Thursday, August 17, 2017

Negative Dimensional Curves

We all understand geometry of integer dimension and fractal geometry describes shapes with positive fractional dimension, but there are only a few descriptions on how shapes could have negative dimensions. In fact I know of three.

1. A negative dimensional space as described by V Maslov considers gaps in a solid as negative dimensional, presumably a solid missing a line is -1 dimensional and a solid missing a plane is -2 dimensional. This seems to be incorrect as it is confusing measure with dimension. If anything one might think of the holes as having negative measure (relative to the solid).

2. A less scientific answer is to say something has negative dimension if it appears to fade away as you look more closely. For example getting closer to a cloud, or to a face painted in pointilism, or perhaps on very close inspection a smile disappears and becomes just connected cells. In other words, for something to reduce in intensity as you look with more detail, you need that something to be a concept rather than just a raw set of points.

3. Benoit Mandelbrot had a definition of negative dimensions when you are describing the intersection of different shapes under a small amount of noise. The intersection of a point and a volume is 0 dimensional, a point and a plane is -1 dimensional, a point and a line is -2 dimensional and between a point and a point is -3D. The idea is that as you look more closely the likelihood of intersection within a reducing size ball reduces at different rates for these cases. It is a nice idea which removes a special case in a certain set operator equation, but it assumes noise so is a somewhat specific definition. It also creates some odd cases such as a 1D plane (if you a intersecting it with a plane) and intersecting more than two sets gives unbounded negative dimension. In fact it is a specific type of idea 2, as the dimension is being defined for a concept (the intersection of sets) rather than just sets.


Instead I propose that we can produce negative dimensional geometry, in this case negative dimension curves, by using the fact that square distance of time quantities is negative. Here we assume the set will embed in the standard n+1 Minkowski space-time. The proper length of time-like curves can decrease with increasing detail level if they tend toward light-speed motions:

Reducing dimension time-like curves
This particular family of curves is the 1+1D Minkowski equivalent of a Koch curve in 2D Euclidean space. In fact, we can think of the Koch curve, the Weierstrass function and this curve to be the same type, only differing by whether the second dimension squared is positive, zero or negative respectively. And accordingly the larger angles increase, make no difference and decrease the dimension respectively, from a 1D line.

The same curves and same dimensions also exist on the complex plane, provided that it is with respect to the real-length of the curve, defined as:



In 2+1 space-time the results represent a point moving around in 2D and are perhaps more visually interesting. Here I follow the replace pattern of the Koch curve in 2D space, but the bend angle doesn't increase the curve length by a fixed percentage (like the Koch curve), instead it is achieved by boosting the motion in order to decrease the proper time by a fixed percentage. As a result the dimension decreases and the measure is of the proper time, meaning we quantify the period of the motion as the time according to the point, in seconds^D where D is the dimension.
This effect is a bit like the internal scattering of light in some medium. Lower dimensions are like there are a greater range of particle sizes, with few that cause large reflections and many that cause tiny course changes. D=1 is like all particle sizes (or scattering angles) are the same.

Saturday, August 12, 2017

Mach's principle

Mach's principle is fascinating. If General Relativity does obey it then GR is telling us something much more profound than just adding gravity and bendy space-time to special relativity.

Mach's principle is that our idea of what a rest coordinate frame is comes from the motion or shape of the bodies in the universe. So if all of the universe was rotating, our rest coordinate frame would also be rotating and it would be equivalent to the whole universe being at rest, as we see it now. Equivalently, if you spin around in space, your arms only separate with centrifugal force if there are stars that you are spinning relative to.

This is a wonderful idea that is telling us something more than Special Relativity. It is saying that space-time itself is only a product of mass-energy. It is not just that mass-energy warps space-time, space-time exists because of a distribution of mass. So flat (Minkowski) space-time does not result from the absence of mass, it is the approximation of space-time when mass is fixed and very far away. To put it a third way, matter doesn't sit on an infinite fabric, the fabric exists only in relating the motion of matter.

