Andromeda paradox and empirical consequences

[my_wan]

The https://www.physicsforums.com/archive/index.php/t-240147.html" occurred to me several years ago when I was still chasing relativity by the tail. I didn't learn that Rietdijk, Putnam, and Penrose had advanced this argument till yesterday. Weird. Anyway there are direct testable consequences if this argument is valid. Not only is it testable but if it holds it would provide a mechanism for measuring astronomical distances through directly controllable parameters.

We obviously couldn't use invading aliens, rather we would need to use any uniquely identifiable event or event sequence. Supernova or more subtle events come to mind.
The Rietdijk-Putnam argument (empirical consequences):
1) The degree to which two nearby recording devices in separate frames disagree on the timing of an identifiable event is a function of the distance of the object being recorded and the relative frame shift of the two recording devices.
2) Given sufficient distance to the event and proximity of the devices, device B could inform device A about an event that device A couldn't know if information is strictly limited to the speed of light.

By knowing the relative frame shift between the devices A and B and the degree to which they disagree on the timing of the event it becomes trivial to calculate the distance of the event for any given frame. The greater the $$\gamma$$ between A and B and the greater the distance to the event the more A and B will diverge in recording time of event. Since we know the $$\gamma$$ between A and B we can easily calculate the distance to the event for a given frame. If the Rietdijk-Putnam argument empirically holds we must give up the notion that that information is limited to the speed of light, irrespective of any notions of hidden variables.

My take:
It will not work. The fault lies with the Rietdijk-Putnam argument and not SRT. Resolving this, at least to my satisfaction, essentially resolved my quest to find problems with SRT. My personal analysis is several years old and would take time to reformulate. In essence the problem stems from mistaking a definition for a physical state. I'll try and illustrate pictorially by assuming a priori that the Rietdijk-Putnam argument is false.

Two frame shifted observers, A and B, observe an event in the Andromeda galaxy. They record the time and compare notes. They notice that the only difference in the timing of the event is directly attributable to the $$\gamma$$ that defines their local disagreement in simultaneity. In fact they may even choose to define the event itself as t=0 without complication. How can this be if the same $$\gamma$$ defines the difference in distance to the event to be a full light day or week shorter for observer B. Simple answer: due to the frame shift, B must define past events on a different time scale that A does. Notice that given the finite speed of light these events are in fact in the past for both observers. The only question is the rate at which the laws of physics allows them access to that information. How far in the past is purely a definition imposed by their respective (frame shifted) points of view. The relativistic interval still holds, as does the Relativity of Simultaneity.

The statement on http://en.wikipedia.org/wiki/Rietdijk-Putnam_argument"

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.

This statement is not only falsifiable but must be false under the principles of relativity. So why does the Rietdijk-Putnam argument remain, or more specifically the above wiki quote, remain unchallenged?

 

[atyy]

Wikipedia's Rietdijk-Putnam article: "Each observer considers their set of present events to be a three dimensional universe but even the slightest movement of the head or offset in distance between observers can cause the three dimensional universes to have differing content."

George Jones brought up on another thread a paper by Dolby and Gull which comments on it: "Also, if Barbara’s hypersurfaces of simultaneity at a certain time depend so sensitively on her instantaneous velocity as these diagrams suggest, then she would be forced to conclude that the distant planets swept backwards and forwards in time whenever she went dancing!". https://www.physicsforums.com/showthread.php?p=1893032#post1893032

 

[granpa]

1) The degree to which two nearby recording devices in separate frames disagree on the timing of an identifiable event is a function of the distance of the object being recorded and the relative frame shift of the two recording devices.
2) Given sufficient distance to the event and proximity of the devices, device B could inform device A about an event that device A couldn't know if information is strictly limited to the speed of light.


if they are near each other then they are seeing the same light from that event so how could one inform the other about things that the other can't see for themselves? they will disagree about the coordinates of the events they are seeing but that's hardly a revelation. if both observers are at the origin at t=0 and the stationary observer calculates the coordinates as (d,d/c) then the other calculates them as (d/gamma, d/(c*gamma)). but they both agree that the light from that event reaches the origin at t=0

 

[Fredrik]

If I understand the Wikipedia article correctly, the R-P argument is trying to prove that the (special relativistic and classical) universe is deterministic, and the "Andromeda paradox" is equivalent to the R-P argument. (Let me know if I have misunderstood what the argument is about). The main point of the argument, when it's stated as the Andromeda paradox, is this (quoted from the Wikipedia article): "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."

