Beads-on-a-string view of Relativistic Gravity

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Beads-on-a-string view of Relativistic Gravity

Postby BurtJordaan on June 15th, 2017, 8:22 am 

In the Acceleration in Space-Propertime (monologue), I have attempted to describe the structure of space-propertime as observed in the reference frame of an accelerating observer, Alice. It can be represented by a 'hyperspace-propertime' (HSPT)[a] diagram, as reposted below.

Image

It looks somewhat like beads, evenly spaced on equal-length taut strings, being rotated relative to each other. Hence the thread's title. To recap, Alice, the right-most bead-string, accelerates to the right at 1g, with each bead representing a propertime step of constant value on her clock. She is however stationary in her own reference frame. The bead-strings behind her represents observers with clocks that are also accelerating so that each remain at a constant (but different) distance behind Alice.

As mentioned in the other post, no observer can achieve that at x = -1 (the left-most bead-string), because it will need infinite acceleration in order to do so. Reminiscent of the what happens at the event horizon of a black hole? Yes, exactly! But before we jump into black holes, here is the structure above in a somewhat controllable 3-D package. Just drag the pointer on the image to rotate it in a direction (I'm not sure what it does on touch-screens).

You may notice that the z-axis is now propertime and that the y-axis is now hyperspace (the 4th spatial dimension). This is more compatible with the specific 3-D package used for the image.[b] There are now 21 beads on each string (20 HSPT steps). By rotating, you can verify that all strings are of equal length and bead-spacing.

What you are viewing is curved spacetime, as viewed by an accelerating observer, or as viewed by an observer sitting on earth's surface in earth's gravitational field. It is this type of gravity that Einstein used in 1911 to predict (half of the true) deflection of light that passes close to the sun. We will get to the other half soon, but like Einstein, we first need to digest the 'gravity of the situation'.[c]

At the surface of earth (or of the sun), gravity is not particularly strong and the constant linear acceleration scenario is extremely close to the gravity scenario. If we consider earth as a (non-rotating) point source of gravity, then unless we were propped up by something, we would all have fallen to the center of earth and been crushed by the singularity that would have existed there, at distance zero - essentially experiencing 'infinite gravity' there.

Using this linear approximation of real gravity, Einstein could have predicted the full deflection of light in 1911, had he considered another (approximately) linear effect - the curvature of space (not spacetime, just space). Notice that the x-axis in the above representation is straight - it is only the rotation (or tilting, if you like) of the bead-strings that cause the HSPT structure change with distance from the singularity.

Here is an attempt to picture spatial curvature on the HSPT structure: this rotatable picture. Note that the bead strings do not start at the x-axis (where y=0), but at a position that is more and more offset in the y-direction, depending on the value of x.[d]

At the singularity (x=-1), the left-most bead-string has been shifted more than a string-length, but it is anyway out of view, inside the singularity, whatever that may mean.

Now, how would this have helped Einstein to predict the full deflection of the starlight (1.75 arc-sec)? The answer is that both the tilting (gravitational time dilation) and the shifting (curved space) of the bead-strings have equal effects on anything, including light, that moves through the HSPT structure. The curvature of space contributes the 'missing half' at the time of the 1911 prediction. The curvature of space was also needed to predict the correct perihelion for Mercury, which Einstein also calculated later, when he had the full theory.

I know this is quite a mouthful, but there is still one more 'trick' op the sleeve of the bead-string visualization - the full 2nd order representation, but that will have to stand over for another post.

Notes:

[a] Hyperspace is a term used when more than 3 spatial dimensions are considered. In order for 4-D spacetime to be curved, one must consider at least one extra spatial dimension into which the normal 4-D spacetime can be curved.

[b] I have used the 3-D graphs part of vis-4.20.0.

[c] Although this is not what I referred to above, the real "gravity of the situation" became very acute when the 1st WW broke out, just as a team of astronomers set out to measure the predicted (half-of true) deflection of light. They got captured in Russia and barely survived the ordeal, but with no results, AFAIK.

[d] The 'true' x-axis will be bent, but the 3-D utility does not support that, AFAIK.
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Re: Beads-on-a-string view of Relativistic Gravity

Postby BurtJordaan on June 19th, 2017, 6:24 am 

BurtJordaan » 15 Jun 2017, 14:22 wrote:Now, how would this have helped Einstein to predict the full deflection of the starlight (1.75 arc-sec)? The answer is that both the tilting (gravitational time dilation) and the shifting (curved space) of the bead-strings have equal effects on anything, including light, that moves through the HSPT structure. The curvature of space contributes the 'missing half' at the time of the 1911 prediction. The curvature of space was also needed to predict the correct perihelion for Mercury, which Einstein also calculated later, when he had the full theory.

