Jonathan's Answer
But the differences need not be vague. Everything can easily be derived from the Lorentz Transformation equations, and there is no mystery to it.
If you have an object receding from you at near the speed of light, it will appear to be moving, at a maximum, of 50% of the speed of light.
Why? (Answer this question for yourself, and if you agree, Upvote, if you don't agree, downvote, and post a comment.)
On the other hand, if you have an object approaching you near the speed of light, there is NO LIMIT to it's maximum apparent speed.
Why? (Answer this question for yourself. If you agree, upvote. If you don't agree, downvote, and post a comment.)
Now, when you look at a fast approaching, or fast receding object are you looking at where the object is now? No. You're looking at where the object WAS when it emitted or reflected the light. That emission or reflection of light is an "event" which happened at a place and time in your perceptions. It has physical coordinates of (t,x,y,z) Space and time.
What happens when you accelerate toward a past event in Special Relativity? It moves away from the observer, and back in time. Again, do the math yourself. If you agree, Upvote. If you don't agree, Downvote and post a comment.)
But yes, if you accelerate toward an event in the past, Lorentz Transformation equations say it moves away and back in time. That's good, because it makes everything consistent with what I said earlier:
As the moving twin is moving away from the sun, he's going to see the sun moving away at less that half the speed of light. When he turns around, he's going to see the image of the sun jump away from him--lurching away from him spatially. And it will also (from his perspective) lurch backward in time... So the emission/reflection event happened much further away and longer ago. So at the "instant of acceleration" is when the earth has suddenly aged in his point-of-view.
(Video explanation added, November 7)
The video explains what is meant by "if you have an object receding from you at near the speed of light, it will appear to be moving at a maximum of 50% of the speed of light."
It also explains why the distance traveled by the earth in the second leg of the inbound frame is much greater than the distance traveled by the earth in the first leg of the outbound frame
Video added November 27, 2015:
In the comments below, I did work through some of the math that I recommended above... I decided to re-post it within the answer, because it's hard to find, looking through the comments.
Alright... I have a lot more Lorentz Transforms for you to do!
I'm trying to give relevance, here, to is the events on the observer's past light cone, and deemphasize the simultaneous events... (See Part 5 below for more discussion of the simultaneous events)
Setup:
Let's imagine a scenario where a deep space-station lies exactly one light-year from earth. A twin goes to visit the space-station. He travels out at 0.866c, stays there for an hour, and then travels back at 0.866 c. These coordinates are in years, and light years, so the hour long visit is just going to be a rounding error somewhere past the 3rd significant digit on this scale. (And we'll pretend that the acceleration can be done quickly without killing the pilot, even though accelerating to .866c in less than a day would be quite a few g's, I'm sure.)
Part I:
Calculate the spacetime coordinates of these events in the earth/space-station frame. (1) the take-off (2) The space-station visit (3) The return home.
Answers
(1) (t,x)=(0,0)
(2) (t,x)=(1.155, 1)
(3) (t,x)=(2.31,0)
Part 2:
Translate these coordinates so that the space-station visit is at (0,0)
Answers
(1) (t,x)=(-1.155,-1)
(2) (t,x)=(0,0)
(3) (t,x)=(1.155,-1)
Part 3.
There are two other events that I think are quite relevant. Let event X be the event that happens on Earth that can be seen on camera at the space-station during the visit. Let event Y be the event where observers on Earth actually see the traveling twin land at the space-station.
Answers:
X: (t,x)=(-1,-1)
Y: (t,x)=(1,-1)
Part 4:
The really interesting event to look at here, from the traveling twin's perspective is Event X. Because he can be watching that event (1) immediately before the visit. (2) During the visit. And (3) immediately after the visit to the the space station. Finally, here, we're doing a Lorentz Transformation on the event at (-1,-1)... Perform a Lorentz Boost of v=+0.866c on event X to find it's location immediately before the visit, and a Lorentz boost of v=-0.866c on event X to find it's location immediately after the visit.
Answers
(1) Before visit: (t,x) = (-0.268 light-years, -0.268 years)
(2) During visit: (t,x) = (-1 light-years,-1 years)
(3) After visit: (t,x) = (-3.73 light-years, -3.73 years)
Finally, let's try to put that into words... What's happened here? You're looking at the coordinates of the same event before, during, and after the visit. What has happened is that the earth's image jumps back, from 0.268 light-years away, to 3.73 light-years away. In fact everything in that direction suddenly becomes 13.9 times as far away at the beginning of the return journey than it was at the end of the outbound journey. This works out to be a factor of 1+v/c1−v/c.
Now, how do you explain this phenomena? Do you explain it as "real" or do you explain it as an "optical illusion?" Yes it is optical. But is it illusion? Is the event really one-light-year away, but it only "LOOKS LIKE" it is .268 light years away, and then 3.73 light years away? To claim that one point-of-view is more valid than the others is to go against the Principle of Relativity.
To the contrary, they are all valid coordinates: The earth is 0.268 light years away for the outbound observer, 1 light year away for the space-station observer, and 3.73 light years away for the returning observer... When you add together these distances 0.268+3.73 you find that the traveling twin literally sees the earth travel about twice as far (more generally, the traveling observes the earth travel γ−βγtimes as far on the way out, and sees the earth travel γ+βgamma times as far on the way in.
Part 5:
Now, is it less "confusing" perhaps to just refer to the simultaneous distance?
For this purpose, let's find the the event which is regarded as simultaneous with the visit during (1) the outbound trip (2) during the visit and (3) on the inbound trip.
(1) A. (t,x)= (0,-0.5) according to the outbound twin
(2) B. (t,x) =(0,-1) according to the space-station twin
(3) C. (t,x) = (0,-0.5) according to the inbound twin
Now, where do those events occur in the space-station frame?
Talk:Twin paradox/Archive 13
Answers:
(1) A. (t,x) = (-.866,-1)
(2) B. (t,x) = (0,-1)
(3) C. (t,x) = (0.866,-1)
Event A is simultaneous with E on the outbound trip. Event B is simultaneous with E according to the twin on the space-station. Event C is simultaneous with E on the return trip.
This seems to wrap up the question neatly and tie it up with a bow: "The total distance of the trip... according to these simultaneous distances is LESS than the "real distance" by a factor of gamma." So it makes "perfect sense" that the time experienced by the traveling twin should also be less by a factor of gamma.
The problems with that explanations of the Twin Paradox that chalk-it-up to length contraction are
(1) It gives the impression that the traveling twin actually "experiences" this length contracted distance through which he is traveling.
(2) it does not not really explain the 1.73 year discontinuity between events A and C,
(3) It sometimes gives the impression that that the discontinuity cannot be explained and remains a paradox (especially when the traveling twin is put in an enclosed capsule so they cannot look outside.)
(4) They gives the impression that simultaneity is an observable. That somehow events A, B and/or C might be marked in some phenomenological way by the traveling twin that he might say "Oh... That was what was happening on earth when I turned around."
A Slower Speed of Light
You can see as you play it that accelerating forward makes objects in front of you appear to move away, and when you accelerate backwards, the objects in front of you appear to come closer.
