New Paradox Discovered, I Think

[Orodruin]

To attempt the sound analogy again: a dragster race is about 300 m long (1000 foot). It takes about 3.6 s to complete it. You start at the finish line and close your eyes. Between the sound of the tires screeching and engines reving to indicate the race has started and the feeling of the dragsters swooshing past you, it takes about 2.7 s. Why? If you had opened your eyes you would have seen the dragsters take off 0.9 s before you heard the sound. The only difference with your example when exchanging sound for light and hear for see is that in the case of light there is no signal that travels faster - so you cannot really do the equivalent of opening your eyes. What you can do is to realize that the sound/light has a finite speed and account for it. When you hear the dragsters starting, you know that it happened 0.9 s ago. When you see the spaceship launch, you know that it happened 2 million years ago.

 

[pbuk]

i'm genuinely asking. how is that possible?

1. A ship takes off from a planet 2 million light years away.
2. The ship accelerates towards us in a negligible amount of time to a speed of very nearly $c$.
3. We see the ship take off.
4. We know that the light from the ship taking off travelled at $c$ (* see note).
5. We know that the planet is 2 million light years away.
6. We therefore know that the light from the take off was travelling for 2 million years.
7. We see the ship arrive here, very shortly after we see it taking off.
8. We therefore know that the ship took a little longer to travel here than the light did - a little more than 2 million years.
9. So we can measure the average speed of the ship by noting that it travelled 2 million light years in a little more than 2 million years, so its average speed was a little less than $c$.

* $c$ is the speed of light in a vacuum; we are ignoring the fact that space is not a vacuum.

 

[pinball1970]

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.

The experts tried to explain it. I read your last few posts, can I ask you a few questions?

When you are looking out into space do you realize you are effectively looking back in time? if you look at the moon you are seeing it was it was just over a second ago, the sun 8 minutes ago, Proxima Centauri 4,2 years ago?

The light you can see from Andromeda was emitted from stars 2 million years ago, you are effectively looking at the past 2 million years ago, today.

If you see the ship launch, that event happened when there were no humans on the earth. The ship took off 2 million years ago and is landing around now, it has been flying through space that whole time.

 

[hutchphd]

The OP has been answered.

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.

As a scientist, why would you talk about other than the "actual" launch event??That would be foolish.

That being said if you would like to add your own explanation for why i'm wrong I would like to hear it.

Because you are addressing scientists in a science forum. QED. finis.

 

[Tomas Vencl]

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?

Take into account the Doppler effect. Your source (ship) is approaching the Earth, so the signals you see (watches on the ship) you see much faster.
https://en.wikipedia.org/wiki/Doppler_effect

 

[Dale]

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.

from our perspective watching the ship leave their planet and journey here that trip is going to take 2 million years to complete

There are a lot of replies already, so someone else may have mentioned it. But this is not correct. You need to actually calculate it and not just make a claim about it.

 

[Ibix]

we observe the ship at launch yes?

We see it two million years after it actually launched, yes.

we continue to watch the ship as it travels accross space.

For ten minutes, yes.

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.

The trip takes two million years; the light arrives at Earth in a ten minute interval, because light emitted later in the trip has less distance to travel so arrives with less delay. As I explained in the post you quoted, you will see light from the launch at some time, light from the half way point five minutes later, and light from the arrival five minutes after that. There's no paradox and we see nothing out of sequence.

I would much prefer to simply discuss the science and my attempt to understand it.

Your error has been explained several times by seven different people, if I count correctly. It is this: we calculate the speed by dividing the distance (two million light years) by the travel time (two million years and ten minutes). We do not divide by the period over which light arrives at us (ten minutes), because this would imply we believe the travel starts at the time the light reaches us, despite knowing that we're looking at something that happened two million years ago at the beginning of the observation and something that's happening right here and now at the end.

 

[Somoth Ergai]

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.

Thank you. This is the crux of my misunderstanding. The problem I was having is not understanding that the ship will appear to travel much faster than light speed as we watch it cover the distance. I acknowledge that this is the case according to the experts. what i'm still struggling with is understanding why. I understand that the light from the ship will take less and less time to reach us as the ship gets closer. but what i'm still failing to get is how that translates to the ship appearing to us to travel not just faster than light but orders of magnitude faster.

Your error has been explained several times by seven different people, if I count correctly. It is this: we calculate the speed by dividing the distance (two million light years) by the travel time (two million years and ten minutes). We do not divide by the period over which light arrives at us (ten minutes), because this would imply we believe the travel starts at the time the light reaches us, despite knowing that we're looking at something that happened two million years ago at the beginning of the observation and something that's happening right here and now at the end.

