## Relativity of motion in hyperbolic geometry

Higher-dimensional geometry (previously "Polyshapes").

### Relativity of motion in hyperbolic geometry

Here is one puzzle that I thought about recently.

Imagine a large body in hyperbolic geometry, like a planet. Let the planet fly through space at constant speed along a straight line.

Of course, only the axis of the planet in the direction of movement moves along a straight line. All other points move along equidistant curves.

But since they are moving along curves, they should feel some sort of force, dependent on their distance from the axis of movement. And so, if you're on surface of such planet, you should be able to measure if the planet is moving and in which direction.

Does that mean that hyperbolic space, unlike Euclidean one, has absolute rest and absolute motion?
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### Re: Relativity of motion in hyperbolic geometry

So, you're thinking of some kind of gravitational field, independent from the body of mass? Produced entirely out of the curvature of space? I'm not familiar enough with hyperbolic spaces, to get a clear enough view of this. But, I do see the concept of taking curved trajectories, and feeling centripetal acceleration.

Let's just say that the planet is hurtling along, pole-first. Does this mean the equatorial region experiences the most of this hyperbolic curvature gravity?
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### Re: Relativity of motion in hyperbolic geometry

ICN5D wrote:So, you're thinking of some kind of gravitational field, independent from the body of mass? Produced entirely out of the curvature of space? I'm not familiar enough with hyperbolic spaces, to get a clear enough view of this. But, I do see the concept of taking curved trajectories, and feeling centripetal acceleration.

Let's just say that the planet is hurtling along, pole-first. Does this mean the equatorial region experiences the most of this hyperbolic curvature gravity?

Yes, something like that (after all, general theory of relativity equates gravitational force with space curvature). In this scenario, equator would indeed feel greatest force.
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### Re: Relativity of motion in hyperbolic geometry

Well, that's interesting. So, if this is the case, then how is this additional acceleration being applied? It seems like it would pull at an equal angle to the planet's gravity. If so, would the hyperbolic curve add or negate some of it?

Or, is it perpendicular, and the force is lengthwise along the ground? Which would be really weird.
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### Re: Relativity of motion in hyperbolic geometry

ICN5D wrote:Well, that's interesting. So, if this is the case, then how is this additional acceleration being applied? It seems like it would pull at an equal angle to the planet's gravity. If so, would the hyperbolic curve add or negate some of it?

Or, is it perpendicular, and the force is lengthwise along the ground? Which would be really weird.

Since the curve is curved inward, towards the axis, I guess it would reduce the gravity. In fact, if the planet's speed surpassed a critical value, it might crumble.

In spherical space, on the other hand, movement would press the equator inward.
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### Re: Relativity of motion in hyperbolic geometry

Btw. a similar effect thus ought occure within spherical geometry too,
as it's only in euclidean space that an equidistant curve interval has the same length
as the corresponding center movement. Just that in spherical embeddings it is
shorter, while in hyperbolic ones it is longer than that.

--- rk
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### Re: Relativity of motion in hyperbolic geometry

Klitzing wrote:Btw. a similar effect thus ought occure within spherical geometry too,
as it's only in euclidean space that an equidistant curve interval has the same length
as the corresponding center movement. Just that in spherical embeddings it is
shorter, while in hyperbolic ones it is longer than that.

--- rk

I have mentioned that in my previous message Marek14
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### Re: Relativity of motion in hyperbolic geometry

I really can't see this happening.

Suppose you are on the surface of a planet of surface H2H (where the earth is S2H), the first two designate the ground and the final H is the direction of fall.

It moves in space. What happens on the ground. One might suppose an equator which is moving at the speed of the planet as Marek says. Further away from the equator, the ground moves ever faster. Think in terms of something like {5,4}, where one of the diameters is the equator, and the perpendiculars are lines of longitude. As the thing moves, the lines of longitude increase at the same rate, but places further away from the equator has to move faster for this to happen.
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### Re: Relativity of motion in hyperbolic geometry

wendy wrote:I really can't see this happening.

Suppose you are on the surface of a planet of surface H2H (where the earth is S2H), the first two designate the ground and the final H is the direction of fall.

It moves in space. What happens on the ground. One might suppose an equator which is moving at the speed of the planet as Marek says. Further away from the equator, the ground moves ever faster. Think in terms of something like {5,4}, where one of the diameters is the equator, and the perpendiculars are lines of longitude. As the thing moves, the lines of longitude increase at the same rate, but places further away from the equator has to move faster for this to happen.

