Relativity puzzle: "half speed" and the twin paradox.

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Relativity puzzle: "half speed" and the twin paradox.

First a definition of terms and numbers we'll be using:

Examples of "half speed" or "half" the relativistic velocity combination are:

.6c relative velocity is the relativistic velocity combination of two frames at .33c.
.8c relative velocity is the relativistic velocity combination of two frames at .5c.
.8824c relative velocity is the relativistic velocity combination of two frames at .6c.
.9756c relative velocity is the relativistic velocity combination of two frames at .8c.
.3846c. relative velocity is the relativistic velocity combination of two frames at .2c.
.2c relative velocity is the relativistic velocity combination of two frames at .1c.

The slopes of the lines of simultaneity x/t for each of these half speeds is the inverse of the slope of their velocity lines x/t so:

.33c half speed slope of simultaneity line is 3.
.5c half speed slope of simultaneity line is 2.
.6c half speed slope of simultaneity line is 5/3.
.8c half speed slope of simultaneity line is 5/4.
.2c half speed slope of simultaneity line is 5.
.1c half speed slope of simultaneity line is 10.

What's interesting to note is that the slope of the half speed perspective's lines of simultaneity intersect the same proper times on the full relative velocity lines as shown in this spacetime diagram (STD):

The thin red line is .6c relative velocity depicted in a Minkowski STD. Alice red is depicted as moving and Bob blue is depicted as stationary when actually both are moving at .6c relative to each other. The horizontal blue lines are Bob's lines of simultaneity from his perspective and the thick red lines are Alice's lines of simultaneity from her perspective.

The thin green line is the half speed (.33c) of .6c. The slanted green lines are the half speed lines of simultaneity which happen to intersect Bob and Alice's proper on-board times. This gives us an indication that, from the half speed perspective, Alice and Bob are ageing at the same rate simultaneously. In fact relativity dictates that any constant relative velocity between two frames yields no age difference between the two participants Bob and Alice. Age difference being a permanent phenomenon is not the same thing as reciprocal time dilation which is due to perspective.

Now let's say Alice stops at the 3 ly mark. She sends a light signal to Bob informing him she has stopped. During the 3 yrs it takes for the light signal to reach Bob, his relative velocity to Alice remains at .6c and her relative velocity to Bob has changed to 0c. This imbalance in relative velocity is verified by the Doppler shift ratio between them. Alice sees a Doppler shift ratio of R=1 and Bob sees R=1/2 which correspond to v=0 and v=.6c respectively.

Upon receiving the news, Bob will be able to calculate Alice has permanently aged 1 yr less than Bob even though they never re-unite. They are separated by distance and separation has a time value associated with it just by the very fact it takes time to traverse a distance. (There is a mathematical formula for converting separation into a time value but I won't discuss it here.)

So let's calculate the age differences for various velocities for when Bob and Alice re-unite or co-locate in an instantaneous present (effects of the delay of the speed of light will be negligible between them). I know, I know, when do I get to the puzzle? I'm still laying the groundwork and setting up the parameters.

If you draw the STD, which I will show later, you get the following results for age difference due to co-location:

At .6c return, Alice's v line intersects Bob's at t=10 and t'=8 so the age diff is 2 yrs less for Alice.
At .8c return, Alice's v line intersects Bob's at t=8.75 and t'=6.25 so the age diff is 2.5 yrs less for Alice.
At .8824c return, Alice's v line intersects Bob's at t=8.4 and t'=5.6 so the age diff is 2.8 yrs less for Alice.
At .9756c return, Alice's v line intersects Bob's at t=8.075 and t'=4.675 so the age diff is 3.4 yrs less for Alice.

This method makes it very hard graphically to establish where the .3846c and .2c return velocities would intersect with Bob to establish their age difference. Well let's just use the way Alice establishes her age difference when she stops and can't co-locate with Bob, with the info contained in her light signal. It turns out this method employs where the half speed lines of simultaneity (green) intersect Alice's return velocity lines (red).

The green lines of simultaneity are labelled with their corresponding half speed velocities which of course have corresponding relative velocities between Bob and Alice.

.9756c (half speed .8c) has a simultaneity line slope of 5/4 and intersects the red v line at t'=4.6. 8-4.6=3.4
.8824c (half speed .6c) has a simultaneity line slope of 5/3 and intersects the red v line at t'=5.2. 8-5.2=2.8
.8c (half speed .5c) has a simultaneity line slope of 2/1 and intersects the red v line at t'=5.5. 8-5.5=2.5
.6c (half speed .33c) has a simultaneity line slope of 3/1 and intersects the red v line at t'=6. 8-6=2
.3846c (half speed .2c) has a simultaneity line slope of 5/1 and intersects the red v line at t'=6.4. 8-6.4=1.6
.2c (half speed .1c) has a simultaneity line slope of 10 and intersects the red v line at t'=6.7. 8-6.7=1.3

There's 1 more line of simultaneity to consider, but it's not for a returning velocity or even a stopped one, it's a separating velocity where Alice does not make a velocity change.