Some people think of this phenomenon as an "optical illusion" but it isn't. The images are coming from the actual coordinates of actual light-reflection events. Are these coordinates observer dependent? Yes. Are these coordinates frame dependent? Yes. Are these differences illusory? NO.
Confusion is when you are unaware of all the facts; or you may have some of them wrong; so they don't all fit together. Surprise is when you see something that you didn't expect.
For people able to accept surprises, putting an emphasis on that surprise can often clear up the confusion.
Most of the literature on the twin paradox does not discuss any observations made by the traveling twin, except for an obsession with the traveling-twin's clock. There is a widespread unwillingness to analyze the twin paradox from the traveling-twin's point-of-view. I think that the traveling-twin's observations are relevant to the twin paradox question, and describing them will greatly increase the "surprise" but greatly decrease the "confusion".
So I'm trying to present the most surprising asymmetry of the problem (other than the well-known fact that their clocks don't match up at the end): Namely, the way the earth JUMPS AWAY from the accelerating twin when he/she accelerates towards it. This fact deserves EMPHASIS because it is SURPRISING. (Not DEEMPHASIS because it is CONFUSING.)
The fact that the earth JUMPS AWAY from the traveling twin on turnaround, is a fact that I expect many readers to find surprising... I strongly suspect that even some people who consider themselves experts on the subject are unaware of this fact, since it doesn't seem to appear anywhere in the literature I've seen; although it does show up quite prominently in the video game from MIT. A Slower Speed of Light
(They do this without words, and it actually looks pretty natural.)
But just to make it clear, this JUMPING AWAY is NOT a result of light-travel-delay-effect
That's not what I said. I said that your emphasis on light travel delay could be confusing to some readers. It is not only normal, it is essential to consider the reference frame of the traveling twin, in order to resolve the paradox. However, while it's certainly interesting to look at what this twin literally sees (which of course has a lot to do with light travel delay), that is not what is meant in general when people talking about different reference frames in SR say "the traveling twin sees X"; it's generally shorthand for "X is the case in the traveling twin's frame", i.e., it's what the traveling twin deduces to have happened after subtracting off the light travel time.
I'm guessing that you are already aware of this, which is why I phrased my comment as "readers may be confused by your choice of emphasis without further clarification" as opposed to "you are wrong".
>> For people able to accept surprises, putting an emphasis on that surprise can often clear up the confusion.
People tend to be able to wrap their heads around travel time delay effects. That "makes sense". What people tend to find far more surprising is that the traveling twinstill "sees" things differently even after subtracting off the effects of light travel time, and that it is this difference that the Lorentz transforms describe. (I've actually had people try to tell me that time dilation goes the other way, i.e., time "goes faster", for obects moving towards you, based on light travel delay reasoning.)
>> Namely, the way the earth JUMPS AWAY from the accelerating twin when he/she accelerates towards it. This fact deserves EMPHASIS because it is SURPRISING. (Not DEEMPHASIS because it is CONFUSING.)
Yes, I think that's well worth talking about, even though it's not necessary for resolving the apparent paradox. Again, I wasn't objecting to that, I was merely commenting on the emphasis on light travel delay, which may lead people to believe that that delay is the source of this (and all other) relativistic effects.
>> The fact that the earth JUMPS AWAY from the traveling twin on turnaround, is a fact that I expect many readers to find surprising... I strongly suspect that even some people who consider themselves experts on the subject are unaware of this fact, since it doesn't seem to appear anywhere in the literature I've seen
It's momentarily surprising, and then you remember that length contraction is momentarily going away, and so of course the Earth must appear to "jump away" (although it then "jumps" right back again).
This is likely not mentioned much in the literature because the turn-around acceleration is often approximated as instantaneous, and so something that happens fleetingly during the acceleration itself is never actually observed. However, if you take the acceleration time to be small but nonzero, I agree that it's a cool side note to discuss.
>> But just to make it clear, this JUMPING AWAY is NOT a result of light-travel-delay-effect
I know that, you know that, but I didn't think it was clear in the original answer, thus my comment.
>> By the Lorentz Transformations this [jumping away] happens whenever you take any distant observed event and accelerate toward it.
Only if you're moving away from that event at the moment. If you're already moving towards them, further acceleration towards them makes them jump towards you.
>> That event jumps away from you backward in space, and backward in time.
I'm not sure what you mean by "backward in time"; if you accelerate towards a distant source, you will both see and "see" time moving at super-speed for that source, with the net effect being that the source "jumps ahead" in time. Perhaps I'm just misunderstanding your meaning, though.
Say an event happened 1 second ago, and one light-second in front of you. (or 1 year ago, one light-year in front of you)
You accelerate toward this event quickly, so you suddenly change your speed by 0.866 c.
Do a Lorentz Transformation on that event by this Delta v. And see if you find that now the event occurred 3.73 (time-units) ago, and 3.73 (distance units) in front of you.
(If you're not familiar with what I'm doing, I could try to show how to do it with more detail. But you should be able to just google Lorentz Transformation and see the system of equations that I'm talking about.)
Also, you don't have to worry about me being unfamiliar with Lorentz transforms, because I'm a physics PhD student and have taught them. :)
I'm trying to give relevance, here, to is the events on the observer's past light cone, and deemphasize the simultaneous events... (See Part 5 below for more discussion of the simultaneous events)
Setup:
Let's imagine a scenario where a deep space-station lies exactly one light-year from earth. A twin goes to visit the space-station. He travels out at 0.866c, stays there for an hour, and then travels back at 0.866 c. These coordinates are in years, and light years, so the hour long visit is just going to be a rounding error somewhere past the 3rd significant digit on this scale. (And we'll pretend that the acceleration can be done quickly without killing the pilot, even though accelerating to .866c in less than a day would be quite a few g's, I'm sure.)
Part I:
Calculate the spacetime coordinates of these events in the earth/space-station frame. (1) the take-off (2) The space-station visit (3) The return home.
Answers
(1) (t,x)=(0,0)
(2) (t,x)=(1.155, 1)
(3) (t,x)=(2.31,0)
Part 2:
Translate these coordinates so that the space-station visit is at (0,0)
Answers
(1) (t,x)=(-1.155,-1)
(2) (t,x)=(0,0)
(3) (t,x)=(1.155,-1)
Part 3.
There are two other events that I think are quite relevant. Let event X be the event that happens on Earth that can be seen on camera at the space-station during the visit. Let event Y be the event where observers on Earth actually see the traveling twin land at the space-station.
Answers:
X: (t,x)=(-1,-1)
Y: (t,x)=(1,-1)
Part 4:
The really interesting event to look at here, from the traveling twin's perspective is Event X. Because he can be watching that event (1) immediately before the visit. (2) During the visit. And (3) immediately after the visit to the the space station. Finally, here, we're doing a Lorentz Transformation on the event at (-1,-1)... Perform a Lorentz Boost of v=+0.866c on event X to find it's location immediately before the visit, and a Lorentz boost of v=-0.866c on event X to find it's location immediately after the visit.
