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Space-Propertime Structure and Acceleration II

Posted August 11, 2020 12:00 AM by Jorrie
Pathfinder Tags: acceleration Propertime spacetime

This is a fanciful little scenario for 'flexing the muscles' of acceleration in space-propertime a little and so giving a better intuitive feel for this representation of relativity. Remember, this is not some 'new theory', just an alternative view of standard Einstein Special Relativity.

Deep Space Astronauts, Alice, Bob, Charlie and Joe are having dinner in their base station (Deep-1), when a 'DeepNote' from their colleague Dot popped up on the Deep-Hologram in the dining hall. Dot is stationed on Deep-2, at a constant 1.66 light years from Deep-1. Dot's 10 year service award for Deep-space service is coming up in the not too far future.

Alice, Charlie and Joe are all due to depart soon in order to relieve Dot and other more distant colleagues, for their scheduled returns to Earth. After dinner, the friends decided to surprise Dot by simultaneous arrival, flyby's and messaging at her 10 year service award function. Here is the broad plan that they came up with (overview graphic one the right, with a more readable one copied below).

Bob will stay at Deep-1, Alice will use her trusted old 1g continuous acceleration spaceship, thrusting halfway to Deep-2, then reversing the thrust for 1g deceleration, in order to come to rest at Dot's station for attending the award function personally. Joe will use the brand new near-instant boost warp-drive, departing simultaneously with Alice, accelerating momentarily to a constant speed that allows him to fly past Deep-2 just when Alice arrives there.

Charlie will use the modern 3g continuous deep space drive to play a relativistic "tortoise and hare" game with Joe - departing somewhat later than the other two and then continuously accelerate at 3g. His departure must be timed so that he will pass Deep-2 exactly when the other two arrive there. The three travelers will be monetarily colocate with Dot at the function.

Not to be totally outdone, 'stay-at-home-Bob' will send congrats using the Deep-hologram system, but also timed so that his message will reach Dot simultaneously with the travelers.

The friends ask Quant-1 (Deep-1's quantum computer) for a proposal on departure times. Here is what Quant-1 came up with.

1. Alice (red curve) is to set off tomorrow at t=T0 on her Deep-synchronized ship clock, accelerating at 1g until t=T0+1.22 yr on her clock and then reverse the thrust. Her maximum speed will be 0.836c relative to Deep-1/2 and she will decelerate until she parks at halt at Deep-2, at T0+2.44 yr on her clock. It will be T0+3.08 yr Dot-time.

2. Joe (gold) is to set off at the same time as Alice, near-instantly warp to a relative speed of 0.534c and fly past Deep-2 at 2.6 yr, also at 3.08 Dot time, as planned. Interestingly, Joe will initially outrun Alice, as she is still accelerating, but somewhere Alice will pass Joe again and then he will catch op again at the moment Alice stops at Deep-2.

3. Charlie (the green hare) is to relax and train in the centrifuge for the upcoming 3g ordeal, until T0+1.12 year when he must set off at a constant 3g acceleration, until he flashes past Deep-2 at T0+1.94 yr on-board (also at T0+3.08 yr Dot time), having reached the incredible speed of 0.986c and still accelerating.

4. 'Bob-the-reference' must send his hologram-message at T0+1.42 yr, exactly 1.66 years before his friends will reach Deep-2. This is how long the signal will take to get there, because sadly, instant quantum data transfer still doesn't work.

Note that everyone arrives at Deep2 at the same coordinate time, as can be seen from their identical space-propertime path lenghts (yes, they are really the same).

Also note that right-hand thick purple vertical line is not only Dot's worldline, but represent the event when her friends arrive. This is an important point: in space-propertime, events are vertical lines, not single points like in a Minkowski diagram. This is the price one pays for the otherwise superb clarity of the uniform scale of space propertime worldlines.

Once this is well digested (?), I will move on to show how space-propertime couples to gravity through Einstein's equivalence principle and other factors.

-J

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#1

Re: Space-Propertime Structure and Acceleration II

08/11/2020 9:06 AM

Four dimensional thoughts in a two dimensional depiction is confusing; why not a four dimensional representation? Your line becomes a plane? Or a dot?

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#3
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Re: Space-Propertime Structure and Acceleration II

08/11/2020 9:54 AM

Not too difficult, but will it help? Tesseract

Fortunately, acceleration in flat spacetime does not require it, but it becomes more of an issue when gravity is considered. Then we need to see curved space AND curved time. I have a plan for that, so watch this space...

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#6
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Re: Space-Propertime Structure and Acceleration II

08/11/2020 10:54 AM

"Your line becomes a plane....? or a dot?"

Going up in dimensions: 0D dots become lines, 1D lines planes, 2D planes cubes and 3D cubes tesseracts, I think...

So what about the 2D circle? I think it goes up like 3D sphere, 4D hypersphere...

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Re: Space-Propertime Structure and Acceleration II

08/18/2020 10:38 AM

Tangentially related to these ideas, in the 1700's, Emanuel Swedenborg, Swedish scientist/philosopher/mystic, wrote about his theory of forms.

"Forms ascend in order from the lowest to the highest. The lowest form is angular, or the terrestrial. The second and next higher form is the circular, because the circumference of a circle is a perpetual angle. The form above this is the spiral, parent and measure of circular forms. The form above this is the vortical, or perpetual-spiral. Next is the perpetual-vortical, or celestial, and last, the perpetual-celestial, or spiritual."

In his own way he goes from 2D to 3D, and beyond, although the geometry of his celestial and spiritual forms escape visualisation.

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#2

Re: Space-Propertime Structure and Acceleration II

08/11/2020 9:51 AM

My brain hurts.

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Re: Space-Propertime Structure and Acceleration II

08/11/2020 9:54 AM

From the beginning, a slow ramp-up....

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#5
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Re: Space-Propertime Structure and Acceleration II

08/11/2020 10:00 AM

"From the beginning, a slow ramp-up...."

It's a marathon lecture. Can you perhaps point us to where he gets to four-dimensional representation?

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Re: Space-Propertime Structure and Acceleration II

08/12/2020 3:15 AM

Well I don't know that it does....but I'll let you know ....

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