The only problem is that, even 100 years after GR, people can't seem to agree whether GR really is 'Machian'. However, on inspection it seems to me that it is:

Wheeler-Einstein-Mach spacetimes

An important paper is Wheeler-Einstein-Mach spacetimes:

"We define the Wheeler-Einstein-Mach (WEM) spacetimes to be those which contain a closed Cauchy surface, are inextendible, and satisfy field equations with a well-posed Cauchy problem, We show that a WEM spacetime can be reconstructed from the "York data" on any given closed (constant mean curvature) hypersurface contained in that spacetime. This result is the strongest and most precise statement to date of Wheeler's version of Mach's principle."

While the Machian view is that the inertial frame can be reconstructed by the distribution of mass-energy in the universe, the authors don't argue over exactly what this means:

"To decide, within these prescribed limits, exactly what should be labeled "mass energy" and used as the Machian input, we eschew philosophical considerations and choose what gives us a nice theorem, i.e., we choose the York data"

they just use the York data:
"Define the "York data" Q, of a given spacelike hypersurface S is composed of the conformal intrinsic geometry, the conformal (transverse traceless) extrinsic geometry, the conformal nongravitational and the mean extrinsic curvature scalar"  (paraphrased)

So based on this, they have found that this large class of GR space-times show no rotation of their universe from their preferred (rest) coordinates. Which is consistent with the idea that your coordinates rotate with the universe:

"Absolute rotation: None of the WEM spacetimes has an absolute net rotation in any well-defined sense."

People have cited the Godel universe as a counter-example, but this paper refutes it clearly:

"Local vorticity (of the sort found in the non-WEM Godel universe and also in the WEM Ryan spacetimes) does not constitute net rotation of the universe in any sense."

The idea that the Godel universe is local vortices is backed up by the wikipedia article on the Van Stockum dust solution:
"Note that unlike the Gödel dust solution, in the van Stockum dust the dust particles are rotating about a geometrically distinguished axis"

Frame Dragging

In the Machian view, the coordinates would rotate with a galaxy if there were no other in the universe. However, since there are others far away, it makes sense that coordinates would rotate partially with the galaxy, and to a lesser degree as one moves out towards the rest of the universe. This does happen and is called frame dragging or the Lens-Thirring effect. Again supporting the Machian idea. 

Rotating Mass Shell

Also there is this paper: Induction of correct centrifugal force in a rotating mass shell
It shows that for a correctly shaped shell over fixed Minkowski space, the flat space-time inside the shell will rotate with the shell:

"Mach’s idea of relativity of rotation is confirmed for a shell-type model of the universe by showing that flat geometry in rotating coordinates, realising correct Coriolis and centrifugal forces, can be continuously connected through a rotating mass shell with not exactly spherical shape and latitude-dependent mass density to an asymptotically Minkowskian outside metric. The corresponding solutions of Einstein’s field equations are given to second order in the angular velocity w but it is plausible that the problem has a solution to any order of w"

I find this a very clear vindication of the principle. Even though it requires a specific shell shape, I expect that requirement becomes less sensitive if you replace it with a more disperse and distant universe.

Ozsvath-Schucking space-times

This metric is also cited as a refutation of Mach's principle. The space-time seems to be parallel gravitational waves. It is said to be an anti-mach metric because it contains no mass but is not just flat Minkowski space. This seems like a weird reason to claim to be non-Machian, for three reasons:
  1. Mach's principle doesn't claim that a universe lacking matter should be Minkowski, since that would preference inertial (rather than nonlinear) coordinates. It claims that a universe lacking matter has completely unconstrained rest coordinates.
  2. As discussed in the WEM spacetime paper above, the problem with infinite space-times is that they exclude a light-cone at infinity, and this may well assert a constraint on the rest frame. Hence the WEM spacetimes are compact.
  3. Mach's principle that the rest frame is derivable from the mass distribution alone was before anyone knew about gravity waves, it seems OK to me to include the dynamics of these waves in resolving the universe rest frame. Besides, gravity waves are a form of non-localised energy. The principle can remain, that if there is no mass or energy (including gravitational waves) then there can be no preferred rest frame (linear or otherwise).

Conclusion

So the way I think of it is that space-time stretches between the masses and so it is impossible to notice any overall motion of the full set of masses. For example if all of the stars in the universe were actually oscillating laterally all at the same time with an amplitude of one metre, then space-time would also oscillate with it, and so the motion would be unobservable, and this goes for any motion. Note that you don't have to invoke magic for this, you need to attach rockets to all the stars and they will give positive and negative pressure values in the stress-energy tensor which bend space-time correctly to physically produce the oscillation.