No. My error was in failing to understand how the apparent trip will take an extremely short time to complete from an observational standpoint. not in misunderstanding the actual speed of the ship or it's actual travel time. If seven people all fail to adequately address the actual problem i'm having that's their problem. And should perhaps try a little harder to comprehend my issue than coming at with an unnecessary attitude.

The experts tried to explain it. I read your last few posts, can I ask you a few questions?

When you are looking out into space do you realize you are effectively looking back in time? if you look at the moon you are seeing it was it was just over a second ago, the sun 8 minutes ago, Proxima Centauri 4,2 years ago?

The light you can see from Andromeda was emitted from stars 2 million years ago, you are effectively looking at the past 2 million years ago, today.

If you see the ship launch, that event happened when there were no humans on the earth. The ship took off 2 million years ago and is landing around now, it has been flying through space that whole time.

If you had read my initial post you would not need to ask me those questions. If you had understood my issue you would see that my error is in thinking our observation of the ship will take 2 million years to complete. this is the entire problem with my thought experiment that nearly everyone save for a few seems to be overlooking, and the reason why I was having a hard time seeing where i was going wrong.

 

[hutchphd]

And should perhaps try a little harder to comprehend my issue than coming at with an unnecessary attitude.

With respect, your thread is titled "New Paradox Discovered, I think" not "Confused about Time Dilation" and you seem(ed) very resistant to contradiction and clarification. FYI.

 

[PeterDonis]

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

No. This is not correct.

What is correct, as @Ibix pointed out in post #3, is that you will see the light from the entire journey of the ship, from launch to arrival, arrive at your location in a much, much shorter time interval than 2 million years. The light signals all still arrive in order--you see the launch, then the journey, then the arrival, all in their proper order. The ship arrives just after the last light signal emitted during its journey.

[Edit: The following is incorrect; see corrected calculation in post #57.]
How much shorter is the time interval in which you see all this? The relativistic Doppler formula tells us that For $v = 0.99999999999$ (in units where $c = 1$), the formula gives for the Doppler factor $f$:

$$
f = \sqrt{\frac{1 + v}{1 - v}} = \sqrt{\frac{1 + 0.99999999999}{1 - 0.99999999999}} = 447214
$$

The time interval $\tau$ during which you see the light signals all arrive is then the journey time $T$ divided by $f$, i.e., for $T$ 2 million years, we have

$$
\tau = \frac{T}{f} = \frac{2000000}{447214} = 4.47
$$

So you will see the light from the launch at Andromeda 4.47 years before the spaceship arrives, and during that 4.47 years you will see all of the light signals emitted by the ship during its journey, in order.

 

[Orodruin]

If seven people all fail to adequately address the actual problem i'm having that's their problem. And should perhaps try a little harder to comprehend my issue than coming at with an unnecessary attitude.

If seven different people have explained it in seven different ways and you don’t get it … there is only one thing that is exactly the same across all of those explanation attempts …

If you had read my initial post you would not need to ask me those questions. If you had understood my issue you would see that my error is in thinking our observation of the ship will take 2 million years to complete.

Have you stopped for two minutes to consider that if so many people who actually know relativity don’t understand your issue, perhaps you did not do such a good job at presenting it as you think you did?

With respect, your thread is titled "New Paradox Discovered, I think" not "Confused about Time Dilation" and you seem(ed) very resistant to contradiction and clarification. FYI.

This! The thread title is confrontative by nature. If OP wanted to learn what was actually wrong, they should have said so rather than claiming to have discovered something fundamentally wrong with relativity (because that is the purpose of claiming a paradox).

 

[PeterDonis]

the light from the launch would arrive about a week before the ship

Actually, it's about 4.47 years. Remember you have to take the square root in the relativistic Doppler formula. See my post #45.

 

[PeterDonis]

how that translates to the ship appearing to us to travel not just faster than light but orders of magnitude faster

"Travel faster than light" is something of a misnomer. In post #45, I showed you the calculations that say that you see the ship leaving Andromeda 4.47 years before it arrives at Earth, and that the ship never outruns any of its light signals. That in itself should be enough to show that there is no "paradox".

If you divide 2 million by 4.47 you get a speed which is much faster than $c$. But calling that "traveling much faster than light" is wrong, and even calling it "appearing to travel much faster than light" is highly problematic.

 

[Dale]

what i'm still struggling with is understanding why. I understand that the light from the ship will take less and less time to reach us as the ship gets closer.

That is the reason why.

what i'm still failing to get is how that translates to the ship appearing to us to travel not just faster than light but orders of magnitude faster

This is called the relativistic Doppler shift, or in cases like this, relativistic blueshift.

