# Traveling at the speed of light paradox?

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If you did hypothetically travel at the speed of light, for some reason people think time would stop, but if time stopped, wouldn't you not be traveling distance over time and therefore not be traveling at the speed of light? In fact, wouldn't you be traveling distance in 0 time making your speed actually instantaneous?

Edited by questionposter
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Work out what happens if you are traveling close to the speed of light with respect to the earth. This will clarify things.

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I think if you could attain the speed of light you would experience all of time instantaneously. You would essential be outside of time. I imagine from light's perspective "now" is both now in our sense and the now of the big bang. I'm not a physicist and am just speculating though, for what it's worth. lol.

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I think if you could attain the speed of light you would experience all of time instantaneously. You would essential be outside of time. I imagine from light's perspective "now" is both now in our sense and the now of the big bang. I'm not a physicist and am just speculating though, for what it's worth. lol.

Well if your outside of time how can you be traveling distance over time?

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If you did hypothetically travel at the speed of light, for some reason people think time would stop, but if time stopped, wouldn't you not be traveling distance over time and therefore not be traveling at the speed of light? In fact, wouldn't you be traveling distance in 0 time making your speed actually instantaneous?

Let's consider this in terms of the twin paradox, so that we can synchronize clocks at a single location and make sense out of it.

Suppose you left earth with a velocity that approaches c, and you get bounced back off a mirror one light year away.

The earthbound twin would observe that you reached the mirror after one year, but it would take one year for that observation to return to earth. Over 2 years, the earthbound twin can watch you approach the mirror (assuming we have additional signals to mark your progress, otherwise I think you'd be dimmed and redshifted to invisibility), and then after 2 years, you return at what approaches the same instant that observations of you reaching the mirror reach the earth. The total time is 2 years, the total distance is 2 LY, your speed is measured as approaching c.

The earthbound twin can calculate that "your time comes to a stop" as the Lorentz factor approaches infinity and the elapsed time according to your clock approaches 0. In this case, the earthbound twin has aged 2 years and the traveling twin aged essentially nothing.

Now from your perspective as the traveling twin, as the Lorentz factor approaches infinity the distance to the mirror approaches 0 and the time to reach it (traveling at a speed approaching c) also approaches 0. The return journey is the same. The distance to your destination has length-contracted to nothing. You travel essentially no distance in no time. Your time continues passing at the usual rate, but the entire journey happens in an instant. Your speed is c calculated using limits. You'd also calculate that the earthbound twin has aged 2 years in that instant. As usual everybody is in agreement about what happened.

You can only come to a conclusion like "you travel non-zero distance in 0 time" if you mix up frames of reference and use distance from one and time from another, or something like that.

Also, light has no "frame of reference" but we can use "limits as v approaches c" instead of "v = c" to avoid problems.

For example, in the above example at v=c it wouldn't matter if you traveled a rest distance of one m and back, or of a billion LY and back. Calculated using only "distance = 0 and time = 0" for the traveling twin, the rest distance is indeterminate. The earth twin could have aged 2/c seconds, or 2 billion years, or any value.

Edited by md65536
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So just that I get this right:

You mean when traveling at the speed of light every kind of journey would end instantaneously at the point you'd like to reach? Or does the instantaneous travel only work when the point of start is the same as the point of the end of your journey?

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Well if your outside of time how can you be traveling distance over time?

Does light have its own frame of reference? I'm just speculating, but try to make sense of space and time in terms of special relativity from the pov of light. What happens at the speed of light?

Edited by Ceti Alpha V
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Does light have its own frame of reference?

What you mean is can one find an inertial frame in which the photon is measured to be at rest? The answer is no. There is no inertial frame in which the photon is at rest.

This is really the course of all the confusion in this thread. Any object that has mass and assuming not subject to external forces, has a rest frame. When one talk about distances or duration as "experienced" by the object one is referring to measurements in this frame. As photons, or anything else travelling at the speed of light (in any inertial frame) do not have a rest frame one cannot really take about them "experiencing" duration or distances.

Edited by ajb
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What you mean is can one find an inertial frame in which the photon is measured to be at rest? The answer is no. There is no inertial frame in which the photon is at rest.

