STELLAR OBSCURA

Black Hole Quantum Gravity Theorem

In order for a theory of quantum gravity to work in needs to be able to deal with gravitational singularities on the quantum scale. In my post, Hyperbolic Perspective and Quantum Gravity I outlined a theory which describes how gravity works on very small scales, but it is clear that something else is needed entirely for it to describe a gravitational singularity on the quantum scale.

The Inter-Dimensional Merry-go-Round

In the wake of his discoveries - regarding length contraction at speeds approaching that of light - the world renowned physicist Albert Einstein attempted to unify this concept with the gravitational spin of the Earth. He knew that the surface of the Earth rotated much faster than its core, and this meant that it must also undergo a more pronounced length contraction. The object he used to imagine this type of geometric contraction was the merry-go-round. He proposed that if a merry-go-round were to spin at very high-speeds, the outer-edge of the disk would undergo a length contraction forcing its edges to curve upward. From this he deduced that space itself was curving and this led to his Theory of General Relativity.

Just as a black hole warps space to an extreme degree we can imagine a merry-go-round that is spinning so fast that the resulting curvature warps it into a sphere. Shifting an object from one dimension into another higher dimension is an excercise in trans-dimensional geometry, which is exactly what we have done here, taking a flat 2-dimensional disk (merry-go-round) and turning it into a 3-dimensional sphere.

In the case of a super-massive star that is about to turn into a black hole, we need to imagine that we are taking a 3D sphere and transforming it into a 4D sphere.

As the star shrinks its surface is becoming more and more length contracted, which corresponds with an increase in the degree of space-time curvature and thusly in gravity. Although we can deduce from this that gravity is at its strongest with respect to the surface of an object, we also know that the force of gravity continues, as in the case of the Earth, down through the mantel to some degree until a null-point is reached at the core. It should be apparent, therefore, that the quantum singularity of a black hole is not 'the centre' of anything, but is in fact only that region of the gravitational field which corresponds to the surface of the star. This means that the entire surface of the super-massive star along with its gravitational field has turned in on itself, and its entire surface area (or circumference) has been reduced to zero. 

You might be wondering where the mass of the star has gone that produces these gravitational effects.When a star collapses it can be said that its entire mass is undergoing a length contraction, in all 3 dimensions. This is a Special Relativistic effect, which implies that the mass of the star is accelerating towards a central point, faster and faster. It reaches the speed of light (the event horizon) and then necessarily exceeds it. Once the star reaches twice the speed of light (at the plank scale) it turns itself inside out, resulting in the gravitational singularity. This effect is known as length expansion and is dealt with in the post titled Direct Relativity.

From then on the singularity can be described as an aperture into the 4th dimension. This is just like the aperture of a camera, or more accurately a camera obscura. In a camera obscura, light is allowed into a receptacle through only one point of entry (the size of a pin-hole). This has the effect of focusing the light at oblique angles creating an inverted (up-side down) image on the back of the black-out receptacle. A gravitational singularity has the exact same effect on the stars mass, only - by virtue of it being a 3-dimensional aperture - it turns the star inside out as well as upside down. In this way the singularity can be seen as a twist in the fabric of space-time, occurring at the Plank Scale.

Although a singularity cannot be said to have any sides, for the sake of argument (and common sense) lets imagine that an asteroid is being dragged towards the right-hand side of a black hole. From a holistic point of view i.e. one that spans all dimensions, the asteroid is actually being dragged towards the left-hand side of the star's mass, because the singularity has inverted the stars orientation. This type of inversion is what makes it reminiscent of the camera obscura.

In Hyperbolic Perspective and Quantum Gravity, I used a hyperbolic tiling to express how quantum particles, which are under a high degree of magnification, do not exhibit a noticeable degree of hyperbolic distortion (i.e. gravity). This same image could be used to describe a cross-section of the Earth's gravitational field; in which the bulk of space-time curvature is perceived at the edges and dead-space or null-point seen at the core. In order to create a singularity, all we need to do is invert the hyperbolic curvature. 

