New Paradox Discovered, I Think

[Somoth Ergai]

TL;DR Summary

Ship travels faster than light while traveling slower than light. The same light is in two different places at the same time.

User
New here but this conundrum is something that occurred to me as I was watching a physics video and I have run out of ways to ask about this. So here I am.

Ok first principle. the setup. let's say we have invented a perfect telescope on earth. such that we can resolve any object no matter how distant in space. we understand that light takes time to travel and as such we know that even though we can resolve objects in distant space, we are not seeing the "true" object as it is. only as it was some time in the past depending on distance.

we turn our telescope to a distant planet in the Andromeda galaxy that is 2 million light years away. we understand that this means anything we observe about the planet in fact happened 2 million years ago in the past.

So far so good

Second Principle. while observing this distant planet we see that there is an alien civilization living on the planet. we observe them build a spaceport. and then a spaceship. they board the spaceship and then launch with a trajectory that will take them to planet earth.

we understand here on earth that this event in fact happened 2 million years ago. meaning that the aliens have actually been in transit toward our planet for the last 2 million years.

we understand that their "real" physical location is 2 million years ahead of the image we see of them departing their planet.

Still so far so good.

Here's the paradox. let's say that the aliens ship in fact travels at very nearly the speed of light. say, 99.999999999% light speed. this means that while the light from the ship is still technically faster, from our perspective the ship essentially arrives at the same time the light from its launch does.

the paradox is this. even though light speed is very fast. and likewise the ship is very fast. from our perspective watching the ship leave their planet and journey here that trip is going to take 2 million years to complete. we would fairly easily be able to track the visual motion of the ship across space during it's two million year journey.

the paradox is that the ship is already here. ahead of all those two million years worth of light from it's journey. meaning that the ship arrived on earth faster than it's own emitted light while simultaneously traveling slower than it's own emitted light.

I understand that the trip is shorter from the perspective of the aliens. that's not the issue. from their perspective the trip isn't just shorter it's nearly instantaneous. they arrive just slightly behind the light emitted at launch. that's all well and good and has nothing to do with the paradox. the paradox is that the rest of the light from their journey is going to take two million years to finally get to us. as we track the image of their ship across space we are going to be watching it for the next 2 million years. this means that the light emitted from their ship is going to take longer to get to us than their ship did.

the paradox is that the ship traveled faster than light while traveling slower than light

So, the ship arrives on earth slightly after we see it launch because by the time we see it launch it has been in transit for 2 million years. since the trip only takes 2 million years that means they arrive just as we see them depart.

the problem is that after we see them depart we still have to observe their entire two million year journey. meaning they arrive on earth 2 million years ahead of the rest of the light from their journey.

meaning if they wanted to, they could come sit at our telescope and observe their own journey across space for the next two million years.

but there's even more to this paradox. because you see, from the perspective of the aliens the trip was almost instant right? meaning from their view they arrived WITH all of the light from their trip. from their perspective their arrival should have appeared to be a bright flash of light as all the light from their two million year journey arrives here at the same time.

leading to a situation where the light from their journey has to be in two different places at the same time.

Because it all arrives WITH the aliens from their perspectivAe, but will take two million years from our perspective. except that they're here. so their light has both reached us and not reached us at the same time. meaning it is in two different places at the same time

 

[Orodruin]

There is no paradox here. Just a mixed soup of misinterpreting relativity of simultaneity and other concepts.

 

[Ibix]

the paradox is that the ship is already here. ahead of all those two million years worth of light from it's journey.

No it's not. The ship is not travelling faster than light, so all light emitted from the ship during its trip will arrive before it. If the $2\times 10^9\ \mathrm{ly}$ trip is done at your specified $(1-10^{-11})c$ then the light from the launch would arrive about a week before the ship and we would see the whole trip over that week.

 

[Somoth Ergai]

There is no paradox here. Just a mixed soup of misinterpreting relativity of simultaneity and other concepts.

well that's nice and all but without an explanation of how or why i'm wrong that just sounds like "nuh uh"

 

[Somoth Ergai]

No it's not. The ship is not travelling faster than light, so all light emitted during its trip will arrive before it. If the trip is done at your specified $(1-10^{-11})c$ then the light from the launch would arrive about a week before the ship and we would see the whole trip over that week.

that would mean that the ship, from our perspective, would travel two million light years in a weak. far exceeding the speed of light. which, would mean we would have to see light traveling faster than itself. This does not resolve the situation.

