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A model of the universe
This article explores the implications of a universe based on a multidimensional circle or hyperbola. Two approaches are used then the results are compared. The first approach starts at Section 4, the second starts at Section 10.

Sections 1. to 3. covering the topic 'Complex components of a simple function' has been included for completeness. If you wish to skip this topic go straight to Section 4.

1. Complex components of a simple function

x² + y² = r² is the equation that describes a circle, radius r, centred on the origin.

Rewrite the equation as y = (r² - x²)1/2 and calculate and plot values of y for (- r) <= x <= r. This gives a graph of a circle in the real x,y plane.

When x <=(- r) and x >= r the graph is a rectangular hyperbola in the complex x,iy plane. See Figure 1.
Figure 1

2. The structure expanded

Another way to explore a graph is to add a constant to the right hand side of the equation. This has the effect of moving the curve up or down the y axis. A curve can then be plotted from the values of the roots of the equation for each value of k.

Alternatively, for this particular equation, you can rewrite the equation as x = (r² - y²)1/2 and calculate and plot values of x for y <= (- r) and y >= r.

A third way is to plot values of y for various imaginary values of x. (ix)

If you do this, you find that the curve has another complex component when y <= (- r) or y >= r.
It is another rectangular hyperbola, this time in the complex ix,y plane. See Figure 2.
Figure 2

3. Is there more than one imaginary direction?

The next question is “Is the ix direction the same as the iy direction?”

The answer can be found by changing the sign of the terms of the original equation. Changing the sign of a term rotates the curve into different planes. (x and y represent positive real numbers)

Consider only the circle part of the curve, for real values of x, y and r.
  • x² + y² = r² defines a circle in the real x,y plane.
  • x² - y² = r² defines a circle in the complex x,iy plane.
  • - x² + y² = r² defines a circle in the complex ix,y plane.
  • - x² - y² = r² defines a circle in the imaginary ix,iy plane.
Since a circle can be defined in the ix,iy plane, the ix direction must be different from the iy direction.

So, the curve defined by the equation x² + y² = r² requires four axes, two real and two imaginary.
By a similar argument the graph of the equation for a sphere, x² + y² + z² = r² can be shown to require six axes, three real and three imaginary. (x, y and z can be positive, negative, real or imaginary numbers)

In general, it seems that a similar equation with n variables requires n real and n imaginary axes to fully display its graph.

4. Space-time

Space-time is generally considered to have four dimensions, three of space and one of time. Mathematically the space dimensions are treated as 'real' and the time dimension as 'imaginary'. A mathematical space with three real dimensions and three imaginary dimensions could be considered to represent a world with three space dimensions and three time dimensions.

What are the implications, for theories about the ‘real’ world, of the structure described in Section 2? How many space and time dimensions does our world consist of?

First some definitions
  • To avoid adding the prefix 'hyper' to the name of a curve to describe a similar curve in a higher dimension, I add the prefix 'four-dimensional', 'six-dimensional' etc. to the name of the three dimensional equivalent. For example, 'four-dimensional sphere' instead of hypersphere.
  • Our perception of the world depends on something I call our Conscious Awareness or ‘CA’. The ‘CA’ seems to move in the general direction of an increasing time (imaginary) axis. I use this concept not for any spiritual reason but as a device to explain what follows.
  • A set of co-ordinates representing n real dimensions and n imaginary dimensions, I call a Matrix.
  • The graph of an equation, with n variables, plotted onto a Matrix, I call the Curve.
  • The four-dimensional Matrix described above contains a four-dimensional Curve consisting of a circle, (the Hub) which is two-dimensional, surrounded by four semi-hyperbolae, (two pairs of Spokes) also two-dimensional.

A Curve, of this type, requires a Matrix of 2n dimensions. Each element of the Curve (the Hub and the n pairs of Spokes) will each have n dimensions.
  • The Hub does not have to be at the origin. Adding a constant 'k', the equation, will move the Curve away from the origin in any direction depending on whether k is real, imaginary or complex.
  • The Curve can be 'rotated' round the Matrix by changing the sign of the terms of the equation, as described previously.
  • The Spokes are infinite in extent, so on the grand scale the Hub can be considered to be a dimensionless point, and the Spokes can be considered to follow their asymptotes.

5. A sphere in a six dimensional Matrix

Consider a ‘CA’ in the six-dimensional matrix, mentioned in Section 4. If the ‘CA’ were to move past a sphere, having one imaginary (time) and two real (space) dimensions, it would be aware of a small circle appearing, getting larger at a reducing rate, then getting smaller and finally disappearing. If the ‘CA’ were to move past a cylinder, along its axis, it would be aware of a circle of constant size.

For the ‘CA’ to be aware of a sphere, rather than a circle, we have to add an extra term to the equation and two more dimensions to the Curve. The ‘CA’ passing a four-dimensional cylinder having one imaginary dimension and three space dimensions, along its axis, would experience a sphere.

As we are aware of three-dimensional objects such as spheres, which persist, this implies that our world requires at least an eight-dimensional Curve.

6. The Universe as part of an eight-dimensional Curve

We perceive the universe as being spherical. If the Curve is defined by the equation (-w² + x² + y² + z²) = r², our ‘universe’ having one time (iw) and three space dimensions (x, y and z) is the Hub (a four-dimensional sphere). The beginning of time is at the centre of the Hub. The size of the universe decreases with time. For space to have a 'beginning in time', and for it to increase in size with time, a constant has to be included in the equation to bring the 'surface' of the sphere to the origin.

If the Curve is defined by the equation - w² - x² - y² - z² = r², our ‘universe’ is one of the Spokes and the extra constant is not necessary. I prefer to choose the simpler scenario (Occam's razor). In this case, the Hub is a four-dimensional sphere in four imaginary dimensions, with each pair of Spokes on a different imaginary axis.

