Acceleration in Space-Propertime

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Acceleration in Space-Propertime

Postby BurtJordaan on May 9th, 2017, 3:59 am 

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.

Epstein acceleration.png
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.

Epstein acceleration1.png
Determination of radius of curvature in SPT


Note that is the Epstein angle of the velocity vector from the axis in SPT diagrams and . Acceleration is indicated by , 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...)

SPT acceleration.png
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 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.
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Re: Acceleration in Space-Propertime

Postby BurtJordaan on May 11th, 2017, 1:50 am 

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).

Image

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

Postby BurtJordaan on May 15th, 2017, 2:00 am 

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.

Epstein 6_100-3.png
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?

Image

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.
Reason: Added PS
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Re: Acceleration in Space-Propertime

Postby BurtJordaan on May 17th, 2017, 4:51 pm 

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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Re: Acceleration in Space-Propertime [12]

Postby BurtJordaan on May 31st, 2017, 4:37 am 

I have been rather tied up for week or so, but here is part [12] of the quest for making relativity "less hard to learn". To refresh memory, here is a copy of the space-propertime diagram of part [11].[a]

Image
BurtJordaan » 07 May 2017, 13:22 wrote:The difference in propertimes at the end of the test is obvious: Alice has recorded just over one year less than Bob. So, was Alice's clock simply slower than Bob's? No, Alice simply moved less through propertime. Actually the real length of the red curve is identical to the length of the blue arrows. It does not look like it, because I had to squash the diagram from top to bottom to about 1:3 in order not have an absurdly tall diagram. Trust me, the red curve is mathematically of identical length to the blue arrow lengths. IMO, this is really the clincher for this method of diagramming space and time for beginners.

It pictures Alice as accelerating away from the inertial Bob at a leisurely acceleration, reaching a relative speed 60% of the speed of light, coasted for just about one year on her clock and then braked sharply (1g) to come to a stop (relative to Bob) in the immediate vicinity of Alpha Centauri, some 4.4 light years from Bob. This happened to be where her old class mate Dot is stationed.

The two girls greeted and then celebrated the event of Alice's arrival by a cuppa and a slice of space cake. While Alice were pressing a few buttons to allow her clocks to set themselves to Dot/Bob's time zone (one hour and a bit ahead of her clocks), Dot watched, smiled and remarked: "Do you still remember how much we at first struggled in Prof Zok's class, trying to figure out this time adjustment through the old-style relativity of simultaneity and synchronization offsets?

"Yea, how horrible with all those accelerations thrown into the mix! It was only in the next year, when he did show us how spacetime intervals work and how they fit into the general spacetime structure, that I did not feel completely lost".

"The same here, but granted, we then had more math skill than during his previous lectures. I remember that we had to use the hyperbolic trigonometry functions to deal with the changes in the spacetime structure. However, the space-propertime view is so simple that we hardly need any of those more complicated math functions."

"Yes Dot, it still beats me why the Prof never mentioned space-propertime as a method of viewing relativity. Especially since one can comprehend acceleration and even gravity quite readily by just looking at the diagrams."

"Did I hear you say 'gravity'? I thought that we should restrict the use to near-flat spacetime, where gravity is too weak to matter. But wait, we are dealing with acceleration. Hmm... through the Einstein equivalence principle ... you may be right."

This little fanciful dialogue may serve to introduce gravity 'gently' into the discussion, as we shall see later on.

Let us take a simple case and let Alice just accelerate away from Bob at 1g for one year on her clock. Using the SPT principles above, Alice will reach a point 0.544 lyr in Bob's frame and Bob's frame clocks will read 1.18 years (roughly 14 months, 4 days and 21 hours). What will this scenario "look like" if we should have chosen Alice's (accelerating) frame as reference?

We know what the respective propertimes are, so a naïve view would be to just straighten the red worldine to reach 1 year and at the same time curve (and "stretch") the blue worldline to the left, reaching 1.18 years at -0.544 light years distance, as pictured below.

Bob-Alice-reciprocal.png
Naïve transformation to Alice's accelerating frame

We know that the naïve view is incorrect, because we have seen that the space-propetime structure of Alice is rotating while she accelerates. But all is not lost, we are just not done yet - we have some rotation to do. For a 1g acceleration, all the horizontal (constant propertime) lines must simply be rotated so that all of them intersect the time-zero horizontal at the -1 ly point as shown below.