Answers
(1) Before visit: (t,x) = (-0.268 light-years, -0.268 years)
(2) During visit: (t,x) = (-1 light-years,-1 years)
(3) After visit: (t,x) = (-3.73 light-years, -3.73 years)
Finally, let's try to put that into words... What's happened here? You're looking at the coordinates of the same event before, during, and after the visit. What has happened is that the earth's image jumps back, from 0.268 light-years away, to 3.73 light-years away. In fact everything in that direction suddenly becomes 13.9 times as far away at the beginning of the return journey than it was at the end of the outbound journey. This works out to be a factor of
Now, how do you explain this phenomena? Do you explain it as "real" or do you explain it as an "optical illusion?" Yes it is optical. But is it illusion? Is the event really one-light-year away, but it only "LOOKS LIKE" it is .268 light years away, and then 3.73 light years away? To claim that one point-of-view is more valid than the others is to go against the Principle of Relativity.
To the contrary, they are all valid coordinates: The earth is 0.268 light years away for the outbound observer, 1 light year away for the space-station observer, and 3.73 light years away for the returning observer... When you add together these distances 0.268+3.73 you find that the traveling twin literally sees the earth travel about twice as far (more generally, the traveling observes the earth travel
Part 5:
Now, is it less "confusing" perhaps to just refer to the simultaneous distance?
For this purpose, let's find the the event which is regarded as simultaneous with the visit during (1) the outbound trip (2) during the visit and (3) on the inbound trip.
(1) A. (t,x)= (0,-0.5) according to the outbound twin
(2) B. (t,x) =(0,-1) according to the space-station twin
(3) C. (t,x) = (0,-0.5) according to the inbound twin
Now, where do those events occur in the space-station frame?
Talk:Twin paradox/Archive 13
Answers:
(1) A. (t,x) = (-.866,-1)
(2) B. (t,x) = (0,-1)
(3) C. (t,x) = (0.866,-1)
Event A is simultaneous with E on the outbound trip. Event B is simultaneous with E according to the twin on the space-station. Event C is simultaneous with E on the return trip.
This seems to wrap up the question neatly and tie it up with a bow: "The total distance of the trip... according to these simultaneous distances is LESS than the "real distance" by a factor of gamma." So it makes "perfect sense" that the time experienced by the traveling twin should also be less by a factor of gamma.
The problems with that explanations of the Twin Paradox that chalk-it-up to length contraction are
(1) It gives the impression that the traveling twin actually "experiences" this length contracted distance through which he is traveling.
(2) it does not not really explain the 1.73 year discontinuity between events A and C,
(3) It sometimes gives the impression that that the discontinuity cannot be explained and remains a paradox (especially when the traveling twin is put in an enclosed capsule so they cannot look outside.)
(4) They gives the impression that simultaneity is an observable. That somehow events A, B and/or C might be marked in some phenomenological way by the traveling twin that he might say "Oh... That was what was happening on earth when I turned around."
This is only true if you wave away the hour spent on the space station as a round-off error. In most contexts, yes, that would be valid, but here, it makes a difference. If the traveler and X are null-separated at (0,0) (the middle of the visit), then they are (slightly) spacelike-separated right before the traveler arrives at the station, and slightly timelike-separated right after the traveler leaves the station. This becomes clear if you imagine a light pulse being emitted at X. A sub-luminal traveler can only cross that expanding sphere exactly once, and the frequency of light detected when they do so is Lorentz-invariant (so all observers will agree on whether they were outbound, visiting, or inbound at the time).
>> Now, how do you explain this phenomena? Do you explain it as "real" or do you explain it as an "optical illusion?" Yes it is optical. But is it illusion?
The LT performs a change from one inertial reference frame to another inertial reference frame. In reality, we're dealing with a single, non-inertial reference frame. During each leg of the trip, the traveler will agree with their corresponding inertial frame about everything they see, but they will not agree with the "inbound inertial frame" about events that occurred during a different leg of the journey. This is subtle, but important. Mathematically, switching reference frames changes everything. Physically, it changes future measurements, but it can't rewrite previous ones. Our traveler will simply disagree with another inbound traveler who's been in inertial motion the whole time, when it comes to events that were observed before they matched speeds. Their two reference frames overlap for a while, but they aren't the same!
>> (1) It gives the impression that the traveling twin actually "experiences" this length contracted distance through which he is traveling.
The traveling twin does measure the station to be only 0.5 ly from Earth during both the outbound and inbound legs, so I'm not sure what you mean by "experience".
>> (2) it does not not really explain the 1.73 year discontinuity between events A and C
Relativity of simultaneity, and/or (pseudo-)gravitational time dilation, depending on your personal aesthetics.
>> (3) It sometimes gives the impression that that the discontinuity cannot be explained and remains a paradox (especially when the traveling twin is put in an enclosed capsule so they cannot look outside.)
Many things in SR give incorrect impressions.
>> (4) They gives the impression that simultaneity is an observable. That somehow events A, B and/or C might be marked in some phenomenological way by the traveling twin that he might say "Oh... That was what was happening on earth when I turned around."
I think the "expanding light sphere" is useful here. It shows why events on the past light cone must be continuous, regardless of what happens with "actual" simultaneity.
After going through all that: if you'll remember, I never said the past light-cone was uninteresting. I merely suggested it might be confusing to talk about that without clarifying that this was not what people generally mean when they talk about SR effects.
Your next paragraph is harder to deal with, because you seem to be subtly trying to reject the results of the Lorentz Transformation(Response below: "not in the slightest"). Your claim (seems to be that the traveling twin would NOT see the same thing; and would not measure the same coordinates of observed events, as a momentarily comoving inertial observer occupying the same point in space at the same moment. (Response below: "Yes they certainly would.")
I hope you'll think more about that.
You said: "Physically, (The Lorentz Transformation) changes future measurements, but it can't rewrite previous ones"
I agree that if the traveling twin records what he is seeing during his journey, the Lorentz Transformation won't go back and modify the tape of what he has recorded.
But the Lorentz Transformation shifts the coordinates of events whether they are in the future or the past. The traveling twin will not disagree with a momentarily comoving inertial observer about the coordinates of events that are currently being observed by temporarily adjacent cameras. (Response below: "Indeed, I explicitly said this.")
Now as for the expanding light sphere concept, we can play with that one too.
Let's say that one year ago, a transmission went out from earth.
(t,x)=(-1,0)
Our intrepid traveling observer accelerates to 0.866c. Where did the event happen now?
The center of the sphere moves to
(t,x)=(-2,1.73) (Response below: "Future observations will be consistent with that, yes")
So the radius of the sphere would now be 2 light years. The back end edge of the sphere is right behind you, just 0.27 light years behind our observer. And the far end of the sphere is 3.73 light years ahead of him. Counter intuitively, the only way he can return to the center of the light-sphere is by accelerating AWAY from the center of it.
Sure, but then it's not a single SR "event". But yes, I certainly agree that in that case it would "still be on" at all three of those points in time.
>> Your next paragraph is harder to deal with, because you seem to be subtly trying to reject the results of the Lorentz Transformation.