If seven people all fail to adequately address the actual problem i'm having that's their problem.

No. if one person misunderstood then it would be reasonable to say “that was their problem”. When seven different experts all fail to address your problem then that is your problem. Your mistake in the analysis was correctly identified in post 3.

 

[PeterDonis]

Your mistake in the analysis was correctly identified in post 4.

Actually it was post #3 by @Ibix.

 

[Somoth Ergai]

No. This is not correct.

What is correct is that you will see the light from the entire journey of the ship, from launch to arrival, arrive at your location in a much, much shorter time interval than 2 million years. The light signals all still arrive in order--you see the launch, then the journey, then the arrival, all in their proper order. The ship arrives just after the last light signal emitted during its journey.

How much shorter is the time interval in which you see all this? The relativistic Doppler formula tells us that For $v = 0.99999999999$ (in units where $c = 1$), the formula gives for the Doppler factor $f$:

$$
f = \sqrt{\frac{1 + v}{1 - v}} = \sqrt{\frac{1 + 0.99999999999}{1 - 0.99999999999}} = 447214
$$

The time interval $\tau$ during which you see the light signals all arrive is then the journey time $T$ divided by $f$, i.e., for $T$ 2 million years, we have

$$
\tau = \frac{T}{f} = \frac{2000000}{447214} = 4.47
$$

So you will see the light from the launch at Andromeda 4.47 years before the spaceship arrives, and during that 4.47 years you will see all of the light signals emitted by the ship during its journey, in order.

OK, I think I think I'm starting to understand. The issue i'm having is wrapping my head around how it could be that we could see an object which is in actual fact traveling slower than c appear to be traveling faster than c. I understand that you're telling me this is the case. I'm needing help understanding why though. conceptually I mean.

 

[Dale]

Actually it was post #3 by @Ibix.

Oops, yes it was.

The OP’s response was (IMO) unacceptable

we would have to see light traveling faster than itself. This does not resolve the situation

”I don’t understand” or “wouldn’t this mean” or “can you show me how” would have all been more productive

 

[Ibix]

Actually, it's about 4.47 years. Remember you have to take the square root in the relativistic Doppler formula. See my post #45.

I think we're calculating different things. In the Earth frame the flight time of light is 2 million years; the flight time of the ship is approximately one part in $10^{11}$ longer. Thus the arrival time of the ship is one part in $10^{11}$ of 2 million years after the light - about ten minutes. The Doppler analysis invokes ticks of the shipboard clock, which I'm not doing.

 

[Orodruin]

I think we're calculating different things. In the Earth frame the flight time of light is 2 million years; the flight time of the ship is approximately one part in $10^{11}$ longer. Thus the arrival time of the ship is one part in $10^{11}$ of 2 million years after the light - about ten minutes. The Doppler analysis invokes ticks of the shipboard clock, which I'm not doing.

To illuminate the difference: The Doppler computation results in how much faster the clock on the ships looks to be running. The result of this includes the time dilation of the rocket. The Doppler factor tells us the ratio between the elapsed rocket clock time and the observation time - not the ratio between the 2 million year travel in the Earth rest frame and the observation time. The former ratio will be much smaller as time dilation at the quoted speed is significant.

Assuming the travel distance in the given frame to be L = 2 million years, the computation for the arrival time difference is simply L/v - L/c = L(1-v/c)/v.

 

[PeterDonis]

how it could be that we could see an object which is in actual fact traveling slower than c appear to be traveling faster than c. I understand that you're telling me this is the case.

Not necessarily. I'm telling you how it can be the case that you see light signals arriving from the ship in a much shorter time interval than they were emitted (where "time interval" here is in your frame, not the ship's frame). But, as I pointed out in post #48, interpreting that observation as the ship "appearing to be traveling much faster than light" is highly problematic.

 

[PeterDonis]

I think we're calculating different things.

No, we're not. We are both calculating the same invariant: the time interval registered by the observer's clock on Earth between seeing the light signal showing the ship departing Andromeda, and the ship arriving at Earth. I say that interval is 4.47 years. You say it's about a week. We can't both be right.

 

[PeterDonis]

The Doppler analysis invokes ticks of the shipboard clock

Oops, yes, I see where I went wrong. The Doppler factor has to be applied to the shipboard elapsed time, not the Earth elapsed time. Fortunately it's easy to fix that: for $v = 0.99999999999$ we have $\gamma = 1 / \sqrt{1 - v^2} = 223607$. So the shipboard elapsed time is $2000000 / \gamma = 8.94$ years. And dividing that by the Doppler factor $f = 447214$ gives $2 \times 10^{-5}$ years.