This argument looks very wrong to me. What's happening in the Andromeda galaxy is described by curves in one region of Minkowski space, and what's going on here is described by curves in another region. A curve is just a continuous function $C:(a,b)\rightarrow M$ where (a,b) is an interval of real numbers and M is Minkowski space. Note that no coordinates are involved. It would be preposterous to think that the existence of two different coordinate systems implies that only one shape of those curves is possible. (I know that's not what they're saying. This is just what would have to be true for determinism to be a consequence of the observation that we would normally associate two specific and different coordinate systems with two observers. It isn't true, so the observation is irrelevant).

A coordinate system is just a function that assigns coordinates to events. The physical thing that corresponds to an inertial frame is a grid of rulers and synchronized clocks. If you decide that you will always label events with the coordinates assigned by the grid that has the same velocity as you, you will get weird effects, like distant events moving backward and forward in "time" when you're moving around. The question is why the **** would anyone want to label events this way? (It wouldn't even be a valid coordinate system, not globally anyway. It would only be valid in some neighborhood of your own world line).

This statement is not only falsifiable but must be false under the principles of relativity. So why does the Rietdijk-Putnam argument remain, or more specifically the above wiki quote, remain unchallenged?

The text you quoted is trivially true in SR (if those people use their co-moving inertial frames), not trivially false. It's just irrelevant, since it's just an assignment of a time coordinate by a couple of different coordinate systems.

 

[Fredrik]

Note also that special relativistic classical mechanics is deterministic. That isn't a consequence of the properties of spacetime. It's a consequence of (the relativistic version of) Newton's second law, which implies that position as a function of time is completely determined by the position and velocity at one time.

 

[JesseM]

2) Given sufficient distance to the event and proximity of the devices, device B could inform device A about an event that device A couldn't know if information is strictly limited to the speed of light.

I agree with granpa, this statement doesn't make sense to me. An observer B can't send a message to another observer A such that when A receives the message she gains information about events outside her past light cone at that moment, since any message sent by observer B is itself limited by the speed of light.

 

[my_wan]

I concur with the statement atyy pointed to in George Jones thread with the http://arxiv.org/abs/gr-qc/0104077" with regard to the dance quote. Thanks. I was having trouble finding any reference to any, except philosophical, issues with the Rietdijk-Putnam argument. Although the arxiv.org paper doesn't mention the Andromeda paradox, the Rietdijk-Putnam argument, or its authors it is clearly taking direct issue with the errors inherent in the Rietdijk-Putnam argument.

http://arxiv.org/abs/gr-qc/0104077]It[/PLAIN] might seem reasonable to suppose that the twin paradox has long been understood, and that no confusion remains about it. Certainly there is no disagreement about the relative aging of the two twins, so there is no ‘paradox’. There is also no confusion over ‘when events are seen’ by the two twins, or over the description by the stay-at-home twin (Alex say), of ‘when events happened’. These aspects of the twin paradox are treated in the standard texts.
However, the description of ‘when events happened’ according to the traveling twin (Barbara say) seems never to have been fully settled. In textbook treatments, Barbara’s hypersurfaces of simultaneity, which define ‘when events happened’ according to her, have consistently been misrepresented or ignored.<snip>

The text you quoted is trivially true in SR (if those people use their co-moving inertial frames), not trivially false. It's just irrelevant, since it's just an assignment of a time coordinate by a couple of different coordinate systems.

No, the wiki statement I quoted is false. The consequences of it being true would imply that if we are walking opposite directions and I look up and see the Andromeda galaxy blow up you will not see it till tomorrow simply because you are walking in the opposite direction. That's more than trivially false, that is empirically false. Whatever philosophical case being made, Block Universe, Eternalism, determinism, etc., is moot when the grounds for the argument is based on empirically false assumptions. Luckily SRT doesn't support the argument either, else good bye SRT.

I agree with granpa, this statement doesn't make sense to me. An observer B can't send a message to another observer A such that when A receives the message she gains information about events outside her past light cone at that moment, since any message sent by observer B is itself limited by the speed of light.

It wasn't meant to make perfect sense. It is, however, a consequence of the assumptions of the Rietdijk-Putnam argument. If the distance to a distant event is foreshortened by even a small percent then over millions of light years even your walking speed would produce a noticeable shift in when the event appeared to occur, due to the finiteness of C. That is fundamentally the assumption of the Rietdijk-Putnam argument.