However, this still does not help us visualise where the star's mass has gone and what is taking place in the fourth dimension. In order to do that I will reduce the number of dimensions down and flatten out space into just 2 dimensions. The grey areas, which denote empty space are merely the result of flattening out the 3-4 D environment out into a 2D image. Such artifacts are regularly observed in map projections, when trying to reduce the curvature of the earth into a 2-d image. As in the case with the map projections no such areas of empty space really exist and motion from one area of space to another can be achieved without undue effort. This then is the fundamental domain of a black hole.

Here we see that the lines of force generated by the gravitational field of the star are being inverted through the singularity at the Plank Scale. What this tells us is that the boundary plain to the four dimensional universe is the 2D limit of the Plank Scale. To understand this same diagram from the point of view of the star in the 4th dimension we need to invert the image. I find this image to have striking correlation to a torus, or a flower. Marko Rodin would say that this is significant. I tend to agree, but I won't be dealing with exactly why this is so until a later date.

So you might be wondering, what is so new about this? Well, physicists generally assume that the mass of the star, while being compacted to an infinite density, remains in this dimension. Others (Marko Rodin for one) have supposed that the singularity may be a gate-way to another dimensions, whereupon the black hole shifts output to produce a white hole. I maintain that when the stars mass falls into the fourth dimension it remains in a static condition; for the obvious reason that there would not be enough mass to support the singularity otherwise.

The reason why the time does not impinge upon it is to do with relativistic laws, but more importantly, because in the 4th dimension time becomes space - and space becomes time. The only way that the star can gain or loose mass (not likely) is through its interaction with the 3D time-based universe. But if the black hole is frozen in time, does that not mean that it should be left behind, as time moves on in our Universe? It should, but because space acts like time in the 4th dimension the mass of the star is continuously dragged along into our time-frame of reference.

The fact that the fourth dimension is intrinsic to the geometry of singularities has been known about in the computer gaming industry for decades. This industry uses a 4-dimensional algebra set, known as the quaternions, to create smooth graphics and to allow for uninterrupted rotations around a 3-D environment. Without the use of the quaternions something known as Gimbal Lock occurs, which is basically defined as a singularity. This means that from a fourth dimensional perspective the singularity does not exist. Refer to the previous two diagrams.

If it has been known that quaternions, and therefor the fourth dimension, are intrinsic to the production and erasure of singularities, then why do physicists insist on thinking of the black hole as being something which exists in our dimension alone? Mathematicians have noted that quaternions/octonions are primarily spatial representations and therefore do not include time as a part of their framework. But I think that they are missing the point. There is no real fourth dimensional space, in my opinion, 4-d space looks exactly like 3-d space only we have swapped one of the spatial dimensions for a time dimension – see post; Within the Octonions for a fuller explanation of this.

As a final way to visualise what is taking place with the stars mass, take another look at the video of the rotating tesseract. Imagine that small cube at the centre of the structure is the uncollapsed star. Then, as the cube rotates, notice how it becomes distorted (length contracted) and finally turns itself inside out.

WITHIN THE OCTONIONS

In the second to last post, I outlined a theory of quantum gravity which utilises hyperbolic geometry. But there are a few interesting elements to this that need to be looked at in more detail. On his site, John Baez gives a very accurate breakdown of what hyperbolic geometry is and how it can be constructed from simple, everyday Euclidian geometry;

Take a bunch of equilateral triangles. Glue them together so that 3 meet at each corner. You get a regular tetrahedron.

Then take a bunch of squares. Glue them together so that 3 meet at each corner. You get a cube.

Next, take a bunch of regular pentagons. Glue them together so that three meet at each corner. You get a regular dodecahedron.

This is fun! We're getting a series of Platonic solids.

