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SPT-Acceleration and the Equivalence Principle

Posted August 17, 2020 3:00 AM by Jorrie

One day after Alice's arrival at Deep-2, she and Dot sat down for the change of shift briefing - recall that Dot must soon return for Home leave. While having a cuppa and a slice of Deep grown space-sponge, Alice pressed a few buttons to allow her ship's clocks to sync to Dot'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 it was with all those accelerations thrown into the mix! Fortunately in the next year he did show us how space-propertime works and it became much more intuitive. And how easily it leads into general relativity!"

This imaginary dialogue might serve as gentle introduction into how gravity works and how space-propertime diagrams can be used to illustrate it.

Take the simplest case that we had before, where Alice just accelerated away from inertial Bob at 1g for exactly one year on her clock. From the SPT principle (or equations), we have found that Alice would reach a point 0.54 lyr distant in Bob's frame, with Bob's clock reading 1.18 years. What will this scenario "look like" if we choose Alice's (accelerating) frame as reference?

Essentially we must straighten Alice's worldline, as her frame is now the reference. This dictates that we must slant all the horizontal lines of constant propertime downward from Alice's position, in such a way that they all converge at 1 lyr to the left on the original distance axis, as pictured on the right.

To Alice's it will initially appear as if Bob accelerates at 1g to the left.[a] Bob is inertial and feels no acceleration (it is a coordinate acceleration, vs. Alice's proper acceleration). Alice feels the proper 1g acceleration, but in her own frame, she is not moving, of course.

This process will curve and compress Bob's worldline from Alice's perspective, so it will not retain it's length, but will appears to be contracted. Bob's worldline will still end at the slanted line of 1.18 yr, as before. Coordinate acceleration can obviously not change Bob's propertime readings.

Can it really be this simple? You bet it can. It embodies what is called the 'Einstein equivalence principle'. That point where all the propertime lines converge is an event horizon. At 1 lyr behind her, from Alice's perspective, the universe 'disappears' into a 'coordinate singularity'[b]. For as long as she keeps up her 1g acceleration, no signal emitted farther than 1 lyr behind her can ever reach her. She will in fact be outrunning light, without going faster then light at any time. A case of a "rocket propelled tortoise against a light-speed hare".

If Alice stops accelerating an start to coast, that light will eventually catch up with her. From Alice's perspective, while accelerating, it looks roughly the same as if Bob was free-falling towards a massive black hole. But this is not quite the full story of gravity yet. It essentially shows only 'time-curvature' and does not have any 'space-curvature' as yet.

There is another issue to solve: Bob's worldline is no longer 'growing at the speed of light', as we had it before. To reinstate this intuitiveness, we can picture an extra (fictitious) spatial dimension, into which Bob's worldline can do some extra curving. We can than essentially have a "5-velocity vector". This way we restore the prior simplicity and Bob's worldline can again extend at the speed of light. Then we have proper curved spacetime, as shown below (copied from Relativity 4 Engineers).

The extra dimension is labeled "hyperspace", allowing Bob's blue worldline in Alice's frame above to curve along two dimensions. Hence allowing it to retain the "growing at the speed of light" property.

There are various ways of picturing this, but my choice eventually fell on a simple 'community edition' of a Javascript 3D graphics utility, Vis.js, into which I could embed the required .js code for calculating and plotting the curve. It sports some animation and user rotation features for the graphic, which I found very valuable for interpreting and understanding the process. Click this link: 3D-Hyper-Space-Proper-Time-Diagram.

It should open a separate window, so it is best viewed side by side with this description. It starts with the three axes: light-year (space), year (time) and 'hyper' (for hyper-space), with two blue dots symmetrically placed around the origin (0,0,0). The right-hand one represents Bob's and Alice's starting positions, 1 ly from the origin and the left-hand one is just a mirror image of Bob's position on the other side of the mass M. This is for easier scaling within the restrictions of the utility and besides, the symmetry looks prettier and is handy when trying to view the scenario from different directions.

I have made Bob free-fall all the way to almost at the central circularity, while Alice uses her continuous propulsion system to hold station at r=1 lyr Schwarzschild radial coordinate from the singularity at r=0.

First just click 'Play' and watch Bob's worldline (and his mirror image) falls into the central singularity of the black hole (BH). The dot separation is constant on the diagram and where Bob falls through the event horizon, the dot colors change to gold. The red dots are Alice's worldline, as she sits at constant distance from the BH and just "moves through time at speed c".

Once the simulation starts over, click 'stop' to pause and dragon the picture to position the eye for different perspectives. Once paused you can also step forward and backwards, or drag the slider for any desired frame. You effectively have a "God's eye view" from anywhere outside space and time.

