[MikeFontenot (OP)]

My coordinate system's horizontal axis gives each instant in the life of the observer (he) who sometimes accelerates (or at least the portion of his life we are interested in, like the portion that includes his trip away from the home twin).  The vertical axis gives the home twin's (her) age, as a function of his age, according to him.

It isn’t a coordinate system if it doesn’t assign 4 coordinates to every event in the region-of-applicability.

What I gave above gives the age of the home twin (her) at each instant in the life of the observer (he) who sometimes accelerates.  Although I haven't been interested in her distance from him, according to him, both the CMIF method and my method DO provide that information, and the two methods give the same value.  For example, if she says that the turnaround point is D lightyears from her, he will say that it is D/gamma lightyears from her.

[MikeFontenot (OP)]

What I gave above gives the age of the home twin (her) at each instant in the life of the observer (he) who sometimes accelerates.  Although I haven't been interested in her distance from him, according to him, both the CMIF method and my method DO provide that information, and the two methods give the same value.  For example, if she says that the turnaround point is D lightyears from her, he will say that it is D/gamma lightyears from her.

The CMIF method and my method give the same answer to the question "What is the distance between the home twin (she) and the traveling twin (he), according to him" ONLY when there is no discontinuity in the CMIF solution.  There is no discontinuity in their separation in the most common version of the twin paradox, where he reverses course at the turnaround point, but comes back at the same speed as he used on the outbound leg.

But if he uses a DIFFERENT speed on the return leg than he used on the outbound leg, the CMIF solution gives a discontinuity in their separation when the speed changes (according to him).  My method never produces a discontinuity.

For example, suppose his speed on the outbound leg is 0.57735 ly/y (gamma = 1.2247).  (That case is advantageous, because the angle of his worldline wrt her worldline (the horizontal axis) is 30 degrees, and can be easily drawn with a 30-60-90 plastic triangle.)  Suppose she is 40 years old when he turns around.  He is 32.66 years old then.  Then suppose his new speed is -0.866 ly/y (gamma = 2.0).  When they are reunited, she is 66.67 years old, and he is 46.0 years old.

For the above case, we can plot their separation at each instant of his life, according to him, for both the CMIF method and for my method.  I call that diagram the "SAAOD", for "Seperation According to the Accelerated Observer Diagram."  Their separation is on the vertical axis, and his age is on the horizontal axis.

The first straight line segment starts at the origin, when he is zero years old, and their separation is zero ly.  It slopes upward at a 30 degree angle.  It reaches its peak when he is 32.66 years old, and their separation is 18.86 ly.

When he changes his speed to -0.866 ly/y, their separation instantaneously drops to 11.55 ly, according to the CMIF method.  Then, from there, their separation declines linearly with a slope -0.866 until their reunion.  (That last straight line segment makes an angle of approximately 41 degrees wrt the horizontal axis).

In contrast, in my method their separation doesn't instantaneously change.  Instead, it decreases linearly with a slope of -2.047 from its peak at 18.86 until his age reaches 38.85 years old, and their distance reaches 6.188. (That endpoint corresponds to when he receives a light pulse that she sends at his turnaround).  Then, at that point, the slope of the line decreases to -0.866 until their reunion.  The two methods coincide along that last segment.

[evan_au]

This Veritassium video suggests that Einstein did address simultaneity at a distance, by saying it is impossible to measure, and defining the convention that the speed of light is identical in every direction.

This video was originally posted by wolfekeeper, in this thread: https://www.thenakedscientists.com/forum/index.php?topic=82170.0

[MikeFontenot (OP)]

This Veritassium video suggests that Einstein did address simultaneity at a distance, by saying it is impossible to measure, and defining the convention that the speed of light is identical in every direction.
[...]

At about the 15 minute point in the video, the author says that it's possible that in some directions, we're seeing extremely distant galaxies as they were in the distant past, whereas in other directions we're seeing extremely distant galaxies as they are now.  It seems to me that that possibility can be discounted because of the fact that what we see in all directions is amazingly uniform, and that seems to contradict our understanding that the universe has greatly changed since the big bang.

