Disagreement on the interpretation of the Andromeda paradox

[J.C.MacSwell]

You do understand that motion is relative, right? You do realize that earthling fixed on Earth moving wrt Andromeda is the same thing as earthling moving wrt. Earth fixed wrt Andromeda.

The amusing thing is that you insist on the latter, when, in fact, Andromeda is moving wrt the Earth.

Sorry, but you are not getting it. We all realize that motion is relative.

 

Forget that I (where did you get that?), or anyone else, insist (again, we don't... it doesn't matter) that Earth is fixed wrt Andromeda. The paradox (and, unlike the twin pseudo paradox, it is a paradox, notwithstanding it is outside and beyond causal/temporal distance) holds for the true/apparent relative velocities of Andromeda and Earth. It is about subtle, seemingly insignificant, changes in velocity of an extremely remote observer/calculator, and how it effects his sense/calculation of what is a simultaneous event at that considerable distance....and it has nothing to do with the relative velocities of Earth and Andromeda.

[xyzt]

Sorry, but you are not getting it. We all realize that motion is relative.

 

Forget that I (where did you get that?), or anyone else, insist (again, we don't... it doesn't matter) that Earth is fixed wrt Andromeda. The paradox (and, unlike the twin pseudo paradox, it is a paradox, notwithstanding it is outside and beyond causal/temporal distance) holds for the true/apparent relative velocities of Andromeda and Earth. It is about subtle, seemingly insignificant, changes in velocity of an extremely remote observer/calculator, and how it effects his sense/calculation of what is a simultaneous event at that considerable distance....and it has nothing to do with the relative velocities of Earth and Andromeda.

For people who are very fixated on a dogmatic interpretation of the paradox, I wrote post 27. In post 2 I presented the Wolfram variant, it seems that it riled a lot of people that refused to accept it as legit. <shrug>

No. What you're describing is true, and is down to the same effect as the Andromeda paradox,

 

Good, can I recommend that you now read post 27? It gives you two different facets of the paradox. Wolfram used a somewhat unorthodox interpretation, perfectly correct nevertheless (you still don't understand it but this is ok). The whole thing is truly trivial, it is about the influence of large cosmological distance ([math]x[/math]) in the way distant observers (two or only one, according to Wolfram) evaluate the temporal separation [math]\gamma vx/c^2[/math].

Edited June 8, 2013 by xyzt

[Delta1212]

For people who are very fixated on a dogmatic interpretation of the paradox, I wrote post 27. In post 2 I presented the Wolfram variant, it seems that it riled a lot of people that refused to accept it as legit. <shrug>

 

 

Good, can I recommend that you now read post 27? It gives you two different facets of the paradox. Wolfram used a somewhat unorthodox interpretation, perfectly correct nevertheless (you still don't understand it but this is ok). The whole thing is truly trivial, it is about the influence of large cosmological distance ([math]x[/math]) in the way distant observers (two or only one, according to Wolfram) evaluate the temporal separation [math]\gamma vx/c^2[/math].

Perhaps it would help if you could link to an example that doesn't use Wolfram's unorthodox approach?

[D H]

This thread needs to be split into two different threads, one containing the original post and perhaps one or two other posts that address the Andromeda paradox. All the rest need to be put into a new thread, "Misinterpretation of the Andromeda paradox".

 

xyzt, you have received negative rep (none from me) because you have derailed this thread with your misinterpretation of the paradox and because of your intransigence.

[xyzt]

This thread needs to be split into two different threads, one containing the original post and perhaps one or two other posts that address the Andromeda paradox. All the rest need to be put into a new thread, "Misinterpretation of the Andromeda paradox".

 

xyzt, you have received negative rep (none from me) because you have derailed this thread with your misinterpretation of the paradox and because of your intransigence.

I am used to negative feedback from people with a limited understanding of the issues, relativity is one of the disciplines that lends itself to such misinterpretations very often. As to your claim of "Misinterpretations of the Andromeda Paradox", I would like, once more , to direct your attention to the Wolfram webpage, generally they are very good at what they are doing. The fact that they use a somewhat unorthodox version doesn't mean that they are incorrect, it only means that the rest of the posters in this thread refuse to make an attempt at understanding the underlying physics. In order to help such people, I created post 27.