Both the original predictions were however made by the first order relativistic gravitational time dilation (tilting) and spatial curvature (offset in the hyperspatial direction) that we have discussed so far. This is because the second order corrections for gravity near our sun is very small, even for light just grazing the surface. Here is another bead-string view of the hyperspace-propertime of the first order relativity used so far.

Hyperspace-propertime1.png
First order general relativistic gravity

Rotatable view available here. If you get confused later by where you are viewing from, just reload the page.

I have changed the bead sizes and also the default coloring scheme, so that each bead-string (the worldlines of static observers) have a color according to the 'depth' into the gravitational field: blue at a distance going to red and eventually black at the singularity at x = -1. Sounds about right, doesn't is? Light from those static observers being more and more red-shifted, until at the singularity, only "black light" can come out, i.e. nothing.

In the diagrams shown so far, we were actually not in a "small correction" domain, because to have the sort of hyperspace-propertime structures shown, it must be quite near a hyper-massive black hole. We normally use the regions around massive black holes to illustrate relativity, because otherwise the effects are too small to be visible on graphics. I have calculated the mass of the black hole required to give the first order effects as in the graphics shown: it comes out at about half the mass of the Milky Way!

Unrealistic as it may be, let's assume such a gigantic black hole can exist. It would have an event horizon "radius" of half a light year and at one light year "distance",[a] its gravity would be about that of earth, i.e. around '1g'. Hard to believe? I will give you the calculations in a follow-on post, should you want to check them - the math just for that is actually pretty easy. ;)

Assuming that my calcs were right, our Alice, originally accelerating at 1g away from Bob, could just as well have sat stationary, one lyr from this hyper-massive black hole (BH), obviously using her rocket engine to maintain station. In this situation, we are forced to use the second order math and the bead string diagram would look like this.[b]

Hyperspace-propertime2.png
Second order general relativistic gravity

Rotatable view available here. If you get confused later by where you are viewing from, just reload the page.

As said above, the event horizon is at -0.5 lyrs from Alice in her coordinates and the black hole singularity is at -1 lyrs. The row of single black dots are worldines inside the BH, but Alice cannot 'look there'. That single red dot at x=-0.5, z=0, is the worldline of Bob, who tried to position himself at the event horizon of the BH. He is in the process of being sucked in by the thing, but as seen by Alice, he will never reach the event horizon - his signal will just gradually become redshifted to 'no signal at all'. So, Alice will never see or hear from Bob again - sad, but hopefully all will learn something from his fate.

Bob will however determine that he has fallen in and he will keep on living for quite some time in his own frame, but will probably in the end be spaghettified and then crushed by the singularity (at x=-1). We do not really know what will happen to Bob once he has crossed the event horizon, but it is almost certain that he will not survive.

Once we have digested this hyperspace-propertime structure, we will attempt to plot Bob's 3D worldline as he falls from Alice's position, through this contorted structure, until he crosses the event horizon. We will also contemplate what he could possibly have done to have saved himself, while still satisfying his curiosity about what's going on near the black hole.

Finally we will also get to see how this extreme curvature influences the path of a light beam and how Mercury's orbit could have gone really wild if it happened to have this monster BH as a parent star.

-=0=-

Notes: [a] The radius and distance of the BH is as observed by Alice in her frame, not directly, but by measuring the redshift of signals arriving from her static helper-observers (the bead-strings).

[b] Alice would have noticed rather easily that the redshift of her helper-observers' signals do not conform to the first order hyperspace-propertime structure, but, provided that she knows some relativity, that they do conform to the second order structure.
Last edited by BurtJordaan on June 19th, 2017, 12:35 pm, edited 4 times in total.
Reason: Typos (probably some more). 2. Added links to rotatable views
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Re: Beads-on-a-string view of Relativistic Gravity

Postby BurtJordaan on June 26th, 2017, 6:31 am 

BurtJordaan » 19 Jun 2017, 12:24 wrote:Once we have digested this hyperspace-propertime structure, we will attempt to plot Bob's 3D worldline as he falls from Alice's position, through this contorted structure, until he crosses the event horizon.

Edit: I discovered an error in the equations for Bob's free-fall HSTP path. It somehow did not 'look right', because the infall time is way too long. After closer inspection, it was diagnosed as quite a significant scale error that also influenced the form of the curve.

Here is an updated (correct, I hope) view of Bob's hyperspace-propertime HSPT path, as he free-falls from rest at Alice's (static) position (x=0), towards the black hole horizon at x=-0.5. It takes Bob around 8.5 months (0.707 yrs proper time) to reach the last static observer, just-just outside the horizon. When Bob-time is converted to Alice-time, it becomes about 15.5 months (the length of her world-line).

