. If this is what you meant, then yes, Lorentz invariance says the final results should be the same, when expressed relative to each frame.

But there is a proviso to this: *all* of the physical objects participating in the experiment must have their motion, etc. specified relative to each frame. I think you recognize this; see further comments below.

As long as the equations are expressed in terms of invariant and covariant quantities, yes. Again, you seem to realize this; see further comments, below.

I'm not sure *all* of the initial conditions could be the same; there would have to be some that varied, because of the relative velocity of the frames.

Yes, except that instead of "guess" I would say "calculate", which has a much stronger connotation of the result being determined by the laws of physics.

Yes.

Yes.

I think you are confusing conditions that specify a scenario with physical laws. For example:

That's not because we can't Lorentz transform the muon's trajectory, or because we can't define what it would mean for the muon to travel FTL. It's because the physical law for muon decay depends on proper time. Obviously any law that includes proper time as one of the invariants on which it depends won't work as it stands if you allow FTL travel. But one could imagine other laws that would not involve proper time.

(Actually, strictly speaking, we *could* use the muon decay law even with an imaginary proper time; the decay law involves an exponential, and you can exponentiate imaginary numbers. When you do, you get sines and cosines; so basically, you would find that an FTL muon "oscillates" between the decayed and non-decayed states. But this is really a side issue; you would still be adjusting the law to accommodate FTL travel, not just using it "as is".)

So if you allowed FTL travel, you would have to adjust some physical laws, yes. For example, as I commented in previous posts, you would have to find a physical law that told you what specific trajectory an FTL rocket would follow. We *assumed* such a law, implicitly, when we assumed for purposes of our scenario that event E and event M were simultaneous in the Earth-Mars mutual rest frame. But to have a full theory of FTL travel we would have to actually find such a law. But so what? We also have to know physical laws to predict the trajectories of non-FTL objects like muons.

Basically, you are observing that some of our current physical laws would have to be modified if we wanted to accommodate FTL travel. That's true, but I don't see how it changes our reasoning about whether to allow FTL travel in the first place; that reasoning is based on the consequences of allowing *any* pair of spacelike separated events to be causally connected (see further comments at the end of this post). Nothing about that reasoning depends on *which* particular spacelike separated events are causally connected in a particular scenario, as I've said before.

No, that's not correct. As above, any "predictive value" over and above relating the coordinates in different frames by Lorentz transformations does *not* come from Lorentz invariance; it comes from the specific physical laws involved, which must be expressible in Lorentz invariant form, but that still leaves a lot of latitude in *which* specific Lorentz invariant/covariant quantities are included in the law. Some quantities (like proper time) cause more problems if you want to allow FTL travel than others do.

It isn't possible. Allowing FTL travel doesn't change which dimensions of the underlying spacetime are timelike and which are spacelike. Therefore it doesn't change along which dimensions you can define a "time" coordinate vs. "space" coordinates. So nothing about spacetime diagrams are constructed can change when you allow FTL travel. That also means that you can't construct a "frame" with a spacelike worldline as its "time" axis; it won't work, regardless of how you try to jigger the behavior of "rods" and "clocks" based on an "FTL mechanism".

All of which gives further reasons to believe that FTL travel is not possible.

Yes, I agree; see above.

No, this is not correct. Coordinates are numbers (4-tuples of numbers) that are used to label events. The events are events that happen to rockets and muons and other things we're interested in. The rods and clocks are imaginary; we use them to help visualize how we could physically measure numbers that were either identical to or closely related to the coordinate numbers we assign. But if you don't believe that the coordinates are "views" about what happens to rockets and muons and so forth, why would you care about coordinates at all?

Nope; this is not true. The coordinates are *direct* descriptions, in a given inertial frame, of the spacetime locations at which things happen to rockets, etc. And the reasoning we use based on imaginary rods and clocks to visualize how we would measure numbers that are either identical to or closely related to the coordinate values works the same if the rockets, etc. move FTL. The only difference is that the worldlines we end up assigning to FTL objects are spacelike instead of timelike; everything else stays the same.