Not in the slightest!
>> Your claim seems to be that the traveling twin would NOT see the same thing; and would not measure the same coordinates of observed events, as a momentarily comoving inertial observer occupying the same point in space at the same moment.
Yes, they certainly would! I explicitly said that they would agree on all observations that happened while they were in fact comoving, but that doesn't mean they agree about the past, frombefore they were comoving.
The Lorentz Transformation deals with transforming between twoinertial reference frames. When you stitch together multiple inertial frames to handle a single non-inertial one, you have to be careful about what is and what isn't valid to say.
>> But the Lorentz Transformation shifts the coordinates of events whether they are in the future or the past.
Yes, but the LT deals with inertial frames. Our traveler's frame istemporarily mimicking an inertial frame, but it's not one. It's still hugely useful to do this, and the traveler's frame will (by necessity) agree with the inertial frame it's currently mimicking for events measured during the time of mimicking, but no such requirement exists for agreement regarding events elsewhere.
>> The traveling twin will not disagree with a momentarily comoving inertial observer about the coordinates of events that are currently being observed by temporarily adjacent cameras.
Indeed. I explicitly said this.
>> Let's say that one year ago, a transmission went out from earth.
(t,x)=(-1,0)
Our intrepid traveling observer accelerates to 0.866c. Where did the event happen now?
Isn't
>> The center of the sphere moves to
(t,x)=(-2,1.73)
Future observations will be consistent with that, yes, if we define the traveler's velocity to be in the positive direction relative to Earth.
>> So the radius of the sphere would now be 2 light years. The back end edge of the sphere is right behind you, just 0.27 light years behind our observer. And the far end of the sphere is 3.73 light years ahead of him.
Sure.
>> Counter intuitively, the only way he can return to the center of the light-sphere is by accelerating AWAY from the center of it.
I'm not sure what you mean by this. There's no way to "return to" a past point in space-time. If you're just referring to a point in space, then that's totally subjective. If you've "returned to" that point in one reference frame, you haven't returned to it in another. So I'm not sure what your point is. In the new comoving inertial frame, the Earth was ahead of them and moving towards them when it gave off the pulse, and then passed by them (so it's not behind them). There's nothing at all weird about having to accelerate away from where Earth used to be in order to catch up to where Earth is now.
>>I'm not sure what you mean by this. There's no way to "return to" a past point in space-time.
I mean "return to the center of an expanding light sphere."
Erik said:
>>So I'm not sure what your point is.
To me, it looks like you got exactly my point. You took the event at (t,x)=(-1,0) and did a Lorentz Boost on it and got
We've agreed on the mathematical result; but in describing it, I'm still not sure whether our difference is semantic (where we just disagree on the meaning of the word "inertial reference frame", or philosophical, (where we really disagree on the nature of time and simultaneity.)
Erik said:
>>When you stitch together multiple inertial frames to handle a single non-inertial one, you have to be careful about what is and what isn't valid to say.
I agree that sometimes you have to be careful about what is and what isn't valid to say... But in this case saying "he switched inertial reference frames when he changed velocity" is a perfectly valid thing to say.
I say the traveling twin is "switching reference frames."
You disagreed with me and said to call the traveling twin's observations a "single non-inertial reference frame."
To me, the amount of effort involved in trying to conceptually stitch together observations in several different reference frames to form a "single inertial reference frame" is not generally worth the effort...Gulstrand-Painle
But regarding our conversation, now you've checked; taking an event at (t,x) = (-1,0) and done a Lorentz boost by +0.866c, and found that event was moved to (-2,1.73). So on the math we are agreed.
But if I understand correctly, you are arguing that the following statement is invalid: "We have switched to a reference framewhere that event happened longer ago, and the sphere is twice as big."
Instead, you suggest the following correction: We have not switched to a different inertial reference frame, but rather, "future observations are consistent with the event being at (-2,1.73)."
So you're saying we haven't switched reference frames, but all of our observations will just look as though we had switched reference frames?
Is this a semantic argument where I'm just using a word that you define differently than me--(maybe your definition of inertial reference frame contains some extra detail* that my definition doesn't have.) or is it a genuine philosophical difference?
In any case, let me make one correction... "CURRENT" observations will be consistent with the event being at (-2,1.73)...
Our "past" observations will be based on whatever inertial reference frame we were in when we made them. Our "future" observations will be based on whatever inertial reference frame wewill be in when we make them. But our "current" observations will be consistent with the inertial reference frame we are in NOW.
(*Editing note: changed the word "garbage" to "detail"... Uncalled for choice of words.)
Colloquially, calling it "switching reference frames" is fine. For most purposes, what you're doing is entirely equivalent to that. I've used that phrasing myself, many times. But, there are times when itmatters that you're just temporarily comoving with a (new) inertial observer, but that your frames don't have the same history. You'll agree on future measurements, but not on past ones, and yournarrative for how future measurements come about will not be the same.
Most of the time, those narratives may not be relevant, but if your entire point is past events jumping around in time, then yes, it matters.
>> So you're saying we haven't switched reference frames, but all of our observations will just look as though we had switched reference frames?
Any new observations will be the same as for the comoving inertial frame (as long as there is no relative acceleration), but that doesn't mean that they map backwards to the same event coordinates. For one observer, there was a large pseudo-gravitational field at some point in the past, while for the other, there wasn't. So, when you both "subtract off" the light-travel delay, you get different things. You can simplify calculations by pretending that there was no field in your past and that history has been rewritten to align with your new frame, but that's a mathematical trick for convenience, it doesn't describe the physical situation.
>> Is this a semantic argument where I'm just using a word that you define differently than me--(maybe your definition of inertial reference frame contains some extra detail* that my definition doesn't have.) or is it a genuine philosophical difference?
An inertial reference frame is a coordinate system that corresponds to a (perhaps hypothetical) observer who never has accelerated and never will accelerate. The term is often used in the context of observers who are just inertial for some period of time, and this isusually fine, because we don't care about the parts where the frames differ. This is not such a case, because you are explicitly drawing attention to events that happened before the relevant leg of inertial motion began. You're effectively running a linear extrapolation backwards, beyond the region where the data follow that line.
>> Our "past" observations will be based on whatever inertial reference frame we were in when we made them. Our "future" observations will be based on whatever inertial reference frame wewill be in when we make them. But our "current" observations will be consistent with the inertial reference frame we are in NOW.
Any observer can measure any events in any reference frame they like. If they switch from one to another, then coordinates of things will change, but that isn't actually a physical change. What'sphysically relevant is the rest frame of the observer, and thatdoesn't involve re-calculating coordinates of past events whenever the observer accelerates. If you're in Denver, and hitchhike to Lincoln, getting picked up (obviously) doesn't mean that you suddenly used to be in Las Vegas, even if the truck that picked you up was. So, while the
If you switch over to the truck's frame entirely, for computational convenience, any jumping around of past events has no physical significance, just like the sudden change in coordinates if you switched to the ISS frame wouldn't have physical significance.
>> (*Editing note: changed the word "garbage" to "detail"... Uncalled for choice of words.)