[charles1948]

This Veritassium video suggests that Einstein did address simultaneity at a distance, by saying it is impossible to measure, and defining the convention that the speed of light is identical in every direction.
[...]

At about the 15 minute point in the video, the author says that it's possible that in some directions, we're seeing extremely distant galaxies as they were in the distant past, whereas in other directions we're seeing extremely distant galaxies as they are now.  It seems to me that that possibility can be discounted because of the fact that what we see in all directions is amazingly uniform, and that seems to contradict our understanding that the universe has greatly changed since the big bang.

Can't the amazingly uniform appearances that we see in all directions, only be ascribed to one of these two theories:

1. The Earth is in a special position, at the centre of an Big Bang-originated Expanding Universe; or:

2. There was no "Big Bang", the Universe has always existed in a Steady State.

I'd put my money on 2, the Steady State theory.  It gets round questions like "But happened before the Big Bang"?

[Kryptid]

Can't the amazingly uniform appearances that we see in all directions, only be ascribed to one of these two theories:

1. The Earth is in a special position, at the centre of an Big Bang-originated Expanding Universe; or:

2. There was no "Big Bang", the Universe has always existed in a Steady State.

Nope. The Big Bang theory, as is, has no issues explaining cosmic uniformity. It's important to remember that the Big Bang didn't happen at one particular location in space: it happened at all locations in space.

[chiralSPO]

This Veritassium video suggests that Einstein did address simultaneity at a distance, by saying it is impossible to measure, and defining the convention that the speed of light is identical in every direction.
[...]

At about the 15 minute point in the video, the author says that it's possible that in some directions, we're seeing extremely distant galaxies as they were in the distant past, whereas in other directions we're seeing extremely distant galaxies as they are now.  It seems to me that that possibility can be discounted because of the fact that what we see in all directions is amazingly uniform, and that seems to contradict our understanding that the universe has greatly changed since the big bang.

Can't the amazingly uniform appearances that we see in all directions, only be ascribed to one of these two theories:

1. The Earth is in a special position, at the centre of an Big Bang-originated Expanding Universe; or:

2. There was no "Big Bang", the Universe has always existed in a Steady State.

I'd put my money on 2, the Steady State theory.  It gets round questions like "But happened before the Big Bang"?

There are (at least) three other options, and I suspect that these options more likely explains what we observe:

3. There are NO special points in the universe (either due to closed geometry or unbounded size), in which case ALL planets (indeed, all points) in the universe would be equally validly the "center" of the universe.

4. There are VERY FEW special points because the universe is finite, but immensely vast, and the earth is deep within a finite region of space that appears uniform, and we are so fat from the edges that our position within the volume is meaningless (ie, basically option three without needing to actually be closed or boundless). As an analogy, I present this single crystal of diamond (https://en.wikipedia.org/wiki/Cullinan_Diamond) which apparently had a mass of 621 g. That means that there are about 3×10²⁵ carbon atoms in a nearly perfect crystalline lattice (every carbon is equivalent to every other carbon except at the very edge, and 99.999% of the atoms are separated from the edge of the crystal by at least 1,000 intervening carbon atoms. Each of those 99.999% carbon atoms could reasonably appear to be the "center" of the crystal when viewed on an atomic scale, and there would be no way for a single carbon atom to "know" where the closest edge was, or even if there is an edge.)

5. There was no big bang. Clearly the universe is NOT in a steady state, so there must be some other explanation. There are several other competing hypotheses, but each has its own host of problems. I am personally intrigued by the idea that our concept of time is very wrong. (this idea is explored somewhat here: https://www.thenakedscientists.com/forum/index.php?topic=73163.msg540694#msg540694)

[chiralSPO]

And as one of the properties of "light" is that it travels at a fixed speed - this is bound to influence, and perhaps dominate, our ideas about how the Universe operates.

But suppose we weren't "visual" beings.  We didn't have eyes.  Then light, and its speed, would mean nothing to us.

Suppose we were intelligent creatures who relied entirely on other media, such as sound-waves, or heat-waves  Or even - though this is taking a science-fictional viewpoint - an ability to detect gravitational-waves?