 

Perhaps it would help if you could link to an example that doesn't use Wolfram's unorthodox approach?

I already did that in post 27, I explained both variants of the paradox for you and the others, I have directed you to reading it, twice.

Edited June 8, 2013 by xyzt

[swansont]

!

Moderator Note

Split from http://www.scienceforums.net/topic/75525-andromeda-paradox/

You can stop negging my posts, if you do not understand the basics, just ask.

I am used to negative feedback from people with a limited understanding of the issues, relativity is one of the disciplines that lends itself to such misinterpretations very often.

!

Moderator Note

I can't speak for the people who have given you negative reputation, but it seems clear that it correlates with your smug and belittling attitude as presented here, which fall a bit short on the civility scale. If you're going to repeatedly complain about being neg-repped, you might try stepping up your game in that regard.

[D H]

Emphasis mine:

If you plug in the velocity of a person walking right past this stationary Earth observer, the velocity will be that of the person working, and [math]t_a[/math] will give you the time on Andromeda that this walking observer measures to be simultaneous to Earth's "now," a time that will be in advance of the time that the stationary observer measures.

The Andromeda paradox is a demonstration of the fact that observers that are very close together moving at only slightly different speeds can have a significant difference in the time they measure to be simultaneous with "now" at very large distances. So that someone walking past you in the street may believe that the events going on "now" in Andromeda are days off from the events that you believe are going on in Andromeda "now" (retrospectively since we have to wait for the light to reach us, obviously) solely because of the slight difference in your relative velocities and the very great distance from here to Andromeda.

 

 

Replace your use of "measure" with "calculate" and you'll have a good description of the paradox. I can calculate my now, but I cannot measure it. And I certainly cannot know what happens in that now.

 

Alternatively, replace your use of "observer" with "omniscient, omnipresent being" and you'll also have a good description of the paradox. The only thing that could measure my "now" at a distance of 2.5 million light years (or any distance, for that matter) is an omniscient, supernatural being.

[Delta1212]

Emphasis mine:

 

 

 

Replace your use of "measure" with "calculate" and you'll have a good description of the paradox. I can calculate my now, but I cannot measure it. And I certainly cannot know what happens in that now.

 

Alternatively, replace your use of "observer" with "omniscient, omnipresent being" and you'll also have a good description of the paradox. The only thing that could measure my "now" at a distance of 2.5 million light years (or any distance, for that matter) is an omniscient, supernatural being.

You're correct, that was an imprecisely chosen word on my part, though I did note in the second paragraph that the observation would have to be made retrospectively since it would need to wait for the light to arrive. Edited June 8, 2013 by Delta1212

[xyzt]

It isn't the best, but there is really nothing unusual about Wolfram's explanation. It doesn't make the mistake of thinking v is the velocity between earth and Andromeda. And, it does have two observers, in so far as this is two observers

That is, of course, not quite correct. The mathematical formalism contains nothing about two observers as already shown in post 2. Only in the last section do the Wolfram people talk about two observers. This is clearly explained in this post .

Edited June 8, 2013 by xyzt

[Delta1212]

That is, of course, not quite correct. The mathematical formalism contains nothing about two observers as already shown in post 2. Only in the last section do the Wolfram people talk about two observers. This is clearly explained in this post .

The formula is being used to calculate the difference in time on Andromeda that two observers calculate as being simultaneous with "now" on Earth. The time only needs to be calculated for one of the observers because Wolfram has set the time for one observer to 0, rendering the result for the other equal to the difference between the two.

 

To break it down:

[math]t_e[/math] is the time on Earth considered "now" conveniently, as they say in the link, set to 0.

 

[math]d[/math] is the distance from Andromeda to the observer.

 

[math]v[/math] is the relative velocity of the observer with Andromeda which, for simplicity's sake and so that they only have to calculate the simultaneous Andromeda time for one observer, they have set to 0 for an observer at rest with Earth. This obviously isn't true to reality, and doesn't have to be done this way, but it doesn't impact the basic paradox and makes the math simpler.

 

[math]t_a[/math] is how much further in the future on Andromeda an observer with velocity [math]v[/math] at distance from Andromeda [math]d[/math] will calculate to be simultaneous to [math]t_e[/math] than an observer also at distance [math]d[/math] but at rest with respect to Andromeda.