So while Newton's relativity and Special Relativity both have no privileged inertial frame (linear velocity), in GR there is no privileged coordinate frame at all, linear or otherwise. Nature is not preferring linear motions, only geodesic ones. And ultimately our idea of what it means to be at rest comes down to a sort of average of the stars around us. Everything is indeed relative.

Inertia

Another point of contention is how Mach's principle effects the inertia or mass of objects. This is worth investigating, by looking at how linear and angular motion changes as the mass of the remaining universe drops down towards zero. For each motion type I look at motion relative to the background universe and motion relative to the presumed natural rest frame.

Linear oscillation

In this case we want to oscillate by applying a sinusoidal force of fixed amplitude. The observer is pictured as the large sphere and the remaining (background) universe is simplified to just a single small sphere.
In this case, as the observer applies its sinusoidal force (e.g. by emitting particles of negligible mass), its large mass compared to the rest of the universe causes substantial frame dragging and the universe mainly follows the same sine wave. Consequently, the observed motion relative to the background universe has reduced.  

The rest point in this set-up is I think roughly at the centre of mass of the two bodies, which is much closer two the observer body. As such, the motion of the two bodies relative to the centre of mass looks like (with amplitudes scaled up):
Relative to this fixed point, the motion is also smaller than for an infinite mass universe. 

In both cases the motion is less than expected and the observer may think that his mass has increased. for zero background mass, the lateral forces cause no motion.

Angular motion

For a low mass remaining universe any angular torque will cause the background universe to rotate somewhat with the observer, so a larger torque is needed to attain a target angular velocity. However, the frame at the observer will rotate mostly with the observer, so the proportion of this relative angular velocity that acts as rotation of the observer reduces.


We define the rest frame as being the point where frame dragging of observer and universe cancel out (see pale blue lines). This is closer to the universe particles as their mass decreases. In coordinates where this cancel point is at rest, the torque required to achieve an angular velocity relative to this rest frame increases with decreasing universe mass. Again, the frame dragging of the observer means that the apparent rotation of the observer is less with corresponding reduction in centrifugal acceleration. You arms separate less when you spin around. 

Conclusion

While it may appear to the observer that its mass and/or inertial moment increases with a diminishing background universe, this is not really the case. What is happening is that the effective mass/inertia are changing because these properties are relative to the universe's mass/inertia. As long as effective mass/inertia and absolute mass/inertia are conceptually distinguished the discussion of inertia remains clear.
In the given rest frames the effective mass reduces because forces applied to the observer act increasingly and oppositely on the universe rather than the observer as the universe mass diminishes.

Both examples demonstrate Mach's idea, that effective inertia only exists in the presence of a background universe.

Scale Dragging and Dark Energy

Is it possible that the expansion producing negative energy termed dark energy is in fact the apparent opposite action on the universe due to contractive frame dragging?

Imagine that a galaxy is contracting under gravity, with a low mass background universe. This background universe will tend to contract with the galaxy due to frame dragging. The equivalent rest frame would be contracting at an intermediate rate. In this rest frame the galaxy would appear to be contracting and the background universe expanding. 

This is just a passing thought, I'm not sure it would actually make sense. 

Thursday, July 13, 2017

general relativity

When you toss a coin, the coin falls down because there is less time at lower altitudes, and time passes slightly more slowly. If we plot altitude against time on a sheet then we expand time horizontally at lower heights, the sheet will splay outwards as shown below. A coin tossed upwards simply follows a locally straight path on this sheet, like an ant walking along it in a straight line. 
We stretched rather than contracted the time direction because we are representing the time dimension spatially, and they have opposite metric signatures. Space and time don't actually bend into a third dimension like the sheet, the curvature is just intrinsic to the space and time; you can just think of time as being less dense at lower altitudes.

The field equation of relativity holds at all points in space and time and is
where Gij is the Einstein tensor describing the bending of space-time, k is a constant and Tij is the stress-energy tensor which represents the density and pressure of matter at that point.

