## Acceleration in Space-Propertime

Discussions on classical and modern physics, quantum mechanics, particle physics, thermodynamics, general and special relativity, etc.

### Acceleration in Space-Propertime

This topic is in response to some questions received in the 'Why is relativity so hard to learn?' topic, but which were deemed too technical to be dealt with there.

Because of the Euclidean nature of space-propetime (SPT) diagrams, initial acceleration can be closely approximated as a rotation at a radius R around some fixed point, which I call the 'center of curvature' below. In the appropriate dimensions, the acceleration is inversely proportional to the radius R.

Clock accelerated from rest in a SPT diagram

Because it this is not Euclidean space that we are dealing with, but rather SPT, the radius R is variable and the center of curvature's position shifts during acceleration, as indicated below.

Determination of radius of curvature in SPT

Note that $\phi$ is the Epstein angle of the velocity vector from the $\tau'$ axis in SPT diagrams and $\large cos(\phi) = sqrt{1-(v/c)^2$. Acceleration is indicated by $\large \b \ddot x$, because the attachments come from the eBook Relativity-4-Engineers, where this is the normal symbol.

Below is a simple algorithm for solving the curve through numerical integration (engineers simply love this method...)

The algorithm for plotting an accelerating clock on a SPT diagram

It is largely self-explanatory, so just a few comments are in order, I hope. The first 2 lines of code after the 'output' line simply rotates the present center of curvature to temporarily sit at a more convenient spot for calculation. The simpler calculation is performed and then the values are rotated back so that the new x,tau coordinate sits in the the correct place. One can't use zero acceleration, but I stick in a value of 1E-12 as close enough to zero for all practical purposes.

This same thing can be done in Rindler coordinates and the point by point results then transformed to SPT coordinates. I find that pretty complex and not very helpful for insight into the problem.

This was very brief, but I'll be happy to provide more information as required.

PS: without me noticing, the spell-checker corrected an error in my spelling of "position" in "center of curvature's position" to "center of curvature's "postillion". :)

Def. of postillion: a person who rides the leading left-hand horse of a team or pair drawing a coach or carriage, especially when there is no coachman. LOL.

On a more serious note, the very first diagram can be a little confusing. That $\Delta t$ should have been next to the curved path to indicate that it means the SPT path length and not the time of the accelerating frame. The second diagram indicates it correctly.

BurtJordaan
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### Re: Acceleration in Space-Propertime

Criticism against the space-propertime (SPT) diagram that it is "event-unfriendly" is valid, because we will plot different observers that are present at the same event at different positions on the APT diagram. The spatial coordinate (x) will be the same, but since each observer will have a different elapsed propertime, the tau coordinate will be plotted at different values, e.g. the 12.2 years for Bob's blue frame and the 11.03 years for Alice's red frame, obviously for the same event (Alice arrives at AC).

This is the price we pay for having the actual path lengths the same for all observers. In Minkowski diagrams, the event is plotted in the same position for all observers, but the path lengths are different, i.e. the price to pay in Minkowski is that the scale differs along the paths of different observers and is difficult to vizulaize.

The way to overcome this drawback of the SPT diagrams is to always choose a fully inertial frame as reference. If we have one or more observers that remain unaccelerated, even if they are moving relative to each other, choose any on of them as the reference frame. If all observers are accelerating at some stage, choose any convenient inertial frame as reference and plot each observer's SPT coordinate against this inertial frame's SPT structure.

In such a case, the event has uniquely defined coordinates, i.e. that of the chosen inertial frame. This can then be used to determine the spacetime interval between that event at any event that is also plotted on this inertial frame. Like for instance the event when Alice has left Bob.

I will program, plot and discuss some multi-observer acceleration scenario in a follow-on post.

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### Re: Acceleration in Space-Propertime

BurtJordaan » 11 May 2017, 07:50 wrote:In such a case, the event has uniquely defined coordinates, i.e. that of the chosen inertial frame. This can then be used to determine the spacetime interval between that event at any event that is also plotted on this inertial frame. Like for instance the event when Alice has left Bob.

I will program, plot and discuss some multi-observer acceleration scenario in a follow-on post.