No, the equator would be moving the fastest -- the ground would move slower as you go away from it.
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### Re: Relativity of motion in hyperbolic geometry

So your planet is just an ordinary S2 type thing like the earth. I thought your plan was to have a H2.

Suppose your planet is about the size of dodecahedral cell in {5,3,4}. The sun can be quite close, eg about the distance of 5 or 10 cells. Under such a situation, the forces exerted on the planet would still be essentially parallel. I did a fairly large calculation on this in 'the universe in a nutshell'. The size of the universe fits into the euclidean sun (ie 864000 miles).
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### Re: Relativity of motion in hyperbolic geometry

wendy wrote:So your planet is just an ordinary S2 type thing like the earth. I thought your plan was to have a H2.

Suppose your planet is about the size of dodecahedral cell in {5,3,4}. The sun can be quite close, eg about the distance of 5 or 10 cells. Under such a situation, the forces exerted on the planet would still be essentially parallel. I did a fairly large calculation on this in 'the universe in a nutshell'. The size of the universe fits into the euclidean sun (ie 864000 miles).

But, the question here is not whether the forces are "essentially parallel". The question is whether there's absolute rest and absolute motion, i.e. whether motion of a body can be detected without relation to any other body.
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### Re: Relativity of motion in hyperbolic geometry

Newtonian relativity is not geometry-dependent. You can't detect where you are, or how fast you are moving in H-space, any more than in E-space. This is evident from Euclid's elements. (hyperbolic geometry only changes postulate IV). Specifically, all right angles are equal tells us that space is the same all over, and even with time H3T (hyperbolic chorix with time), is not that much different to E3T (newtonian space).

The galilean principle of inertial frames hold here, although if you understand the coordinate system, you should not have much difficulty with the metric system,

You can detect proper motion if there are plenty of fixed stars around, but unless you suppose there is lots of stars, you would not have to move too far to get out of the universe. Suppose the constant of radius is 4000 miles (6400 kms if you use them). This means that the radius of curvature of space is roughly the same as that of the earth. The earth would be about the size of a cell of {5,3,4}, such as you see in the Knot-knot poster. If the sun were as bright as it is here, it would be say, 64000 kms away. It would be larger, about 30000 kms across, but we would see it only the size we see the sun now.

Unlike the present sky, which stays reasonably constant, something like Sirius, a bright star in one season, would be invisible six months later. The sky does not stay constant. It waxes and wanes as you move closer and further.

Bonola's "Non-Euclidean Geometry" has a supplement by John Bolyai "The science of Absolute Space" which is worth the read.

zB https://archive.org/details/scienceabsolute00bolyrich which is the quoted book itself.
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### Re: Relativity of motion in hyperbolic geometry

wendy wrote:Newtonian relativity is not geometry-dependent. You can't detect where you are, or how fast you are moving in H-space, any more than in E-space.

Yes, but if a body is moving through H-space, not all parts of the body can move along a straight line (unlike Euclidean space), and so it seems that you CAN detect how fast you are moving by measuring "tidal forces" so exerted. That's the base of this topic. I know that you shouldn't be able to detect how fast you are moving, but this thought experiment says otherwise, so there has to be something I missed.

The size of the body is irrelevant (as long as it's not zero): no matter how small the tidal force is, it is still, in principle, detectable.
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### Re: Relativity of motion in hyperbolic geometry

In principle, if the thing is in orbit, it's in orbit, and you would detect the motion from that.

If something is moving say, in a straight line, the bits away from the centre-line of motion will be moving faster than the bits near the C/L. You expect that. What would happen is that the energy would redistribute so that, say if the poles are moving twice as fast as the bit on the C/L, they would have four times the energy.

The point is that the body is still in inertial motion, and you can't detect inertial motion from being still.

I'm not sure if tidal forces is appropriate here, that requires a sun or something to do the pulling.
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### Re: Relativity of motion in hyperbolic geometry

wendy wrote:In principle, if the thing is in orbit, it's in orbit, and you would detect the motion from that.

If something is moving say, in a straight line, the bits away from the centre-line of motion will be moving faster than the bits near the C/L. You expect that. What would happen is that the energy would redistribute so that, say if the poles are moving twice as fast as the bit on the C/L, they would have four times the energy.

The point is that the body is still in inertial motion, and you can't detect inertial motion from being still.