Surely then our interpretation of the Universe might be completely different.

I hope you see what I'm driving at - that our fixation on "light" and its invariant speed  may be leading us astray.

It's not just the speed of light. It's the speed of any massless particle, and is the unreachable limit of any massive particle. It's not just special because of light. The speed of sound waves depends highly on the medium through which they travel, and they are unlikely to be of much cosmological importance, because sound doesn't really propagate through the vacuum of space.

[MikeFontenot (OP)]

Can't the amazingly uniform appearances that we see in all directions, only be ascribed to one of these two theories:

1. The Earth is in a special position, at the centre of a Big Bang-originated Expanding Universe; [...]
[...]

I think the Big Bang is regarded as having occurred "everywhere at once", even though that "everywhere" was infinitesimally small initially.  I.e., no location in the current universe is farther from the Big Bang than any other place.

[Halc]

At about the 15 minute point in the video, the author says that it's possible that in some directions, we're seeing extremely distant galaxies as they were in the distant past, whereas in other directions we're seeing extremely distant galaxies as they are now.

No mention of 'distant galaxies' was made. This is special relativity being discussed, and while it might be meaningfully applied to nearby galaxies, it certainly does not apply to the distant ones, which illustrates that at cosmological scales, the Minkowski metric cannot be applied.

That said, the video mentioned stars 'hundreds of light years distant', and I disagree with little of it despite the often vague wordings and careless switching of implied frame references being done.
It said that we could see Mars as it is 'now', which is false for the same reason that no massive object can move at light speed.  But it can be arbitrarily close, so the time between the Mars even and our observation of it could be any arbitrarily small time, but not zero time as they seem to emphasize. We cannot see Mars at it is 'right now'.

It seems to me that that possibility can be discounted because of the fact that what we see in all directions is amazingly uniform

The point of the video was that a difference in the one-way speed of light would have zero effect on what you measure, so discounting the possibility due some measurement seems only to demonstrate that you didn't get that point.
OK, the video was posted first to the thread which was specifically about the ability to measure the one way speed of light, and only posted here because Einstein knew it couldn't be measured and was very careful to define simultaneity by a chosen convention and not by any assertions that there is a way to know the ordering of a pair of events, even relative to a specific frame of reference.

that seems to contradict our understanding that the universe has greatly changed since the big bang.

Special relativity has never been applicable to the universe at large.

[MikeFontenot (OP)]

(my post #41)
[...]
In contrast, in my method their separation doesn't instantaneously change [when he changes is velocity from 0.57735 ly/y to -0.866 ly/y].  Instead, it decreases linearly with a slope of -2.047 from its peak at 18.86 until his age reaches 38.85 years old, and their distance reaches 6.188. (That endpoint corresponds to when he receives a light pulse that she sends at his turnaround).

I have just verified that the above segment is indeed a straight line.  The verification is similar to what I did for the linearity of the middle segment in the age correspondence diagram (the "ACD"), which I described in Section 8, titled "Pulses Partly in Both Halves of the Minkowski Diagram", of my monograph.

On the Minkowski diagram, first find the vertical line descending from the traveler's (his) worldline at his turnaround point, down to the horizontal axis (her worldline).  Mark the half-way point on that vertical line, and call it "point Q".  Then draw a light pulse being sent by her, at such a time in her life that the pulse passes through that half-way point, and continues on until it reaches his worldline.  Mark that point on his worldline as point R.  His age is 35.75 then.

According to her, their separation at the turnaround is 23.094 ly.  And according to her, that half-way point is at distance 11.547 ly from her.  According to a perpetually-inertial observer traveling at 0.57735 ly/y (with gamma = 1.2247), that distance is 11.547 / 1.2247, or 9.43 ly from her.  Similarly, a perpetually-inertial observer traveling at -0.866 ly/y (with gamma = 2.0), concludes that that pulse travels a distance 3.09 ly from point Q until it reaches the traveling twin at point R.  So the traveling twin (he) then concludes that the distance between him and her at the instant he receives that pulse is 9.43 + 3.09 = 12.52 ly.  If, on the SAAOD (Separation According to the Accelerated Observer Diagram), you plot the separation 12.52 ly when he is 35.75 years old, you will see that it does indeed lie on the midpoint of the previously described straight line.