Next, take a bunch of regular hexagons and glue them together so that three meet at each corner. Now the angles of the hexagons add up to 360 degrees, so we don't get a Platonic solid. Instead, we get a tiling of the plane. It looks like a honeycomb that stretches out forever in all directions.

Next, let's take a bunch of regular heptagons and glue them together so that three meet each corner. Now the angles add up to more than 360 degrees, so we get a tiling of the "hyperbolic plane". The hyperbolic plane is like the opposite of a sphere, since it's saddle-shaped at every point instead of bulging out at every point. In fact the sphere and the hyperbolic plane are the two most symmetrical forms of non-Euclidean geometry. The sphere is "positively curved", while the hyperbolic plane is "negatively curved".

We can input this series into a table (of sorts) to get a better sense of how the geometry develops;

Tetrahedron;……3D, 4 triangles,…...3 meeting at each vertex

Cube;……………3D, 6 squares,……3 meeting at each vertex

Dodecahedron;...3D, 12 pentagons, 3 " " "

Hexagonal tiling; 2D, ∞ hexagons,…3 at each vertex

Hyperbolic plane;4D, ∞ heptagons,..3 at each vertex

What is interesting here is that we go from 3 dimensions, back to 2, and then straight to 4. This corresponds with what I wrote in Direct Relativity, where an object that is accelerating beyond the speed of light goes from 3 dimensions, to 2 and finally to 4. In this sense we can think of the above series as representing what happens to space as an object accelerates, and shows that it essentially "turns itself inside out". Acceleration is equivalent to an increase in energy, this energy reaches its limit at the hexagonal 2D plane. But, were it possible to go beyond this limit more dimensions would have to be added. This means that extra-dimensions equal an increase in energy/information.

7,3 hyperbolic tiling shows 7 linesconverging on an axis.

When we think of 4-dimensional space, we often think of the three dimensions of length, breadth and height, along with a fourth spatial dimension at right angles to each of these. But what the above series shows us is that when we introduce the fourth dimension this breaks the continuity of the first 3 dimensions, into something more like six. This break in continuity occurs so that each axis remains equidistant from one another around a central point. We must therefore think of each of these 3 dimensions as being comprised of two separate values, much as we do in daily life. Height is broken into; up and down, breadth into; left and right, and length; into forwards and backwards. In reality there is no break in continuity. But from the point of view of an outside observer, like ourselves, the inclusion of a fourth spatial dimension results in an apparent disconnection of these dimensions.

The 4/7 dimensions of hyperbolic geometry

 and the quantum scale

Time is often thought of as the fourth dimension, but it should also be noted that, in some sense, time is just another spatial dimension. So the heptagonal (7,3) tiling is an accurate description of 4-dimensional space-time and the hyperbolic curvature that creates gravity. We should also note that whereas the 3 spatial dimensions each have 2 components; up/down, forwards/backwards etc. the time axis only has 1. This lack of symmetry may account for the reason why time only moves in one direction; from past to future, and never the other way.

If we were to remove one of the seven dimensions, it does not matter which, the other six will rearrange themselves into the infinitely tiled hexagonal plane of only two dimensions. Lets say, for argument sake we can remove the "up" dimension. This results in time taking the place of the up dimension, and therefore time no longer serves the function of duration in this instance. Without time we also lose the use of the 3rd dimension, because as I stated in Direct Relativity, our conscious is a 2-dimensional plane, and can only access the third dimension by moving into it (and time is a function of motion). Despite this, the hexagonal plane will continue to possess the appearance of being 3-dimensional, due to the fact that its structure still contains all the necessary data for l, b and h implicit in its geometry.

The hexagonal tiling still has all the informationof 3-dimensions implicit within it. In fact, becausethe cubes can be viewed in two different waysit may even imply a 4th dimension.