Alice-propertime-frame.png
Proper transformation to Alice's accelerating frame

Logically, the blue spacetime path of Bob gets altered by this rotation and guess where Bob ends up? Just below the 1.2 year slope of constant propertime - where else? Can it really be this simple? You bet it is. And what's more, this smoothly converts to general relativity, with gravity solidly in the picture. Bt that will the subject of the next part, because this part needs to be properly digested and analyzed.

Firstly, that intersection point of all the propertime lines is called a 'coordinate singularity', also called an 'event horizon'. This is because from Alice's perspective, the universe behind her "stops" at -1 lyr, in the sense that no signal can reach her from beyond that, provided that she keeps up her acceleration. It is like a supercharged tortoise that accelerates to a speed approaching that of the hare, so given enough head-start, the tortoise always stays ahead, no matter how long the race.

If Alice stops accelerating, signals from farther than 1 lyr behind her will eventually reach her. I must stress that this coordinate singularity only exists for the accelerating Alice and it is not a black hole that forms in her wake, as I've seen some popular writings say for effect. It is true that in Alice's accelerating perspective, space and time gets "compressed" behind her and in effect "uncompressed" or "stretched" in front of her.

In the next part we will use the equivalence between gravity and acceleration to show that this very same graph can also be used for gravity, and that the idea of a black hole then acquires some reality. ;)

-=0=-\

[a] This was originally intended for the thread "Why is relativity so hard to learn?", but I have decided to shift it to the physics section, where it really belongs.
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Re: Acceleration in Space-Propertime [13]

Postby BurtJordaan on June 3rd, 2017, 4:50 am 

BurtJordaan » 31 May 2017, 10:37 wrote:In the next part we will use the equivalence between gravity and acceleration to show that this very same graph can also be used for gravity, and that the idea of a black hole then acquires some reality. ;)


Here is a side by side comparison between Bob's inertial frame and Alice's accelerating frame, where Alice experiences a 1g proper acceleration to the right.

Alice-Tom-propertime-frame.png
Comparison between Bob's inertial frame and Alice's accelerating frame.

In the accelerating frame (right), Alice's space-propertime (SPT) path has been 'straightened' and in the process her reference frame had to be distorted so that all the horizontal grid lines converge at -1 lyr. This is just the inverse of the 1g acceleration (if it was 2g, the converging would have been at 1/2=0.5 lyr).

This also means that Bob's SPT path had to be "compressed", both in distance and time, as seem by Alice's accelerating frame. It can also be viewed as if, due to her acceleration, Alice "sees" the original rest coordinates as tilted into a 3rd dimension (into the screen), so that Bob's SPT path is still straight and of the same length, but has moved around something resembling part of a cone and is then projected back onto the screen. More about that below.

The most interesting aspect is that Bob's clock would keep on reading his normal time and can be read off the chart along the sloped grid lines. Let us for a moment imagine that Alice's spacecraft was 0.33 lyr long, with Alice in the front and her colleague Tom riding in the back. Bob would then have passed Tom when Alice reached 1yr on her clock. Tom could have peeped into Bob's cockpit and have read his clock, which would have said 1.18 yr. However, Tom's own clock will read 0.67 yr at that event.

Now we have three clocks that all started with 0, but giving 3 different reading for essentially the same event. Clocks running at different rates? No, not quite - it is rather a case of clocks running at the same universal rate, but have covered different SPT paths, just like we had before.

This is easily comprehended if we consider an additional spacetime dimension for the right-hand diagram, directly into the screen, so that the SPT paths can be tilted (or curved) into that 3rd dimension. The SPT paths of all three (Alice, Bob and Tom) are of equal length in this 3-D spacetime. These paths are tilted into the screen at angles depending on their distances from the (coordinate) singularity at x = -1.0. Even Alice's path is already tilted relative to observers to the right (in front) of her, but more about that later.