This is really the course of all the confusion in this thread. Any object that has mass and assuming not subject to external forces, has a rest frame. When one talk about distances or duration as "experienced" by the object one is referring to measurements in this frame. As photons, or anything else travelling at the speed of light (in any inertial frame) do not have a rest frame one cannot really take about them "experiencing" duration or distances.

Indeed.

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Einstein figured out a lot of stuff based on imagining moving at the speed of light, but it was more about figuring out the problems with this than suggesting it's possible. See http://www.pitt.edu/~jdnorton/Goodies/Chasing_the_light/ -- not sure if the linked information is any good but I couldn't find anything else easily.

Because the speed of light is invariant and relative to any observer, even if you travel at 0.9999c relative to something else, light will still move away from you at c. If you try to catch up, it will just keep traveling at c relative to you. Another way to think of this is that you're always at rest relative to yourself, and any light you observe is moving relative to you.

So just that I get this right:

You mean when traveling at the speed of light every kind of journey would end instantaneously at the point you'd like to reach? Or does the instantaneous travel only work when the point of start is the same as the point of the end of your journey?

As v approaches c, the Lorentz factor approaches infinity, which means that lengths contract to (approaching) 0 in the line of motion. Any distance becomes vanishingly small. You're still traveling at a speed of (approaching) c, just over a tiny distance.

The Lorentz transform "breaks down" at v=c... there is a division by 0.

This is okay because traveling at the speed of light is impossible anyways.

There are differences between how a moving object behaves, and how light behaves. You can imagine "being" light, but you can't imagine that you're still yourself! Light doesn't experience the universe as we do. It does not receive incoming light or information.

However, I think that considering light's behavior using "v approaching c" is probably okay. I think you can say that photons do not age. This would mean:

- There is no way to define a clock for a photon, in which a photon could experience time.

- A photon is exactly the same at the end of its journey, as it was when it started.

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Einstein figured out a lot of stuff based on imagining moving at the speed of light, but it was more about figuring out the problems with this than suggesting it's possible. See http://www.pitt.edu/...sing_the_light/ -- not sure if the linked information is any good but I couldn't find anything else easily.

Because the speed of light is invariant and relative to any observer, even if you travel at 0.9999c relative to something else, light will still move away from you at c. If you try to catch up, it will just keep traveling at c relative to you. Another way to think of this is that you're always at rest relative to yourself, and any light you observe is moving relative to you.

As v approaches c, the Lorentz factor approaches infinity, which means that lengths contract to (approaching) 0 in the line of motion. Any distance becomes vanishingly small. You're still traveling at a speed of (approaching) c, just over a tiny distance.

The Lorentz transform "breaks down" at v=c... there is a division by 0.

This is okay because traveling at the speed of light is impossible anyways.

There are differences between how a moving object behaves, and how light behaves. You can imagine "being" light, but you can't imagine that you're still yourself! Light doesn't experience the universe as we do. It does not receive incoming light or information.

However, I think that considering light's behavior using "v approaching c" is probably okay. I think you can say that photons do not age. This would mean:

- There is no way to define a clock for a photon, in which a photon could experience time.

- A photon is exactly the same at the end of its journey, as it was when it started.

What about skipping the speed of light entirely? What about going at 99.9999% the speed of light, and then instead of reaching 100%, you just jump to 101% without having to go at the speed of light at all?

Also, why aren't photons subject to all this time dilation stuff? Why don't photons contract infinitely?

Edited by questionposter
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What about skipping the speed of light entirely? What about going at 99.9999% the speed of light, and then instead of reaching 100%, you just jump to 101% without having to go at the speed of light at all?

One, it's impossible. Two it doesn't make sense.

The Lorentz factor would then have an imaginary number. This doesn't correspond to reality. If you can make sense of how your idea would work, you'd have to describe it with something that is not special relativity.

To me it sounds something like "If I can't catch up to the car in front of me no matter how fast I go, then why don't I just pull in front of them and not worry about catching up?"

If the amount of energy it takes to reach c is infinity, what would it take to go even faster?

Also, why aren't photons subject to all this time dilation stuff? Why don't photons contract infinitely?

Because photons aren't observers in a frame of reference.

Length contraction is described in terms of a particular frame or observer.

In the twin paradox example, a traveler approaching c moving from earth to a mirror 1 LY away would see the distance between earth and mirror contract.

The earthbound twin would be at rest with the earth and mirror and not observe any length contraction between the earth and mirror.