FreeFallBob4-100.png
Bob's Hyperspace-Propertime (HSPT) path in strong gravity


You can again rotate the view through this link. The rotated view is highly informative - give it a try! Here is a bit of explanation of what the above view represents. The flat x-y plane represents 4D hyperspace (of which only 2 dimensions can be shown. The x-z plane (the 'back wall' of the open box), representing 4D space-propertime.

As before, the straight bead-strings represent the tilted worlines of static observers at various constant distances from the horizon, as they have progressed at constant rates through the HSPT structure. They all have the same HSPT (proper) lengths. Their colors indicate their approximate signal-redshifts, as observed by Alice, the right-most bead-string. The curved multicolored bead-string is obviously the path of Bob, as he falls through regions of increasing spacetime curvature, towards the black hole event horizon at x=-0.5. His propertime is plotted as read off Bob's clock locally successive static observers, as he falls past them.

Now for the clincher: Bob's bead-string is of the same proper length as all the other bead-strings - exactly 1.29 lyrs (although Bob has aged only 0.707 yrs during that fall). Alice would have aged 1.29 years though. It is a bit difficult to judge from the above view (or the rotations), due to the necessary scale differences of the axes and the 3-D projections, but the math that has produced them tells us that they are of the same proper length.

This is super-significant - it tells us that in 5D (i.e. 3+1+1) HSPT, the same principles as in special relativity (SR) hold locally. The proper movement through HSPT is still the same for all observers, whether static or in free-fall near a black hole. It just does not 'look like that' for Alice. For Alice the space and time near the black hole seem contracted, to such an extent that for her, space and time (and Bob) "vanish" at the horizon (x=-0.5). Bob's signals just redshift to extinction and Alice never observes him crossing that horizon.

In a way, the same is true for an observer static just barely outside the horizon, like the bright-red, left-most one in the diagram (using forces approaching infinity to keep him there) - Bob flashes by at a speed extremely close to that of light and vanishes. So such an observer does not really see Bob 'falling in', but he can reasonably deduce that Bob has - and so can Alice...

Bob himself will continue seeing signals from the observers outside, but more about his view in a later post.
Last edited by BurtJordaan on July 2nd, 2017, 11:27 am, edited 7 times in total.
Reason: Typos (probably some more) + replacing faulty graphics. See text.
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Re: Beads-on-a-string view of Relativistic Gravity

Postby BurtJordaan on June 28th, 2017, 7:44 am 

BurtJordaan » 26 Jun 2017, 12:31 wrote:For Alice the space and time near the black hole seem contracted, to such an extent that for her, space and time (and Bob) "vanish" at the horizon (x=-0.5). Bob's signals just redshift to extinction and Alice never observes him crossing that horizon.

This is the origin of the popular science description of how Alice observes Bob - that Bob's time seems to 'stop' at horizon and that Bob seems to stop falling in as well, as if 'frozen' there in space and in time. In reality she would only see Bob's clock- and infalling rates approaching zero, but never reaching it. But his signals may become undetectable to Alice after some time.

Bob, on the other hand, will still see the signals from Alice as he crosses the event horizon. The contorted hyperspace-propertime through which signals propagate will cause Bob so observe Alice's signals as increasingly blueshifted up to certain point, from where they will be graduallly redshifted, until at the horizon, where he will find that they are at the original Alice-frequency again. So Bob has a way of detecting exactly when he crosses that one-way membrane, called the event horizon of the black hole.

What happens when Bob is inside the horizon is not easy to describe with only one spatial dimension, so let it stand over for another time.

-=0=-

PS: I have corrected the two broken links in the OP above (they were accidental and not an effort to lead readers to the website). I mention this here, because in order to easily understand this thread's conclusions, readers should look at the rotatable views of the above three posts in chronological order. They build up to the final view.
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Re: Correction: Beads-on-a-string view of Relativistic Gravi

Postby BurtJordaan on July 2nd, 2017, 3:19 pm 

I have blundered in the simulation code of Bob's hyperspace-propertime (HSPT) path in the second-last post before this one. To avoid confusion, I have updated that post with the latest (correct, I hope) diagram and link to the corrected 3-D rotating view.

The erroneous simulation indicated that Bob will take 3.6 years of own propertime to fall the 0.5 lyr to the event horizon. Given the huge gravitational pull so close to a black hole, this does not make sense and I should have caught that out earlier. It took some time to verify that the correct travel time on Bob's own clock would be about 0.71 years (8.5 months) before he crosses the event horizon. The profile of Bob's HSPT path now also has the expected shape: something resembling a slightly distorted circle segment in the x-z (space-propertime) plane, depending on the 3-D viewpoint, e.g.