.6c (half speed .33c) has a simultaneity line slope of 3/1 and intersects the red v line at t'=8. 8-8=0 age difference.

So we see we got all the right answers plus 3 more right answers. But we can get so many more using this method. There are areas of this graph that seem incomplete. What happens if Alice just slows down a little when she makes her velocity change. What happens to the age difference if she speeds up? These questions are mathematically solvable, but does relativity allow them to be solvable? That's the puzzle.

Here's the STD with velocities where Alice slows down and doesn't stop or return.

We can see that separating velocities also yield results for age difference:

.2c (half speed .1c) has a simultaneity line slope of 10/1 and intersects the red v line at t'=7.3. 8-7.3=.7
.3846c (half speed .2c) has a simultaneity line slope of 5/1 and intersects the red v line at t'=7.6. 8-7.6=.4

Would anyone believe this nice clean hyperbolic line joining the points of intersection is some meaningless coincidence? If this method yields final age difference results, can it predict age difference results as they unfurl for each year Bob ages? (The answer is yes mathematically but what does relativity have to say about it? What happens if you extend the hyperbola to separating velocities of greater than .6c? Is that allowed under relativity? Here's the STD:

So if Alice accelerates to .8c away from Bob she will turn out 8-8.6=.6 yrs older than Bob.
And if Alice accelerates to .8824c away from Bob she will turn out 8-9=1 yr older than Bob at the time he gets her message that she is accelerating away.

Does anyone have a problem with this because you should.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

Here is the STD showing the smooth progression of age difference between Bob and Alice for every Bob year in the transition period between constant relative velocities. The derivation of this STD uses the same relativistic tools outlined above. How would the classical relativistic method for determining age difference be able to achieve the same results? I'm just using 2 simple relativistic rules:

1. Age difference can't occur during constant relative motion which can be determined by matching rx and tx doppler shift ratios of broadcast clock readings. This leaves a window of time where age difference can occur. Bob and Alice do not continue ageing differently after that window is closed.

2. Since 1 is true, the window is not closed by co-location of the participants but by reception of delayed information about the change in relative velocity. The co-location of participants occurs outside the window of ageing difference so it irrelevant and only confirms the total age difference that had been established earlier.

Last edited by ralfcis on November 21st, 2018, 11:12 am, edited 2 times in total.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

P.S. It seems counter-intuitive that Alice is ageing faster than Bob if she accelerates away from him yet is still covering vast distances in very little of her time. She has only aged faster than Bob for 4 Bob years but the rest of the time is ageing at the same rate as Bob while still travelling space at Yv which is mathematically traversing Bob's space in her dilated time. Yv is the inverse slope of the red velocity lines if you use the time units printed with those lines (t'-axis) as opposed to Bob's time units on the horizontal t-axis.

P.S. It's also counter-intuitive that Alice's increase in speed has a similar effect on age difference as if Bob had been the one who had increased his velocity relative to Alice. This probably points to some deeper meaning.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

I just figured out what that deeper meaning is. When Alice accelerates away from Bob from .6c to .8824c, Alice is accelerating away by another .6c from her former relative velocity. The velocity combo law says .6c +.6c =.8824c. This causes Alice to age 1 yr more than Bob which can also be stated that Bob ages 1 yr less than Alice. This yields the same result as if Alice had remained at .6c and Bob had been the one to change his velocity to .6c away from earth towards Alice. This would be 0c relative to Alice which means he and Alice would be relatively stopped and Bob will have aged 1 yr less by the time his light signal reached Alice at her time t'=8.

Similarly if Alice's turnaround back to Bob at .6c results in her ageing 2 yrs less than Bob, Bob can equivalently age 2 more years than Alice if she remains at .6c away from him and he takes off at .9692c towards her. But if he only takes off at .8824c towards her, he will age 2 yrs less than her.

This is how relative velocity works which the Minkowski STD has trouble depicting. Bob is depicted as stationary but he is not actually stationary. But if you used the Loedel STD where both Alice and Bob are depicted as moving, you'd have to depict them as moving .33c away from each other to maintain the actual .6c relative velocity between them. You'd have to develop a Loedel STD where the relationship to the background grid is half speed of the relative velocity. So when you depicted .33c, you'd know it was a half speed depiction of .6c relative velocity.