Addendum: Photons are like point-particles without any size. I don't know if it's fair to consider them subject to length contraction, but if you did you might say that they're already infinitely length-contracted, according to any observer. But uh... don't quote me on that! This kind of involves imagining photons as ordinary matter, and will probably lead to more false conclusions than useful ones.

Perhaps it is just better to say: Photons are point-particles and so wouldn't be affected by length contraction.

Edited by md65536
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Because photons aren't observers in a frame of reference.

Length contraction is described in terms of a particular frame or observer.

Also, relativity is based on the postulate that light will always be observed to travel at the same speed by any observer, so if length contraction and time dilation caused anything else to occur, special relativity would be internally inconsistent.

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What about skipping the speed of light entirely? What about going at 99.9999% the speed of light, and then instead of reaching 100%, you just jump to 101% without having to go at the speed of light at all?

One, it's impossible. Two it doesn't make sense.

The Lorentz factor would then have an imaginary number. This doesn't correspond to reality. If you can make sense of how your idea would work, you'd have to describe it with something that is not special relativity.

As far as I know there is nothing in special relativity which prevents objects that move faster than the speed of light, nor should a physicist balk at seeing imaginary numbers (for one, they come up when solving simple things like a mass-spring-damper system) as long as the observables you predict are real.

We already have quantities that act like a speed faster than light and ways to deal with them, we call them space-like intervals (of which the concept of distance is a sub-set).

Such objects are known as tachyons, and are generally considered not to exist due to there being zero evidence for them, or for theories that produce them. If they did exist they would be able to carry information back in time, this would break causality (and I think also the laws of thermodynamics).

On top of this, you get other weird things, like the slower they move the more energy is required, and they cannot move slower than the speed of light. I also have no idea how an interaction between tachyons and normal matter would work.

I also hear that they are incompatible with newer physics (somewhere between QED and string theory I hear talk of models producing tachyons and their rejection due to the fact this would cause runaway inflation) although I do not understand the rationale behind this.

Also, relativity is based on the postulate that light will always be observed to travel at the same speed by any observer, so if length contraction and time dilation caused anything else to occur, special relativity would be internally inconsistent.

This would be fine, if you were somehow able to hop over the speed of light I'm pretty sure you'd just find that the light moved at the speed of light in the opposite direction.

Edited by Schrödinger's hat
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I think that when you travel at the speed of light, the time doesn't stop

otherwise, how come that photons change position with time, and have change of energy, ..etc

but maybe because photons are too fast, everything else seem relatively very very slow, and speaking

about observing other things at that speed, it might be impossible,

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I think that when you travel at the speed of light, the time doesn't stop

otherwise, how come that photons change position with time, and have change of energy, ..etc

but maybe because photons are too fast, everything else seem relatively very very slow, and speaking

about observing other things at that speed, it might be impossible,

Photons don't change (they have different energies in different frames, but in any given frame they aren't changing) -- well, the wavefunction of the light changes over time, but let's not get into quantum for now -- any photon with a given energy-momentum and spin is completely indistinguishable from every other photon with those properties. They also have no size that I know of.

At any rate, the concept of travelling at the speed of light is outside the realm of special relativity. It's a bit like asking, 'how green is a second?'. The answer is simply outside any model we have.

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As far as I know there is nothing in special relativity which prevents objects that move faster than the speed of light, nor should a physicist balk at seeing imaginary numbers (for one, they come up when solving simple things like a mass-spring-damper system) as long as the observables you predict are real.

I think though the observables would also be imaginary. I think you'd get imaginary distances.

But maybe that itself allowed with GR? And objects inside black hole event horizons??? (It's easy to consider real-world examples that are outside the scope of SR simply by considering gravity. GR doesn't violate SR but it handles situations that SR doesn't.)

[...]to carry information back in time, this would break causality

I always thought that a violation of causality was a violation of SR. Certainly you can get paradoxes (http://en.wikipedia.org/wiki/Special_relativity#Causality_and_prohibition_of_motion_faster_than_light)...

Or can you allow violation of causality without violation of SR through interpretations like alternate realities and MWI?

To me it sounds something like "If I can't catch up to the car in front of me no matter how fast I go, then why don't I just pull in front of them and not worry about catching up?"