 

 

 

If you leave Earth and Andromeda traveling at their respective velocities (which is fine) and plug Andromeda's velocity into [math]v[/math], then you have found the difference in the time that an observer on Earth at rest with respect to Earth calculates to be simultaneous on Andromeda with "now" with the time that an observer also on Earth but at rest with respect to Andromeda calculates to be simultaneous with "now."

 

 

This is, of course, an example of the Andromeda paradox, but the paradox doesn't require such extreme speeds to work and certainly doesn't require the exact relative velocity of the Andromeda galaxy.

 

If you want to illustrate the paradox using the more typically used walking speeds without assuming Earth and Andromeda are at rest, you would need to calculate [math]t_a[/math] once for [math]v[/math] equals Andromeda's velocity for the observer standing still on Earth and once for [math]v[/math] equals Andromeda's velocity plus the walking speed of the observer, and then subtract the first from the second.

 

By setting Andromeda's speed relative to Earth to 0, they set the value of [math]t_a[/math] for the first case to 0, and since subtracting 0 from the [math]t_a[/math] calculated by the walking observer would have no effect, they can strike out all of the math from the stationary observer and find the difference between the two observers' calculated simultaneous times on Andromeda solely by calculating the time for the walking observer.

 

So yes, the Wolfram mathematical formalism does include two observers, but they've zeroed one out to simplify the math.

[xyzt]

 

 

 

This is, of course, an example of the Andromeda paradox, but the paradox doesn't require such extreme speeds to work and certainly doesn't require the exact relative velocity of the Andromeda galaxy.

Thank you for (finally) admitting it. No one said that the explanation requires "extreme speeds" (BTW: 130km/s isn't extreme). What it was said, time and again, is that the paradox requires that [math]vx[/math] is of the order of [math]c^2[/math]. See my post 2, the one that generated all the flurry of answers.

 

 

By setting Andromeda's speed relative to Earth to 0, they set the value of [math]t_a[/math] for the first case to 0, and since subtracting 0 from the [math]t_a[/math] calculated by the walking observer would have no effect, they can strike out all of the math from the stationary observer and find the difference between the two observers' calculated simultaneous times on Andromeda solely by calculating the time for the walking observer.

 

So yes, the Wolfram mathematical formalism does include two observers, but they've zeroed one out to simplify the math.

 

This is not how it is done, I have already shown you how to do this correctly:

 

Now, if one insists on having two observers, the math gets just a tad more complicated but the outcome is just the same. Say that two earthlings observe the event [math](x_a,t_a)[/math] on Andromeda. The earthlings are in relative motion with speed [math]v[/math] along what they call the [math]x-axis[/math]

 

The observers have speeds [math]v_1=0[/math] respectively [math]v_2=v[/math] wrt Andromeda, so:

 

[math]t_{e2}-t_{e1}=\gamma(t_a+v_2 x_a/c^2)-\gamma(t_a+v_1 x_a/c^2)=\gamma vx_a/c^2[/math]

Edited June 8, 2013 by xyzt

[Delta1212]

That was the best way I can explain it. If you still aren't able to grasp the mistake you're making then I'm going to have to let someone else make the attempt because apparently what I'm saying isn't getting through.

[D H]

Wrong, wrong, and wrong, xyzt. The velocity of the Andromeda galaxy with respect to us has absolutely nothing to do with the Andromeda paradox.

 

Why are you so intransigent on this? Google (or Bing, if that's your preference) is your friend. Use it. Look for other resources than that one somewhat lousy one you insist on using (and misinterpreting).

 

Just to reiterate, the velocity of the Andromeda galaxy with respect to us has absolutely nothing to do with the Andromeda paradox. Not one single thing. The Andromeda paradox is about how two co-located observers have different surfaces of simultaneity, and how those different surfaces inherently result in significantly different interpretations of "now" at very remote locations.

 

 

This post, along with the four preceding posts, do not belong in this thread. They belong in that other thread that expands on xyzt's complete misunderstanding of the Andromeda paradox.

[xyzt]

That was the best way I can explain it. If you still aren't able to grasp the mistake you're making then I'm going to have to let someone else make the attempt because apparently what I'm saying isn't getting through.