The deeper meaning here is that for any depiction of Alice never returning to Bob and closing off the spacetime path, there is a depiction of Bob making a move that closes off the spacetime path for Alice and yields a result for her age difference that relativity previously did not allow. Alice ageing 1 more yr than Bob by going .6c faster relative to her former relative velocity yields the same result as Bob ageing 1 yr less less than Alice by him accelerating to .6c relative to his former self (depicted as stationary) and Alice maintaining her relative velocity.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

I've made a math error in my last post. Let's say all jumps in relative velocity are in .6c units. If Alice makes a jump of 1 .6c unit away from Bob, she will age 1 yr more than Bob. This is similar to Bob making a jump of 1 .6c unit towards Alice (after 4 yrs depicted as stationary) where he will age 1 yr less than Alice. If he makes a jump of two .6c units towards Alice, he will age 2 yrs less than Alice just as if Alice had made a jump of 2 .6c units away from Bob.

The delta is the same but the values are not. In the first instance Bob is 8 and Alice is 9 while if Bob initiates the change he is 7 while Alice is 8. The distance they're separated are also quite different.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

My last post is a lie. I've made some detailed calculations and STD's and relativity doesn't quite work the way I described for Alice or Bob accelerating away from each other at another .6c unit. Alice or Bob will age .8 yrs more (not 1 yr) for 1 more .6c unit away from each other and 1.35 yrs (not 2) for 2 more .6c units away from each other. You can only fudge the expected answers by altering the green half speed velocity lines. For example, Bob will age 2 yrs more turning away from Alice if his half speed velocity lines of simultaneity have a slope of 5/13, not 1/3 as expected. So the age difference doesn't follow the same pattern once you accelerate away from the initial relative velocity. This must have a really deep meaning but I don't know what it is. I will show the STD's in the next few posts.
I'm also getting some weird results when both make various velocity changes when their on-board proper time clocks turn 4. Most of the results come out as expected but there are mathematical anomalies that I don't yet understand. I'll show the unexpected results in more STD's.

P.S. I can see where 5/13 comes from in my example. 2 units of .6c past .6c equals .9756c. So the velocity combo law between .6c and .9756c is (.9756-.6)/(1-.9756*.6) = .902. The half speed of .902 is 3.846 or 5/13. So that works out. So I was wrong to assume it should have been the lines of simultaneity for .33c, they should have been the lines of simultaneity for .3846c which yields the expected result that if Bob or Alice accelerate two .6c units away from their .6c relative velocity, they will age 2 yrs more than the other. I need to check the results for one .6c unit acceleration.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

Sorry, I have so far failed at putting equations describing age difference when either Bob or Alice accelerate away from each other in .6c units past their initial .6c relative velocity. All the slopes of the half speed lines involve prime numbers. For example when Bob turns away from Alice in a .6c increment, the slope of the half speed line is 1/3 * 1/5 * 1/13. If he turns away at two .6c increments, the slope of the half speed line is 5/13. If Alice turns away in a .6c increment, the slope of the half speed line is (6+7/9)/11. The slopes of the velocities also include prime numbers. .6c is 3/5, .8824c (one .6c unit past .6c) is 15/17 and .9756 (two .6c units past .6c) is 40/41. There's a pattern here but it's taking too long for me to work it out so I have to put this aside for now.

The math between the bounds of Alice continuing on at .6c, slowing down, stopping or returning back at near c works fine. But what happens when Bob or Alice both make changes towards each other? Does this introduce new problems? I've looked ahead and weirdness does ensue and it could lead to significant insights.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

Here's the STD of Bob making the velocity changes. The ones where he accelerates away from Alice are also included and show what I expected the results to be based on the pattern of velocity changes where he doesn't accelerate away and what I think the actual results should be to be consistent with Alice's results when she ages the same number of years less than Bob. For example, if by turning away from 0 to -.6c Bob ages 1 yr more than Alice, it should match Alice turning toward Bob from .6c to 0 and her ageing 1 yr less than Bob.

ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

Here's the STD with Alice making the changes in velocity away from Bob. More prime numbers in the slopes of the corrected half speed lines. .6c in green is corrected to 1/9 * 61/11. .8c in green is corrected to 73/89 c. Strange how prime numbers are popping out of this relativity problem.

ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

Now we're going to look at Bob and Alice making velocity changes when both their on-board clocks read 4 yrs.

Alice goes from .6c to 0 relative to earth. Bob takes off at .6c from earth towards Alice. Bob and Alice will cross paths when they're both 8 so no age difference has occurred as a result of the velocity changes.

They also notified each other via yellow light signals of the changes. The light signals would have reached Bob and Alice when they would have both been 8 but since they were moving towards each other the light signals reach them when both are 6. The green half speed lines verify no age difference has occurred. This is logical because if Bob hadn't changed, Alice would have aged a year less than Bob and vice versa so the changes cancel each other out.