I'm not sure if tidal forces is appropriate here, that requires a sun or something to do the pulling.

But not only will the bits away from the centre-line of motion be moving faster, they will also be moving on curved paths. Shouldn't there be a force associated with moving along a curve?

Let's assume a simpler moving body: a long bar.

The bar is moving in H2, perpendicularly to its axis. Without external forces, both ends move in a straight line at start -- but as those straight lines are divergent, they are pulled back by the strength of the material.

Now imagine you pull a bolt in the middle of the bar and separate it in two halves. If it wasn't moving, both halves will stay together. But if it was moving, each half is now free to continue along its own straight line, and they will start to diverge.
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### Re: Relativity of motion in hyperbolic geometry

Now imagine you pull a bolt in the middle of the bar and separate it in two halves. If it wasn't moving, both halves will stay together. But if it was moving, each half is now free to continue along its own straight line, and they will start to diverge.

Now that makes more sense. I can see that happening. The natural tendency to diverge at speed ought to be caused by an acceleration. The more time spent diverging, the greater distance they reach.

Which opens another question, though: what is the rate of curvature/acceleration on the curved lines in Hn (while in motion in above experiment)? Constant, variable, degree-3?
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### Re: Relativity of motion in hyperbolic geometry

I realized one thing, though.

If the whole spacetime is hyperbolic (H4), then two points that are not moving with respect to each other will grow more distant as time passes. In the bar example, the force on the bar would exist whether it was moving or not, and would be probably always the same (assuming special theory of relativity).

On the other hand, if time is Euclidean, the spacetime would be H3 x E1 hybrid which is not isotropic, so we would expect that moving in different spacetime directions WILL be distinguishable.
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### Re: Relativity of motion in hyperbolic geometry

Let's suppose a planet is moving on a straight line. The points of the planet move at v. cosh(r/R), where r is the radius from the line of motion, and R the radius of space. This is always greater than one, and the further-out points experience a throwing-out force countered by gravity and electric bonds. The force increases with the square of speed F = mv²/r. Were the same situation happening in S3, the tendency would be for everything to crash to the centre as speeds increase, and one relies on electrical bonds only.

If one supposes c to be constant, then the model E3J (minkowski-geometry) applies. But i don't think anyone has solved the source-free EM field space in H3. This is what JC Maxwell based his assertions on. In other words, the only kind of relativity i am familiar with H3 is H3T (absolute time). This could be projected into H4, but if you assume H3 is an equidistant space, then it would forever expand and become E3J over time, because R = R_0 cosh t/T
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### Re: Relativity of motion in hyperbolic geometry

The point is that we can detect the rotation of the earth by the colarisis force, but the force due to the orbit on the sun is to faint to feel.

So yes, you could detect the fact your frame of reference is being accelerated, in order to keep in a straight line, but all that it tells you is that you are in orbit around a straight line. It's a long way from an absolute motion.

Also Einstein-spacetime is not E4 + E3J (there is at least one complex axis), and the coversio factor between E and J is the speed of light,
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### Re: Relativity of motion in hyperbolic geometry

wendy wrote:Let's suppose a planet is moving on a straight line. The points of the planet move at v. cosh(r/R), where r is the radius from the line of motion, and R the radius of space. This is always greater than one, and the further-out points experience a throwing-out force countered by gravity and electric bonds. The force increases with the square of speed F = mv²/r. Were the same situation happening in S3, the tendency would be for everything to crash to the centre as speeds increase, and one relies on electrical bonds only.

If one supposes c to be constant, then the model E3J (minkowski-geometry) applies. But i don't think anyone has solved the source-free EM field space in H3. This is what JC Maxwell based his assertions on. In other words, the only kind of relativity i am familiar with H3 is H3T (absolute time). This could be projected into H4, but if you assume H3 is an equidistant space, then it would forever expand and become E3J over time, because R = R_0 cosh t/T

That depends on how you define the successive "instants" of time, I think.

Let's use H2 as a base, with one space and one time dimension.

If we consider a horizontal line through origin the instant t=0, then one way to introduce time is to say t=1 is "all points on the line, 1 second in the future", t=2 is "all points on the line, 2 seconds in the future", and so on. That's what you seem to be doing, and the effect is that the successive time instants are equidistants whose curvature will flatten to approach zero, i.e. a horocycle.

This time is not homogeneous, though -- there is a special instant of extreme curvature (as the curvature would start to flatten if we go to the past as well).