I can't really comment more on most of the rest of your post since it just seems to be circling around the same misconception as above. But I do want to comment on this:

No, this is wrong. SR (I assume here that "SR" includes "Lorentz invariance") holds that all frames are equal, period. There is no wiggle room. What "FTL travel is possible" means, in SR terms, is "spacelike separated events can be causally connected". And all the stuff I have been describing is just showing how letting spacelike separated events be causally connected leads to physically unreasonable results. That's really all there is to it; there's no point in squirming around trying to find an "interpretation" that will somehow make FTL consistent with SR by jiggering with stuff. "Spacelike separated events can be causally connected" is a simple, well-defined proposition, and it has unreasonable consequences. That's it.

Saw

PeterDonis, I appreciate your patience in maintaining this conversation. I see that none of us is going to persuade the other, because we both circle, as you say, around our respective conceptions (or misconceptions). So it will be understandable if any of us drops out at any moment. In the meantime, I am pleased to make some responses.
So Lorentz invariance, as you understand it, means that the two views should be ok, but in fact only one is and the other is not.
That is a non sequitur. At leat there is another possibility: Lorentz invariance, as you understand it, is wrong.

We are dealing with the typical reductio ad absurdum. The absurdum is that there are two physically inconsistent elements. On this we agree, but we differ in calling those elements “situations” or “explanations”. You are assuming that relativity, when stressed under the tension of FTL, brings about two contradictory situations; hence you kill the messenger, you rule out FTL travel. Instead I assume that relativity, in face of FTL travel, only seems to offer contradictory explanations; hence I infer that those explanations are not really contradictory, so I reinterpret them.

So far, so good. The two approaches could be valid. Which one is more straightforward? I tend to think that mine.

Please consider this example. I had two friends, a Spaniard and an Englishman, who had a common girlfriend and they went on and on like this because the girl rejected marriage. However, one day the English guy found her marrying the Spaniard. He said: our respective claims on the girl are both legitimate, none of them should be privileged, but the girl cannot have more than a husband in our legal system. So I must be dreaming: this scenario is impossible. Well, think this if you wish, but nine times out of ten you will be wrong and simply lose the girl.

In any case, leaving jokes aside, the key for choosing one approach or the other is scrutinizing the concept of Lorentz invariance. For this purpose, my epistemological approach is quite a down-to-earth one, apparently not clashing with scientific method. I am saying that the meaning of our concepts (in our discussions, the spacetime coordinates associated to events) is determined by why and how they are in practice set up, that is to say, for which practical purpose and on the basis of which empirical measurement activities.
What? I did not say that at all. I clearly said that coordinates have a practical purpose, not other than precisely predicting what happens with rockets and muons. And I also said that the coordinates contribute to this goal by constituting a “mirror” or at least a “clue” about what happens with muons and rockets. But one thing is being a clue about an object and the other is being the object itself. I tend to use a simile to explain this. The slipper helps you find Cinderella, but the latter is not the former. Conflating the two is intellectual fetishism. Therefore, if you apparently have two conflicting clues, what you do is not assuming that both clues should be at any rate valid (and thus infer that you are dreaming); instead you simply behave like a good detective and reinterpret such clues.
Why period? You mean, “for no reason and hence without any domain of applicability”? If it were so, it would not be a scientific theory, but a sort of ideology.

Finally, I have to reject your last remark, for the same reasons:
Right, but insisting that this…
… is patently wrong. In my thought experiment, I did not include any unreasonable result, did I? If I did, please mentally wipe it out because that was not my intention. I explained that a very judicious person took a rocket and flew from the Earth to Mars. What is unreasonable in that? Of course, the problem is that there is a certain observer whose coordinates or clocks reading show a lower value for event M than for event E. What does that mean? Under a dirty and realistic interpretation, it means that this guy carried out some measurement operations that give out such outcome. Nothing more. So what this guy should be told to do is deducting that his measurements are not the best account of the story, probably because they are faced with a tough challenge (FTL travel) for which those data are no valid clue. That is all.