Let's use the same logic where Alice makes two .6c unit changes towards earth and Bob only makes one .6c unit change towards Alice. You'd think Alice would end up 1 yr less than Bob because Alice's move would have rendered her 2 yrs less than Bob if he didn't change velocity and Bob's move would have rendered him 1 yr less than Alice if she hadn't have changed with a net result of Alice ageing 1 yr less.

See if you can spot the problem.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

The problem is me theorizing without checking out the math first. Alice and Bob re-unite where their velocity lines cross over and their age difference is 6.4-5.6=.8 yrs, not 1 yr as I had theorized. The fact is there are no .6c units of change. As Alice's velocity change sweeps from .6c away from Bob, to stationary with Bob, to .6c towards Bob, to .8824c towards Bob (formerly know as three .6c units towards Bob from Alice's original velocity line) to .9756c towards Bob (formerly known as four .6c units towards Bob), Alice's age difference goes from 0, to -1, to -2, to -2.8 to -3.4. It's not a linear progression and never has been.

So, in the above example, if Alice moves 2 units towards Bob and Bob moves 1 unit towards Alice, that's a total of 3 units which is outside the "linear" range of a maximum 2 unit change. I think I also drew the green lines wrong so I'll have to correct the STD above.

All that stuff about velocity changes away from each other was also wrong and didn't need those prime number correction factors. I wish I could erase it but no one cares anyway.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

So at this moment I don't know for sure the math behind age difference outside of Bob being depicted as stationary and Alice depicted as moving at .6c and then making relative velocity changes shown in this STD:

We don't yet know what happens when Bob or Alice accelerate away from each other outsideof this cone but we do have a tool to find out. The tool is we can depict the relative velocity between Bob and Alice as both going at .33c away from each other, or Bob going at .8824c and Alice going at .9756c. These velocities are relative to the background cartesian coordinates that enables us to fake a depiction of .6c relative velocity between Bob and Alice. Bob is not stationary in the above depiction. He is moving through space relative to Alice. The depiction can't show that. But the depiction we choose must give the same results as any other. So if there's linearity for up to two .6c jumps relative to each other in the Minkowski depiction, that rule is valid for the Loedel depiction and for the one where Bob is going at .8824c and Alice at .9756c. We can stitch those depictions together so long as we don't make a jump of more than two .6c units in a depiction. That will tell us the math for as many .6c jumps as we like. We can then do the same for jumps of .33c units or .8c units to verify the behavior is consistent. That's a lot of work ahead.

P.S. The other tool that's available to us is that whether Bob or Alice make the jumps, we must get the same results for age difference. Bob ageing one more yr than Alice is equivalent to Alice ageing 1 less year than Bob.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

Ok, so I've been kicked off the physics forum for math mistakes, fair enough until I can get my act together. It's good in a way because I no longer have to leave it up to the reader to come to his own conclusions about relativity based on the evidence (math) I'm presenting. Here's what I've been saying in plain English.

1. Relativity begrudgingly allows the establishment of age difference using light signals between two participants relatively stopped at a distance from each other. The distance separation causes problems that co-location of the participants to establish age difference does not.

a) Distance can be converted into time so there is still a hidden time difference between the two separated participants that is not recorded by their clocks. It comes into play in the next velocity change but does not if they remain stopped wrt each other.

b) The separation allows the perspective of other moving frames to see them as having age differences different than the one they see.

I have not verified for myself whether this is true and am just repeating what I think relativity is saying. I think these niggling points are irrelevant to the establishment of age difference for velocity changes between the two participants themselves.

c) If the exchange of light signals is good enough to establish age difference between the participants when they relatively stop, it is good enough to establish age difference for any change in relative velocity between the two. The perspectives of other frames can be worked out separately.

2. To relativity, time is not universal, it's all about perspective. Hence, past, present and future are a "persistent illusion" while a perspective present is reality. To me, perspective is an illusion as in a man 100 yds away from me is not really as tall as my thumb. You can use the illusion of perspective to establish real results but perspective itself is illusion.

The universal instantaneous present is not the same as the Newtonian concept. It is the simultaneity of proper time, not the simultaneity of perspective time. Age difference occurs at the proper time level which then affects perspective time.

Since relativity does not recognize a universal instantaneous proper time present, I had to substitute the half speed perspective present in my discussions. The two have the same slopes for their lines of simultaneity.

3. Using the half speed lines of simultaneity I was able to establish the following truths

a) Only the information between the two participants needs to co-locate to establish age difference between the two.

b) Age difference does not require the participants to only move towards each other and re-unite or relatively stop. The velocity change can also be a slowing down between the two. Relativity can't calculate an age difference as a result of them relatively slowing down.

c) I am currently working on the age difference for when they speed up relative to each other. Relativity can't calculate that either.

d) math supercedes theory.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

You're aware that general relativity (the 1915 theory) deals with accelerated frames, right?