[aujlaakaran0]

I've heard that Einstein wrote a paper in 1907 (2 years after his SR paper, and 8 years before his GR paper) that may have discussed the above result.  I'm currently trying to find that 1907 paper (translated into English, of course).  I'd like to know if he used CMIF (co-moving inertial frame) simultaneity in getting his result, and whether he made any assumptions in justifying that choice.

[MikeFontenot (OP)]

I've heard that Einstein wrote a paper in 1907 (2 years after his SR paper, and 8 years before his GR paper) that may have discussed the above result.  I'm currently trying to find that 1907 paper (translated into English, of course).  I'd like to know if he used CMIF (co-moving inertial frame) simultaneity in getting his result, and whether he made any assumptions in justifying that choice.

That is a quote from one of MY posts, not from aujlaakaran0, so I don't know what's going on here.  I DID find a copy of that 1907 Einstein publication, and have read the part about clocks in the front and rear of a rocket, but I haven't been able to understand it very well.  He doesn't appear to be assuming that an accelerating observer always agrees with the perpetually-inertial observer who is momentarily co-located and stationary wrt to himself.  But Einstein also is doing the analysis only for an arbitrarily small acceleration.

[MikeFontenot (OP)]

In what follows, I refer to the perpetually-inertial twin as "she", and to the twin who instantaneously changes his velocity as "he".

I've worked out an example that shows an instantaneous negative ageing for her, according to him, when CMIF (co-moving inertial frame) simultaneity is used ...i.e., she instantaneously gets YOUNGER according to him, when using the CMIF simultaneity method. And I'll show the results for this example for my simultaneity method, which gives no discontinuities and gives no negative ageing.

First, here's a description of the Minkowski diagram (with time tau on the horizontal axis and separation X on the vertical axis). That always has to be determined first.

The two "twins" in this case aren't really twins ... they are just babies who were born at the same time but 20 lightyears (ly) apart, and with zero relative velocity. This situation continues until they are both 40 years old. We represent this initial situation by drawing a horizontal line on the diagram at the point X = 20 on the vertical axis, extending from tau = 0 to tau = 40. That is the initial segment of his worldline. At the end of that first segment, write 40 immediately above the end of that segment of his worldline to show his age , and vertically below there write 40 immediately below the horizontal axis to show her age.

Then, he instantaneously changes their relative velocity to v = 0.57735 ly/y, and continues that velocity for the rest of their lives. This causes his worldline to slope upward toward the right at a slope of 0.57735 (and an angle wrt the horizontal axis of 30 degrees). Label that point where the second segment of his worldline starts as point T.

Next, draw a 45 degree line starting at the point 40 on the horizontal axis, and sloping upward to the right, representing the worldline of a light pulse that she transmits when she is 40, and that is moving toward him. We then write an equation giving X as a function of tau for that light pulse, and then we write another equation giving X as a function of tau for the upward sloping segment of his worldline. Then, we set those two equations equal (force their X values to be equal). The result gives the value of tau where those two lines intersect ... label that point Q. That point is vertically above the point tau = 87.32 on the horizontal axis. Write that value just below the horizontal axis, vertically below that point of intersection.

We also need to determine their separation according to her (the value of X) when she is 87.32. The answer is 47.32 ly.

Next, we need to plot two lines of simultaneity (LOS's) that show what "Now" is for him. (The LOS's for her are just vertical lines). His LOS's (anywhere for him when his velocity is 0.57735) have slope 1/v = 1/o.57735 = 1.73, and they make an angle of 60 degrees wrt the horizontal axis. The first LOS we need goes through point Q. That line intersects the horizontal axis at the point tau = 60. That is determined by writing the X(tau) equation for that LOS, and solving it when X is set to zero. So this tells us that when he is 78.63 years old, she is 60, according to him.