My main question in the OP was why I couldn't find where anybody took issue with the Rietdijk-Putnam argument directly. As atyy via George Jones pointed out people have taken issue with the inherent assumptions if not the Rietdijk-Putnam argument itself.

I'll be going through the reference list of that arxiv paper.

 

[granpa]

It wasn't meant to make perfect sense. It is, however, a consequence of the assumptions of the Rietdijk-Putnam argument. If the distance to a distant event is foreshortened by even a small percent then over millions of light years even your walking speed would produce a noticeable shift in when the event appeared to occur, due to the finiteness of C. That is fundamentally the assumption of the Rietdijk-Putnam argument.
.

notice this at the bottom of the wiki page:

Notice that neither observer can actually "see" what is happening on Andromeda at any given moment because light from Andromeda takes about two million years to reach earth. The argument is not about what can be "seen", it is purely about what different observers consider to be contained in their instantaneous present moment.
lets put it this way:
reality doesn't change just because you change speed. you change. your perspective changes. things look different because of your new perspective if you draw a spacetime diagram of all the events and draw in the 2 different coordinate systems then you should be able to see that the positions and relationships of the events don't change just because you change coordinate systems. that's kinda the whole point of a spacetime diagram.

in that sense changing coordinate systems in relativity is no different than changing coordinate systems in Euclidean geometry.

 

[JesseM]

It wasn't meant to make perfect sense. It is, however, a consequence of the assumptions of the Rietdijk-Putnam argument. If the distance to a distant event is foreshortened by even a small percent then over millions of light years even your walking speed would produce a noticeable shift in when the event appeared to occur, due to the finiteness of C. That is fundamentally the assumption of the Rietdijk-Putnam argument.

It's true that even changing your walking speed will change the time that the event occurs in your instantaneous rest frame at that moment, but I don't see what this has to do with the idea that there is any way for you to actually know the outcomes of events if the light from them has not had time to reach you. Just because an event has already happened in your reference frame doesn't mean there's any way for you to know about it at a time when the event doesn't lie within your past light cone.

No, the wiki statement I quoted is false. The consequences of it being true would imply that if we are walking opposite directions and I look up and see the Andromeda galaxy blow up you will not see it till tomorrow simply because you are walking in the opposite direction. That's more than trivially false, that is empirically false.

The wiki statement is only talking about the time-coordinates assigned to events in your current inertial rest frame, not about when you actually see the events. If the two people are passing next to each other when the light from an event reaches one of them, then of course that is also the moment that the light reaches the other one, but they will disagree about which past events on their own worldlines (or which events on a clock at rest on the Earth) the events in the Andromeda galaxy were simultaneous with. On the other hand, if they are not next to each other when the light from the event reaches one of them, then of course it will reach the one who's closer to the Andromeda galaxy first, but if the two observer's paths cross at some time and at much later times they each see the light from two successive events in the Andromeda galaxy A and B, it is quite possible for one observer to retroactively say that A happened "at the same moment" that the two observers' paths were crossing, while the other observer says that it was B that happened "at the same moment" their paths were crossing. This is consistent with the wiki statement:

If one of the people were walking towards the Andromeda Galaxy then events in this galaxy might be hours or even days advanced [i.e. he retroactively says the later event B was happening 'at the same time' the two observers were passing each other] of the events on Andromeda for the person walking in the other direction. [i.e. the other person retroactively says the earlier event A was happening 'at the same time' the two were passing]

 

[atyy]

Penrose presents the 'paradox', but not as a paradox! It actually seems his intention is to give a vivd picture of the relativity of simultaneity. :rofl:

Emperor's New Mind, p260:
http://books.google.com/books?id=oI...X&oi=book_result&resnum=5&ct=result#PPA260,M1

 

[Fredrik]

No, the wiki statement I quoted is false. The consequences of it being true would imply that if we are walking opposite directions and I look up and see the Andromeda galaxy blow up you will not see it till tomorrow simply because you are walking in the opposite direction.

That's not what the Wiki quote means. They're just saying that if you and I are walking in opposite directions (one of us towards Andromeda) and we both use our rest frames to assign coordinates to events, we will disagree about which event at a particular location in the Andromeda galaxy is simultaneous with (e.g.) the event where we meet. This is how simultaneity works in SR (when we insist on using co-moving inertial frames). Their statement doesn't in any way contradict the fact that light from the explosion would reach us at the same time.