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Re: Relativity puzzle: "half speed" and the twin paradox.

And what does that have to do with what I'm saying? When I say speed up or slow down it's a jump to a higher or lower velocity. I'm not dealing with GR only SR. Acceleration has no place here but a jump in velocity does. Whether the jump to a new velocity is due to acceleration or a clock handoff makes no difference to the results I'm discussing.
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Re: Relativity puzzle: "half speed" and the twin paradox.

I forgot to mention point #4

4. This discussion also showed how age difference progresses. Relativity demands the spacetime path comes to an end in order to establish total age difference. I have the math now that shows this requirement is not needed and that age difference progresses in a smooth transition and not in a lump sum discontinuity.

This is all really significant stuff to problems with relativity that almost no one knows even exist or will even consider.
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Re: Relativity puzzle: "half speed" and the twin paradox.

I also forgot to mention the most important point:

5. Age difference is not caused by acceleration forcing participants out of constant relative velocity or by the fact that the time travelled through space subtracts from the time on the "moving" clock. Both participants are moving through space relative to each other and both clocks look like perspective time passage (which is ageing) is slower for the other participant. One can accelerate from the other and little permanent age difference starts and one can handoff clock info to another returning ship without either ship experiencing age difference with the earth. Only the clock info handed off experiences age difference with no acceleration involved.

What causes permanent age difference is a participant initiating a change in relative velocity to which the other participant is unaware until the delayed info of the change reaches him. The reason there is little age difference occurring at the start of acceleration is the delay of velocity change info is relatively small. The age difference only occurs during that delay of information and it is not about perspective time but occurs at the instantaneous proper time simultaneity level. That's what my math has alternately proven to relativity's method of establishing permanent age difference and it has areas of application that relativity can't touch. This truth should be obvious and one day it will be.
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Re: Relativity puzzle: "half speed" and the twin paradox.

Ok, here we go, the determination of age difference due to Alice or Bob increasing relative velocity. This has nothing to do with acceleration, it has to do with Alice or Bob making a change in velocity that increases their relative velocity away from each other resulting in the initiator ageing faster. The question is how much faster. The technique was outlined in the post before last and is exploratory and therefore exhaustive. Please tune out until I arrive at the answer and I'll let you know what I find out at the end.

Here's the first STD. It is a composite of Alice's change of relative velocity from .6c to 0c and Bob's instantaneously simultaneous change from 0c to .6c. I'm no longer going to hide the fact there are 2 types of simultaneity. There's relativity's perspective simultaneity and the new proper time instantaneous simultaneity which is the purple line between t=4 and t'=4. It has the same slope as the half speed (.33c) perspective simultaneity which is the parallel green line.

Bob and Alice's velocity changes intersect at t=t'=8 so they have aged the same when they re-unite. But if you've been paying attention, you know I've introduced a method to determine age difference based on a half speed perspective simultaneity from the downward pointing .33c velocity line between Bob and Alice's new relative velocities. Hence when the light signals are received, they both know their proper times are t=t'=6 which confirms there is no age difference between them as a result of their velocity changes.

Now we've learned previously that age difference is linear for no more than two .6c jump changes between Bob and Alice, We really want to see what happens when Bob makes a ,6c jump away from Alice when she makes a .6c towards Bob.

I've spent a couple of days on this problem and it requires tons of calculations to solve. I'm terrible in arithmetic and always make mistakes entering values into the calculator so I just don't have the time right now to do all these calculations. The STD's are just masses of spaghetti lines.

Anyway the final answer to the problem above after Alice's light signal reaches Bob when he's 12 is Alice will have aged 2 yrs less than Bob. Bob's change of velocity away from Alice adds 1 yr to his age and Alice's instantaneously simultaneous change of velocity towards Bob subtracts a yr from her age. So the total is 2 yrs age difference in Alice's favor. How age difference unfurls in the intervening yrs is what takes up a lot of time to figure out.

What's really interesting about this result is that at no time in the instantaneous present did Bob or Alice break from maintaining a relative velocity of .6c. Hence, you would have expected the end result would have ended with no age difference between the two. But the cause of age difference comes from the delay of information of velocity changes and this affects the proper times in the instantaneous present. Maybe someone out there can get out their light cones 4-vector math and see what result relativity arrives at for this problem. I'm being derisive because we all know the solution to this problem is outside the capabilities of the theory of relativity.

I'll just post the STD with the final answer as no one will read it or understand it anyway. If I do enough of these I'll eventually be able to figure out a shortcut but what's the point. No one can understand what my argument is let alone understand the basic high school algebra supporting it. I won't bother cleaning it up or explaining it or include the STD's to show how age difference unfurls.

ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

The above post is no more than a guess based on a single example that requires many more examples to see if the methodology is sound. But it is encouraging as it fits in with the methodology employed to determine age difference for changes in velocity that involve a turnaround or a slow down. It's easiest to see the consistency if you deal in terms of .6c units.

1. Bob is depicted as stationary and Alice is moving at one .6c unit away and then makes a change at the proper time of 4 to make a velocity change one .6c unit towards Bob. She ages 1 yr less than Bob. If she makes a change of two .6c units towards Bob, she ages 2 yrs less than Bob. If she makes a change of three .6c units towards Bob, she ages 2.8 (not 3) yrs less than Bob. If she makes a change of four .6c units towards Bob, she ages 3.4 (not 4) yrs less than Bob. In order to age 4 yrs less than Bob, she would have to go at c towards Bob.

2. If Bob makes moves towards Alice, at the proper time of 4,the same results are achieved as in 1. If he moves one .6c unit towards Alice, he ages 1 yr less. Two units, 2 yrs less. Three units, 2.8 yrs less. Four units, 3.4 yrs less.

3. If Alice doesn't change and Bob moves away from Alice one .6c unit, Bob ages .8 yrs more than Alice. But if Alice also makes a change of one .6c unit when both their proper times are 4, Bob ages 1 yr more than Alice and she ages 1 yr less than Bob for a total age difference of 2 yrs.

4. This answer coincides with the .33c half speed line of simultaneity and with the fact that after the light line from Alice reaches Bob at 12, there can be no further accumulating age difference between the two. Any other slope of green line would violate this rule.

As an interesting side note I've introduced an instantaneous proper time line of simultaneity which has the same slope as the half speed perspective line of simultaneity. This instantaneous line of simultaneity joins Bob and Alice when their proper times are 4. From other perspectives, the velocity changes are not simultaneous and either Bob or Alice can initiate the velocity change first. (I say they change simultaneously from the instantaneous proper time line of simultaneity and the half speed time line of simultaneity.) However, there is no dispute that Alice receives Bob's light signal before Bob receives hers because there is no possible perspective from which Bob can receive her light signal first.

The result in 3. is the same result as Bob not moving and Alice moving two .6c units towards Bob.

This should mean if Bob moves two units away from Alice and Alice stays at 1 unit towards Bob, we should see Alice ageing 2.8 yrs less than Bob if the math gives consistent results. We can't do this graphically with accuracy so we're going to have to verify this algebraically.

We are talking about finding the intersection point of 3 lines. The slope of the blue velocity line for Bob is 15/17 ( two .6c units which is .8824c). The slope of the green line of half speed simultaneity is .6c. The slope of the yellow light signal line is 1. The light signal line should intersect each of the two other lines where they intersect each other.

I made so many calculation errors that I had to draw the solution graphically and the answer is 20 - 17.2 = 2.8 (Alice ages 2.8 yrs less than Bob for a 1 unit change towards Bob who makes a change of 2 units away from Alice same as if Alice had made 3 units change towards Bob) as predicted so the methodology looks consistent so far.

ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

It looks like a simple law for calculating age difference is becoming clear (but must be rigorously verified):

Final age difference is the point in proper time where the last light signal intersects a participant's velocity line minus where the half speed line of simultaneity, between the two participants, emanating from that point intersects the proper time of the other participant's velocity line. It looks as though this rule can also be used for intermediate age difference results before the final cumulative result. Relativity can only calculate a tiny subset of scenarios.

This may turn out to be true for all cases or it could be the first step to a law that is applicable for all cases. Either way, I'm wondering what the relevance of the theory of relativity will be after this. When the math is this elegant, consistent and predictive, it cannot be denied. Of course for those unfamiliar with algebra, none of what I'm saying is visible in these posts.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

Yikes I'm having a lot of trouble with how the age difference unfurls year by year. I haven't yet figured out how to handle both participants making overlapping changes. This'll be very satisfying once I figure it out and maybe some deeper revelation will come of it. I love this stuff.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

For all those sticking with SR/GR, maybe they can find someone on this planet to solve the problem I've proposed using relativity. The problem again is what is the final age difference between Bob and Alice if they start at .6c separating and at the proper time of t=4, Alice makes a change of .6c towards Bob and Bob makes a change of .6c away from Alice. For simplicity ignore acceleration and assume their pre-programmed flight plan never allows them to waiver from their original .6c relative velocity. The answer is not zero age difference.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

I don't want to be dragged in again (because SR/Ralfitivity is utterly boring), but yes, it is not zero - it is anything, depending on who measures/observes the "age difference". And there can be a zillion different observers with different answers/observations.

BurtJordaan
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Re: Relativity puzzle: "half speed" and the twin paradox.