Next, we need to determine how old she says he is when she is 60. We know that according to her, he ages slower than she does by the factor gamma = 1.2247 (once he has changed his velocity to 0.57735). So according to her, while she ages from 40 to 60, he ages from 40 to 56.33. Mark that age on his worldline.

We also need to determine their separation according to her (the value of X) when she is 60. The answer is 31.55 ly.

Next, we do the same thing for the LOS that goes through the point T where his worldline starts sloping upward. The result is that her age when he changes velocity is 28.45, according to him. He was 40 then.

From the above information, we can draw the Age Correspondence Diagram (the ACD), which is a plot of her age (on the vertical axis), according to him, versus his age (on the horizontal axis).

During the first segment, their relative velocity is zero, so they each agree that they are ageing at the same rate. Therefore the first segment of the ACD is just a line of slope 1, sloping upward to the right, making a 45 degree angle wrt the horizontal axis. This first segment is the same, regardless of whether you are using the CMIF simultaneity method, or my method. Label the end of that segment point T.

In the CMIF method, at point T, when he changes velocity from zero to 0.57735, he says that she instantaneously gets younger by 11.55 years, from 40 to 28.45. So, for the CMIF case, we draw a vertical line downward from point T, of length 11.55 ly. Then, the next (last) segment slopes upward forever at a slope of 1/gamma = 0.8165.

What does the plot look like after the point T in the case of my simultaneity method? It is a straight line between the point T and the point Q. Point Q is where his age is 78.64 and her age is 60. It is a point on the third segment of the CMIF line we determined above. Point Q is where he received the pulse from her, and it is the end of the "Disagreement Interval" (DI) between him and the a perpetually-inertial observer who is co-located and co-moving with him. So, after point Q, the ACD for my method coincides with the CMIF method for the rest of their lives. I.e., after the end of the disagreement interval (DI), the CMIF method and my method agree thereafter in this example.

So, as was claimed, with my method, the ACD has no discontinuities, and no negative ageing (i.e., the ACD plot never slopes downward).

I personally prefer the CMIF method, because of its simplicity, and because I'm not bothered by discontinuities or by negative ageing. But for those people who ARE bothered by those characteristics of the CMIF method (and in my experience, that's a LOT of people), my method offers a safe refuge.

[MikeFontenot (OP)]

    There is another argument that (according to the CMIF simultaneity method) the home twin's (her) negative ageing must occur, according to the accelerating twin (he). Suppose in the standard twin paradox scenario that he instantaneously changes his velocity at the intended turnaround from v to -v. That causes an instantaneous increase (according to him) in her age, say by +T years. But suppose that he immediately changes his mind, and instantaneously changes his velocity from -v back to +v. The net effect on her age (according to him) of those two back-to-back instantaneous velocity changes is the same as if he never changed his velocity at all: his world line just continues at the constant slope +v, with no turnaround. But that requires that her first instantaneous age change of +T must be canceled by her second instantaneous age change of -T. If negative instantaneous age changes aren't allowed, then positive instantaneous age changes can't be allowed either.

[MikeFontenot (OP)]

For a given perpetually-inertial observer (the "home twin", her), there is only one answer to the question, "How old is that distant person right now?", that agrees with her own measurements. The way she can make those measurements was described in detail by Einstein: a collection of equally-spaced clocks are laid out that are stationary with respect to her (and with respect to her perpetually-inertial mother or grandmother), and all those clocks are synchronized via light signals. There is a perpetually-inertial observer at each clock. Those observers keep a record of every person who goes flying past them, recording the time on that person's clock, and what their own clock shows at that instant. They then send that information to the home twin (which she receives after a long delay) . For the case where she and the traveler (he) were born simultaneously when they were momentarily co-located, and were both perpetually inertial with a relative speed of 0.866 ly/yr (giving gamma = 2.0), those measurement results tell her that when she was 80 years old, he was 40 years old. That result is exactly what the time dilation equation (the TDE) told her, long ago. The TDE told her that result immediately, without the long delay that the measurements took. ANY other answer will disagree with what the measurements say. There is only one correct answer to that original question.