Obviously, you have explained it incorrectly and I pointed out your mistakes. On a different note: the language of physics is math. Until you learn how to express yourself in a mathematical formalism, you aren't doing physics. English composition isn't physics.

Wrong, wrong, and wrong, xyzt. The velocity of the Andromeda galaxy with respect to us has absolutely nothing to do with the Andromeda paradox.

 

Why are you so intransigent on this? Google (or Bing, if that's your preference) is your friend. Use it. Look for other resources than that one somewhat lousy one you insist on using (and misinterpreting).

 

Just to reiterate, the velocity of the Andromeda galaxy with respect to us has absolutely nothing to do with the Andromeda paradox. Not one single thing. The Andromeda paradox is about how two co-located observers have different surfaces of simultaneity, and how those different surfaces inherently result in significantly different interpretations of "now" at very remote locations.

 

 

This post, along with the four preceding posts, do not belong in this thread. They belong in that other thread that expands on xyzt's complete misunderstanding of the Andromeda paradox.

I think you are deeply confused, I post again, bolded this time, in the hope you will understand :

 

Now, if one insists on having two observers, the math gets just a tad more complicated but the outcome is just the same. Say that two earthlings observe the event [math](x_a,t_a)[/math] on Andromeda. The earthlings are in relative motion with speed [math]v[/math] along what they call the [math]x-axis[/math]

 

The observers have speeds [math]v_1=0[/math] respectively [math]v_2=v[/math] wrt Andromeda, so:

 

[math]t_{e2}-t_{e1}=\gamma(t_a+v_2 x_a/c^2)-\gamma(t_a+v_1 x_a/c^2)=\gamma vx_a/c^2[/math]

 

The math is very simple and the definitions are very precise, I do not understand why you have so much difficulty and you keep misrepresenting.

Edited June 8, 2013 by xyzt

[michel123456]

I don't understand clearly , again:

Would that mean that;

1. the 2 observers co-located would observe the same event but after making calculations would conclude point 2 ?

2. after 2,5 million years, the 2 observers non-co-located would observe different events that after calculations happened when they were co-located?

 

(edit) changed "non-co-located" instead of "non-located"

Edited June 8, 2013 by michel123456

[xyzt]

I don't understand clearly , again:

Would that mean that;

1. the 2 observers co-located would observe the same event but after making calculations would conclude point 2 ?

2. after 2,5 million years, the 2 observers non-located would observe different events that after calculations happened when they were co-located?

Explanation is very simple:

 

Say that two earthlings observe the event [math](x_a,t_a)[/math] on Andromeda. The earthlings are in relative motion with speed [math]v[/math] along what they call the [math]x-axis[/math]

 

The observers have speeds [math]v_1=0[/math] respectively [math]v_2=v[/math] wrt Andromeda, so:

 

[math]t_{e2}-t_{e1}=\gamma(t_a+v_2 x_a/c^2)-\gamma(t_a+v_1 x_a/c^2)=\gamma vx_a/c^2[/math]

[Delta1212]

I don't understand clearly , again:

Would that mean that;

1. the 2 observers co-located would observe the same event but after making calculations would conclude point 2 ?

2. after 2,5 million years, the 2 observers non-located would observe different events that after calculations happened when they were co-located?

If I understand you correctly, I believe you've got it, yes.

 

The two co-located observers will agree that "now" on Earth is the same, and they will see the same light from Andromeda, but they will disagree on what is "currently" happening in Andromeda, but which they won't be able to see for about 2.5 million years.

[J.C.MacSwell]

If I understand you correctly, I believe you've got it, yes.

 

The two co-located observers will agree that "now" on Earth is the same, and they will see the same light from Andromeda, but they will disagree on what is "currently" happening in Andromeda, but which they won't be able to see for about 2.5 million years.

...and to add to that, the paradox is that if one assumes a "now" with certain but as yet unknown events taking place on Andromeda, where the other, at the same time and location on Earth but their "now" being days earlier on Andromeda...then what does that say about uncertainty, or inevitability, wrt the future in general?

[Iggy]

This post, along with the four preceding posts, do not belong in this thread. They belong in that other thread that expands on xyzt's complete misunderstanding of the Andromeda paradox.