Gij and Tij are both 4x4 tensors (very similar to matrices) but they are symmetric and this means they represent a scaling along four orthogonal principle axes. For the spatial dimension this can be seen as stretching a sphere into an ellipsoid:


If we work in this principle inertial frame (the rest frame) then the field equation just has diagonal entries, e.g:
ρ is the density of matter and px,y,z are the pressures, which are equal in most idealised cases, such as an idealised fluid.
Scaling by k gives the Einstein tensor Gij, where each diagonal entry represents the scalar curvature of the 3D volume orthogonal to that entry's axis. For example the first entry means the 3D spatial curvature is kρ since the spatial volume is orthogonal to the time axis.

This 3D scalar curvature (or 3D Ricci scalar) is proportional to the difference between the surface area of a small sphere around that point, compared to the expected surface area of a sphere in Euclidean space. A positive scalar curvature (as is the case when mass is positive) has a smaller than expected sphere area, and parallel paths converge. In the case of the first entry this means that mass causes space to have spherical curvature, like a 3-sphere.

This is the simplest explanation of General Relativity; the stress-energy tensor represents the scalar curvature of the 3D volume orthogonal to each principle axis.

For earth, the density ρ is 5,500 kg/m^3 and since k is the rather small 2e-43, the spatial curvature G00 is 1.1e-39. This results in the earth being 2mm smaller in radius and with 35 fewer hectares of ground than if it were in flat space. 

A closer look

However, this definition isn't very helpful in telling us how space and time distort with each other, since the scalar curvatures on each axis combine together in their effects. In order to get actual differential equations we expand out the field equation definition to:

where Rij is the Ricci tensor which represents the 4D curvature of space-time and R is the 4D Ricci scalar. Working in the mass's rest frame, the Ricci tensor is just a linear transform of the stress-energy tensor:

This transform can be graphed in 2D when the pressure is equal in all directions (using units where k=1):

For any given mass and pressure of a body the blue arrows show the directions of increasing contraction with respect to time and with respect to space. Lets look at this curvature more carefully.

Each element in Rii defines the rate of change of contraction of a small orthogonal 3D volume V with respect to the element's axis. For R00 this 3D volume is just the spatial volume (written as Vxyz): 
as given by John Baez's relativity tutorial. For R11 it is a small time period t multiplied by the perpendicular area yz:
Note that Vtyz isn't exactly a volume, while increases in the yz directions increase it, increases in the time period do the opposite, due to the negative metric signature:
and equivalently on the other two spatial axes. This is for a volume at rest, otherwise we subtract volume acceleration in flat space from the left hand side. The negative sign is because these are contractions.

In standard units the equations just contain a few more constants:
and
Noting that the first term in the stress-energy tensor is actually energy density rather than just density, and since e=mc^2, that is why we use ρc^2.
So for these "normal" units of density and pressure, the pressure has almost zero effect and also changes with respect to space are tiny compared to those with respect to time. 
For example, for earth ρ is 5,500 kg/m^3 and earth's pressure is 2e11 Pascals, so the pressure term is 4e-8% of the energy density term. 
So the rate of volume contraction is 0.0005% per second squared. 
whereas the rate of Vtyz contraction with distance is 5e-21% per metre squared.

Therefore curvature per metre squared is roughly 1e-15% of the curvature per second squared, for earth.

Example of space above earth:

The Ricci tensor in space is zero. This means no volume Vxyz  acceleration with time, and no time x area (Vtyz) acceleration with distance x. This does not make space Lorentz flat, it can still bend and preserve 3-volume and that is what it does near earth:
- With respect to height, the horizontal area contracts at lower altitudes (being 35 hectares less than in flat space at ground level), and the other coordinate must expand to compensate (we draw it spatially as expanding), but since this is the negative signature time period, it must also reduce, and it does, causing gravitational time dilation.
- With respect to time an initially static volume's height expands since gravity is slightly stronger closer to earth, but its width and length get shorter, as it moves downward where area is slightly contracted.
In both cases the expansion and contractions cancel out to give zero volume change.

This diagram shows how objects move above ground, causing vertical expansion, and below ground, causing vertical contraction. In solids the pressure forces resist the inwards pull. Therefore relative to these straight yellow lines, the ground surface is in fact accelerating upwards.

Spatial curvature is much less pronounced than curvature with time. This bend is irrespective of speed.
  


For more details try out my cheat sheet:

and some other helpful links:

Wednesday, May 24, 2017

diamond square fractals

I have previously worked on fractal automata which produce animating patterns in 2D and 3D. A special case of this is where only the larger scale affects the smaller scales, giving static images. One shortcoming with these automata and images is that they have square or cubic symmetry, so features are just at right angles, which gets a bit dull, this was one reason I made this cubic modification for 2D images, giving a hexagonal symmetry. 