 

[my_wan]

That's not what the Wiki quote means. They're just saying that if you and I are walking in opposite directions (one of us towards Andromeda) and we both use our rest frames to assign coordinates to events, we will disagree about which event at a particular location in the Andromeda galaxy is simultaneous with (e.g.) the event where we meet. This is how simultaneity works in SR (when we insist on using co-moving inertial frames). Their statement doesn't in any way contradict the fact that light from the explosion would reach us at the same time.

Take another look at http://en.wikipedia.org/wiki/Rietdijk-Putnam_Argument" . In the image it says:

A car moving past a stationary person will have a different set of things that are simultaneous. At the distance of the Andromeda galaxy the present instant for the stationary person might contain a meeting where a space-admiral is deciding whether to invade earth. In the present instant for the person in the car the Andromedian fleet is already on the way!

Right under the image it says:

The 'paradox' consists of two observers who are in the same place and at the same instant having different sets of events in their present moment.

Notice in the image the space-admiral is deciding whether to invade on monday and the fleet leaves on tuesday. Notice also the person in the car sees the fleet on the way while the person next to the passing car sees the admiral deciding our fate. By the claims of this argument there is 24 hours difference between what the person in the car sees as opposed to what the person next to the car sees. If this were true then it only takes a couple of minutes for the car to stop and tell the person that the fleet is on the way. This information is forbidden, by the laws of physics, to the person on the side of the road because it is information faster than the speed of light in his frame of reference. The laws of physics will not allow him that information till the following day, irrespective of whether it actually already occurred in some location or frame of reference or not.

Even the milder version you stated is false when you said, "we will disagree about which event at a particular location in the Andromeda galaxy is simultaneous with (e.g.) the event where we meet." If the proximity of our location is sufficiently close that it may be considered for practical purposes the same location we will not disagree on what events are manifest at that point. We certainly can disagree on how to interpret the simultaneity of those events with respect to external points. Never is simultaneity of events manifest at a particular point in space in question. Else it becomes possible to circumvent the speed of light to transmit information. The Relativity of Simultaneity only becomes an issue when you are trying to decide the simultaneity of two separate and seperated events.

 

[JesseM]

Notice in the image the space-admiral is deciding whether to invade on monday and the fleet leaves on tuesday. Notice also the person in the car sees the fleet on the way while the person next to the passing car sees the admiral deciding our fate.

You misinterpret the image, it does not show what either person is seeing, only what distant events are simultaneous with the event of the two people crossing paths in their own respective inertial rest frames. If an event 4 light-years away from me in my frame is simultaneous with me turning 30 in my frame, of course I won't actually see the event until I turn 34! The wikipedia article is pretty explicit about this:

Notice that neither observer can actually "see" what is happening on Andromeda at any given moment because light from Andromeda takes about two million years to reach earth. The argument is not about what can be "seen", it is purely about what different observers consider to be contained in their instantaneous present moment.

Even the milder version you stated is false when you said, "we will disagree about which event at a particular location in the Andromeda galaxy is simultaneous with (e.g.) the event where we meet." If the proximity of our location is sufficiently close that it may be considered for practical purposes the same location we will not disagree on what events are manifest at that point. We certainly can disagree on how to interpret the simultaneity of those events with respect to external points.

Isn't that exactly what Fredrik said? If in my frame the event of my clock reading 10 seconds is simultaneous with the event of your clock reading 10 seconds and these two events happen when we meet at the same position in space, then of course it must be true in your frame as well that the event of the two clocks reading 10 seconds were simultaneous. But in your frame the event of our meeting may be simultaneous with a clock on another planet reading 30 seconds, while in my frame the events of our meeting may be simultaneous with that same distant clock reading 90 seconds...I believe this is what Fredrik meant when he said "we will disagree about which event at a particular location in the Andromeda galaxy [for example the event of the distant clock reading 30 seconds vs. the event of the distant clock reading 90 seconds] is simultaneous with (e.g.) the event where we meet."

 

[my_wan]

The wiki statement is only talking about the time-coordinates assigned to events in your current inertial rest frame, not about when you actually see the events.<snip>

You can't avoid the notion of "seeing" when you could easily replace the alien invaders (event) with a Supernova event. A person in a car can certainly "see" a Supernova event. If the motion of the car changes what events occur in your instantaneous rest frame at that moment (by 24 hours relative to the guy next to the road) it can certainly pull over and tell the guy by the road a Supernova will be visible in a few hours.

