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?
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.
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
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.
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).
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.
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.
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.
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
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