Relativity and simple algebra II

I'm confused. A rocket going out on a long elliptical round trip is the same thing as a linear twin paradox scenario so at what point does the ellipse become circular enough to qualify as HKX reciprocal time dilation with no permanent age difference between the twins? This is contradictory to me saying the HKX is the same as the centrifuge time dilation example (except that the participants are separated at the start in the centrifuge). Since I'll never find anyone who understands the question, let alone the answer, I guess I'll never have these contradictions of circular motion resolved. I'll stick with linear for now and come back to this way in the future.
So sad. All these geniuses on physics forums and not one can produce a counter Minkowski diagram of either the centrifuge or HKX scenarios.

You are lost in your Loedel diagrams world

That tells me there is great confusion in understanding the difference between Loedel diagrams, Loedel velocities and Loedel lines of perspective simultaneity. They are not the same things. Add to this the inability to understand relative velocity or even basic high school algebra means I'm going to have to shorten and lower the bar on my explanations. I've only produced one Loedel diagram here to illustrate the concept of how true relative velocity would be drawn on a spacetime diagram. I'm not lost in any Loedel diagrams world because I just don't use them. Where most people are lost is in the significance of Loedel (half speed) velocities and Loedel lines of perspective simultaneity and their window into the underlying proper time based relativity. I guess I'm going to have to explain the terms "line", "perspective" and "simultaneity" and how they differ from how I and SR use them in my next post.
I posted this on my old forum in case someone there can answer my questions on circular motion and SR.

I was introduced to the new wrinkle of circular motion that no one can answer my questions to. There are two variants to this circular motion scenario:

One is the relative velocity and resulting time dilation of the circumference of a centrifuge to its center. This is similar to the HafelKeating Experiment (HKX) if there was a guy at the center of the Earth conducting it. The other is the HKX where two planes leave an airport at the north pole in opposite longitudinal orbits around the Earth neglecting gravitational effects.

Now on this forum we discussed closing speed as not being relative velocity. An example of closing speed would be orbits parallel to yours on Earth where the orbiting satellite maintains the same distance from you. There is no linear vector between you and the satellite so the relative velocity between the two of you should always be zero. This should be the same relative velocity between a rotational centrifuge and its center. But here is some irrefutable math from John Rennie on the PSX who proves that assertion not true.

"Suppose you're whirling about a pivot with velocity v at a radius and I'm watching you from the pivot. I'm going to measure your position using polar coordinates (t,r,θ,ϕ), and in polar coordinates the line interval is given by (I'm leaving c in the equation this time):

Note that this is just the flat space, i.e. Minkowski metric, in polar coordinates. We're using the flat space metric because there are no masses around to curve spacetime (we'll assume you and I have been on a diet :). We can choose our axes so you are rotating in the plane θ=π/2θ=π/2, and you're moving at constant radius so both dr and dθ are zero. The metric simplifies to:

We can simplify this further because in my frame you're moving at velocity v so dϕ is given by:

and therefore:

In your frame you're at rest, so ds2=−c2dt′2ds2=−c2dt′2, and equating this to my value for ds2 gives:



which you should immediately recognise as the usual expression for time dilation in SR. Note that the centripetal force/acceleration does not appear in this expression. The time dilation is just due to our relative velocities and not to your acceleration towards the pivot."

One of the blowhards on my new forum who doesn't understand that reciprocal time dilation is not the same as permanent age difference caused by the twin paradox, insists that for every revolution, the time difference between the center and the guy on the circumference is accumulating. Basically the centrifuge has become a Jules Verne time machine.

So I drew a Minkowski diagram (Md) to support reciprocal time dilation between the two but it fell apart when I tried to stop the centrifuge to compare clocks to establish permanent age difference between the two. The fact remained no matter how many orbits the centrifuge performed, when stopped, the only separation between them was always just the length of the radius. In normal linear twin paradox, the total age difference greatly increases the farther the separation between them when a change in velocity is made. So I asked the blowhard to draw me an Md that supported his theory. Of course these philosophers have no way to back up their opinions with math so he left in a huff calling me a mentally ill liar. I got an even worse reaction on the PSX to this question and I'm not allowed to ask any more questions.