Our ‘universe’ would then be just one four-dimensional Spoke (a four-dimensional hyperboloid) of the eight-dimensional Curve. That is, one of the eight Spokes each having three space dimensions and one time dimension (the axis). Let's see what some of the properties of such a ‘universe’ might be.

7. The evolution of the universe

The 'beginning in time' in this universe is the point at which it links to the Hub. At this time the rate of expansion is momentarily infinite. This could explain 'Inflation'. The rate of expansion slows down with time, eventually becoming almost constant as the four-dimensional hyperboloid becomes more like a four-dimensional cone. The expansion continues forever. See Figure 7.


Figure 7
Figure 7
The horizontal axis represents the three space dimensions of our 'universe'.
The vertical axis represents our time dimension.
The dotted red line represents one of the four time dimensions of the Hub.
The solid blue curve represents a Spoke (our 'universe').
The dotted blue curve represents the companion Spoke (a complementary 'universe').
The time t0 is the beginning of time in our universe.
The black circles represent the increasing size of our universe at times t1 to t5.
At t0, the universe is expanding at an infinite rate (the blue line is horizontal).
At t3, the rate of expansion is slowing down.
By t5, the expansion is continuing but at an almost constant rate.

8. Relevance to current theories

At any instant of time, our universe (not to be confused with the 'observable' universe) is the surface of a four-dimensional sphere, and has no thickness in either the fourth space dimension or any of the time dimensions. Call it a three dimensional sheet.
  • Deformation of the sheet into the Spoke's own time dimension is an explanation of mass and gravity. (Einstein)
  • Deformation of the sheet into the fourth space dimension is an explanation of charge and electrical interaction. (Maxwell)
  • These two deformations taken together constitute the theory of Kalusa.
  • Deformation of the sheet into the three remaining time dimensions could be an explanation of some of the properties of the three generations of elementary particles and their interactions. (The Standard Model)
  • Particles have a combination of properties. For example, an electron has mass and charge. So, a particle may be a deformation of space-time in more than one dimension.

9. Summary so far

  • Where this model differs from others is that, in this case, what we call 'the universe' is part of a larger structure, the Curve. Our universe exists as a four dimensional element of an eight-dimensional Curve.
  • Although we perceive 'our universe' to be spherical, it could actually be part of a four-dimensional hyperboloid. This part has a time axis and three space dimensions.
  • The fourth space dimension and the other three time dimensions are not 'curled up' (as they are in the Kalusa-Klein model). They are 'outside' our universe except where our space-time is deformed into them.
  • This type of model offers the possibility of bringing together Einstein, Maxwell, Kalusa and the Standard Model.
  • It proposes that the universe had a beginning, and offers an explanation for Inflation Theory.
  • This particular model predicts that the universe will go on expanding forever.

10. The ‘Light Cone’

The second approach starts with the concept of the ‘Light Cone’. A Light Cone is a graphical representation of the way light gets to, and leaves, a point in space-time. The vertical axis represents the time dimension and two ‘horizontal’ axes represent two of the three space dimensions. The assumed paths of rays of light approaching or leaving the point lie on the surface of a cone. See Figure 10.

Figure 10
An assumption inherent in this representation is that the universe has no beginning (All time-lines are parallel).

11. The ‘Light Cone’ and the ‘Big Bang’

When you include the ‘Big Bang’ in the scenario, the time-lines are not parallel. The time axis then has a beginning. Time-lines radiate out from the ‘Big Bang’. A ray of light coming from the past can no longer follow a straight line, but must spiral outwards from the ‘Big Bang’, keeping a constant angle relative to the local time-line. Each infinitesimal area of space is perpendicular to its own time-line, but at any instant all areas of space are contiguous. There is no longer a single time dimension.

Imagine a space-time diagram for a universe beginning with a ‘Big Bang’.

For clarity, Figure 11 is based on a vertical section through Figure 10.

Figure 11
  • This diagram shows that the principle of the ‘Light Cone’ holds only on a very small scale. For example, the region circled at the top. On a larger scale it can be seen that light spirals outwards from the ‘Big Bang’.
  • In this ‘section’ there are two time dimensions. Space is also two dimensional, as it curves round, from the circled region at the top to the perpendicular regions at the sides. At any point in this space-time, except at the origin, an ‘inhabitant’ would experience only one space dimension and one time dimension.
  • Likewise, an inhabitant of a space-time with three space and three time dimensions would only experience two space and one time dimension.
  • A space-time, of this type, that is perceived as having three space and one time dimension would actually have four space and four time dimensions.

12. Comparing the results of the two approaches

The two approaches are actually complementary. The eight dimensional Curve of the mathematical approach was based on the assumption that our part of the ‘universe’ had one time dimension. The Light Cone/Big Bang approach shows that our awareness, of a space-time with three space dimensions and a single time dimension, is an illusion.

13. A consequence for Cosmology

If light travels in spiral paths, one has to consider the possibility that these paths may have crossed previously, since the universe became transparent, maybe more than once. If this were so, objects at these crossover points would appear, to us, to be ‘smeared out’ over the whole sky if the space-time distances - geodesics - were the same in all directions. Objects just before and after a crossover would appear magnified to a lesser extent.

14. In conclusion

In this article I have tried to show how a particular complex curve might be a representation of the large scale structure of the universe.

Research in the late 1990s, indicated that the rate of expansion of the universe is actually accelerating. This would appear to rule out a universe based on a multidimensional sphere or hyperboloid.

Many curves have real and imaginary components. A model of the universe based on such curves cannot ignore the extra imaginary dimensions. I think that the principle of a universe with many time dimensions is worthy of further investigation.

Mike Holden - Sep 2005
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