But there is an alternate approach. Once again, we define a horizontal line through origin as the instant t = 0, but t = 1 will be defined as "line perpendicular to the worldline of origin, 1 second after t=0".

In this system, each time instant is a line, so the curvature doesn't change with time. However, time becomes strongly relative to observer. In the first model, you would see the space expand exponentially, but keep a synchronized time; in this model, the expansion would be so strong that distance between two points could become infinite in finite time ("time instants" of two points would eventually completely cease to intersect). The farther a point is, the faster you would perceive its time, eventually seeing the whole future of a distant point compressed into a finite interval right before it disappears in the infinity.

Next, we could adjust the standard relativistic spacetime diagrams and display them in a projection to see how speed of light would look in this world and how moving bodies would perceive time.
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### Re: Relativity of motion in hyperbolic geometry

I tried to create a simple construction in Cinderella, but I'm not sure what to use for light lines. Angle bisectors (45-degree lines) don't really seem to work in hyperbolic plane.

I attach my preliminary file, but as I said, it doesn't really work.
Attachments Hyperbolic Relativity.zip
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### Re: Relativity of motion in hyperbolic geometry

I suspect that this structure of hyperbolic universe that expands and whose curvature doesn't change with time, might be, in fact, what's called "de Sitter universe"...
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### Re: Relativity of motion in hyperbolic geometry

If you take equidistants from a plane (ie a dividing space), in H-space, then it progresses towards a horocycle, and this rather quickly. For example, one can suppose a time scale t, where the distance between two points on the same surface is 10^t. After 40 t, you have something bigger than our universe in circumference. In other words, H3H1 with equidistants would be a model for a big bang, not the universe as we see it. Even so, with this model, even though the circumference might be 10^40 t on the girth, the distance inside is no more than 80 t (with suitable time travel), nearly every point on the surface would be 80 t distant.

Relativity is just that everyone sees the same thing. It has no bearing on c, for example. Newtonian relativity models E3T (euclidean space + absolute time), whereas special relativity is E3J (three euclidean space + complex-euclidean time, so c is constant).

If one were to suppose some sort of 'hyperbolic space' with euclidean time, then one could get something like H3T to work. This is the usual first-model idea of hyperbolic physics: hyperbolic space and euclidean time (on a separate line). E3J is something that does not affect us, because v²/c² << 1. Even the solar system is big enough for v²/c² < 1 (ie an anomaly in Mercury's perihelion, the usual answer is to lift the needle over to the next track).

There is no problems with changing curvature. "Flat" means that it has the same curvature as space. So if space expands, collinear relations do no disappear, they become like geodesics on a balloon. But whereas an equidistant on a sphere is smaller than the equator, and the shortest distance lies opposite the equator to the equidistant (latitude-lines), the shortest distance in H space lies entirely inside the space, and it is possible to have a convex region with an infinite number of right angles (all of the cells of {5,4} adjacent to a line).

Hyperbolic space treads a thin area between euclidean space (ie H3 -> E3 as 1/R² -> 0 ), and some monstrous thing that i have seen once or twice. In essence, the space around a point is not S2 but E2, and that one can stand beside a super-nova and not feel even the faintest of heat, since matter would dissipate over the matter of a few inches.
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### Re: Relativity of motion in hyperbolic geometry

It all comes down to quadrics.

In Euclidean spacetime, the interval between two events is sqrt((time distance)^2 - (space distance)^2). Physically, this is the "proper time", i.e. how much time you'd see pass if your worldline is the straight line connecting those events. If you draw a line of constant interval (for example, all events you can reach in 1 second of your subjective time), their locus will be a two-sheet hyperboloid in four dimensions. The two-sheetedness is why we have absolutely separate past and future. Limit speed of light is a consequence of the fact that this hyperboloid has asymptotes.

The author Greg Egan has a book series where he explores consequences of changing the - sign to + (http://gregegan.customer.netspace.net.a ... 00/PM.html) -- this would mean that moving at high speeds will cause MORE time to pass for you, not less, and that there is no speed limit analogical to speed of light: locus of constant intervals will be a circle or an ellipse has no asymptotes. There is also no intrinsic difference between space and time: your time direction might be space direction of someone else.

There is also a third option, a quadric degenerated to two hyperplanes. In such universe, time would pass at the same rate, no matter how fast you'd move. This is the Newtonian universe.

In hyperbolic space, it's harder to build the precise equations. But we CAN introduce quadrics.