Remember, all this is just how Alice is observing things in her accelerated frame of reference. She essentially observes curved spacetime all around her. I will try to find a simple 3-D package that can take my data points and present them in a 3-D view from different perspectives. Anyone knows of a simple, preferably free-on-web utility that can do this? It will be cool way for comprehension of what's going on.

Now for the (slow) transition to gravity. The scenario sketched above is exactly equivalent to Alice sitting at a constant distance from a mass concentration that causes a uniform gravitational field[a] in her vicinity, trying to accelerate her towards the singularity at 1g. Alice has a rocket to keep her stationary in this position in the field, and she feels the same continuous 1g acceleration as before. The inertial Bob will simply be free-falling towards the singularity and feel no g-forces, as before. Tom, in the back of the rocket will actually feel an acceleration of 1.5g, because he is only 0.67 lyr from the singularity. He would have felt the same 1.5g acceleration in the non-gravity situation. Everything is simply equivalent.

With gravity having entered, the singularity now becomes real, well sort-of... It is not yet Einstein's gravity (we will get to that in the next part), but unless Bob has concealed a reserve rocket somewhere, he will eventually (in his own frame) reach the singularity, but first being stretched ("spagettified") and then crushed. What then happen to the atoms and particles of his body, we do not quite know...

In the next part we will see that in the 1912-14 period, Einstein was essentially still viewing gravity like the principles stated above, but he used the math, not SPT diagrams, AFAIK. He still did not consider spatial curvature in his equations, which came later. He finished the first version of his full theory of general relativity (GR) in Sep. 1915 (published Nov 1915).

-=0=-

Note [a]: Uniform gravity is a first order approximation to full relativistic gravity. It simply means that higher order terms in the full equations are ignored. It works perfectly fine when the gravitational field is not too intense, like in most places in the solar system.
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Re: Acceleration in Space-Propertime [13]

Postby BurtJordaan on June 10th, 2017, 12:24 am 

I found a free package with some 3-D functionality. I have transferred the data from my spreadsheet to their "Playground" and it produced the charts below. From my previous post:

BurtJordaan » 03 Jun 2017, 10:50 wrote:This is easily comprehended if we consider an additional spacetime dimension for the right-hand diagram, directly into the screen, so that the SPT paths can be tilted (or curved) into that 3rd dimension. The SPT paths of all three (Alice, Bob and Tom) are of equal length in this 3-D spacetime. These paths are tilted into the screen at angles depending on their distances from the (coordinate) singularity at x = -1.0.

First the same view as in the right hand one of my previous post, but here pictured for constant distance behind Alice (steps 0.1 yr) and for constant movement (0.1 years) through 5-D space-propertime. Only 2 dimensions are shown, 1 space and one time.[a]

Viscanvas1-1.png
Normal 2-D Space-Propertime view


As we shall see below, the propertime steps are getting smaller to the left, but that's because more of the unit-length steps go into the extra dimension. Below is a plot of the same data-set, slightly rotated so that the 5th (hyper-space) dimension becomes visible in 3-D. It now shows 2 spatial dimensions and one of propertime.

Viscanvas2-1.png
3-D view of Space-Propertime (2 spatial dimensions)


Note that the 'singularity' at distance -1 is not singular in the z-direction. It actually has the same length (one unit) than all the other sloped lines. The effect of the coordinate singularity for proper acceleration is two-fold: no signal can ever reach Alice if emitted at or 'behind' the "singularity". Secondly, there cannot be any object there that accelerates with Alice so as to maintain a constant distance behind her[b] - because it would have needed infinite proper acceleration.

This latter fact is the same as what happens at the event horizon of a black hole. It would also require an infinite thrust to maintain static there. Since that is impossible, the rocket would simply fall in to the hole.

I hope this makes the structure of accelerating reference frames a little easier to comprehend. Please ask if the the brief explanation is not clear enough. I will rather answer specific questions than write verbose...

Notes

[a] 5-D spacetime is just a mathematical and visualization tool and we do not think that such a dimension need to exist. All that we know is that effects of gravity and acceleration are much easier to calculate and visualize if we pretend that it exists. As you know, particle physicists utilize 11 dimensions to unravel the mysteries of their trade.

[b] A constant distance behind an accelerating Alice would mean accelerating so that a taut string remains taut, but is not stretched.
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