 

[granpa]

You can't avoid the notion of "seeing" when you could easily replace the alien invaders (event) with a Supernova event. A person in a car can certainly "see" a Supernova event. If the motion of the car changes what events occur in your instantaneous rest frame at that moment (by 24 hours relative to the guy next to the road) it can certainly pull over and tell the guy by the road a Supernova will be visible in a few hours.

how many times do we have to tell you that that isn't how it works. events that occur at the same place at the same time do so for all observers regardless of velocity. learn to draw a spacetime diagram.

 

[JesseM]

You can't avoid the notion of "seeing" when you could easily replace the alien invaders (event) with a Supernova event.

You still misunderstand, it has nothing to do with brightness, it has to do with the fact that the light from these events hasn't had time to reach them at the moment they're passing. If a supernova 4 light-years away in my frame is simultaneous with my 30th birthday in my frame, then of course I'll see this event (probably get roasted by it) when I turn 34, but there's no way for me to see it when I turn 30 because the light needs time to cross those 4 light-years and reach me. The wikipedia diagrams are showing which events at Andromeda are simultaneous with the event of the two people crossing paths in each of their own inertial rest frames, but the light from either event has not had time to reach them at the moment they cross paths so they couldn't see them at that moment no matter how bright they were.

 

[my_wan]

You misinterpret the image, it does not show what either person is seeing, only what distant events are simultaneous with the event of the two people crossing paths in their own respective inertial rest frames. If an event 4 light-years away from me in my frame is simultaneous with me turning 30 in my frame, of course I won't actually see the event until I turn 34! The wikipedia article is pretty explicit about this:

Yes wiki was explicit in that regard. It makes no difference that the event has already technically occurred in some location or frame of reference. The fact remains that information about that already occurred event is limited to the speed of light.

 

[JesseM]

Yes wiki was explicit in that regard. It makes no difference that the event has already technically occurred in some location or frame of reference. The fact remains that information about that already occurred event is limited to the speed of light.

Yes, that's the whole reason they can't actually see the events at the moment they cross, because the information is "limited to the speed of light"! The diagram is only showing which events are occurring at the same moment they cross in that "technical" sense that the distant event is assigned the same time-coordinate as the time-coordinate of their crossing in some frame of reference, that's what is meant by "simultaneity" in relativity.

 

[my_wan]

Let me restate it this way then. We have a driver and a hitchhiker. Two million years ago two million light years away in the cars frame a Supernova went boom. He looks up and says wow check that out! About this time he sees the hitchhiker and pulls over. Now because the hitchhiker is in a slightly different frame of reference the distance is two million light years + 24 light hours away. The laws of physics forbids him information about the Supernova event for 24 more hours, else he received information faster than the speed of light. Yet here's the driver pulling over giving him exactly that, perhaps even with video on his cell phone. No.

 

[granpa]

the driver the hitcher and the light from the supernova are at the same place at the same time. this is true for all observers regardless of velocity. you just need to study up a bit more. all beginners have trouble with relativity of simultaneity.

 

[my_wan]

the driver the hitcher and the light from the supernova are at the same place at the same time. this is true for all observers regardless of velocity. you just need to study up a bit more. all beginners have trouble with relativity of simultaneity.

That is exactly the point I have been repeating. It is the basic truth that the Rietdijk-Putnam argument violates. Are you now assumming I'm claiming that the Rietdijk-Putnam argument is valid. How has my point here suddenly become the answer to my point that I somehow missed?

 

[JesseM]

Let me restate it this way then. We have a driver and a hitchhiker. Two million years ago two million light years away in the cars frame a Supernova went boom. He looks up and says wow check that out! About this time he sees the hitchhiker and pulls over. Now because the hitchhiker is in a slightly different frame of reference the distance is two million light years + 24 light hours away.

If the car is moving relative to the Earth, then in the car's inertial rest frame (which has always been moving at the same constant speed) both the Earth and the Andromeda galaxy are in motion, so if the distance between the Earth and the Andromeda galaxy is 2 million light years in this frame, that is not the same as the distance between the position where the supernova occurred and the car's position when the light reaches it, so the supernova cannot have occurred 2 million years ago in this frame. Also, of course whatever the time is in this frame, the time would be different in the frame of the hitchhiker who is at rest relative to the Earth. It will work out so that if one frame predicts the light reaches them when they cross paths, the other frame predicts it too.