As an extra check, we can note that the difference in arrival times at Earth, calculated by your method, is the distance times a factor $(1 / v) - 1$; this factor is, as you say, $10^{-11}$. Multiplying this by $2 \times 10^6$ gives $2 \times 10^{-5}$ years.

Note that this is not even a week; it's $631$ seconds, or about $10.5$ minutes.

 

[Somoth Ergai]

Not necessarily. I'm telling you how it can be the case that you see light signals arriving from the ship in a much shorter time interval than they were emitted (where "time interval" here is in your frame, not the ship's frame). But, as I pointed out in post #48, interpreting that observation as the ship "appearing to be traveling much faster than light" is highly problematic.

Could you please explain how we could see all of the light from the ships journey in a short time frame. in chronological order. and it not have the appearance of traveling faster than light? I'm beginning to wrap my head around the mechanism of the doppler effect. but I don't see how that does not translate to the object appearing to move faster than it actually is moving.

if we observe the ships journey from start to finish and we track it's movement across space, and the time of travel we observe from earth is less than the time it would take light to travel that distance. how is that not faster than light?

I promise i'm not actually trying to be difficult. it just doesn't make sense to me.

 

[pervect]

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

You already noted that the light is faster than the ship. Thus, the light should reach the Earth first, followed by the ship. This comes simply from the relation velocity = distance * time, or time = distance / velocity. As of yet, this doesn't even involve relativity.

You can put specific numbers in the formula if you like. It's unclear why you think there is a contradiction here. Really, all we can suggest is that you apply some math, do some calculations, and present your argument a bit more rigorously. "Show your work" would be the short answer.

 

[Ibix]

Note that this is not even a week; it's $631$ seconds, or about $10.5$ minutes.

Yes - as noted above, I initially used 2 billion light years and got a week, which turns out to be 10,080 minutes. 10.5 minutes is the correct figure.

 

[PeterDonis]

Could you please explain how we could see all of the light from the ships journey in a short time frame. in chronological order. and it not have the appearance of traveling faster than light?

What does "appearance of traveling faster than light" mean?

The light signals that you see don't tell you how far away the ship was when they were emitted. So you can't calculate a speed just from the light signals alone. You have to bring in other information. What other information would you use, and why would it lead you to describe what is going on as "appearance of traveling faster than light"?

In other words, as things stand now, nobody has anything to explain, because nobody has shown why "appearance of traveling faster than light" is even a valid description to begin with. We have to have some argument for why that is even a valid description first.

if we observe the ships journey from start to finish and we track it's movement across space

How do you "track its movement across space"? As above, the light you see doesn't have any distance information in it.

the time of travel we observe from earth is less than the time it would take light to travel that distance

But it isn't. It can't be, because the light arrives before the ship. So the ship's time of travel is longer than the light's time of travel.

 

[Vanadium 50]

Have you stopped for two minutes

Interestingly, that's about the time between a message being posted and the OP's reply, and therefore an upper bound on how long he spends thinking about these messages.

Honestly, I would drop relativity completely. The same conceptual "paradox" exists with a fast-moving car beeping its horn every second or someone pitching baseballs out a car.

 

[Orodruin]

Honestly, I would drop relativity completely. The same conceptual "paradox" exists with a fast-moving car beeping its horn every second or someone pitching baseballs out a car.

I did bring up the sound analogy …

 

[Vanadium 50]

I know, but I suggest we get this straight first, before overturning a century of physics.

 

[Bandersnatch]

If you divide 2 million by 4.47 you get a speed which is much faster than . But calling that "traveling much faster than light" is wrong, and even calling it "appearing to travel much faster than light" is highly problematic.

And yet, that's almost verbatim how it's called in astronomy, in the context of AGN jets. Because that's how they look like at a first glance. No need to bash the OP for using the same nomenclature as textbooks do, as long as they leave with an understanding that this is isn't really what it says on the tin.

 

[Somoth Ergai]

How do you "track its movement across space"? As above, the light you see doesn't have any distance information in it.

Doesn't it? I'm sorry I don't understand how that statement could be accurate. the only way we can tell where anything is at all is by measuring the light emitted from it against it's surroundings. How else do we even know how far away the planet is?

Are you saying that objects moving very fast don't have distance information while slower moving objects do? Again, i'm sorry but I'm not understanding your point.

What does "appearance of traveling faster than light" mean?

What I mean is, 1) if we know how far away the planet they are departing from is (2 million light years). and 2) we can determine how far away the back ground stars are. And 3) we can can follow the ship as it makes it's journey. then we should be able to "see" how quickly the light emitted from the traveling ship
"appears" to cross the distance of space and calculate the velocity. I believe this is the same way we determine how quickly anything else is moving in space.