I appreciate it. I know that disagreements like this can get frustrating, on both sides.
That puts it fairly succinctly. We seem to be agreed that the Lorentz Transformation causes coordinates of events, both future, past, and present, to "jump around" in space, and time. You don't seem to have any argument with the math... just the words I'm using.
>> there are times when it mattersthat you're just temporarily comoving with a (new) inertial observer, but that your frames don't have the same history.
When, exactly does it matter? My body consists of oodles and oodles of atoms that have been blasted out of supernovae billions of years ago... They are temporarily co-moving along with my body. Now, the path the atoms have taken may affect the half-lives of the isotopes.. It might affect their ages, and their experiences... But it does not affect the coordinates of events in history relative to me. The only thing that affects that is my current position, velocity, and facing. The path my atoms took to arrange themselves to my current position, velocity, and facing is irrelevant.
>>If you're in Denver, and hitchhike to Lincoln, getting picked up (obviously) doesn't mean that you suddenly used to be in Las Vegas, even if the truck that picked you up was. So, while the x=0 point of the truck's inertial frame corresponds to Vegas at some point in the past, the x=0 point of your rest frame doesn't, even though you and the truck are now co-moving.
Aha... Very good example. All observers will obviously agree that you were not in Las Vegas. But that is not because Las Vegas was never here. It's because when Las Vegas was passing here, you weren't here. You arrived in the truck after Las Vegas passed by it, and stayed in the truck. But if you use the truck's reference frame, and it moves with constant velocity, then Las Vegas was definitely "here" before you got here.
By the way, a few years ago I put an article about this particular topic on my website called Galilean-Transformation. (Skip the first several paragraphs and read the "simple seeming question")
You, as a conscious observer, have some acceleration history. If you want to measure past events in your rest frame, you can't use your current comoving inertial frame. Or, rather, you can use whatever frame you like, it just won't be your rest frame.
>> But if you use the truck's reference frame, and it moves with constant velocity, then Las Vegas was definitely "here" before you got here.
This is my entire point. The truck's reference frame is not the same as your reference frame at all times, they just happen to overlap for some range of times because you are comoving.
>> (Skip the first several paragraphs and read the "simple seeming question")
For the first question, both A and B are true in your rest frame. I agree with you on #2.
I think it really highlights the subtleties of the issue involved.
"Let's say you have a trolley that goes by your house every morning. It travels in a perfectly straight line travelling at exactly 5 miles per hour. It never speeds up, and it never slows down. It passes by your house, and you have to really run to catch up and get on it. There are two events in this little thought-experiment. (1) You get up in the morning, and (2) three hours later you get on the trolley. Now, obviously, before you get on the trolley, you are at your house, so you got up "Here" and "three hours ago."
Once you are on the trolley, from your new perspective, onboard the trolley, where did you get up this morning?
A. At your current position
B. 15 miles in front of the trolley
C. 15 miles behind the trolley
My answer is Answer B. There's no question to me here. When the observer jumps onto the trolley, his perspective changes, and the event jumps from "here" to "15 miles away."
Now, don't get me wrong... I fully realize how preposterous this answer sounds. But I am doubling down on it because
#1, it's correct, mathematically.
#2, it's intuitively surprising
#3, when you work with events in the observer's past-light-cone, it actually matters; in predicting what the traveling twin sees.
#2 This has little bearing on correctness, one way or the other, in the context of SR.
#3 If you want to calculate what they see using an inertial frame, then yes, that's the one to use. But that is not their actual rest frame; it's just comoving with their rest frame at that moment. They will see the same things, but they will infer different event locations in spacetime from those same observations, because one is in an inertial frame and the other is not. All of this is fine.
>>An inertial reference frame is a coordinate system that corresponds to a (perhaps hypothetical) observer who never has accelerated and never will accelerate. The term is often used in the context of observers who are just inertial for some period of time, and this is usually fine, because we don't care about the parts where the frames differ. This is not such a case, because you are explicitly drawing attention to events that happened before the relevant leg of inertial motion began. You're effectively running a linear extrapolation backwards, beyond the region where the data follow that line.
Based on that definition, I'll offer this modification:An inertial reference frame is a coordinate system that corresponds to a hypothetical observer who never has accelerated and never will accelerate. A real observer can be said, for instance to be "in the house's inertial reference frame" if he is momentarily moving at the same velocity as the house... and "in the trolley reference frame" if he is momentarily moving at the same velocity as the trolley. They will agree (except, possibly for translation, and rotation) on the coordinates of all events in the universe for that single moment that they are momentarily comoving. And this is true even if theyexplicitly draw attention to events that happened before the observer entered the reference frame of the house or the trolley.
>>Only if he abandons his own rest frame for a comoving inertial frame. It is in no way unambiguously mathematically correct; it depends entirely on whether or not you choose to abandon your rest frame.
The problem specifically says "Once you have boarded the trolley, from your new perspective..." The specification of the problemforces the observer to abandon the house reference frame and adopt the trolley reference frame.
>>#2 This has little bearing on correctness, one way or the other, in the context of SR.
To put it more mathematically, the matter in question is whether events with coordinates where t<0 are in the (functional) domain of the Lorentz Transformation equations. Is your claim that they are not in the domain? My claim is that they are in the domain. This question DOES have bearing on correctness in the context of SR, if you are claiming that the results of Lorentz Transformations are incorrect or ambiguous for all events with t<0!
>>they will see the same things, but they will infer different event locations in spacetime from those same observations, because one is in an inertial frame and the other is not.
They will see the same things, but they will infer different locations in spacetime because one of them embraces the Principle of Relativity, and the other does not.
If I ask the observer right after he jumps on the trolley, "When and Where did you get up this morning" he will probably say, pointing to his house, "Here, three hours ago."
No, they will only agree on events that occurred during the time that they were comoving, unless the accelerating person abandons their own rest frame in favor of the inertial house frame.
>> The problem specifically says "Once you have boarded the trolley, from your new perspective..." The specification of the problem forces the observer to abandon the house reference frame and adopt the trolley reference frame.
You're speaking as if those are their only two options. If you're talking about what the observer experiences, then you're talking about that observer's rest frame. It is neither the nouse frame nor the trolley frame. It is comoving with the house frame for a while, and later it is comoving with the trolley frame, but it is neither.
If you instead decide to use those inertial reference frames (house and trolley) as a convenient shortcut, then you have to use the one that is relevant at the time of the event in question. From the traveler's perspective, there is no "change of frame", just a uniform gravitational field.
>> To put it more mathematically, the matter in question is whether events with coordinates where t<0 are in the (functional) domain of the Lorentz Transformation equations. Is your claim that they are not in the domain? My claim is that they are in the domain.
This has nothing to do with our actual disagreement. We entirely agree on the spacetime coordinates of the events in the house inertial frame and in the trolley inertial frame. We are disagreeingon whether or not it makes sense to treat the traveler as if they hadalways been in the trolley frame. As I've said, any observer can use any frame they like, but if you want this to represent the traveler's rest frame, then the trolley frame can't be blithely applied to events that occurred before the traveler got on the trolley; it's outside of the validity of that particular approximation (approximating the traveler's rest frame as temporarily inertial).