 

I second that. All of the following posts as well. I feel bad for the OP, or anyone else bumping into this thread, looking for an honest explanation of the Andromeda Paradox.

 

---

 

 

Now, if one insists on having two observers, the math gets just a tad more complicated but the outcome is just the same. Say that two earthlings observe the event [math](x_a,t_a)[/math] on Andromeda. The earthlings are in relative motion with speed [math]v[/math] along what they call the [math]x-axis[/math]

 

The observers have speeds [math]v_1=0[/math] respectively [math]v_2=v[/math] wrt Andromeda, so:

 

[math]t_{e2}-t_{e1}=\gamma(t_a+v_2 x_a/c^2)-\gamma(t_a+v_1 x_a/c^2)=\gamma vx_a/c^2[/math]

 

The math is very simple and the definitions are very precise, I do not understand why you have so much difficulty and you keep misrepresenting.

 

You have three variables for velocity (you should only have one). You have three variables for time (only need two). You don't say which velocity belongs in the gamma factor. Unless you explain every variable and give a sample scenario with numbers there is just no way to unravel what you are trying to do with that equation.

[D H]

...and to add to that, the paradox is that if one assumes a "now" with certain but as yet unknown events taking place on Andromeda, where the other, at the same time and location on Earth but their "now" being days earlier on Andromeda...then what does that say about uncertainty, or inevitability, wrt the future in general?

 

Nothing.

 

What the Andromeda paradox does say is that the point in space-time that I label as (r=2.5e6 light years, t=0) might be labeled differently by you, even if you and I are right next to one another. What it says with regard to free will, determinism, and all that philosophical claptrap-- not one thing.

 

Adding the Andromedan invasion fleet to the paradox is not science. The vote by the Andromedan council, the launch of the invasion fleet, and its subsequent recall -- that knowledge requires omniscience and omnipresence. Posit omniscience and omnipresence and of course we're back to square one in the age old debate of free will versus determinism. Restrict knowledge to that accessible to science and the council, the launch, and the recall all vanish. That knowledge is unknowable to science.

[Iggy]

 

Here is a very good explanation as to why the effect depends on the distance to Andromeda.

Basically, it all stems from the Lorentz transform for time:

 

[math]t_a=\gamma(t_e+vx_e/c^2)[/math]

 

Above, [math]t_e[/math] represents "when" the Andromeda effect happened as measured in the Earth frame and [math]x_e[/math] represents "where" the Andromeda effect happened as measured in the Earth frame.

[math]t_a[/math] represents "when" the Andromeda effect happened as measured in the Andromeda frame.

[math]v[/math] represents the relative speed between the Earth and Andromeda.

 

Quite a few posters have missed the fact that the above is a variant of the "Andromeda paradox", one with only one observer, NOT two. Nevertheless, the "paradox" works just the same and the above explanation is perfectly correct.

I wrote post 27. In post 2 I presented the Wolfram variant<shrug>

 

You did *not* present the 'Wolfram variant'.

 

You said this:

 

[math]v[/math] represents the relative speed between the Earth and Andromeda.

 

Wolfram said this:

 

Assume, for simplicity, that the galaxy and the Earth remain momentarily at this fixed distance, with no relative motion.

 

Those things are mutually exclusive.

[J.C.MacSwell]

Nothing.

 

What the Andromeda paradox does say is that the point in space-time that I label as (r=2.5e6 light years, t=0) might be labeled differently by you, even if you and I are right next to one another. What it says with regard to free will, determinism, and all that philosophical claptrap-- not one thing.

 

Adding the Andromedan invasion fleet to the paradox is not science. The vote by the Andromedan council, the launch of the invasion fleet, and its subsequent recall -- that knowledge requires omniscience and omnipresence. Posit omniscience and omnipresence and of course we're back to square one in the age old debate of free will versus determinism. Restrict knowledge to that accessible to science and the council, the launch, and the recall all vanish. That knowledge is unknowable to science.