By rotating to various vantage points, one can convince yourself that the worldline increment per step is indeed the same for Alice and Bob, all the way until Bob reaches the dreaded central singularity. He may never actually reach it, but to answer that requires a theory of quantum gravity, which mankind has not quite found.

In a follow-on post, I will say something about the numerics around this simulation, e.g. what the mass and size of the BH must be, at what distance Alice sits from the horizon and how long it takes Bob to meet his demise, if that is what will happen.

-J

Notes

[a] Bob's acceleration as observed in Alice's frame will gradually decrease to near zero as he 'falls' closer to the coordinate singularity of the event horizon.

[b] A coordinate singularity means that the choice of coordinates creates an event horizon and an apparent black hole, which can be removed by a change of coordinates.

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#1

Re: SPT-Acceleration and the Equivalence Principle

08/17/2020 6:11 AM

Is it possible that Black Holes as represented don't exist, but are just an observational anomaly due to curved space/time...?

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#2
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Re: SPT-Acceleration and the Equivalence Principle

08/17/2020 12:49 PM

The observational evidence for the existence of BH's is very strong, albeit indirectly. The spacetime curvature does not go singular at the gravitational horizon - it only does so at the center, where GR breaks down. I would not label that as an observational anomaly.

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Re: SPT-Acceleration and the Equivalence Principle

08/18/2020 12:01 AM

I definitely appreciate your efforts here, and have made more than a trivial attempt to understand the concepts, but the principal result has been confirmation that I'm an old-school Classical Physicist!

Thanks for trying to help me!

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Re: SPT-Acceleration and the Equivalence Principle

08/18/2020 3:16 AM

Hi DK, I'm glad it helped a little bit. I attempted to make Einstein "as simple as possible, but not simpler".

I found that most engineers have no problem with thinking in 3D, so I have "developed" these Space-Proper-Time diagrams, originally introduced by Epstein, further by the introduction of the constancy of the incremental space-propertime path length for all observers, even in a strong gravity environment.

Outside of the horizons of black holes, the geometrical properties of these diagrams are "cast in stone" (Einstein), but inside there is some doubt. I will elaborate on that in a later Blog post.

Feel free to ask any question in connection with this geometric representation.

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Re: SPT-Acceleration and the Equivalence Principle

08/18/2020 9:39 AM

Well what I mean is we view the black hole as a dark sphere, at least that is the only way I've seen it represented, but isn't it possible that this is just the appearance from afar and the actual shape is totally different, but is not apparent, but the result of the extreme warpage of space and the time distortion that makes it appear as a black sphere....?

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#6
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Re: SPT-Acceleration and the Equivalence Principle

08/18/2020 11:09 AM

The "dark sphere" was never actually observed, and maybe never will. The closest we came is some 2 years ago when "https://www.jpl.nasa.gov/edu/news/2019/4/19/how-scientists-captured-the-first-image-of-a-black-hole/" was released by the Event Horizon project. They had to subtract a lot of light that came from between us and the center of the galaxy M87. From the image it appears as if we are not looking at it from normal to the accretion disc, so it is somewhat elliptical, but likely symmetrical around the disc axis.

Apart from that, general relativity (gr) predicts what possible forms the horizon can take quite accurately, depending on the mass and the spin rate of the BH. Spin flattens out the sphere. It is possible that gr does not give the full picture, but relativist are pretty confident that (up to just-just outside of the event horizon), it is right.

Quantum gravity is expected to come into play somewhere near the internal singularity, but we don't really have a workable theory for that. The expectation is that (up to just-just outside of the singularity) the spacetime curvature should be as per gr. The BH that I have used is not spinning, i.e. a so-called Schwarzschild BH. Super-massive BH's are not expected to spin at any appreciable rate.

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Re: SPT-Acceleration and the Equivalence Principle

08/18/2020 11:16 AM

In other words, that the appearance from the outside at an extreme distance gives one appearance, and the view from the inside is totally different, not only in shape but in time and distance perspective relative to what is true on the outside...that the appearance on the inside could be a long tube shaped area of space that is not visible from the outside....sort of like the Doctor Who phone booth...an entire warped universe existing in another dimension....?

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#8
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Re: SPT-Acceleration and the Equivalence Principle

08/18/2020 12:13 PM

OK, I see what you are after. You don't even have to go inside and look out. Just go and park a spaceship, engines blazing, just outside of the horizon and the picture will be totally different from the distant view. Paul Sutter's Ask a Spaceman amusing short videos have some explanations:

https://www.youtube.com/watch?time_continue=314&v=TLEjjFv2ocA&feature=emb_logo

https://www.youtube.com/watch?time_continue=489&v=mooGr_PAwdM&feature=emb_logo

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