 

[Ibix]

that would mean that the ship, from our perspective, would travel two million light years in a weak

No it doesn't. When I see the light from the launch arriving I can see that it was emitted two billion light years away so I know it started two billion years ago. When the ship arrives a week later I therefore know that it travelled two billion light years in two billion years and a week - so slower than light.

You seem to be aware of the need to correct for light travel times. You just don't seem to be applying that knowledge to the process of calculating the ship's speed.

 

[PeroK]

that would mean that the ship, from our perspective, would travel two million light years in a weak. far exceeding the speed of light. which, would mean we would have to see light traveling faster than itself. This does not resolve the situation.

You can accurately measure the speed of a train using the idea that light signals travel from the train to the observer almost instantaneously. But, if the object itself is moving at a significant fraction of the speed of light, then you have to take the time delay of light signals into account when calculating an object's speed.

This is, in fact, how the speed of light was first estimated by Romer in 1676. He measured that the orbits of Jupiter's moons were out of step depending on their varying distance from Earth:

https://en.wikipedia.org/wiki/Rømer's_determination_of_the_speed_of_light

 

[Somoth Ergai]

No it doesn't. When I see the light from the launch arriving I can see that it was emitted two billion light years away so I know it started two billion years ago. When the ship arrives a week later I therefore know that it travelled two billion light years in two billion years and a week - so slower than light.

You seem to be aware of the need to correct for light travel times. You just don't seem to be applying that knowledge to calculating the ship's speed.

million not billion but who cares. the point is, if the ship travels at near light speed it means it arrives slightly behind the light from its launch. the thing is, if we are observing the launch from the start that entire trip is going to play out over 2 million years plus whatever change. it has to. we observe the ship traveling at near light speed means it's going to take 2 million years for the trip to complete. meanwhile, the ship has alrready been traveling for 2 million years by the time we see it launch. since the real, "physical" trip takes 2 million years, that means the ship must alrready be here. or will be in about a week as you say.

the only way for all of the light from the ships trip to reach us now is for the trip, from our perspective, to take much less time than the full 2 million year journey.

 

[Somoth Ergai]

You can accurately measure the speed of a train using the idea that light signals travel from the train to the observer almost instantaneously. But, if the object itself is moving at a significant fraction of the speed of light, then you have to take the time delay of light signals into account when calculating an object's speed.

This is, in fact, how the speed of light was first estimated by Romer in 1676. He measured that the orbits of Jupiter's moons were out of step depending on their varying distance from Earth:

https://en.wikipedia.org/wiki/Rømer's_determination_of_the_speed_of_light

I mean, that's great and all but that doesn't really address why i'm wrong about this.

 

[PeroK]

the only way for all of the light from the ships trip to reach us now is for the trip, from our perspective, to take much less time than the full 2 million year journey.

You can also check out this:

https://en.wikipedia.org/wiki/Superluminal_motion

 

[PeroK]

I mean, that's great and all but that doesn't really address why i'm wrong about this.

The speed of an object does not depend on when light signals from that object arrive at an observer. The speed of an object is the actual distance it travels divided by the time interval. In this case, the speed of the spaceship is just less than $c$. As measured by someone on Earth. You have the calculation essentially in your original post: Andromeda is 2 million light years away and the ship took 2 million years to travel from there to Earth. The ship's speed, by definition of speed, is approx $c$. Whether you can see the ship or not doesn't alter its measured speed.

 

[Ibix]

the only way for all of the light from the ships trip to reach us now is for the trip, from our perspective, to take much less time than the full 2 million year journey.

There is a difference between what we directly see (the first light gets here a week before the ship) and what we conclude (the trip took two billion years and a week). Many sources are sloppy about the distinction, but it's the second one that is the measurement.

Consider a row of bouys between here and Andromeda. Each one has a light sensor on it and a lamp which is turned on when the sensor receives a laser pulse. Andromeda fires a laser pulse at us. Each bouy turns on its lamp as the pulse passes, but the light from the lamps travels at the same speed as the laser. The light from each lamp turning on will therefore arrive at us simultaneously with the laser pulse. So we see the whole trip of the laser pulse on zero time. The speed is infinite according to you! So why can't I get an instant response from Andromeda by sending a laser message?