 

From Wikipedia: (last line bolded mine)

Roger Penrose^[4] advanced a form of this argument that has been called the Andromeda paradox in which he points out that two people walking past each other in the street could have very different present moments. If one of the people were walking towards the Andromeda Galaxy, then events in this galaxy might be hours or even days advanced of the events on Andromeda for the person walking in the other direction. If this occurs, it would have dramatic effects on our understanding of time. Penrose highlighted the consequences by discussing a potential invasion of Earth by aliens living in the Andromeda Galaxy. As Penrose put it:

"people pass each other on the street; and according to one of the two people, an Andromedean space fleet has already set off on its journey, while to the other, the decision as to whether or not the journey will actually take place has not yet been made. How can there still be some uncertainty as to the outcome of that decision? If to either person the decision has already been made, then surely there cannot be any uncertainty. The launching of the space fleet is an inevitability. In fact neither of the people can yet know of the launching of the space fleet. They can know only later, when telescopic observations from earth reveal that the fleet is indeed on its way. Then they can hark back to that chance encounter, and come to the conclusion that at that time, according to one of them, the decision lay in the uncertain future, while to the other, it lay in the certain past. Was there then any uncertainty about that future? Or was the future of both people already 'fixed'?"^[5]

^{Literally...that, and the Invasion story, was part and parcel to the Andromeda Paradox, regardless of whether the questions are scientific or not, or what you, I, or anyone else thinks about it ...it's simply a historical fact, not my opinion, and not something I made up}

Edited June 9, 2013 by J.C.MacSwell

[xyzt]

You have three variables for velocity (you should only have one). You have three variables for time (only need two). You don't say which velocity belongs in the gamma factor. Unless you explain every variable and give a sample scenario with numbers there is just no way to unravel what you are trying to do with that equation.

You missed the trivial facts that [math]v_1=0[/math] , [math]v_2=v<<c[/math]. I don't know where you get the third variable for velocity but you are clearly wrong.

Now, for time: you have the time for the Andromeda event ([math]t_a[/math]) and the two different coordinate times for the two Earth observers ([math]t_{e1},t_{e2}[/math]).

This is a very simple exercise, I don't understand why you have so much difficulty with it.

You don't say which velocity belongs in the gamma factor.

People who understand relativity know which speed, not velocity goes with [math]\gamma[/math]. Since you had so much difficulty, let me make it plain for you:

 

 

[math]t_{e2}-t_{e1}=\gamma(v_2)(t_a+v_2 x_a/c^2)-\gamma(v_1)(t_a+v_1 x_a/c^2)=\gamma(v_2) v_2x_a/c^2+t_a (\gamma(v_2)-1)=\gamma(v) vx_a/c^2+t_a (\gamma(v)-1) \approx \gamma(v) vx_a/c^2[/math]

 

[math]\gamma(v) \approx 1[/math] since [math]v<<c[/math]

 

so:

 

[math]t_{e2}-t_{e1} \approx vx_a/c^2[/math] as already correctly explained in post 2. Based on our prior interactions I thought you could do these simple calculations all by yourself. I would appreciate next time you challenge so hard and claim that you found mistakes that you ask first, I will be more than happy to explain to you.

Edited June 8, 2013 by xyzt

[swansont]

!

Moderator Note

More posts moved from the other thread, because apparently telling people to limit that discussion to no relative motion of earth and the Andromeda galaxy was too confusing.

[Iggy]

This is a very simple exercise, I don't understand why you have so much difficulty with it.

 

People who understand relativity know which speed, not velocity goes with [math]\gamma[/math]. Since you had so much difficulty, let me make it plain for you:

 

 

[math]t_{e2}-t_{e1}=\gamma(v_2)(t_a+v_2 x_a/c^2)-\gamma(v_1)(t_a+v_1 x_a/c^2)=\gamma(v_2) v_2x_a/c^2+t_a (\gamma(v_2)-1)=\gamma(v) vx_a/c^2+t_a (\gamma(v)-1) \approx \gamma(v) vx_a/c^2[/math]

 

[math]\gamma(v) \approx 1[/math] since [math]v<<c[/math]

 

so:

 

[math]t_{e2}-t_{e1} \approx vx_a/c^2[/math] as already correctly explained in post 2. Based on our prior interactions I thought you could do these simple calculations all by yourself. I would appreciate next time you challenge so hard and claim that you found mistakes that you ask first, I will be more than happy to explain to you.

 

I know you're suspended, but that shouldn't stop you from reading this reply and responding when and if you return.