FreeFallBob4-1-70.png
Corrected HSPT path for Bob


The length of Alice's worldline is about 1.29 lyr or 15.5 months. You can see that more clearly if you rotate the figure by using the updated 3-D program, as linked to above.

The original gravitational acceleration is 0.5g at Alice's static position relative to the hole. It then increases to over 6000g for the last static observer that Bob passes (which is where the simulation stops, because the next step would have been past the event horizon). No static observer could possibly be there, hence it cannot be calculated in the same fashion. Another type of coordinate system is needed for that. More about that in a future post.

Sorry for the misinformation originally posted, but that is what can happen if you develop stuff in "public" view. ;)
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Re: Beads-on-a-string view of Relativistic Gravity

Postby BurtJordaan on July 12th, 2017, 2:40 am 

While looking at the perspective from the in-falling Bob's perspective, I discovered that the coordinate system that I have used in the calculations and in the 3D graphs is not quite compatible with Bob's own time against the (static) space of Alice - which was the objective. This is a roundabout way of saying that I have blundered again! :(

Without wasting time on justifying the blunder, here is the corrected HSPT diagram:

FreeFallBob5-1-100.png
Bob's free-fall Hyperspace-Propertime (HSPT) path (corrected)

As before, the x-axis is Alice's space coordinate, the z-axis represents propertime (for all participants) and the y-axis pictures space curvature into an extra space dimension, making x-y a hyperspace-plane. The bead-strings are again all of the same length, with Alice's the right-most blue one, Bob's the multicolored curved string and the slanted mono-color ones represent static observers, equally spaced according to Alice. The black beads just represent the radius of the black hole, not static observer bead-strings, because there cannot be any such inside the horizon.

The (new) rotatable 3D view will help to make everything a little clearer. Also note that I have put the black hole center at x=0, as is usually done in gravitational discussions. This did not change the physics in any way, because it is only radial distances that matters.

The coordinate system now have the proper "free-fall from a finite distance profile"[a], for Bob starting out static relative to Alice, at twice the horizon radius, before he takes the 'rocket-less plunge'. We have to stop Bob's fall a short distance before he would pass through the horizon. Bob's free-fall time will be about 1.75 years on his clock, while Alice can later deduce that it took some 2.4 years on her clock.[b] Recall that the static observers can read Bob's clock directly as he passes them, but Alice cannot. There is one last static observer situated just a short distance (0.001 ly or 0.2%) outside the horizon - it is represented by the leftmost red bead-string, lying flat on the x-y plane.

One can mathematically track Bob's x-z profile to inside the horizon, in fact all the way down to just short of the singularity at x=0. But then one cannot show Alice and her team of observers' bead-strings, because they would all have lengths approaching infinity. That's why no static observer, no matter how close to the horizon, can actually observe Bob falling through the horizon. In their local frames, he takes a time approaching infinity to get there, i.e. according to them he never gets there.

How confident can we be that the 'new view' and the numerical values are correct this time? Never 100%, but pretty close to it. I have checked and rechecked the results against detailed calculations made according to the equations from http://www.physicspages.com/2013/09/09/falling-into-a-black-hole/. It is based on Thomas Moore's,"A General Relativity Workbook", a pre-graduate level text.

Although the presentation that I have used is non-standard, the physics is standard. I'm actually amazed that, complex as it may be, GR can be viewed in such a straightforward way. Yes, the equations can be intimidating, but as soon as one gets the hang of curved space and time, it makes a rather pretty and (almost) simple picture.

-=0=-

Notes:
[a] It is a variation of free-fall coordinates, which deals with free-fall time from infinity in Schwartzschild space, here adapted to free-fall time from a finite Schwartzschild distance against a local space (Alice's).

[b] Since Alice can be in two-way contact with her squad of static observers, she can obtain the values that they have read off Bob's clock as he fell past them. Since she also knows the Schwartzschild distance to each of her squad, she can easily determine the curve that Bob follows in her x-z plane. More on 'real distance' versus Schwartzschild distance later.
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Re: Beads-on-a-string view of Relativistic Gravity

Postby BurtJordaan on July 16th, 2017, 2:57 pm 

BurtJordaan » 12 Jul 2017, 08:40 wrote:Notes:
[a] ...
[b] Since Alice can be in two-way contact with her squad of static observers, she can obtain the values that they have read off Bob's clock as he fell past them. Since she also knows the Schwartzschild distance to each of her squad, she can easily determine the curve that Bob follows in her x-z plane. More on 'real distance' versus Schwartzschild distance later.