 

[Somoth Ergai]

You can also check out this:

https://en.wikipedia.org/wiki/Superluminal_motion

There is a difference between what we directly see (the first light gets here a week before the ship) and what we conclude (the trip took two billion years and a week). Many sources are sloppy about the distinction, but it's the second one that is the measurement.

Consider a row of bouys between here and Andromeda. Each one has a light sensor on it and a lamp which is turned on when the sensor receives a laser pulse. Andromeda fires a laser pulse at us. Each bouy turns on its lamp as the pulse passes, but the light from the lamps travels at the same speed as the laser. The light from each lamp turning on will therefore arrive at us simultaneously with the laser pulse. So we see the whole trip of the laser pulse on zero time. The speed is infinite according to you! So why can't I get an instant response from Andromeda by sending a laser message?

wrong. we will see a succession of the lamps turning on one after the other. the light might travel at the same speed but it's departure time is staggered by the separation of each bouy. it absolutely would not all arrive here at the same time. in order for that to be true it would have to appear that all events in the universe happen simultaneously. which is obvious nonsense.

 

[Ibix]

million not billion but who cares.

Indeed, my mistake. It should be two million light years and about ten minutes' difference in arrival times.

 

[Somoth Ergai]

The speed of an object does not depend on when light signals from that object arrive at an observer. The speed of an object is the actual distance it travels divided by the time interval. In this case, the speed of the spaceship is just less than $c$. As measured by someone on Earth. You have the calculation essentially in your original post: Andromeda is 2 million light years away and the ship took 2 million years to travel from there to Earth. The ship's speed, by definition of speed, is approx $c$. Whether you can see the ship or not doesn't alter its measured speed.

riight so. let me ask this because this is kind of the whole point of contention for my problem. imagine we are watching the ship depart right? from launch to arrival the ship travels at just slightly less than c. we observe this entire trip from our telescope. how long does the trip take from our vantage point?

 

[Ibix]

wrong.

It is you who is wrong. I would suggest doing some maths instead of waving your hands.

the light might travel at the same speed but it's departure time is staggered by the separation of each bouy.

And the flight times are reduced by exactly the correct amount to cancel out the start time offsets. Hence the simultaneous arrivals.

it absolutely would not all arrive here at the same time. in order for that to be true it would have to appear that all events in the universe happen simultaneously. which is obvious nonsense.

That is a nonsense conclusion, yes. That should make you reflect on your reasoning given that all the light does arrive simultaneously. That was, in fact, exactly why I suggested the scenario.

 

[Ibix]

how long does the trip take from our vantage point?

Two million years and ten minutes. Because we're able to see that oir first measurement is two million years late.

 

[Orodruin]

wrong. we will see a succession of the lamps turning on one after the other.

Wrong. If you do not understand this very basic example, then there is simply no way for you to understand the resolution to your ”paradox”.

the light might travel at the same speed but it's departure time is staggered by the separation of each bouy.

… in exactly such a way that the light arrives at the observer at the same time. The light from closer buoys may be emitted later, but it has less travel time and arrives at the same instant.

If the buoys all flashed simultaneously (in the buoy rest frame) the signals would arrive at different times, but this was not the setup.

it absolutely would not all arrive here at the same time.

See above. This assertion is simply wrong.

in order for that to be true it would have to appear that all events in the universe happen simultaneously.

No. This inference is simply false. Only light from events on the past lightcone of the observer would appear at the observer at the same time.

which is obvious nonsense.

… because your inference is nonsense, what results from it is nonsense. This is not very surprising.

 

[Somoth Ergai]

It is you who is wrong. I would suggest doing some maths instead of waving your hands.

And the flight times are reduced by exactly the correct amount to cancel out the start time offsets. Hence the simultaneous arrivals.

That is a nonsense conclusion, yes. That should make you reflect on your reasoning given that all the light does arrive simultaneously. That was, in fact, exactly why I suggested the scenario.

Two million years and ten minutes. Because we're able to see that oir first measurement is two million years late.

right. so if we're going to be sat here watching the ship travel for the next two million years, yet in actuality the ship has already been traveling for two million years across two million light years of space at a rate of just under c. then the ship has essentially completed its jjourney buy the time we see it depart correct? i'm not concerned about the additional 10 minutes. for all intents and purposes the ship arrives at nearly the same time the light does.

 

[PeroK]

riight so. let me ask this because this is kind of the whole point of contention for my problem. imagine we are watching the ship depart right? from launch to arrival the ship travels at just slightly less than c. we observe this entire trip from our telescope. how long does the trip take from our vantage point?