Yes, that is the correct answer in relativity. It is indeterminate. But I come up with an exact answer based purely on mathematics. The same math relativity uses. So that can only mean my math is at fault or the fault is at the beginning. So all you need to find fault is prove the information delay of a change of velocity is not sufficient to establish age difference. I've already stated participants stopped separated in space can be perceived to have different age differences based on outside perceptions but the perceptions of each other are the same. The difference is age difference is not a matter of perspective, it is permanent. It can't be gotten rid of once you account for the illusion of perspective. The outside observers are observing reciprocal time dilation while the two participants are observing permanent age difference. I've separated the two very different phenomena mathematically. Yes rigorous proofs are boring but that doesn't mean they're automatically wrong. And if one takes the time to understand them and they disprove a long established theory, there's nothing more interesting.

P.S. So far as the math goes, I come up with the same answers as relativity within the small domain relativity can definitavly determine age difference where it is not confused with reciprocal time dilation. That domain is round trip flights. But when I extend that domain past where relativity can go, I get the same consistent answers. Is that just coincidence or shouldn't be investigated as it's boring? If I were an expert I'd already have answers to these.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

Ralf, the boring part is that your arguments are so incomplete, and hence full of holes, that it seems to be a gross waste of time trying to follow them. Just a few posts back you admitted that much (well, almost):
Yikes I'm having a lot of trouble with how the age difference unfurls year by year. I haven't yet figured out how to handle both participants making overlapping changes. This'll be very satisfying once I figure it out and maybe some deeper revelation will come of it.

Take your little (imprecise) scenario, first make it precise and then add to it:

"Then after the same (predetermined) time on their own respective clocks, Alice makes a change of 0.3c towards Bob and Bob makes a change of 0.3c towards Alice, both as observed from the others signal Doppler shift."

They can then leisurely determine that they have actually aged by the same amount during the whole experiment. Why?

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Re: Relativity puzzle: "half speed" and the twin paradox.

I have totally and completely answered that question a few posts back with an STD and everything "Postby ralfcis on December 11th, 2018, 1:40 am". That's an easy problem to solve because it's in relativity's comfort zone. A theory is not only judged by the problems it can solve but by the solvable problems it can't. Try reading my posts instead of skimming over them for key words that have nothing to do with the context in which I use them (like BraininVat did with acceleration). Yes it's a pain because so many beginners have the same questions and the same counter-theories but if that were true in my case you'd be able to knock down my never before seen arguments quite easily. I'm in the process of charting new waters and each advance into the unknown takes some reckoning. To say my theory is incomplete because I haven't yet solved the latest advance does not mean I won't be able to because the foundation it's based on is somehow flawed in some general unspecified way. Relativity says there are no solvable intermediate steps to how age difference progresses yet I've proven that's plainly untrue. I'm having trouble with the next level of difficulty for now. Your counter-arguments have been vague, general and personal and you don't even want to discuss it on the simple algebra I'm presenting. That's your choice but even if you find a specific reason you don't agree you'll still be grateful for the different and mathematically sound perspective I'm presenting here. These are not the ravings of a stream of consciousness brain salad loon that permeate these forums.

P.S. Are you taking over from Braininvat because if you are I'm in deep do do.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

After sleeping on it, I've decided that age difference has nothing to do with time dilation. Age difference is a part of set theory defined by the interaction of two sets: an arithmetic progression of numbers that represent unit changes in velocity and the hyperbolic progression of numbers that represent age difference. So in the example I've been using, the change set of 0,1,2,3, n (it actually starts at -inifinity and limits at a formula for the time light travels through time plus the converted time value of the distance separation) corresponds to the age difference set of 0,1,2,2.8,3.4,4. So what does this mean in a numerical example.

When neither Bob nor Alice make a change at a distance from each other, there is no age difference. No time dilation perspective makes any difference to this fact, only the change number of 0 does. Einstein's fanciful idea that perspective is reality holds no water. Relativity's fanciful explanation for there being no age difference, because the reciprocal perspectives cancel each other cancel out, has no mathematical basis or basis in physical reality.

Only the addition of change numbers causes age difference, not the subtraction of reciprocal time dilation numbers. What does that mean? If Alice makes velocity changes of 1,2,3..n units, any units Bob changes towards her subtracts from her units. Any units Bob makes away from her, adds to the change number. The resulting change number corresponds to a number in the age difference set.

So if Alice makes a unit change towards Bob and he makes a unit change away from her, the change number =2 which corresponds to and age difference of 2. If Bob or Alice make another similar unit change, the number = 3 which corresponds to an age diff of 2.8 in our example. If Alice makes an infinite number of velocity changes towards bob, her change number of infinity will correspond to an age diff number of 4 in our example which when added to Bob's change number of 1 will yield a total age diff of 5. If Bob and Alice make the same number of unit changes towards each other, the final change number =0 which means no age diff as if they had never made a unit velocity change. The math supports relativity but also applies way outside relativity's domain. Which are you going to believe, the theory that only partly follows the math or the math itself?
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Re: Relativity puzzle: "half speed" and the twin paradox.