And now that I think about it, saying "I don't see how that's possible" is not good evidence that it's impossible. However, you probably wouldn't be able to pass a car that you can't catch, by speeding up, nor would you be able to travel faster than light by speeding up. But if say your car were able to "jump" ahead without changing speed, it might be possible.

But I still don't think it's possible.

If you're only trying to understand observed reality or relativity, I think it's safe and beneficial to assume that causality cannot be violated. If you want to change our understanding of reality, perhaps there will come a reason to reevaluate it.

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One, it's impossible. Two it doesn't make sense.

The Lorentz factor would then have an imaginary number. This doesn't correspond to reality. If you can make sense of how your idea would work, you'd have to describe it with something that is not special relativity.

To me it sounds something like "If I can't catch up to the car in front of me no matter how fast I go, then why don't I just pull in front of them and not worry about catching up?"

If the amount of energy it takes to reach c is infinity, what would it take to go even faster?

Because photons aren't observers in a frame of reference.

Length contraction is described in terms of a particular frame or observer.

In the twin paradox example, a traveler approaching c moving from earth to a mirror 1 LY away would see the distance between earth and mirror contract.

The earthbound twin would be at rest with the earth and mirror and not observe any length contraction between the earth and mirror.

Addendum: Photons are like point-particles without any size. I don't know if it's fair to consider them subject to length contraction, but if you did you might say that they're already infinitely length-contracted, according to any observer. But uh... don't quote me on that! This kind of involves imagining photons as ordinary matter, and will probably lead to more false conclusions than useful ones.

Perhaps it is just better to say: Photons are point-particles and so wouldn't be affected by length contraction.

What about traveling faster than light but not by using kinetic energy? Maybe there's some unknown force which allows matter to accelerate faster than light, like dark energy, which I assume is different than the energy we already know.

Edited by questionposter
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I think though the observables would also be imaginary. I think you'd get imaginary distances.

space-like intervals represent real distances, remember it's $(\Delta s)^2 = (\Delta (ct))^2 - (\Delta x)^2$

A negative interval squared (imaginary interval) merely represents more distance than time.

So this would work if you were considering a world-line (ie. particle which is 1 dimensional at any given time)

You wouldn't be able to use formulae for things like length contraction and time dilation, but these are simply derived from the Lorentz transform.

I do not know how to think about non-pointlike objects in this context. Perhaps they would just fall apart because the interactions that bind them would not be able to travel between events in the object's 'worldline'?

But maybe that itself allowed with GR? And objects inside black hole event horizons??? (It's easy to consider real-world examples that are outside the scope of SR simply by considering gravity. GR doesn't violate SR but it handles situations that SR doesn't.)

The distance from an object inside an event horizon to one outside is just undefined (rather than imaginary) as far as I know. You could probably consider a geodesic running the other way though. GR is a bit beyond me, but I believe the concept of distance gets a little shaky once you consider gravity, you have to talk about path length if you want any meaningful results.

I always thought that a violation of causality was a violation of SR. Certainly you can get paradoxes (http://en.wikipedia....ster_than_light)...

Or can you allow violation of causality without violation of SR through interpretations like alternate realities and MWI?

Well, I was restricting my thinking to SR in isolation (ie. events happen in Minkowski space) rather than special relativistic physics. I'm sure tachyons would cause paradoxes all over the place.

If you're only trying to understand observed reality or relativity, I think it's safe and beneficial to assume that causality cannot be violated. If you want to change our understanding of reality, perhaps there will come a reason to reevaluate it.

Agreed

What about traveling faster than light but not by using kinetic energy? Maybe there's some unknown force which allows matter to accelerate faster than light, like dark energy, which I assume is different than the energy we already know.

None of our theories hint as to this being possible, and we've seen no evidence which hints that such a mechanism exists.

It would also cause paradoxes and allow time travel in some sense or another. So we put it in the bin with all the other ideas that we have no reason to consider right now, until we find a good reason to do so.

Actually, there's one exception ...kinda...in GR. It's not really moving faster than light, kind of side-stepping the limit by bending spacetime just so.

It has its own problems (such as requiring infinite energy -- for one -- and possibly requiring that you can already move at FTL speeds ), and would probably have all the same paradox issues.

Edited by Schrödinger's hat
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None of our theories hint as to this being possible, and we've seen no evidence which hints that such a mechanism exists.