One answer is that "vantage point" has no meaning in physics. You can easily calculate that light from the departure event reaches Earth only a short time before the ship arrives. But, that is irrelevant to any definiion or measurement of speed. You could invent a new term of "apparent speed" and that would be very large (almost infinite) in this case. But, your "apparent speed" is not what physicists mean when they use the term "speed".

When we say speed of light in vacuum, we mean something precise in terms of that word speed: which is distance/(time of travel). It is not distance/(time interval between light signals).

 

[Somoth Ergai]

One answer is that "vantage point" has no meaning in physics. You can easily calculate that light from the departure event reaches Earth only a short time before the ship arrives. But, that is irrelevant to any definiion or measurement of speed. You could invent a new term of "apparent speed" and that would be very large (almost infinite) in this case. But, your "apparent speed" is not what physicists mean when they use the term "speed".

When we say speed of light in vacuum, we mean something precise in terms of that word speed: which is distance/(time of travel). It is not distance/(time interval between light signals).

you haven't answered my question. i'll ask a different way. is it correct that as we observe the ship as it launches from its planet, that it will take two million years to complete its journey to earth?

secondly. is it also correct that since the ship has already been traveling at near c for the past 2 million years, that by the time we observe its launch it is essentially already here?

 

[Orodruin]

I strongly suggest you consider the same kind of reasoning but using sound instead of light. Your argument is not contingent on light being the messenger or even the existence of time dilation:

A jet fighter flies from A to B at mach 2. The flyby takes 2 seconds. T sound from the jet emitted at A reaches B 2 seconds after the jet reaches B. Hence the jet is travelling back in time.

This is the essence of your argumentation.

 

[Orodruin]

you haven't answered my question. i'll ask a different way. is it correct that as we observe the ship as it launches from its planet, that it will take two million years to complete its journey to earth?

No. When you actually see the ship leaving it has already been travelling for 2 million years. You can conclude that the launch occurred 2 million years ago when you see the light because you know the launch occurs 2 million ly away. The ship will take 2 million years + change to arrive at the Earth from the moment it was launched. By the time you see the launch, it is therefore just the change left before it arrives

secondly. is it also correct that since the ship has already been traveling at near c for the past 2 million years, that by the time we observe its launch it is essentially already here?

Not also as the first statement was incorrect. See above.

 

[Somoth Ergai]

I strongly suggest you consider the same kind of reasoning but using sound instead of light. Your argument is not contingent on light being the messenger or even the existence of time dilation:

A jet fighter flies from A to B at mach 2. The flyby takes 2 seconds. T sound from the jet emitted at A reaches B 2 seconds after the jet reaches B. Hence the jet is travelling back in time.

This is the essence of your argumentation.

no, it isn't. i never at any point suggested that anything traveled backwards in time. and in your analogy the jet is traveling faster than sound. which means that yes, the jet arrives in time to hear it's own arrival. that's the problem. that's fine for sound. but for light it's a huge problem.

No. When you actually see the ship leaving it has already been travelling for 2 million years. You can conclude that the launch occurred 2 million years ago when you see the light because you know the launch occurs 2 million ly away. The ship will take 2 million years + change to arrive at the Earth from the moment it was launched. By the time you see the launch, it is therefore just the change left before it arrives


Not also as the first statement was incorrect. See above.

explain what we see as we observe the ship launch from its planet. and its subsequent journey. right? we're watching the aliens get on the ship, and we watch the ship take off. it we observe the ship travel at near c. we track the ship across space as it makes its journey. how long does it take for us to watch the entire trip and its arrival on earth?

the fact that we can deduce that the ship in actual fact departed 2 million years ago is irrelevant to this question.

 

[PeroK]

yi'll ask a different way. is it correct that as we observe the ship as it launches from its planet, that it will take two million years to complete its journey to earth?

Yes, the journey time (as measured on Earth) is 2 million years.

secondly. is it also correct that since the ship has already been traveling at near c for the past 2 million years, that by the time we observe its launch it is essentially already here?

Yes.

 

[Ibix]

right. so if we're going to be sat here watching the ship travel for the next two million years

We aren't.

The ship sets off at time zero. It emits a light pulse. That pulse arrives at Earth at time two million years.

Half way here the ship emits another light pulse. It does this at time one million years and five minutes and the light takes one million years to reach Earth, so arrives at two million years plus five minutes.