If I could only draw animation I'm sure most of you would then get my point.Let's say cartoon Alice starts running away from cartoon Bob starting with the standard puff of smoke and streak lines away from him. Bob thinks Alice is trying to beat him in a surprise race by getting a head start. When he realizes this, he too takes off in a puff of smoke after her but as soon as he does, she stops running away relative to Bob based on what she sees of his televised image to her (the Doppler shift ratio is 1 whereas it was 1/2 before she stopped which is a relative velocity change from .6c to 0). They end up with no age difference between them still looking like cartoon kids just as if Bob had never taken off after Alice and she had never stopped. However if only one of them had made the change in relative velocity, one of the cartoon kids would be entering puberty without the other. The one who makes the velocity change doesn't age as much.

Now Jorrie's and relativity's argument is that instead of Bob chasing after Alice, he races away from her thinking that's the way to beat her at her own game effectively changing the imaginary finish line. As soon as he starts to do this (she's clairvoyant and knows he was going to do this when both their watches said 4 pm), Alice realizes he is putting himself in grave danger and she stops. The direction he's going puts him in a magical world where everything is possible and simultaneously indeterminate only because they will never meet again. In the cartoon, Bob turns into a flower firmly stuck in the mane of a unicorn.

But in ralfativity world, this doesn't happen. Bob skips puberty and turns into a handsome prince forever older than Alice.

The end.

Epilogue: Even though they never met again, Alice eventually received a transmitted selfie from Bob, figured out her age and his when the selfie was taken and realized ralfativity had been right all along.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

To clarify, Alice is always clairvoyant and makes her changes in the instantaneous proper time present relative to Bob when she knows both their proper times are the same. Otherwise the delays in the speed of light would introduce perspective time. This instantaneous simultaneity of when velocity changes will happen on their clocks can be agreed to beforehand but for simplicity the cartoon made the changes appear spontaneous even though the cartoon is scripted.

Also Alice's stop is considered a relative velocity decrease from distance increasing over time to relatively not increasing over time (even though it's also not decreasing over time). Similarly a stop from Alice going towards Bob would be a relative velocity increase in that 0 is greater than a negative velocity.
ralfcis
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Re: Relativity puzzle: "half speed" and the twin paradox.

I'm done drawing up legal contracts with endless disclaimers every time I try to describe relativistic events. So here is my new language to describe the above cartoon.

Bob and Alice separate at 1 relative velocity unit (RVU). 1 RVU = .6c in this example. (An RVU can't be drawn on a graph or STD without setting up a background frame (often wrongly confused as a preferred frame) depicted as Cartesian coordinates. The resultant depiction is a compromise as relative velocity can't be drawn as two moving participants.) At a pre-determined proper time (T=4), Bob changes to +1 RVU and Alice to -1 RVU. The resultant age difference (RAD) is 0 yrs. If neither had made a change, the RAD would have remained at 0. If one or the other had made their change, the RAD would have been -1 yr for the initiator of the velocity change.

In the above example if T=4 (T is proper time) and 1 RVU =.6c and Bob changes to -1 RVU and Alice changes to -1 RVU, the RAD is +1 yr for Bob and - 1 yr for Alice for a total of -2 yrs for Alice (Alice ages 2 yrs less than Bob). This is equivalent to Bob not changing and Alice changing - 2 RVU's (relativity is allowed to calculate this result as RAD = -2yrs for Alice). I recently bought a book on mathematics and have just nullified relativity by "combinatorial proof" (equivalency can't yield inconsistent results). All equivalent RVU's must have the same RAD result. So an RVU = -2 can't have a RAD = -2yrs and RAD = indeterminate.

Relativity has no answer for the other scenarios because it has made up a bunch of arbitrary rules based on mistaken assumptions about the nature of time and time dilation to disallow the answers. I have explained why relativity must do this so that its foundations don't crumble to dust but maybe someone else has some mathematical or physical backing for these rules that I'm not aware of. The debate would be so interesting.

P.S. If Alice had not done her change and Bob had - 1 RVU, the resultant RAD would have been +.8 yrs for Bob which means Alice aged .8 yr less than Bob. (I need to work out the math here as to why this is true as I can't yet give you the corresponding RVU for this RAD result.) Relativity bans calculating both this scenario and its equivalent so the answer of .8 yrs was arrived at in the STD following the method I used to arrive at other valid results. This is called proof by mathematical induction in the book I'm reading.

P.S. Sorry I've edited this so many times because every time I read it I find a new arithmetic error.
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