There is just one more question. The only difference between the HKX and centrifuge scenarios is the participants begin separated in the centrifuge and begin together in the HKX. Otherwise they could be both modelled as circular motion. Except the HKX is free to expand into ever larger and more pointed elliptical motion which would at some point be the same as the linear twin paradox scenario which ends in permanent age difference, not reciprocal time dilation like the centrifuge scenario. So I also asked for an Md for the circular HKX which I assume is not an example of reciprocal time dilation but is always an example of the round trip twin paradox. Of course, no one understands this question either.
What's the difference between a line and an axis? A coordinate system is defined by its axes. Each axis will be labelled by single units where as lines will have multiple coordinate labels from the axes. Lines are defined by their slopes and intersections. In the simplest Md in SR, there are 4 axes: ct, ct', x and x' for two coordinate systems. My math uses only 3 axes (no need for x') so each line can have 3 different slopes and their reciprocals (v=x/ct, Yv=x/ct' and Y= ct/ct') resulting from 3 different coordinate combinations ((ct,x), (ct',x) and (ct,ct').

There's more but let's discuss the obvious elephant in the room; how can c be the same from all perspectives if time dilation is not balanced out by length contraction, how can I have no x' axis in my math? Let's say the moon is 1.5 light seconds from Earth and they build a road to it on which you can drive your fast car. Your car has an odometer on it calibrated to an Earth mile at very slow speed. Now according to SR, if you drive your car at .6c to the moon, your odometer will read 1.5/1.25 = 1.2 ls because your speed has length contracted the road between Earth and moon. Your clock says it took you 2 seconds to travel that distance which is correct since .6c = x/t = x'/t' =1.2/2. The Earth says the distance to the moon is 1.5 ls and at .6c that should have taken you 2.5 secs so length contraction must exist.

Unfortunately, that magic odometer doesn't exist nor does any method to actually measure length contraction so in the mean time we need to figure out how you drove 1.5 ls in 2 secs at .6c, the math just doesn't add up. Let's use light to check this all out, after all light is the same from all perspectives. The Earth says the light will reach the moon in 1.5 secs but using Alice's clock that 1.5 secs is 1.5/1.25=1.2 secs her time. Her velocity using Earth's distance/her time = 1.5/2 = x/2 = x/(t/Y) = Yx/2.5 and her perspective on c using her time = 1.5/1.2 = x/1.2 = Yx/1.5. So using arithmetic, her ratio to c is .6c because the Yx's cancel out. This means if her velocity using her time is Yv=.75c and light's velocity using her time is Yc = 1.25c, she still sees c as c because her ratio to c is still .6c. This is what length contraction does in SR to maintain the velocity's ratio to c the same from all perspectives. It's just a question of whether you group Y with x or with v. Light still beats you to your destination no matter how big your Yv is.


Notice I have 2 sets of units on my light line where as SR has no units on its light line. The blue units are Bob's perspective of c using his clock. The red units are Alice's perspective of c using her clock. More on this diagram coming up.
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What most people who blindly believe that length contraction is a necessity for keeping c constant from all perspectives don't realize that the equations that limit c to c don't mention length contraction anywhere:

$$ c^2 = v^2 +v_t^2$$ addition of velocity through space to velocity through time
$$w =(v+ u) / (1 + vu/c^2)$$ addition of two different velocities through space
$$c^2= ((v+u)^2+ v_t^2u_t^2) / ( 1 + vu/c^2)^2$$ addition of two different velocities through space and their velocities through time.

They also can't fathom that using different clocks to measure c does not mean that if that clock gives a velocity result of Yv that numerically exceeds c, it does not mean that c from that perspective can be physically exceeded by that velocity. Light will always beat any Yv velocity to its destination from any perspective. The relationship of that velocity to c using one clock will always equal the relationship of that velocity to c using that perspective's clock.
I've been really struggling with how to define lines of simultaneity. I'm prepared to move far away from Einstein's definition of simultaneity using his clock sync method to set all clocks within a frame to the same time. The problem is those clocks do not share the same present within a frame because they are separated by distance, there is a delay of present reality between them. Constant velocity between frames changes that fixed delay into a rate of delay which makes it look like those clocks are ticking at a different rate but they aren't. All clocks tick at the same universal proper time rate but it's subject to perspective.

My definition of "simultaneity" stems from a a mathematical relationship between velocity lines and lines of perspective (which SR calls lines of simultaneity or now slices). Lines of perspective have the reciprocal slope to velocity lines but how this physically relates to simultaneity I don't know yet. The clocks on my lines of perspective aren't sync'd, they just tick off time at the same proper time rate and light signals are used to assign meaningful labels to those clock readings. The only significant clock readings are at the endpoints of a line of perspective where it intersects velocity lines. (Again, I don't understand why those endpoints define a line of perspective that has the reciprocal of the slope of the velocity lines.) The endpoints almost never share the same time reading (except in Loedel lines of perspective) despite Einstein's clock sync method to try to make them so.

Here is an Md with blue perspective lines of simultaneity from v=0 and red perspective lines from v=3/5c.


The slope of the v=0 line is oo (infinity) so the slope of its lines of perspective are 0. The slope of the v=3/5c line is 5/3 so the slope of its lines of perspective are 3/5. The ratio of the endpoints are Y but more importantly the ratio of the length of time between two lines of perspective intersecting both velocity lines is Y= dt /dt'. (t is from the perspective and t' is what that perspective sees.) This is SR's formula for reciprocal time dilation.
For example, from the blue perspective dt= 2-1 and dt'=1.6 - .8 and from the red perspective I use the labels dt' and dt'' for dt and dt' so dt' = 2=1 and dt'' = 1.6 - .8. so Y = 5/4 from both perspectives.

If you find lines too confusing, and who doesn't, let's derive a new formula for Y without using velocity.

As I've said I use a slightly different main formula for relativity because I overlay both perspectives as reference perspectives in one Md instead of have both separated into an Md and a reverse Md. In order for this to work I assign a Y0 and a Y1 to each velocity line as if they were separate. Y0=1 for the v=0 line and Y1=5/4 for v=3/5c. So the main formula is

$$(Yct')^2 = (Yct)^2 - x^2 $$ so
$$Y^2 = x^2/ ((ct)^2 -(ct')^2)$$
also $$Y^2 = (ct)^2/(ct')^2$$ and plugging this Y^2 above yields
$$Y=x/ sqrt((ct)^2- (ct)'^2)$$

For example Y0=1.5/sqrt (2.5^2 -2^2) =1 and Y1=1.5/sqrt(2^2-1.6^2) = 5/4

While SR only employs 2 perspectives, I also use the Loedel perspective and perspective lines to and from light lines. Why is coming up.
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Despite Einstein's attempt to establish a present based on perspective, there are only 2 instances in SR that are a true shared present without delay. One is when two clocks are side by side either stopped or moving past each other. The other comes from the main equation in SR $$(ct')^2 = (ct)^2 - x^2 $$ that establishes hyperbolic lines that intersect all velocity lines at the same present time. Einstein didn't know if atoms existed let alone atomic clocks and their universal accuracy that did not require sync. They ticked off proper time that was connected by those hyperbolic lines into a universal instantaneous god's-eye present.

You can see from the last post that the lines of perspective show that each calculates the other time ticking at .8 of their own time rate due to 1/Y or $$ v_t=c/Y$$. Using the pink and yellow light signals in the same way as the lines of perspective, they can actually see through DSR=1/2 the other's time appear to slow to 1/2 of their own. See is different from calculate. But using the green Loedel lines of perspective whose slope is $$ v_h$$ (half speed 1/3c of 3/5c), they can calculate that perspective sees both clocks tick at the normal rate of DSR=1. They are not reciprocally time dilating. The endpoints of the Loedel lines of perspective are the same proper time between the two velocities whereas the hyperbolic lines intersect the same proper time for all velocities. These Loedel lines are a window into the underlying universal present even for clocks that are separated. Here is the Md :


So what's the speed of the yellow light line from Alice to Bob from Bob's perspective? It's x/t = .75/(2-1.25) = Y0 c =c
What's the speed of the yellow light line from Alice to Bob from Alice's perspective? If Alice takes 1 yr to travel .75ly, light takes .6 yr to travel the same distance back .
x/t' = .75/.6= 1.25c = Y1c.
What's the speed of the pink light line from Bob to Alice from Alice's perspective? If Alice takes 2 yr to travel 1.5 ly, light takes 1.2 yr to travel the same distance at Y1c =1.25c = x/t' =1.5/1.2.
What's the speed of the pink light line from Bob to Alice from Bob's perspective? It's x/t = 1.5/(2.5-1) = Y0 c = c.
I just noticed there is a third proper time present and it doesn't involve a Loedel perspective and it is also independent of clock separation. If, for example, a planet was 3 ly away in the same reference frame as Earth (0 reative velocity), the lines of perspective between them would both be horizontal and have the same proper times at both endpoints. They are in the same instantaneous present, calculable but hidden by the delay of reality between them. Things that occur at 3 pm on both planets don't just happen at the same clock reading, they happen instantaneously at the same time that can be verified once the light signals are post processed on both problems. This is the only example of Einstein's clock sync method that could work to define a shared present rather than an artificially set shared clock reading. So I see relativists who told me clocks must be co-located in SR to establish age difference were wrong all along.
"on both problems"? I think I meant from both participants. Anyhoo I gotta go back to page 9 and continue answering all of James R's questions. I think he's lost interest in trying to save me but let's slog on.
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Before I continue I have a question. Before I came here had anyone ever heard of the concept of permanent age difference due to the twin paradox. No one on any forum I've been on had. They all thought age difference was due to time dilation or maybe a tiny minority may have known that some special kind of age difference happens when the twins re-unite. However they didn't seem to know that the age difference was not reciprocal and that all perspectives (not just the 2) could agree that one was younger than the other. They had no idea that age difference was permanent as they had never seen an Md where the constant age difference existed outside the point of re-unification except maybe if both twins came to a stop. They had no idea reciprocal time dilation was the paradox but accepted the counter-intuitive notion that both could see the other as being younger and both notions were equally valid realities and not illusions of perspective. The twin paradox, on the other hand, was the paradox because it did determine one was indeed younger than the other which is against the norm in relativity. They bought that the determination was made by acceleration which was outside SR's realm of constant velocity and that had to be the reason for the twin paradox. I'm sure in the centrifuge experiment people believe it's reciprocal time dilation that accumulates with each revolution. Even experts like John Renni probably believe that although I'd never expect an honest answer to a direct question on a physics forum. Is anyone going to answer this direct question?
I get the feeling my last post is like a monolith, so far no one agrees with it or disagrees with it they just don't know what to make of it.
Before I go back to page 9 and continue answering James' questions, I think a long series of Md's is in order to show what's the cause of apparent time slowing by a factor of gamma in reciprocal time dilation and what is the cause of apparent time slowing and speeding up in the Doppler Shift Ratio DSR. Information coming from reality is transmitted at a rate of information transfer c (called time) at a speed through space also called c. Distance causes a delay of information and velocity causes a rate of delay of information and, some could argue, a change of received rate of information transfer. You are getting less and more delayed information over time if you're moving away and more information at less delay if you're approaching. Your velocity is not affecting the transmitted rate of information transfer so there's no slowing of time itself going on here. Clocks provide the information but that can be distorted by motion, the underlying reality is not distorted. SR says clocks report the distortion of reality itself by motion. In fact, you're not even getting a change in the received rate of clock information, your velocity is using distance separation and simultaneity to reformat the size of information packets. The information is stored until it can be disseminated as I'll show in this series of Md's. Relativity preserves the rule of conservation of information, none can be created or destroyed.
Before I came here had anyone ever heard of the concept of permanent age difference due to the twin paradox.
Of course, every acurate description of the twin states that.
No one on any forum I've been on had
I can only assume you didn't understand people in the other forums.
They all thought age difference was due to time dilation or maybe a tiny minority may have known that some special kind of age difference happens when the twins re-unite.
The age difference is due to time dilation and it is seen when the twins reunite and the age difference is permanent.

You don't know what you are talking about and there is no reasoning with you because you're a crank.
Of course, every acurate description of the twin states that.

I can only assume you didn't understand people in the other forums.

The age difference is due to time dilation and it is seen when the twins reunite and the age difference is permanent.

You don't know what you are talking about and there is no reasoning with you because you're a crank.
...though, strictly speaking, I would say it is because there is no reasoning with him that he is a crank, rather than the other way round. :biggrin:
I haven't seen any cranks who can back up their arguments with math to those who can't. Easy to reason with that.
Ok a brief overview for the algebra impaired. Algebra is about the slopes and intersections of straight lines. In relativity, the slopes are lines of perspective, the reciprocals are velocity lines and the intersections between the perspective and velocity lines form various relativistic concepts and terms.

The gamma function (Y) is the ratio of these two points of intersection. The velocity through time is the reciprocal of that. Permanent age difference is the difference between the intersections of the Loedel perspective and velocity lines (reciprocal time dilation is not). The relativity of simultaneity (RoS=vx/c^2 in SR) is the difference between an intersection between a line of perspective and the Loedel intersection on the same velocity line. DSR is the ratio of the points of intersection between a light line and a velocity line.

Algebra may be far too out of reach for the philosophers on physics forums so let's use arithmetical examples to illustrate the above concepts from my last Md:

The slope of Alice's velocity line is ct/x = (2.5-0)/(1.5-0) = 5/3.
so the reciprocal v=x/t = 3/5.
The slope of Alice's line of perspective = ct/x = (2.5-1.6)/(1.5-0) = 3/5.
The time intersections of this line with the velocity lines is t=1.6 and t'=2 so Y = 2/1.6 = 5/4 from Alice's perspective and $$v_t = 4/5c$$ .