When it arrives the ship emits another light pulse. It does this at time two million years and ten minutes and the light arrived instantaneously because it has zero distance to travel.

So we conclude that the trip took two million years and ten minutes, but all the light arrived at Earth in the last ten minutes.

What part of this are you not comprehending?

 

[Somoth Ergai]

Yes, the journey time (as measured on Earth) is 2 million years.

Yes.

well, you and Orodruin seem to be at odds as he says i am wrong on both counts. so which is it?

that being said, this is why there is a paradox. if the travel time as measured by earth is two million years, yet the ship is already here, that means the rest of the light from their journey has yet to arrive.

We aren't.

The ship sets off at time zero. It emits a light pulse. That pulse arrives at Earth at time two million years.

Half way here the ship emits another light pulse. It does this at time one million years and five minutes and the light takes one million years to reach Earth, so arrives at two million years plus five minutes.

When it arrives the ship emits another light pulse. It does this at time two million years and ten minutes and the light arrived instantaneously because it has zero distance to travel.

So we conclude that the trip took two million years and ten minutes, but all the light arrived at Earth in the last ten minutes.

What part of this are you not comprehending?

the part where from our perspective on earth a 2 million light year trip that takes place over the course of two million years, actually appears to only take ten minutes.

again. we observe the ship at launch yes? we continue to watch the ship as it travels accross space. the issue of where the ship actually is, is irrelevant to this point. the issue is the apparent travel time. if we watch the ship launch and from launch to arrival it travels at near c, it must necessarily take 2 million years for the trip to complete. what part of that are you not comprehending?

 

[Baluncore]

that being said, this is why there is a paradox.

There is no paradox. There is only the misunderstanding of the science, by a beginner.

This is an attitude problem. Show some respect to science. When things do not make sense, you should be able to identify your misunderstanding, by studying the subject to which you are being introduced.

 

[PeroK]

again. we observe the ship at launch yes? we continue to watch the ship as it travels accross space. the issue of where the ship actually is, is irrelevant to this point. the issue is the apparent travel time. if we watch the ship launch and from launch to arrival it travels at near c, it must necessarily take 2 million years for the trip to complete. what part of that are you not comprehending?

By your method of calculation the speed of light must be infinite. There is no delay between the light from the launch event in Andromeda and that light being observed on Earth. The delay for the ship is a few minutes, say, but there is no delay for the light itself. The speed of light, therefore, by your calculation is distance/zero, which is either infinite or undefined.

After all, if the light from the launch event got here before the ship, then the light must have been travelling faster than the ship.

So, if the ship is travelling faster than light, then the light is travelling faster than light. And, that is a paradox!

 

[Somoth Ergai]

By your method of calculation the speed of light must be infinite. There is no delay between the light from the launch event in Andromeda and that light being observed on Earth. The delay for the ship is a few minutes, say, but there is no delay for the light itself. The speed of light, therefore, by your calculation is distance/zero, which is either infinite or undefined.

After all, if the light from the launch event got here before the ship, then the light must have been travelling faster than the ship.

So, if the ship is travelling faster than light, then the light is travelling faster than light. And, that is a paradox!

That's not what I'm saying. I'm acknowledging that there is a very clear difference between the "actual" launch event of the ship and the time we "see" the event. by the time we see the event happen the ship theoretically should have very nearly completed it's journey since the actual event took place 2 million years ago. I do not see how I have implied that light speed should be infinite in anything I have said. In fact I have very specifically said the light must take two million years to reach us as the actual event of the launch must have happened 2 million years ago.

my issue is this. say we mark the moment we see the launch as t = 0. we continue observing the spaceship from second to second. so the next second of the journey of the spaceship will be t = 1 and so on until the ship reaches us.

what i am saying is that when the trip finally concludes the final time of t will be two million years. plus the other ten minutes or whatever to account for the slight difference between the ship speed and c.

one person says this is a correct interpretation and another says it isn't. my current, albeit amature, knowledge of physics and intuition tells me this has to be correct. if it isn't, I am failing to understand how or why. if it is, I don't see how it does not lead to a paradox

 

[Somoth Ergai]

There is no paradox. There is only the misunderstanding of the science, by a beginner.

This is an attitude problem. Show some respect to science. When things do not make sense, you should be able to identify your misunderstanding, by studying the subject to which you are being introduced.