 

I'm afraid your explanation didn't do much for me. The trouble starts with the first thing you say in post 2 from the other thread:

 

...having two observers... Say that two earthlings observe the event [math](x_a,t_a)[/math] on Andromeda.

 

The Lorentz time transformation (which you're about to try to use) transforms from one coordinate system to another. But, you attribute these coordinates, above, to both earth observers. The point of the Andromeda paradox is that the t coordinate of an event at a significant distance is different in two different frames.

 

Your most recent post in the other thread:

 

You missed the trivial facts that [math]v_1=0[/math] , [math]v_2=v<<c[/math]. I don't know where you get the third variable for velocity but you are clearly wrong.

 

The three velocities are [math]v_1[/math], [math]v_2[/math], and [math]v[/math]. If [math]v_2=v[/math] then it sounds like you're just being redundant.

 

Now, for time: you have the time for the Andromeda event ([math]t_a[/math]) and the two different coordinate times for the two Earth observers ([math]t_{e1},t_{e2}[/math]).

 

Ok...

 

[math]t_{e2}-t_{e1}=\gamma(t_a+v_2 x_a/c^2)-\gamma(t_a+v_1 x_a/c^2)=\gamma vx_a/c^2[/math]

 

What you just said means...

 

[math]t_{e2} = \gamma(t_a+v_2 x_a/c^2)[/math]

 

and

 

[math]t_{e1} = \gamma(t_a+v_1 x_a/c^2)[/math]

 

This could only make sense if you have a third frame. If [math]t_a[/math] is a coordinate in a third frame then [math]t_{e1}[/math] and [math]t_{e2}[/math] would be the same coordinate in two different frames. The third frame, however, would need to have its origin in the same place as the other two frames. So, now we have three earth observers. There is no need for that, I assure you.

 

This is a very simple exercise, I don't understand why you have so much difficulty with it.

 

People who understand relativity know which speed, not velocity goes with [math]\gamma[/math]. Since you had so much difficulty, let me make it plain for you:

 

That doesn't help.

 

If you evaluate your equation numerically, the only way it works is if [math]t_a[/math] is zero. The further from zero, the less correct. In fact, the left hand side and right hand side would disagree by exactly the value of [math]t_a[/math]. So, if [math]t_a[/math] is half c (t_a = 150,000,000) then the equation would be off by 150,000,000 seconds. Nobody wants to be nearly 5 years off, so...

 

If [math]t_a[/math] has to be zero then:

 

[math]t_{e2}- \textcolor{red}{t_{e1}} = \gamma(t_a+v_2 x_a/c^2)- \textcolor{red}{\gamma(t_a+v_1 x_a/c^2)} =\gamma vx_a/c^2[/math]

 

the red terms HAVE to be zero. No point in subtracting zero from anything, and you told us v = v₂, so...

 

[math]t_{e2} = \gamma(t_a+v_2 x_a/c^2) [/math]

 

That would be the correct equation. Like I said, one variable for velocity and two for time. The second post in the other thread explains the variables and what they mean in the context of the Andromeda paradox:

 

 

[math]t = \frac{t'+vx/c^2}{\sqrt{1-v^2/c^2}}[/math]

 

where,

• t is the time between present instants at a distance of x by the stationary earth observer
• x is the distance to the Andromeda galaxy (2.3 x 10²² meters)
• v is the velocity between the stationary earth observer and the earth observer who is walking past him toward the Andromeda galaxy (1.3 m/s)
• t' is zero (the present instant as defined by the walking earth observer)

[math]t = \frac{0+(1.3)(2.3 \times 10^{22})/299792458^2}{\sqrt{1-1.3^2/(299792458)^2}}[/math]

 

[math]t = 332682 \ \mbox{seconds} = 3.8 \ days[/math]

 

3.8 days separate the present instant in the Andromeda galaxy between two observers on earth who have a relative walking speed. That is the Andromeda paradox.

 

The equation simply transforms the t coordinate in one frame to the t coordinate in another (given distance and velocity). If t' is zero then t is the change in time between the two.

 

When you say "Now, if one insists on having two observers, the math gets just a tad more complicated", it really doesn't. It is nothing more than one Lorentz transform. If you have one observer who changes velocity then the math and the equation is the same.