The physics term for what I loosely referred to as "real distance" is the 'proper distance', which in the case of Alice would be measured by a "non-stretchable" measuring tape, along the curve formed on the x-y (space-hyperspace) plane by the bottom ends of the slanted bead-strings, as shown here from above the plane. It is actually just Bob's curved HSPT path as projected down onto the x-y plane.

BobsPathHS-77-1.png
A birds-eye view of spatial curvature near a BH

One can easily see that due to the curvature, the distance increments along the curve are not the same as the x-values increments marked on the x-axis. The curve's increments are getting longer, the closer we are to the horizon of the BH, until they will diverge to approach infinitely long at the horizon.

This means that we cannot extend that tape to (or even to very near) the horizon at 0.5 ly. Its lower end will be ripped off and swallowed by the BH, or it will be ripped away from whatever it is anchored to and will then completely fall into the BH. So how do we determine the regualr x-distances on the graph?

We simply avoid the black hole by putting the tape around it in a circle (at Alice's static position) and measure the circumference. Dividing that by 2 pi then gives a radius, which we call a 'circumferential radius', to distinguish it from the unmeasurable proper radius. It is a little bit of a 'cheat', but it allows as a lot of insight-giving calculations and presentations. This is exactly what Karl Schwartzschild have used in 1915 to derive the first exact solution of Einstein's general relativity equations.[a]

I have called it "a little bit of a cheat" because we cannot make a static tape circle at (or inside) the horizon. But because of the regularity of the x-coordinate values that the circumferential method provides on the outside, it is reasonable to think that they may stay meaningful inside the horizon as well. We will later see how useful the concept really is, when we extend the HSPT path of Bob right up to the x=0 (well, almost).

This then is how "real distance" around a black hole relates to Schwartzschild distance.

-=0=-
Notes:
[a]
Schwarzschild_coordinates In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordinate chart on a static and spherically symmetric spacetime, which is adapted to these nested round spheres. The defining characteristic of Schwarzschild chart is that the radial coordinate possesses a natural geometric interpretation in terms of the surface area and Gaussian curvature of each sphere. However, radial distances and angles are not accurately represented.
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Re: Beads-on-a-string view of Relativistic Gravity

Postby BurtJordaan on July 18th, 2017, 6:12 am 

BurtJordaan » 16 Jul 2017, 20:57 wrote:I have called it "a little bit of a cheat" because we cannot make a static tape circle at (or inside) the horizon. But because of the regularity of the x-coordinate values that the circumferential method provides on the outside, it is reasonable to think that they may stay meaningful inside the horizon as well. We will later see how useful the concept really is, when we extend the HSPT path of Bob right up to the x=0 (well, almost).

As justification for this argument, take a look at this 2D Schwartzschild_space-Propertime (x-z) view of Bob falling in.

FreeFallBob5-Inside-4-77.png
Bob's HSPT path in 2D (x-z), until he reaches the singularity. External clocks have been stopped when Bob reached the horizon. The y-axis 'goes into' the screen.

It looks completely reasonable, but I must confess another 'cheat': I have stopped the clocks of Alice and her team of observers just before Bob crossed the horizon, otherwise their clocks had to run to infinity and the diagram would have been useless. Bob's clock ticked on merrily and his curve smoothly reaches the singularity at x=0 (or thereabouts) in 2.22 years after he has left Alice.

What happens on the x-y (hyperspace) plane when we suppress z? With another (minor?) "cheat", we can in fact plot it.

FreeFallBob5-Inside-5-77.png
Bob's path on the x-y (hyperspace) plane, with z 'coming out of' the screen.

The 'minor cheat' is again to make things view-able. Bob's y-coordinate will go to infinity by the time that he reaches the singularity, and my system cannot plot that. So we perform a crude "renormalization" on its value to bring it back to reasonableness.[a] The important thing is that Bob do reach the region of the singularity in a finite time on his clock, as shown on the x-z plane above.

When we rotate the graph so that we can see all axes together, it looks great, but somewhat confusing.

FreeFallBob5-Inside-3-77.png
The full 3D picture of the curved HSPT, with Bob's trajectory.


Here is the usual rotatable view if you want to get a clearer (or perhaps more confusing ;-) picture.

I'm working on a few enhancements, among them an effort to animate the growth of the bead strings. Until then, enjoy...(?)

-=0=-

Note [a]: Renormalization is a common technique in physics to get around infinities. Andrew Hamilton did the same thing when he depicted Schwarzschild geometry in his page on the subject.
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