It is obvious that since time is the fourth spatial dimension it needs the other three to build on. But what is less obvious is that without time, space, as a 3-dimensional construct, also ceases to exist in a useful manner. Therefore, time and space are inextricably linked in this geometric fashion. The more intense the curvature of space the less time is seen to pass, because time is becoming space.

The hyperbolic plane shows a description of space-time viewed from the quantum scale. At this magnitude space appears flat, but in the distance the accumulative effect of matter is shown to curve space-time. Since Einstein related space-time curvature to acceleration, it appears that from the point of view of the quantum world, space is being accelerated at the macro scales. Acceleration is further linked to time dilation and therefore time ceases to exist in the meso-macro scale for all particles and hadrons in the quantum world. Could this account for some of the strange behaviour, as seen by us, on the quantum scale? I think it must.

As already noted, the hyperbolic plane is a description of how matter interacts with space-time and not anything else. One of the key principles of matter is that it is always changing. Time is the rate of change. Therefore, matter is time, or more accurately an expression of space-time. We could redraw our seven axial model of space-time, in which the primary spatial dimensions are preserved and the 4th/7th dimension becomes bent into a loop (see Fano plane). This looped time is essentially matter. In this regard, we could ask does matter create the curvature of space-time or does space-time curvature create matter? It is a chicken-and-egg phenomenon.

The Fano Plane Order 2could be used to describe thesix directions of space along withthe looped dimension of time,which is also matter.

Just like with normal perspective, when we zoom in on a particle that particle grows in size making all the other atoms and particles appear smaller. By zooming out again, all their relative sizes are made uniform, making all of space appear flat again. Continuing to zoom out at would warp space in the opposite direction, giving space the appearance of one of the Platonic Solids or simply a sphere. Again this is related to Scale Relativity where being zoomed in on an object at the quantum scale is equivalent to stopping, and zooming out the equivalent of accelerating. From this perspective flat space-time i.e. what we normally perceive space as, is equivalent to space traveling at the speed of light (see Direct Relativity). 

The heptagonal hyperbolic plane is linked to the number 168, and by extension to the Klein Quadric, PSL (2,7), the Fano Plane and Octonions. We have seen already that the Fano plane relates to space and time, as does the number 168 (24 x 7). The fact that the numbers 24 x 7 (as in 24 hours, 7 days a week) should crop up in our formulas for space time implies that our local description of time, as denoted by the rotation of the Earth around the Sun and the geometric division of the circle into 360º, is somehow intrinsic to the nature of higher-dimensions and the universe on a cosmological scale. How could this be, when we have so long been told that Earth and our solar system are no longer the centre of the universe? The answer, I think, lies within the Octonions.

We have already looked at space as a 7-dimensional construct, at least at very small scales, but the 8-dimensional algebra of the Octonions would ask us to look again. Octonions have 7 dimensions plus 1; this extra dimension is a projective point, which I maintain may be linked to consciousness. Reality, as a whole, must be projected from the conscious mind of the individual perceiving it, just as the Buddhists and mystics have been saying for thousands of years. This in itself would explain why the structure of time and space in our locale are inextricably linked with the underlying mathematical laws that the human mind has been developing over the ages.


We already spoke of time as resulting from the asymmetry of the 7 dimensions of hyperspace. But, if consciousness represents the eighth dimension then shouldn't it provide the symmetry needed to prevent the forward motion of time? Well in some ways it does; consciousness provides us with a memory with which we can skip through the events of our lives without a great deal of difficulty.


However, the arrow of time is still very much a persistent factor in our daily lives, and cannot be dealt with so easily. A more comprehensive view then would suggest that the eighth dimension is a fractal dimension containing all of the previous 7 dimensions on repeat down towards infinity. This makes sense because consciousness is the only place where we can be aware of dimensionality and of the passage of time. It is also clear that consciousness, and life in general, is a self-referencing organism that would sit well alongside our definition of a fractal system.

For more information on how the number 168 relates to time and the speed of light see my first post; Scale Relativity.