Four Astronauts, Alice, Bob, Charlie and Joe are having dinner in their space station (Deep-1), reading a email from their colleague Dot, who is stationed on Deep-2, situated at a permanent 1.66 light years from Deep-1. 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 make their respective journeys a spacetime exercise and at the same time to surprise Dot by simultaneous flyby's and messaging at the anniversary of her 10 year service on Deep-2. Here is the broad plan that they came up with.

Alice will use the trusted old 1g continuous acceleration rocket, thrusting halfway to Deep-2, then reversing the rocket in order to come to rest at the Dot's station so that she can attend the anniversary personally. Joe will use the brand new near-instant boost drive, departing simultaneously with Alice, accelerating momentarily to a constant speed that he maintains all the way and onward to his destination. It must be calculated so that he flashes past Deep-2 exactly when Alice arrives there.

The space-propertime diagram below depicts the broad plan with some details on timing. The dynamic details will be given after the rest of plan has been described in more detail.

Deep1/2 Space-Propertime Scenario

Charlie will use the modern 3g continuous deep space drive to play a relativistic "tortoise and hare" game - departing somewhat later than the other two and than continuously accelerate at 3g. His departure time shall be timed so that he will also pass Deep-2 precisely when the other two arrive there. The hare must take care not to lose!

In other words, the 'race' will need a 'photo-finish' and may be so close that it can be considered a draw. Not to be totally outdone, 'stay-at-home-Bob' will send congrats using ordinary light, but also so timed that it will reach Dot simultaneously with the travelers.

The numbers were crunched out by Quant-1, Deep-1's quantum computer, which is pretty good at "what-if" scenarios. The friends fed it a variety of options and then settled on the following numbers for their planned surprise visit.

1. Alice (green curve) sets of at t=0 on her synchronized ship-stopwatch, accelerating at 1g and then at 0.882 yr, reverses thrust (which the Qunt-1 said is 1.2 yr on the Deep-1 stopwatch). She will reached a speed of 0.836c relatively, before starting the slowdown. She will cruise to a halt at Deep-2 at 1.44 yr (3.08 yr Deep-2 time).

2. Joe (gold) also sets of immediately, near-instantly reaching a relative speed of 0.534c and fly past Deep-2 at 2.6 yr, also at 3.08 Deep-2 time, as planned.

3. Charlie (the green hare) relaxes and trains in the centrifuge for the upcoming 3g ordeal, until 1.12 years after the others have departed. He sets off at a constant 3g, until he flashes past Deep-2 at 1.94 yr (also at 3.08 yr Deep-2) having reached a speed of 0.986c and still accelerating.

4. 'Bob-the-reference' sends his congrats message at 1.42 yr, exactly 1.66 years before Quant-1 told him that the friends will reach Deep-2.

In reality, these insight-giving curves were all plotted by my simple spreadsheet, just by adding 'delta t' = 0.02 year long spacetime vectors, at the proper angle for each friend's present velocity relative to Deep-1. I have illustrated the method in the OP (graphic copied below for convenience). This is way cool, but it would have been even cooler of I had the software to animate the worldlines. Anybody interest is grafting such an animation package?

Any questions on the scenario?

PS: Sorry for all the typos; I fixed all the ones that could see. Like most engineers, I'm better at designing the thing than writing it all up. ;)
Last edited by BurtJordaan on May 15th, 2017, 12:30 pm, edited 1 time in total.

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### Re: Acceleration in Space-Propertime

I have programmed a crude Space-Propertime with Acceleration HTML/Java script to show how the path-lengths in space-propertime grow at the same rate. That rate is loosely speaking equal to c, depending on the coordinate system used. It is called the 4-velocity in relativity. The growing lines and curves all end up with the same lengths (5 lyrs).

Unfortunately, it has zero other utility (such as pausing, or asking for other parameters) than to re-run it by refreshing the page. The squares are 1 lyr a side, with the blue vertical the reference clock, the black horizontal a light pulse, the green curve a clock accelerating from rest with a constant 0.4g proper-acceleration and the red curve a clock starting at a relative speed of 0.2c and with constant 1g proper-acceleration.[a]

The three clocks all show different elapsed propertimes (tau), with the reference clock (blue) showing 5 years. The light pulse is obviously not a clock and shows nothing, but it ends up 5 lyrs from the origin.

I thought this may help a little in our struggles to understand relativity... ;)

Note [a]: Proper-acceleration is as measured on-board with an accelerometer. In the reference frame, the observed acceleration will decrease over time.

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