 A tweet by someone called R4_Unit showed a different way to achieve a similar goal. Use the diamond-square algorithm which is used for plasma fractals and hilly height fields, but use automata rules rather than random offsets. R4_Unit applied this in 3D, and has since submitted his code here, and a user called Softology has added his own implementation to his Visions of Chaos software. In 2D the diamond-square algorithm gives a pseudo octagonal symmetry, quite similar to automata rule type 7 that I made in the 2nd link above which gave nice results:
But with a rule that is more complicated than diamond-square. The 3D diamond-square is actually cube-irregular_octahedron-octahedron and doesn't have nearly the simple symmetry of diamond-square, so here I investigate what the results look like in 2D.

For 4 parent points there are 2^4 parent combinations and the central point is either black or white for each, giving 2^16 rule sets. However, once you constrain it to square symmetry, this drops down to only 64 rule sets, which I draw using a 4x4 start grid:  
0001
0010
0011
1111

And for the symmetric start grid:
0000
0110
0110
0000
five along one up seems like the 'nicest' in both cases.

For more than two colours the number of combinations explodes. One way to contain that is to demand 'bit symmetry' which in the two-colour case means the rule sets acts the same if black and white are swapped. Unfortunately it is not possible to have a two-colour bit symmetric rule with this diamond-square approach, because there are an even number of identical parents. If two parents are white and two are black then there is no child colour that acts the same if black and white are reversed.

The generalised 'bit symmetry' for n colours means the rules are the same if the n colours are permuted in any way. Then for three colours it is possible to have bit symmetric rules, and there are only three of them:
Overall the rule is to choose the most popular colour of the parent. If this is a tie, then choose the least popular. The choice occurs when there are three parent corners of a single colour. Do you choose the colour there is none of, one of or three of among the parents? The above shows each respectively.

The third is the simpler rule (as it is subsumed by the first criteria), and gives more solid results. Here is a close up of the only complex feature it seems to generate:
These teardrop shapes are all identical. While they are polygonal, each protrusion seems to have an infinite number of edges, in fact it is a tree. So each teardrop is a tree. 

In fact, these are the only three bit-symmetric rule sets possible for the diamond-square algorithm for any number of colours, and none exist in 3D at all (assuming cubic symmetry), so they are quite unique.

However, a square lattice is not the only scalable lattice, the only other I know is the triangular lattice. And in this lattice bit-symmetric automata occur only with four colours, and there are only two of them:
The rule is the same as for the square lattice case, but the choice is when two of your parents are the same colour, do you choose the colour there is one of or two of? Again, the more solid results come from the latter case. It is quite a wonderful shape, with almost rounded looking blobs that belie its triangular lattice frame. These used a randomised 4x4(ish) start pattern, and here is higher resolution with an 8x8(ish) start pattern:
The rule for this and the teardrop above are incredibly simple, pick the most popular parent colour, if it is a tie then pick the least popular. 

Here are three simplest cases:
The apparent fifth colour is a nowhere-dense mix of the four colours and primarily takes the shape of a modified Koch snowflake, by contrast the dense colours are all finite sided polygons.  


Wednesday, April 26, 2017

New basic fractal curves

The three most well known 2D fractal curves are probably the Von Koch curve, the Levy C curve and the Dragon curve. They are all generated from the same simple replacement rule of one oriented line segment with two in a v-shape:
The angle of the V-shape controls how rough the resulting curve is. At 60 degrees we get:
 Levy:
Koch:
Dragon:

What doesn't seem to have been described before is that there are three other fractal curves from this basic construction. In fact, since an oriented line segment can be rotated 180 degrees, reflected around its length, or both, there are four options per child line, so 16 possible fractal curves. However many just give reflected or rotated versions of the same curve, leaving just six unique fractal curves. The remaining three curves look like this, for bend angle 30, 60 and 90 degrees:

new curve 1:


new curve 2:


This has a critical point, just before it starts overlapping, at about 85.5 degrees:
new curve 3:

This last curve is especially interesting because, like the Von Koch curve, it curls into a solid triangle without any overlap, here is is at 85.5 degrees: