Relativity and simple algebra

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Permanent time difference due to the twin paradox.

If Alice goes out at 3/5c and turns around at 3ly, there will be a stretch of time when Bob will not be aware that Alice has changed their relative velocity. Relative velocity is the cause of time dilation which is normally reciprocal because the relative velocity is reciprocal. But a change of relative velocity made at a distance apart means there is an imbalance in relative velocity because Bob continues to see Alice going away from him at .6c for a time while Alice sees immediately sees Bob approaching him at -.6c which are two different relative velocities. So time will be permanently affected between the two by Alice aging 2 years less than Bob when they re-unite and that 2 yr difference will continue until one of them makes another velocity change.
 
Do you understand reference frames? Do your equations or derivations use the idea of different reference frames?

Each participant can be set as the reference frame or the stationary frame.
 
I'm just presenting a new math framework based on 2 equations and a single algebraic/geometric unit that can re-construct relativity from simple algebra. It's a math exercise, not a philosophical one. It's like Euclid's geometry based on real geometric shapes was superseded by Hilbert's geometry which didn't require that (not implying that I'm either of them).
 
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Have I answered all your questions or would you like to wait until I present the rest of the math?
 
Oh yes you said you didn't see x' mentioned anywhere. I deal with the main equation that has 3 axes in it: ct', ct, x. I superimpose coordinates on the ct' axis that represent the coordinates for 3 different velocities: v, v' and $$v_t$$. I deal in velocities and only break them up into their time and space components at the end if requested. I don't use coordinate rotation to create ct' and x' coordinates. The x' axis is the same as the line of simultaneity from the 'moving' perspective which is 1/v with a v slope= ct/x. Again this math deals primarily in velocities, not their space and time constituents (except at the end).
 
Damn found some transcription mistakes

[ tex]w_t= c / Y_w = c / (Y_vY_u(1 + vu/c^2)) [/tex]
[ tex] Y_w = c / (Y_vY_u(1 + vu/c^2)) [/tex]

[ tex]Y_ww = (v+u)Y_uY_v [/tex]

[ tex]DSR_v= Y_u/Y_w[/tex]
 
Ok let's begin with relative velocity.

When Alice takes off from Earth at 3/5c, she has no on-board speedometer that reads .6c. Her DSR (Doppler Shift Ratio) reading relative to Earth is .5 and this is what tells her she's moving .6c relative to Earth. Her relative velocity to other ships moving would be different relative velocities. The thing about relative velocities to everything is they are just as valid if you made them all relative to one common thing such as the Earth.

Consider all cars on Earth being equipped with radar guns instead of speedometers. You could measure your velocity relative to every car. You could assume, despite the Earth's momentum, that you can't tell if your car is moving relative to a stationary Earth or your wheels are spinning the entire planet underneath them like a giant hamster cage. You could extend that view to thinking, from your perspective, the sun orbits the Earth and the entire universe speeds past a proton in the LHC. With adjustments to gravity and acceleration effects, your calculations would be pretty close to those of a more realistic perspective of relative velocity.

(Before I continue, consider this very important fact: space (and substances like water or glass) is an electromagnet medium with permittivity and permeability properties that define the speed of the electromagnetic wave that propagates using that medium. Space is made up of a distributed capacitance and inductance that can be compared to a material medium's inertia and elasticity that allows it, via Hooke's law, to allow propagation of mechanical waves. The equation for EM wave propagation is very similar to Hooke's law for mechanical wave propagation.

The important difference is material has no relative velocity to a vacuum's electromagnetic medium. If you blasted a vacuum bottle into space, the relative velocity of the bottle to the vacuum it contains is the same as the bottle's relative velocity to the vacuum it's speeding through. It's as if the enclosed bottle's behaving like an open ended cylinder. Since light is an EM wave travelling through an EM medium, the bottle also has no velocity relative to light.

This is a bit simplistic because the Fizeau experiment suggests that the EM medium of water limits the speed of light to .75c. This means if you blasted a water bottle into space, the velocity of the bottle could be added to the medium and hence to the light that travels through that medium. It would act as any wave through a medium moving relative to an outside observer.)

Anyway, relativity uses spacetime diagrams (Minkowski, Epstein or Loedel) to try to depict relative velocity. However they are incomplete without considering each perspective as stationary so two diagrams are needed to depict relative velocity. I have gotten around this and can provide both perspectives in a single diagram because I don't use coordinate rotation in my math.

The closest relativity can come to depicting it in a single diagram is using the Loedel diagram. 3/5c leaving Earth could be depicted as two half speed 1/3c ships leaving Earth in opposite directions. A more precise description is that a single ship would leave Earth at 1/3c and somehow the Earth would blast away at 1/3c from a faceless point in space from where the ship and Earth once stood together. If you wanted to maintain Earth as the common background frame, the depiction of the ships leaving it at 1/3c would translate to a .6c relative velocity depiction as 1 ship being the stationary frame, the other leaving it at .6c and the Earth between the two leaving the blast off point at half speed 1/3c. Although a spacetime diagram is supposedly about the two ships, Earth is also a participant and so is this non-physical cartesian coordinate background which is the true reference frame in empty space.

The Earth is travelling at what I call the Loedel or half speed $$v_h$$ velocity through space. Wiki defines it as an intermediate perspective between two equal velocities of opposite sign. (I'm paraphrasing because this is difficult to find on Wiki.) It is completely ignored by relativity but is extremely important to this math exercise. If you want I can provide how all this post looks using spacetime diagrams but I've found no one knows how to read them so they get intimidated or suspicious I'm trying to pull a fast one on them. Plus they take a lot of work.
 
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No further questions so I guess I've lost everyone? Ok let's get deeper into the math.

https://photos.app.goo.gl/pZyjCkU9dM6sQw1s7

Here is the Loedel diagram of Alice leaving Bob on Earth at .6c. It would not be the same diagram as Earth being the reference frame and Bob and Alice leaving Earth at 1/3c in opposite directions. In fact the reference frame is empty space here represented by the half speed Loedel velocity $$v_h$$. Nothing is referenced to this stationary reference frame and everything is referenced to the Earth/Bob velocity 0f -1/3c. There are no rules in math where a reference frame needs to be stationary. Breaking this rule means I can set any velocity axis as the reference simultaneously on one spacetime diagram.

It does alter the main equation slightly though. It introduces gammas with subscripts: $$Y_o$$ for the reference frame and $$Y_1$$ for the subsequent frame depending on what perspective you choose. If you want to look from Bob`s perspective the main equation is

$$(ct')^2 = (ct)^2 - Y_ox^2 $$

from Alice's perspective it's

$$(ct')^2 = (ct)^2 - Y_1x^2 $$.

What I'm doing mathematically is layering the spacetime diagrams into one diagram. Relativity doesn't do this but again this thread is a math presentation that could be applied to relativistic physics to arrive at the same answers.

You may also notice that I've layered 2 separate scenarios for Alice, one where she turns back to Bob at t'=2 and the other where she just keeps going.

I've developed a quick formula for calculating $$v_h $$ but it's only applicable to Minkowski diagrams where the reference frame is stationary. It doesn't strictly apply to a Loedel diagram where $$v_h=0) $$ always. There's no need to develop a more general formula as I'll show Minkowski mostly doesn't get into the trouble of having empty space as a stationary reference frame.

$$v_h=Yv/(Y+1) $$.

Notice the thin green curved line which represents the hyperbolas generated by the main equation. They intersect all the velocity lines at the same time on their universally accurate atomic clocks which all beat at the same time rate within their frames. Notice the thick green line which is the line of simultaneity from the Loedel perspective. It also intersects the two blue velocity lines at the same proper time. It's not physically correct to call the Loedel line of perspective simultaneity the line of proper simultaneity but I use it as such mathematically. It seems to me that the hyperbolas generated by the main equation suggest proper time simultaneity but I've never gotten a straight answer to this question.

The two thin red lines are Bob and Alice's perspective lines of simultaneity. If I added more perspectives, their lines of simultaneity would look like a hysteresis eye around the thick green line which is the only one that joins 2 proper times as endpoints. The Loedel line of simultaneity (which I've tried to coin) is not discussed in relativity but to me, mathematically, it has significant attributes especially that it provides an unambiguous perspective of proper time.

Notice there are two light lines emanating from Bob and Alice signalling each other simultaneously from the Loedel perspective at t'=2. I will show how light lines are used to stitch together the simple graphical construct that will be the mathematical building block of graphically representing all the physical phenomena of relativity. But for now, I will show how the light signals are affected when this Loedel diagram is depicted as Minkowski and reverse-Minkowski diagrams.
 
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Ok I'll use the quote feature from now on but I don't think that's the main reason my posts are incomprehensible.
 
ralfcis said:
Ok I'll use the quote feature from now on but I don't think that's the main reason my posts are incomprehensible.
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I can't speak for others, but I'm not about to fight through them to figure out what you're saying and what someone else is saying. It's really in your best interest to make comprehension as seamless as possible for your readers. I've asked moderation if there's any way they can help you repair the thread.
 
It does need repair, I've really messed up the equations page too. The last forum I left there was no time limit on correcting past posts. At the end, the red ink corrections made it hard to read. I should've just erased all the past mistakes.
 
DaveC426913 said:
That wasn't what I was referring to.
Letting us know that you've tried to post this on other sites, and have either left or been banned, raises red flags.
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I know, I was just trying to diffuse your implications. If you have evidence of what you're saying as opposed to what I clearly said, please present it. Is this what you'd rather this thread be about? You want to know why I chose to leave?
 
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