It would also cause paradoxes and allow time travel in some sense or another. So we put it in the bin with all the other ideas that we have no reason to consider right now, until we find a good reason to do so.

Actually, there's one exception ...kinda...in GR. It's not really moving faster than light, kind of side-stepping the limit by bending spacetime just so.

It has its own problems (such as requiring infinite energy -- for one -- and possibly requiring that you can already move at FTL speeds ), and would probably have all the same paradox issues.

Well it says that things like time dilation wouldn't apply, which I guess makes sense since it's different than just kinetic energy which would increase an object's relative mass. I think there also might be quantum teleportation, kind of like that thing that was in the Douglas Adams books where the ship went through every possible point and ended up somewhere, but there might be a way to control it like creating a narrow dumbbell shape in the same relative region for all the atoms of a ship as to cause it to appear in one of those regions, and if you don't appear in the right one, just try it again till you get in the other, there's a 50% chance. I think it would require more energy depending on how far away something was, and it would need to be fast.

Edited by questionposter
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I think that when you travel at the speed of light, the time doesn't stop

otherwise, how come that photons change position with time, and have change of energy, ..etc

but maybe because photons are too fast, everything else seem relatively very very slow, and speaking

about observing other things at that speed, it might be impossible,

You gotta think on terms of relative motion. Say you are travelling at a constant speed of 87 percent the speed of light (and going in a constant direction) relative to me. So you are in what's called uniform motion, i.e. no change in speed or direction. But per Einstein's Principle of Relativity, from your point-of-view, you are at rest and I am moving at 87 percent the speed of light. So for a velocity v of 0.87, according to the Lorentz Factor, you observe my time running two times slower than yours.

The Lorentz factor is the square root of (1 - v^2) where v is velocity as a fraction of the speed of light .

Now what if you are in uniform motion at a v of 99.5 percent the speed of light relative to me? Then per the Lorentz factor, you see my time running ten times slower than your time. And if you go at 99.999 percent the speed of light, you see my time running 224 times slower than yours.

So the closer your speed gets to the speed of light, the slower you see my time running. At the speed of light (v equals 1), the Lorentz factor is 0. So if you could travel at the speed of light, you would see my time running infinitely slower than yours. This can be interpreted as you seeing my time as STANDING STILL. But nothing with mass can travel at the speed of light.

And the experts say that a reference frame traveling at the speed of light (like a photon) does not constitute a legitimate inertial (unifrom motion) reference frame -- I think because we have to divide by zero and get infinity.

And, oh by the way, no matter what uniform speed you travel at relative to me, you observe your time running normally and my time running slower. AND I see my time running normally and your time running slower. Time is relative.

Edited by IM Egdall
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space-like intervals represent real distances, remember it's $(\Delta s)^2 = (\Delta (ct))^2 - (\Delta x)^2$

A negative interval squared (imaginary interval) merely represents more distance than time.

So this would work if you were considering a world-line (ie. particle which is 1 dimensional at any given time)

You wouldn't be able to use formulae for things like length contraction and time dilation, but these are simply derived from the Lorentz transform.

The Lorentz factor would be imaginary, as would the result of the Lorentz transform, no?

Do you mean that intermediate imaginary numbers are fine, because the space-time interval can still come out to be real, even with imaginary time and spatial coordinates?

Time dilation and length contraction wouldn't apply, and time paradoxes could be constructed, but the math still works out and describes a situation that can be made sense of, and is compatible with SR even if it corresponds to no observed phenomenon?

Edited by md65536
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The Lorentz factor would be imaginary, as would the result of the Lorentz transform, no?

Do you mean that intermediate imaginary numbers are fine, because the space-time interval can still come out to be real, even with imaginary time and spatial coordinates?

There are a few details, but you got the gist of what I was trying to say.

For one, this kind of transformation is not in the Lorentz group. There's no boost which makes you move faster than light. It'd have to be some other kind of thing.

I guess in this makes it not really in the realm of SR.

My logic was as follows:

1) Have something moving less than c

2) Something mysterious happens st. v>c. You could, say, change from a dX/dtau with a positive magnitude squared* to one with equal negative magnitude squared and with the same direction in its spatial components.

3) What would SR say about the situation afterwards.

