BurtJordaan
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Why is relativity so hard to learn?[10X]
May 5th, 2017, 12:13 am

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OK, back to Alice's mission. Her plan has changed a little and she will now first stop in the region of Alpha-C (AC) for some observations. It is way too late to achieve the rocket braking at the original very low (0.075g) acceleration, so Alice asked her INS computer to get them into a 'parking' orbit at a more comfortable 1g retro-burn. Her INS and my trusted spreadsheet 'know' that at -1g, it will take 0.69 years to get the coasting ship to zero velocity relative to Bob, and that it must start to brake 0.2 lyrs (on their rotated map) short of the parking position. At just over 11 years into her mission, Alice comes to rest in Bob's spacetime structure again.

During the 0.69 years of deceleration, Alice's tilted structure has gradually rotated back to align with Bob's structure once again. While their structures were at an angle to each other, space and time were different for them. Here is a summary of the space and time accounting columns:

Stage/event _______ T_Bob yr __ D_Bob lyr __ T_Alice yr__ Alice g's __v_rel/c
Start burn 1 _________ 0 ________ 0 ________ 0 ________ 0.075 ___ 0
End burn 1 ________ 10.07 _____ 3.38 ______ 9.3 _______ 0 _______ 0.6
Duration coast 1 ____ 1.29 _____ 0.77 ______ 1.03 _______ 0 _______ 0.6
Start burn 2 _______ 11.36 _____ 4.15 _____ 10.33 ______ -1 _______ 0.6
Duration burn 2 _____ 0.75 ______0.25 ______ 0.69 ______ -1 _______ 0.6
End burn 2 ________ 12.11 _____ 4.40 ______ 11.02 ______ 0 _______ 0

(The spacing may be a bit random - does anyone know how to make a proper table using the 'table-tag' above in this editor?)

Alice and Bob are now separated by a distance of 4.4 lyrs. How do they test whether the final figures are correct? That's now apart from asking an accountant to check the addition! The simplest way is for Bob to have another observer, say Dot, stationed in slow orbit around AC. Bob and Dot are in the same inertial frame (spacetime structure), just separated in space and they could have synchronized their clocks long before Alice arrived. If Alice 'parked' her ship near Dot, they can directly compare their clocks and confirm the final times above. There are other methods as well, but let us only consider this one method for now, because it's simple and works superbly.

Alice and Dot can also confirm that their clocks now 'run the same', so we can safely assume that they are now in the same spacetime structure. Before the long journey, Alice and Bob were together and in the same spacetime structure. During the journey, Alice was in a spacetime structure of her own, which changed orientation almost continuously relative to Bob's structure. But now Alice's structure is lined up with Bob and Dot's structure and the only observable difference between them is that less propertime has elapsed for Alice then for the other two.

The root cause of the difference is that Alice has actively changed her spacetime structure by expending a lot of energy in terms of fuel. We pictured it as a rotation of Alice's structure in relative to Bob's, so that she covered spatial distance at the cost of covering time. This the nature of relativistic spacetime.

I am working on a space-propertime diagram to illustrate the complete situation, but that will have to stand over for the next part. In the meantime, I will leave you to ponder the following statement and decide if it is correct: it does not matter in which direction Alice did her one-way journey, or what Bob's speed relative to the 'universe at large' is, the outcome of experiment would have been the same.

Last edited by BurtJordaan on May 6th, 2017, 12:43 pm, edited 3 times in total.
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