>> This question DOES have bearing on correctness in the context of SR, if you are claiming that the results of Lorentz Transformations are incorrect or ambiguous for all events with t<0!
I am making no claims of the sort. I have explicitly said the opposite. I'm not sure how we're going to resolve this if you don't understand what my objection is.
>> They will see the same things, but they will infer different locations in spacetime because one of them embraces the Principle of Relativity, and the other does not.
No idea what you mean by this. I'm applying Relativity throughout.
>> If I ask the observer right after he jumps on the trolley, "When and Where did you get up this morning" he will probably say, pointing to his house, "Here, three hours ago." x=0,t=−3 And when I point directly away from his house and say... "No, you got up 15 miles that way, three hours ago" x=15,t=−3 obviously people will look at me like I'm a loon. But that isn't because the question is ambiguous. It's because I'm drawing my inference by using the Galilean transformation equations to get the correct answer. He is drawing his inference by not using the Galilean transformation, and he does not get the correct answer.
No, you both get the correct answer, in different reference frames. This is totally fine. Someone riding on another trolley, going the opposite direction, would give yet another answer. Your rest frame and his rest frame are not the same, they're just comoving at the moment. A Galilean transformation assumes inertial motion over (at least) the entire time interval in question.
If you agree with the statement above, then our disagreement is all semantics, and we're just disagreeing on the meanings of words.
If you disagree, then I will continue to try to understand your objection. For starters, I don't understand what you mean by the traveling twin's "rest frame" Is this something you can show me a diagram of, or describe in greater detail? Because right now, I'm not at all sure it's a valid or well-defined concept.
Also, please explain what you mean by "during the time they were comoving" You mean between the planes of simultaneity defined at the instant of their meeting and the instant of their departure?
If you ask "how far away is the Earth?", then the answer will increase and then decrease down again over the course of the acceleration (length contraction, where
>> For starters, I don't understand what you mean by the traveling twin's "rest frame" Is this something you can show me a diagram of, or describe in greater detail? Because right now, I'm not at all sure it's a valid or well-defined concept.
It is a (not necessarily inertial) reference frame in which the traveler is always at rest. It's basically "following the experience of the traveler", except focused on events rather than perceptions (which can get muddled up by light propagation delay).
>> Also, please explain what you mean by "during the time they were comoving" You mean between the planes of simultaneity defined at the instant of their meeting and the instant of their departure?
"Comoving" as in "moving together (as one)" as in "at rest relative to one another".
Let's just review the exact agreement we had here... (I copied the full text of the relevant comment up into my answer)
We said, from the point-of-view of the traveling twin visiting the space-station 1 light year away, and traveling at 0.866c to get there, the image of the earth (and all associated events) were at these coordinates:
(1) Before visit: (t,x) = (-0.268 light-years, -0.268 years)
(2) During visit: (t,x) = (-1 light-years,-1 years)
(3) After visit: (t,x) = (-3.73 light-years, -3.73 years)
Do you still agree with those coordinates?
And do you agree that this event "jumped" due to the acceleration of the traveling twin?
If you choose to do your calculations in an inertial frame, instead of the traveler's rest frame (which will often make sense), then there will be a jump (as seen). If you instead want to know about the actual experience of the traveler, the actual rest frame, the coordinates of past events don't change; after all, according to the traveling twin, he isn't doing anything, it's the rest of the Universe that's accelerating!
But I think I understand how you are thinking of your construction... Here is the link to WWoods diagram on Wikipedia:File:Twin 5.png
My impression of your "rest frame" is that you want to take these diagrams, and chop them along the lines of simultaneity and stitch together the result like so:
One reason I would object to that is that in this “rest frame of the traveling twin” all of the events that happen on earth between Event A and Event C are unrepresented. Also, if the traveling twin were to look in the opposite direction, some events would cross the line of simultaneity in the other direction, and happen twice.
Let me suggest another version of this "traveling twin's rest frame."
As a solution, I would offer the following modification.
Instead of stitching together the rest-frame along the line of simultaneity, what if we stitched it together along the line of x=-ct (e.g. the locus of events that are actually observed by the traveling twin at the instant of his acceleration when he looks toward earth.)
For that, we'll start with another diagram
File:Three frames.JPG
This diagram does not share the same labels as the earlier one. Instead, event B here is the event-(stream) on earth observed by the traveling twin at the time of the turnaround.
Instead of chopping the two diagram along the line of simultaneity, I chop it along the line EB and combine them like this.
We still have this line from beginning-to end of the trip for the traveling twin represented as a straight line. But now it shows all of the events on earth observed by the traveling twin instead of skipping a bunch.
Also it has the advantage of actually showing, rather than hidingthe discontinuity in the motion of the earth in the reference frame(s) of the accelerating observer.
By using
Now, just to be clear, though. Just because I can stitch together parts of two diagrams to make the world-line of an accelerating observer look straight, I still think it is probably better to just say he is "switching inertial reference frames" because even those events that are far in the observer's past are still in the locus of events affected by the Lorentz Transformation.
There's nothing "mine" about it. I did not invent non-inertial reference frames, nor did I invent the idea of using them here. It's one of the two standard ways of resolving the Twin Paradox, the other being to show that certain aspects of it must be equivalent to the Three Brothers thought-experiment (with Earth, Outbound, and Inbound brothers; the Outbound and Inbound brothers sync clocks as they pass each other). Einstein himself preferred the non-inertial rest frame resolution (Twin paradox).
>> One reason I would object to that is that in this “rest frame of the traveling twin” all of the events that happen on earth between Event A and Event C are unrepresented.
This is only in the (unphysical) case of instantaneous acceleration. If the acceleration is just very very rapid, events on Earth occur at super-speed due to (pseudo-)gravitational time dilation.
>> Also, if the traveling twin were to look in the opposite direction, some events would cross the line of simultaneity in the other direction, and happen twice.
That's quite interesting (I hadn't thought about that possibility), and worth looking more closely at, to see what is and isn't happening there. But, it is not in and of itself problematic as long as all such events are outside the traveler's causal influence / light cone.
>> [diagrams]
I'm not entirely sure that I followed what you were getting at with your diagrams, but that's probably my fault, because I don't have the time to work through your diagrams at the moment. I am also somewhat reluctant to go too far down some alternate path, when my original definition seems unambiguous to me. But, this is open to discussion.
>> Just because I can stitch together parts of two diagrams to make the world-line of an accelerating observer look straight, I still think it is probably better to just say he is "switching inertial reference frames" because even those events that are far in the observer's past are still in the locus of events affected by the Lorentz Transformation.
The LT tells you how to switch between two inertial frames, and is valid for all times and places. That does not, however, govern how the traveler sees events, because the traveler is not inertial. The math works, it's just not an accurate model of the person we're trying to talk about! That is not to say that it isn't a useful thing to do, you just have to either restrict your usage to time intervals in which there is no acceleration, or, if you do include a period of acceleration, you have to remember that what you're doing is a useful mathematical trick, and does not literally represent the traveler's actual experiences.