The slope of Bob's velocity line is ct/x= (2.5-1.6)/0 = oo (infinity).
So the reciprocal v=x/t = 0.
The slope of Bob's line of perspective = ct/x = (2.5-2.5)/(1.5-0) = 0.
The time intersections of this line with the velocity lines is t=2.5 and t'=2.5 so Y = 2.5/2.5 = 1 from ABob's perspective and $$v_t = 1/1 c$$ .

The Loedel half speed velocity comes from the velocity combo equation $$w =(v+ u) / (1 + vu/c^2)$$ where v=u and w = 3/5c in this example. So $$v_h = 1/3 c$$.
The slope of the Loedel perspective's velocity line is ct/x = (3-0)/(1-0) = 3/1.
so the reciprocal $$v_h = 1/3 c$$.
The slope of the Loedel line of perspective = ct/x = (2.5-2)/(1.5-0) = .5/1.5 =1/3.
The time intersections of this line with the velocity lines is t=2 and t'=2 so Y = 2/2 = 1 from the Loedel's perspective of Alice or Bob. Both experience no reciprocal proper time dilation during constant relative velocity.

The Loedel perspective strips the hysteresis of RoS away. If you want to calculate Alice's RoS from Alice's perspective at t'=2 it's (2-1.6) = .4.
Bob's RoS from Alice's perspective at t'=2 it's (2.5-2) = .5.
Alice's RoS from Bob's perspective at t=1 it's (1.25-1) = .25.
Bob's RoS from Bob's perspective at t=1 it's (1-.8) = .2.

For DSR, the pink light line from Bob to Alice intersects the velocity lines at t=1 and t'=2 so DSR = 1/2 from Alice's perspective.
The yellow light line from Alice to Bob intersects the velocity lines at t'=1 and t=2 so DSR = 1/2 from Bob's perspective.

I feel I'm losing you, too much overwhelming arithmetic. Where are these numbers coming from? What is their real significance? We didn't have to use calculators in our philosophy courses. Are numbers even real or just constructs of the human mind? Weren't numbers created so god could keep track of the number of human souls he was admitting to heaven? If science can't answer this question it has no use. No one seems to question the validity of these questions on a physics forum but my algebra is black magic and the devil's work. Right rocket man?

The significance of these numbers will be seen in the next posts.
Oops I made 2 math errors in the above post. RoS is not the time difference between the Loedel intersection and each perspective intersection, it is the total of those two or the time difference between the two perspective's intersections. So in my example the RoS from Alice's perspective is .4+.5 = .9 and from Bob's perspective it's .2+.25 = .45. Also time dilation and DSR are ratios of deltas of points of intersection, not ratios of points of intersection so you need two lines intersecting the velocity lines. This becomes apparent when we discuss the twin paradox and the lines of perspective are no longer of constant slope while the time dilation remains a constant value in this example.

Also I find there is a general reading comprehension issue with the title of this thread. People read as far as the first word but redact the subsequent words. This is not a thread about relativity, this is a thread about relativity and simple algebra. So if you're out of practice with simple algebra, this thread will make no sense to you. Unless your criticisms are based in simple algebra, they have no relevance here. If your understanding of relativity can't expand beyond textbook or recital of wikipedia articles, then your reasoning that "Einstein said so" or "SR is still valid for over 100 years" or personal attacks have no validity here. You got problems with what I say, back them up with some math because this is a math thread, not an "Old Man Yells at Cloud" thread.

So here we go chickadees, more math fer ya. This one took me several of days to figure out. It's the same example of a round trip journey at 3/5c. Alice returns 2 yrs permanently younger than Bob, yet the amount of information that Bob sent out to Alice must be conserved when they re-unite otherwise they won't be in the same proper time present even though Alice lost 2 yrs getting there. You'd think Bob sent out 2 more years of info and this would show up as an imbalance of time at re-unification but it doesn't. How does that happen? I could just tell you that SR is counter-intuitive and that should be a satisfactory explanation for anyone believing in SR but I'm going to share what I figured out.

This is super hard to understand so don't bother if you've been lost so far. I've hurt people's feelings many times because I express no gratitude for responses that are off topic and do not contribute any insight. However I'm more than willing to clarify or answer questions even though I still have a backlog to answer. I drew an Md but it's so full of lines it's difficult to see any white space so I'll have to present the results over the next few weeks in sections. I don't consider this a waste of time but if you consider reading it a waste of time, that's really your problem, not mine. Just put me on ignore and get over your wounded pride.