I did not begin with the attitude. I give what I am given. And i will not ever veer from that. I would much prefer to simply discuss the science and my attempt to understand it. That being said if you would like to add your own explanation for why i'm wrong I would like to hear it.

 

[Orodruin]

no, it isn't. i never at any point suggested that anything traveled backwards in time. and in your analogy the jet is traveling faster than sound. which means that yes, the jet arrives in time to hear it's own arrival. that's the problem. that's fine for sound. but for light it's a huge problem.

The mach 2 was simply to point out what would be the conclusion in a case where you apply your reasoning to a case where the speed is larger than the signal speed. The jet is travelling back in time as much as the 2 Mly journey takes 10 minutes (ie, not at all).

Let the jet travel at 0.99 mach and you have the exact analogy.

Yes, the journey time (as measured on Earth) is 2 million years.

I believe he is using ”observed” as sees. The journey time is 2 million years, but it will not take 2 million years from when the light from the launch reaches the observer until arrival.

well, you and Orodruin seem to be at odds as he says i am wrong on both counts. so which is it?

We are not. We are answering two different questions as your question is inherently ambiguous. This may be why you are confusing yourself.

My no to the second question was because of the formulation with ”also” conditioning the answer on the first reply being yes, which it is not.

actually appears to only take ten minutes.

no it doesn’t. Not if you properly account for light travel time. We agreed that there was no paradox in the jet fighter arriving to hear its own sound. The resolution here is similar.

if we watch the ship launch and from launch to arrival it travels at near c, it must necessarily take 2 million years for the trip to complete. what part of that are you not comprehending?

The problem is that you are not comprehending why you are simply incorrect. Part of it may be due to the inpreciseness of regular language and the misunderstandings it leads to. Realize that you are conversing with people who actually understand the theory on a level which id not just descriptive with words, but know the actual underlying math.

 

[Somoth Ergai]

The problem is that you are not comprehending why you are simply incorrect.

This is entirely accurate. I am not getting how i'm wrong. Even though I know I am. your saying that, from our perspective, as we watch this ship travel the distance from a planet 2 million light years away, all the way to earth, that it's apparent travel time will be 10 minutes. that necessarily means we would have to observe the ship traveling many times faster than light in order to cover that distance in that amount of time. You're absolutely correct that i'm not able to get my head around how that could be.

i'm genuinely asking. how is that possible?

 

[Motore]

Let's try it like this:
You are on the earth and watching in the direction of the Andromeda galaxy. At time t=0, you see (actually see) a departure of the ship from Andromeda as the light when the ship left Andromeda reaches you. The first conclusion should be: ok, I see the ship left Adromeda, so it must have left (here you account for light travel time) 2 million years ago according to me (you know Andromeda is 2 million light years away). You turn your gaze away and back again and at t=10 mins you see (actually see) the ship next to you. So you conclude that it arrived to earth 10 mins after departure, and you already know that the departure happened 2 million years ago, so the time it took the ship to travel from Andromeda to you is 2 million years + 10 minutes. So you have the time and the distance traveled so you can calculate the speed: 2 million light years/(2 million years + 10 minutes) = 0.9999999905 c
Is everything till here clear?

 

[Bandersnatch]

Three was a link supplied earlier, to the concept of 'superluminal motion' (in astronomy it means a very specific type of observables - rather than being a simple synonym for going faster than light). It's almost exactly the scenario at hand, and well worth working through.
It is concerned with looking at jets emitted from active galactic nuclei and resolving the issue of them appearing to travel over the speed limit. The main difference from the spaceship scenario is that we're looking at the sideways component of the motion (because that's what's actually measurable). The spaceship is in this sense a purer version, since we assume we can see it in all detail and measure its exact radial distance as it travels.
But to reiterate: this happens in nature; it's not just a matter of thought experiments that one may be worried to have not constructed properly.
The resolution is the same in both cases - and involves taking account of signals from an approaching source being received in shortened intervals. I.e. you will 'see' the spaceship cover the entirety of the distance in extremely short time, and you will observe processes on board speed up (unrelated to relativistic effects!), and if you were to take a stopwatch and try to ascertain its speed by dividing the distance you see it cover by the time on your stopwatch - you'd end up with superluminal velocities. But it would be a mistake of omission. Of ignoring the fact that the spaceship, or the jet, 'chases' its earlier signals (as seen from Earth), bunching them all up so that its entire history arrives to be observed almost at the same time.