You are correct, your definition is unambiguous.
>>There's nothing "mine" about it.
I don't agree. Whether you inherited this idea from someone else, or you came upon it independently, your original definition is your idea...
You asked me earlier, to understand your objection.
If I am to understand your objection I would like to be able to credit you with your own idea, acknowledge it, and put it into my own words, so that I can show you precisely how it differs with my own.
Your idea is not wrong... It is not ambiguous. But I still find it misleading, and I want to explain why.
In my words, your idea is To create a "noninertial frame" we can stitch together spacetime diagrams so that we show all eventsoccurring simultaneously with the traveling twin's observations.
That part is unambiguous...
It was also unambiguous when you said that the earth will actually jump away to its full uncontracted distance, and then return to its contracted distance during the acceleration.
It was also unambiguous when you said that the earth ages quite suddenly as its simultaneous distance increases and decreases suddenly.
I don't disagree with you on these things. I agree with you on all of these issues. I merely point out the underlined words above "simultaneously", "during", and "as". These all imply simultaneity. We have an agreed upon, well-defined and unambiguous definition of simultaneity.
However, even though your description is well-defined, and unambiguous, I don't think that it is that helpful in describing what is really going on.
(1) No observer will have any empirical experience of the earth suddenly aging when the traveling twin accelerates.
(2) No observer will have any empirical experience that the earth suddenly jumps away to it's uncontracted distance, and returns to it's contracted distance during the acceleration of the moving twin.
(3) In the case of instantaneous acceleration, some events disappear from the diagram altogether, and some happen twice.
(4) In the case of non-instantaneous acceleration, all events beyond the Rindler Horizon will progress backwards in time from the future to the past.
But I am preserving this idea of taking the traveling twin's world-line and making it straight.
However, I want to make a change to it. Instead of stitching together events which are simultaneous with the observer's observations, I want to stitch together the events which are in the past-light-cone of the observer's observations
This corrects the problems above
(1) The earth leaps away from the traveling twin in agreement with his empirical experience.
(2) The earth does not suddenly age during the acceleration, but instead because it travels further on the second leg of the journey.
(3) All of the events cross the past light-cone only once, even in the case of instantaneous acceleration-there is a discontinuity in their position but not in their existence.
(4) All of the events cross the past light cone from the future towards the past
>>I'm not entirely sure that I followed what you were getting at with your diagrams,
Here, this video might help:
I'm not sure what you mean by "all events occurring simultaneously with the traveling twin's observations". What I would consider to be the "natural" interpretation of that description would be incorrect (it ignores light propagation time, which non-inertial frames do not), but I'm open to being corrected on what you meant.
>> However, even though your description is well-defined, and unambiguous, I don't think that it is that helpful in describing what is really going on.
You're welcome to find it unhelpful. Einstein, Born, myself, and many others would disagree with you there, though.
>> (1) No observer will have any empirical experience of the earth suddenly aging when the traveling twin accelerates.
The accelerating twin does! Also, empirical experiences of length contraction vary wildly (due to light propagation delays), but it is still considered to be a useful concept and description.
>> (2) No observer will have any empirical experience that the earth suddenly jumps away to it's uncontracted distance, and returns to it's contracted distance during the acceleration of the moving twin.
The accelerating twin does!
>> (3) In the case of instantaneous acceleration, some events disappear from the diagram altogether, and some happen twice.
If you take the instantaneous limit, then the events don't disappear, they just go by infinitely quickly. This is because, for any non-zero acceleration interval, they go by at finite speed. The limit is well-defined.
>> (4) In the case of non-instantaneous acceleration, all events beyond the Rindler Horizon will progress backwards in time from the future to the past.
Formally, such events are "elsewhere", not "future" or "past", no? If it's outside the light cone, I'm fine with it.
Also, doesn't the same thing happen with your description, except without a mechanism?
>> But I am preserving this idea of taking the traveling twin's world-line and making it straight.
Not just straight; specifically the line
>> Here, this video might help
I'll take a look later.
Sounds accurate.
>>Not just straight; specifically the line
x=0.
Agreed
>>I'm not sure what you mean by "all events occurring simultaneously with the traveling twin's observations".
Observations are events. Events which occur simultaneously with those observations are also events. (Edit... Ohh, I understand the confusion... Here, I mean "the event of the observer observing... Not the events being observed."
>> (it ignores light propagation time, which non-inertial frames do not),
Your idea ignores light propagation times,.
My idea uses the past light-cone of the observer to take into account the light propagation time.
>>Also, doesn't the same thing happen with your description,
For a given observer, distant events can cross the plane of simultaneity (t=0) over and over and over again, as an observer accelerates back and forth. But once an event crosses a future light-cone into the present, or crosses a past light-cone into the past, it can never pass that light-cone again.
The exact same mechanism is involved... There's no change in the theory. It's just a choice of which events we view as "important". Your version uses planes of simultaneity (t=0) with the traveling twin's observation as "important" and I'm choosing to treat the past light cone (t=-r/c) of those same observations as "important". There is no quantitative difference between our two ideas.
>>except without a mechanism?
><The accelerating twin does! Also, empirical experiences of length contraction vary wildly (due to light propagation delays), but it is still considered to be a useful concept and description.
In many situations, length contraction is a great concept to highlight... But when discussing the twin paradox, I think it is not the best concept to highlight. In the train-and-tunnel thought-experiment... In the muon half-life experiment, it works. It isn't broken... And it is great for explaining the twin paradox so long as nobody asks "What does the traveling twin see." Because the traveling twin never sees that length contraction. The only time you really see length contraction is when an object is passing by, perpendicular to your line of sight:
I'm confused as to what events you are saying are simultaneous with a given observation.
>> The only time you really see length contraction is when an object is passing by, perpendicular to your line of sight
I'm not sure what you mean by that. You can see something moving towards you at a certain speed for a certain period of time, and then hit you. That tells you how far away it started. You then notice that this distance is contracted relative to what you measured at rest relative to the object.
Moving on to actual content:
At around 6 mins, you start drawing light-cones. That's all well and good in inertial rest frames, but not so much in the traveling twin's rest frame; gravitational fields affect the trajectories of light beams, and that includes the pseudo-gravitational field experienced by the traveling twin. A standard Minkowski diagram will not accurately describe this.
I address this issue in the video:
"If this were a good representation of an inertial reference frame, for instance it stands to reason that here, the traveling twin should actually see three images of the earth. One image from earth going along path A-B, one as it goes along path B-toC, and one as it goes from C-to-D. If you want to call me out on this, you may feel free to do so, I am deliberately abusing this construction... We already defined events A to B to C to be simultaneous with event E... The coordinates of those events were valid during the turnaround but they are no longer valid between during the return trip."
But I want you to call ME out on this. I don't want you to keep saying "Born and Einstein did it, so it is okay." I know they did it. I know it is not okay. I know why it is not okay to do it that way. And the more important point is, I know how to fix it.
Sorry I didn't have much patience for your comments last night, and ended up deleting three of them and one of my own. Despite the fact that the video lasted only 12 minutes, I spent more than 8 hours making it, and I wasn't really in the mood for "I watched the first six minutes and you're wrong".