PeterDonis

I think that's an acceptable way of putting it.

Which would not refute what I said; I said "Lorentz invariance says this scenario is not possible". Obviously if Lorentz invariance is wrong then one can no longer use it to say the scenario is not possible. I have already said that several times. But Lorentz invariance has been verified by countless experiments, so it is certainly not just wrong. It is possible that Lorentz invariance is violated in ways we can't (yet) measure; we won't know for sure until we can make more accurate measurements, over a wider domain.

This only works if you exclude "Lorentz invariance" from "relativity". But how do you justify that, when Lorentz invariance is a fundamental feature of the theory? Both SR and GR include Lorentz invariance (local Lorentz invariance, in the case of GR, but that's enough for what we're discussing here). You can't "reinterpret" SR or GR to change that; the Lorentz invariance of SR and GR makes definite physical predictions, which for the scenario we've been discussing are the ones I've described. There's no way to alter those predictions without altering the theory.

So if you are admitting the possibility that Lorentz invariance is wrong, you are basically saying we can't use SR or GR to analyze this situation. If that's the case, all bets are off: we can't say anything about it at all unless you can offer some alternative theory that matches all the experimental predictions of SR and GR in the domains where they've been verified, but also allows violation of Lorentz invariance and consequent FTL travel in a situation like we've been discussing.

I don't think I disagree with this as a general statement, in so far as I can parse a meaning from it at all. But for any specific case, how you actually make the connections between concepts and empirical measurements is theory-dependent. If you fix the theory, you fix the connection; conversely, if you want to change the connection, to change, for example, how an observer would "interpret" the light signals he receives that seem to indicate physically unreasonable consequences under standard SR, you have to change the theory.

Of course not. The reason is that it is required by the logical structure of the theory; you can't change it without changing that logical structure, and hence changing the theory. The domain of applicability is at least as large as the domain of all the experiments that have been done that have confirmed the theory's predictions. It may be larger than that; we won't know for sure, as I said, until we've done more experiments over a wider domain. Obviously nobody has yet done direct experiments with rockets traveling at relativistic velocities, nor has anyone actually observed spacelike separated events that are causally connected. So the domain we're discussing here is definitely not part of the domain in which SR and GR have been experimentally confirmed. I've never suggested otherwise.

I didn't say you did. I said only that what you did specify in that thought experiment, combined with the logical structure of SR, leads to physically unreasonable consequences. If you want to reject the conclusion, you have only two alternatives:

(1) Change the specifications of the thought experiment so that, combined with the logical structure of SR, they do not lead to physically unreasonable consequences. The only way to do that is to eliminate the FTL travel--i.e., to ensure that no pair of spacelike separated events are ever causally connected.

(2) Change the theory: stop using standard SR and start using some other theory, with a different logical structure, that leads to different consequences, physically reasonable ones, when combined with the specifications you gave. I have no objection to taking this option in principle, but it doesn't mean much unless you have such an alternative theory. It doesn't seem like you do; and without it, we can't have a useful discussion because we don't have a set of common premises to start from.

You appear to think that there is a third option: keep the specs as they are, keep SR as it is, but somehow "reinterpret" things so the physically unreasonable consequences don't happen. That's not a possible option: the predictions of physically unreasonable consequences, which I have spent quite a bit of time now elucidating, don't depend on "interpretation". They are straightforward logical consequences of SR plus the assumption that any pair of spacelike separated events can be causally connected. Here "SR" does include the physical meaning we assign to coordinates in particular inertial frames in which particular observers are at rest. But as I said above, that's part of the theory; you can't change it without changing the theory's predictions, which means changing the theory.