Time dilation and length contraction wouldn't apply, and time paradoxes could be constructed, but the math still works out and describes a situation that can be made sense of, and is compatible with SR even if it corresponds to no observed phenomenon?

Yeah, that's pretty much what I was trying to say. Or it at least works out for a little while. If you thought about it for too long either your head or the universe (in your model which includes tachyons) would probably explode.

It'd do all sorts of weird things, like if you were to watch it, you'd only see it after it'd passed you. Then you'd see it in two places (if you could see it...don't know how it would interact with light), moving away from you in opposite directions.

I don't even know how to think about objects that aren't point-like in this context. Depending on how close 1/v was to 0, they'd do some sort of weird length-dilation until they took up the entire universe (in that direction) for a few moments -- as measured by your currrent frame. What you'd see would be more like the previous paragraph.

Trying to think about other things, such as how doppler shifted would it be -- ie. what would you actually see rather than where would you see it? I'm not sure if SR has anything meaningful to say.

@moderators: could we move this thread to speculations? Although it's fun to talk about this model of FTL, I don't think it has a place in the physics forum.

*is magnitude squared the right word? I shall specifiy with maths just in case:

This thing: $\frac{dX}{d\tau}^T \left[\begin{matrix}ct&0&0&0 \\ 0&-1&0&0 \\ 0&0&-1&0 \\ 0&0&0&-1 \end{matrix}\right]\frac{dX}{d\tau}$

Edit: Fixed typo and changed the sign of my metric to be consistent with text.

Edited by Schrödinger's hat
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@moderators: could we move this thread to speculations? Although it's fun to talk about this model of FTL, I don't think it has a place in the physics forum.

I think the thread so far is better here; speculations doesn't tend to be treated very seriously.

The discussion seems to have evolved to "Traveling at the speed of light is prohibited by SR, but what implications does relativity have for hypothetical FTL speeds?" I think this is useful because it shows the problems with the concept within SR, which are not there in classical physics, and would be ignored with open speculation.

It'd do all sorts of weird things, like if you were to watch it, you'd only see it after it'd passed you. Then you'd see it in two places (if you could see it...don't know how it would interact with light), moving away from you in opposite directions.

I don't even know how to think about objects that aren't point-like in this context. Depending on how close 1/v was to 0, they'd do some sort of weird length-dilation until they took up the entire universe (in that direction) for a few moments -- as measured by your currrent frame. What you'd see would be more like the previous paragraph.

Okay this next part belongs in speculations! And I'm in way over my head.

I picture imaginary magnitudes as negative distances. Treating yourself as a point observer, these would be distances with a magnitude that projects "into you", not along normal spatial axises, but along imaginary axises.

My intuition says that even a point particle at an imaginary distance, say equivalent to $\sqrt{-d^2}$, would take up area in your imaginary field of vision, and look like a spherical shell of radius d at a distance of d (I guess you'd intersect its surface). But uh... it's inverted or something. Anyway it would get bigger the farther it is from you, until it took up half the imaginary universe at a distance of infinity.

Yes, my head exploded, and that was the result.

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Okay this next part belongs in speculations! And I'm in way over my head.

I picture imaginary magnitudes as negative distances. Treating yourself as a point observer, these would be distances with a magnitude that projects "into you", not along normal spatial axises, but along imaginary axises.

My intuition says that even a point particle at an imaginary distance, say equivalent to $\sqrt{-d^2}$, would take up area in your imaginary field of vision, and look like a spherical shell of radius d at a distance of d (I guess you'd intersect its surface). But uh... it's inverted or something. Anyway it would get bigger the farther it is from you, until it took up half the imaginary universe at a distance of infinity.

Yes, my head exploded, and that was the result.

Huh? Negative interval squared (imaginary interval) just means more (real) distance than (real) time. At no point did I suggest distance squared would be negative.

Putting an imaginary number in x, y, or z is has no meaning that I can think of without further context.

If we try and use the length contraction and time dilation equations we will get imaginaries, but if we were to take each point on the object's world line (assuming you came up with some consistent way to do our magical v<c --> v>c transform in a consistent way over the whole thing) and apply further lorentz transforms it would work fine.

I think this is the same sort of thing as having to be careful and manually input signs/add 2*n*pi when dealing with functions such as asin or picking the right sign of your square roots.

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