Nobody is claiming it is, other than you, implicitly, by trying to represent it with a standard Minkowski diagram. As you noted, that doesn't work. But all that means is that you can't treat non-inertial frames as if they're inertial. That should not be a surprise to anyone.
>> I don't want you to keep saying "Born and Einstein did it, so it is okay." I know they did it. I know it is not okay. I know why it is not okay to do it that way.
There's nothing wrong with what they did. I brought them up in the hopes that you would take what I was saying more seriously, once you knew that the person who came up with the theory in the first place did it that way.
Of course, physics has advanced since Einstein. Einstein is not automatically right about everything. However, I think he can be trusted to correctly apply his own theory!
The Rindler coordinate system is represented by a standard Minkowski diagram.
>>As you noted, that doesn't work. But all that means is that you can't treat non-inertial frames as if they're inertial. That should not be a surprise to anyone.
Did you really only watch the first six minutes of the video? If you watch to the end, you'll see I showed in the video TWO ways of working with the Minkowski diagrams. One which doesn't work, and one which does work.
Yes and no. Remember, you drew in linear light cones! Null geodesics in Rindler coordinates aren't lines. (Plus, this frame isn't even uniformly accelerating.)
>> Did you really only watch the first six minutes of the video? If you watch to the endy, you'll see I showed in the video TWO ways of working with the Minkowski diagrams. One which doesn't work, and one which does work.
I saw both ways, although I'm not sure I followed your reasoning on the second... things seemed oddly patched together, and it didn't seem quite like what I had described. Importantly, no amount of cutting and pasting a finite number of inertial frames will allow you to represent an accelerating one without discontinuities.
The Rindler coordinate system... Hmmm... Perhaps we should discuss the Rindler coordinate system in detail sometime. To my understanding, the Rindler coordinate system consists of the curved world-lines of observers arranged in a "Born Rigid" pattern.
If you want to represent a geodesic in a Rindler coordinate system, though, I don't think I agree that geodesics can be said to be curved... Rather, they are MOVING.
The only things not moving in the Rindler coordinate system are the "family of Rindler observers" and their hyperbolically curved-space-time paths.
>>I saw both ways, although I'm not sure I followed your reasoning on the second... things seemed oddly patched together, and it didn't seem quite like what I had described.
The "second" example was chopping the Minkowski diagram along the line of the speed of light from event F to event E. It wasn't intended to be what you had described
The "first" example was chopping the Minkowski diagram along lines of simultaneity. It was meant to be exactly what you described. Not just you, but Born and Einstein.
>>There's nothing wrong with what [Born and Einstein ]did. I brought them up in the hopes that you would take what I was saying more seriously, once you knew that the person who came up with the theory in the first place did it that way.
I don't really know what else to do to take them seriously. I am presenting the case in the video as I understand it as fairly as possible.
Max Born died in 1970, before I was born, and Einstein passed on before that. I can't really say whether they would have agreed with me.
I know that both of those men probably were much more brilliant than I am, but I have access to better technology, so for instance, I know that neither of them ever wrote a java application that does animated Lorentz Transformations on the fly. Neither of them had access to Mathematica, or Flash. They weren't able to create animations and simulations of their concepts the way I have been able to. So I think it is quite possible that they may have overlooked some things that I have seen. They never got to play "A slower speed of light". They never read Mike Fontenot's "Current Age of Distant Objects." They never read "Relativity Visualized" by L. Carrol Epstein. They never seriously considered the gist of "Relativity, Gravitation and World Structure" by A. E. Milne.
All of these guys are pointing to something that Einstein overlooked. And you're not going to be able to show me a link to a website where they overlooked it. Because if they had considered the problem, and devoted some time to it, they probably would have figured it out. But they never considered the problem.
>>Importantly, no amount of cutting and pasting a finite number of inertial frames will allow you to represent an accelerating one without discontinuities.
Yes, that is important... My point was, though, that if you cut and paste along the light-cone, instead of the line if simultaneity, you can have this unavoidable discontinuity in the correct place.
(Also, as you pointed out earlier, in the limit as the delta V approaches instantaneous, there was going to be a continuous curve; not a true discontinuity.)
"Chopping"? Me, Born, and Einstein? The effects of gravitational time dilation are continuous, not discrete. If you choose to take the instantaneous-acceleratio
>> I don't really know what else to do to take them seriously. I am presenting the case in the video as I understand it as fairly as possible.
You seemed to be approaching the whole thing in terms of trying to explain why it was wrong, instead of trying to understand why it was right. Perhaps I was mistaken in that impression, but hopefully that explains why I "called in reinforcements", as it were.
>> I know that both of those men probably were much more brilliant than I am, but I have access to better technology, so for instance, I know that neither of them ever wrote a java application that does animated Lorentz Transformations on the fly.
Such things can be helpful, but are in no way necessary for analyzing the situation we're talking about. Their analysis was quite sound.
>> Neither had Mathematica, or Flash. They weren't able to create animations and simulations of their concepts the way I have been able to. So I think it is quite possible that they may have overlooked some things that I have seen.
Given that their analysis straight-forwardly works, and that you kept trying to draw straight light cones on diagrams representing non-inertial frames, I find this unlikely.
>> They never got to play "A slower speed of light". They never read Mike Fontenot's "Current Age of Distant Objects." They never read "Relativity Visualized" by L. Carrol Epstein. They never seriously considered the gist of "Relativity, Gravitation and World Structure" by A. E. Milne.
And more's the pity, but it doesn't stop them from knowing how to analyze this situation.
>> All of these guys are pointing to something that Einstein overlooked. And you're not going to be able to show me a link to a website where they overlooked it. Because if they had considered the problem, and devoted some time to it, they probably would have figured it out. But they never considered the problem.
You have yet to demonstrate such a "problem", so I'm not sure what you're talking about.
If you want to bring Born and Einstein into this, please don't invoke gravitational time dilation in a discussion of the twin paradox.
(If you want to claim that gravitational time dilation or pseudo-gravitational time dilation is involved in the twin paradox, then I'll discuss that with YOU.
But you'll have to defend that idea yourself.)
>>You seemed to be approaching the whole thing in terms of trying to explain why it was wrong, instead of trying to understand why it was right. Perhaps I was mistaken in that impression, but hopefully that explains why I "called in reinforcements", as it were.
It was your claim that your method was "unambiguous" that made me made me take you seriously.
Unfortunately, your objections about Born and Einstein disagreeing with me are really ambiguous. I'm not able to take ambiguous objections seriously.
>>you kept trying to draw straight light cones on diagrams representing non-inertial frames
First off, when I "tried" to draw straight light cones on diagrams representing non-inertial frames, there was no "try" about it. I just drew an arrow.
Then I explained why the process failed when the diagram was chopped by lines of simultaneity.
I followed that by explaining why the process succeeded when the diagram was chopped along the light-cones.
(Same video, if you care to see it again)
I don't need to demonstrate the problem, because you're already pointing it out when you say "You can't draw a