How would hyperbolic universe with interval governed by various (central) quadrics look? Let's look at 2D universe with one space and one time direction:

Case 1: Convex hyperbola

This is analogue of Einsteinian spacetime. Convex hyperbola, seen from its center, has two asymptotes. Moving at high speed would slow time down.
But here's the first major difference. In Euclidean spacetime, if you move for 1 second of subjective time, you can end up in arbitrarily distant future. Not so in hyperbolic spacetime, and that is because a convex hyperbola has a special line that connects the two ideal points on one of its branches.

This will depend on how exactly the hyperbola is defined, but it's for example possible that if you move for 1 subjective second, you can never end up 3 seconds in the future. You can move as far in space in that 1 second as you wish, but you will always move less than 3 seconds in time. After 3 seconds for the observer at rest, all possible positions for 1-second move will be in his past.

How would speed of light look in this universe? Well, let's have a look at a picture: hyperbola.png (10.96 KiB) Viewed 4112 times

This is in Klein projection; blue lines are intersecting straight lines while green and red are two possible lines of constant intervals.

The points where the blue lines intersect the horizon divide the universe into "future", "past" and two spacelike areas which are inaccessible. If you move at "reasonable" speeds, you will hit the green line, than the red (no idea what your actual proper time for them would be). But you could also move so fast that you would, for example, hit the green line, but NOT the red. Or you could not even hit the green line. You'd hit the horizon first.

What would that mean, physically? Well, horizon is infinitely far away, so that would mean you'd cover infinite distance in finite time. That seems counterintuitive, but remember that this spacetime would be constantly expanding. Move fast enough and you'll end up in a region of universe that is causally disconnected from your previous one. Your trip was one-way and from the point of view of the static observer, this is exactly as if you no longer existed: nothing you do can affect him anymore.

I have to conclude that the projection of hyperbolic plane does NOT show the whole spacetime: only the part that has causal links (at least potential ones) to the point in the middle.

Case 2: Ultraparallel lines

This is the hyperbolic version of Newtonian universe. Picture: parallel.png (4.28 KiB) Viewed 4112 times

Here, there is no limit speed. Since even in the previous case we were able to get to infinite speed fairly easily (and to higher, causality-breaking speeds as well), this is scary.

You can still move at "supracausal" speeds (and it's easier), but you have no "spacelike" regions of spacetime that you wouldn't be able to enter.

Case 3: Circles

This is the hyperbolic version of Greg Egan's Orthogonal universe. Looks very similar to the normal one.

Case 4: Ellipses, equidistants and concave hyperbolas

This is the "crazy" hyperbolic version of Greg Egan's Orthogonal universe. Picture: ellipse.png (8.93 KiB) Viewed 4112 times

There's no exclusive time dimension, but the two dimensions are not equivalent! In one dimension, travel is "easier" then in the other; specifically, there's only one speed at which you will never hit the horizon.

Let's come back to Case 1. How would travel at supracausal speeds look like?

First of all, EVERY speed is actually supracausal. No matter how slow you're moving away from a stationary observer, you will eventually pass his horizon. The apparent speed will drastically increase with distance, due to expansion of space.

Now, if we imagine horizontal line through the origin (a "snapshot" of the universe at one point of time, defined by origin), each point will see the others recede at increasing speed, eventually vanishing beyond the horizon. Light signals sent from these points will reach the origin's worldline at time growing much faster than their original distance. A signal from one limit point would only reach the origin in infinitely distant future, and light signals beyond this point will never reach the origin at all.

But let's say you're on a ship starting from a certain planet to another, at speed that will take it past the horizon in two years of subjective time (right at the time the destination planet crosses it). What would you see then?

Thanks to the expansion of space, your home would recede faster and faster. Eventually, the speed would become supraluminal and you'd stop getting any new signals. The precise interplay of speed of light and curvature of space would determine when the last signal comes. But you'd definitely lose it before two years pass.

In the direction you're flying INTO, however, you would see signals coming at accelerated speed. On planets further than your goal, you'd eventually see their whole history (infinitely compressed). Even though you're flying towards them, the expansion is powerful enough to put them beyond your horizon.

On your destination planet, expansion and your speed will exactly cancel. But infinite time has passed on that planet since you started. This seems to be the cause of causal disconnect: somehow, it is possible in this universe for transfinite time to pass. You can see the whole infinite history of another object, and there's yet more time...
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