Let me give a numerical example. To make things easy, assume the distance is 2 million light years in the Earth/galaxy rest frame, and assume we have a ship moving at 0.6c in the direction of the galaxy in this frame which sees the light from the supernova at the moment he crosses path with the hitchhiker at rest on Earth. In the hitchhiker's frame, if the light reaches him at the moment the ship is passing him, then if he calls his own position x=0 light years and the time that the light reaches him as t=0 years, the coordinates of the supernova must have been x=2 million light years, t=-2 million years. So, we can do the Lorentz transformation to find the coordinates of this same event in the ship's frame, assuming that the ship's frame assigns the event of crossing paths with the hitchhiker coordinates x'=0, t'=0. The Lorentz transform here is:

x' = gamma*(x - vt)
t' = gamma*(t - vx/c^2)
with gamma = 1/sqrt(1 - v^2/c^2). For v=0.6c, gamma = 1.25. So if the supernova has coordinates (x=2 million light years, t=-2 million years) in the hitchhiker's frame, the coordinates in the ship's frame are:

x' = 1.25*(2 million l.y. - (0.6 l.y/year)*(-2 million years)) = 1.25*(2 million l.y. + 1.2 million l.y.) = 4 million l.y.

t' = 1.25*(-2 million years - (0.6 l.y./year)*(2 million l.y.)/(1 l.y./years)^2) = 1.25*(-2 million years - 1.2 million years) = -4 million years.

So, in the ship's frame the supernova happened 4 million light years away 4 million years ago, so it makes perfect sense he'd be seeing the light now too.

Another way of thinking about these numbers: in the ship's frame, the distance between the Earth and the galaxy is shrunk by a factor of sqrt(1 - 0.6^2) = 0.8, so it's only 1.6 million light years from Earth. And in the ship's frame the galaxy is moving towards him at 0.6c, so 4 million years before he saw the light, the galaxy must have been at a distance of (1.6 million light years) + (0.6 light years/year)*(4 million years) = 1.6 million light years + 2.4 million light years = 4 million light years. So if the light is just reaching the ship now, it must be true that the supernova happened 4 million years ago in the ship's frame.

But what if we don't want to just assume the light is reaching the ship at the moment it crosses paths with the Earth, as I did in that last sentence? Well, in the Earth's frame the ship is moving at 0.6c and the distance is 2 million l.y., so it'll take another (2 million)/0.6 = 3.333... million years for the ship to reach the Andromeda galaxy. If a clock in the Andromeda galaxy is synchronized with the Earth's in their rest frame, then it read -2 million years when the supernova occurred and it will read 3.333... million years when the ship reaches it, so the clock elapsed 5.333... million years between the supernova happening and the ship reaching it. In the Earth's frame the ship is moving at 0.6c so its clock is slowed by a factor of sqrt(1 - 0.6^2) = 0.8, so the ship's clock only elapsed 3.333... million years*0.8 = 2.666... million years between passing Earth and reaching the galaxy. Meanwhile in the ship's frame it is the Andromeda galaxy's clock that is slowed down by a factor of 0.8, so if the Andromeda galaxy clock measured 5.333... million years between the supernova and the ship reaching it, the supernova must have actually happened 5.333... million/0.8 = 6.666... million years before the ship reached the Andromeda clock. And since the ship's own clock reads t' = 2.666... million years when it reaches the Andromeda clock, that means in the ship's frame the supernova must have happened at t' = 2.666... million - 6.666... million = -4 million years, the same number we found from the Lorentz transform. And again, if the galaxy was 1.6 million light years away at t'=0 in the ship's frame, and is moving towards the ship at 0.6c, then 4 million years ago it must have been (4 million)*0.6 = 2.4 million light years further out than it was at t'=0, so it must have been a total of 1.6 million + 2.4 million = 4 million light years away.

 

[my_wan]

The crux of the difficulties here stem from a disagreement over the assumptions of the Rietdijk-Putnam argument as presented in the wiki article. That last post by granpa at least tells me that his grasp of SR is solid and forced me to look at my presumptions about what the wiki article says. My apologies for being so dense but the Rietdijk-Putnam argument did in fact avoid the issue of locally observer simultaneous events. They dealt strictly with distant events deemed simultaneous via relativity. Yes I was told this and still missed it. So we can agree on the foundations of SRT, and on the fact that I screwed up the interpretation of the Rietdijk-Putnam argument.

My apologies...