It follows then, necessarily, that 1) a photon crossing that distance (2 million light years) would take two million years from point A (their planet) to point B (Earth). And 2) that if the time it takes the ship to cross that distance, again from our perspective, is less than the time it would have taken a photon to cross that distance then the ship must, as a matter of logical non contradiction, "appear" to cross the distance faster than light.

I believe I am beginning to understand how the doppler effect makes this true, but I'm not fully there yet. however if you're saying that this is not the case then I am still quite lost.

 

[PeterDonis]

that's almost verbatim how it's called in astronomy, in the context of AGN jets.

Yes, I know the terminology is used; I'm just saying that one has to be very careful in interpreting what it means. Scientists often use language to communicate with each other that, for them, is not a problem because they all understand the actual physics and don't get misled--but which can still be very misleading to a lay person who is not aware of all the complications lurking underneath what seems like a simple use of terminology.

 

[PeterDonis]

How else do we even know how far away the planet is?

How do we know how far away the Andromeda galaxy is? Do we know that just by looking at the light coming from that galaxy?

The answer is no, we don't. We have to calculate how far away it is by combining the information in the light we see from it with other information from other sources. In the case of the Andromeda galaxy, we have to measure the periods of Cepheid variable stars we see in that galaxy, and then use a relationship for Cepheid variables between the period and the luminosity, which we have to obtain by observing Cepheid variable stars in our own galaxy, to calculate the luminosity of the Cepheids we see in the Andromeda galaxy. Then we combine that with the apparent brightness of those Cepheids to calculate their distance.

This is the sort of thing I was referring to when I said the light we see, by itself, doesn't tell us distances (or times). So we can't just "track the distance" by looking at the light. We have to already know the distances some other way and then calculate things.

we can determine how far away the back ground stars are

What "background stars" are you talking about? The ship is coming straight from Andromeda to us. The only "background" we are seeing it against is the Andromeda galaxy itself. We certainly don't have "background stars" placed at convenient intervals all along the way that send us a signal when the ship passes them.

I don't think you have thought through very carefully what we would actually be observing in the scenario you have described.

a photon crossing that distance (2 million light years) would take two million years from point A (their planet) to point B (Earth)

In the Earth's rest frame, yes. But note that this statement is only true in the Earth's rest frame.

the time it takes the ship to cross that distance, again from our perspective, is less than the time it would have taken a photon to cross that distance

But it isn't. You have already been told this, repeatedly. But let me restate it again.

Call the time in the Earth's frame when the ship leaves Andromeda time zero. A light signal is emitted from the ship towards Earth at the same instant.

That light signal arrives at Earth at time 2 million years.

The ship itself arrives at Earth at time 2 million years plus about 10.5 minutes (per the calculations done in earlier posts).

In other words, the ship arrives after the light. So it can't possibly take less time than the light takes to travel the same distance. If it did, the ship would arrive before the light, not after it.

Now let's talk about the "speed much faster than light" calculation. How is that calculated? It's calculated by taking the distance of 2 million years and dividing it by 10.5 minutes, i.e., the time elapsed on Earth between the light arriving and the ship arriving. But this same calculation would say that the light itself travels with infinite speed, since the time between the light arriving and the light arriving is, well, zero. So here again the ship does not travel faster than the light does. It travels faster than $c$ by this calculation, but the light travels faster than $c$ too--in fact infinitely faster. This is why I say that describing this as "the ship appearing to travel much faster than light" is misleading--because you would then have to describe the light itself as traveling infinitely faster than light.

 

[Dale]

It follows then, necessarily, that 1) a photon crossing that distance (2 million light years) would take two million years from point A (their planet) to point B (Earth). And 2) that if the time it takes the ship to cross that distance, again from our perspective, is less than the time it would have taken a photon to cross that distance then the ship must, as a matter of logical non contradiction, "appear" to cross the distance faster than light.

You said “that distance” 3 times and “the distance” once. In this problem there is no one unique distance. There are varying distances ranging from 2 million light years to 0. There is light that travels all of those varying distances. I think a big part of the problem is that you are fixated on a single distance in a problem whose distance varies.

 

[Chicken Squirr-El]

Take a look at these spacetime diagrams of the twin paradox from wikipedia:
https://upload.wikimedia.org/wikipedia/commons/a/a2/Rstd4.gif

The top half of the right image is the ship traveling towards you in your frame of reference. The bunched up blue lines at the top are what you see through your telescope from launch to arrival and demonstrates why you see all the light signals of the entire trip during a short interval at the very end.