The Remarkable Theorem

Jack Dikian

September 2011

Ever since I read Flatland: A Romance of Many Dimensions by the English schoolmaster Edwin Abbott my mind turns to the idea of higher dimensions, and whether we humans have the capacity to visualize the fourth dimension. I don’t mean using time as a fourth dimension viz a viz Special Relativity – rather trying to imagine the existence of a 4-dimensional being looking back at us and our world.

Abbott‘s in his 1884 satirical novella wrote pseudonymously as "A Square", in the fictional two-dimensional world of Flatland to offer pointed observations on the social hierarchy of Victorian culture. A 3-dimensional being, of course, is could see everything in their world, and all at once.

In the same way, a 4-dmensional being looking back at us could look inside our stomach, and remove, if they want to the lunch we just had without cutting through our skin, just like we can remove a dot inside a circle (flatland) by moving it up into the third dimension perpendicular to the circle, without breaking the circle.

Then years later, I learned about Carl Friedrich Gauss and his Theorema Egregium (the remarkable theorem in Latin). How for example, can an Ant (say a 2-dimensional being) stuck on the surface of our curved world, and can’t stand back to see the curvature of our planet ever realize that the surface is curved.

The theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space. An absolutely amazing insight! This however only applies to curved surfaces which are 2-dimensional.

It would take a brilliant student of Gauss, Bernhard Riemann at the age of just 26 to develop and extend Gauss's theory to higher dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. That is generalizing Gauss’ work to describe the curvature of space in any dimension. Again, how do we, non-mathematicians, visualize a curved 3-dimensional space. What encapsulates it? The genius of Riemann was to show that we don’t need to step into the fourth dimension to tell if space is curved. We can do it form the inside.

Albert Einstein, as we know, came along and used the theory of Riemannian manifolds to develop his General Theory of Relativity. In particular, his equations for gravitation are restrictions on the curvature of space. He took the mathematics of Gauss and Riemannian and used it to develop a revolutionary picture of our physical world showing that we live in the curved worlds of Gauss and Riemannian.

So we get to finally that gravity is not a pull downwards but rather an object falls following the simplest path through bend space. Of course, Einstein didn’t stop there and showed that the presence of mass that bends space.

2001: A Space Odyssey & An Explanation Of What Happens In A Vacuum

Jack Dikian

August 2011

I first watched Stanley Kubrick's 2001: A Space Odyssey in the mid 90’s and I remember being struck by the power of its visual imagery. In the enduring years I probably saw the film another 2 or 3 times. Yesterday evening I watched again, this time thinking about how and why this film appealed so much to of one of the greatest mathematicians of the twentieth century, Paul Adrien Maurice Dirac.

Paul Adrien Maurice Dirac, held the Lucasian Chair of Mathematics at the University of Cambridge, and shared the Nobel Prize in physics for 1933 with Erwin Schrödinger, "for the discovery of new productive forms of atomic theory."

As I read about Dirac, one learns that he hardly spoke unnecessarily. In fact people who knew Dirac well coined the word “a Dirac” meaning, amusingly the smallest number of words spoken in an hour and still be involved in a conversation. Interestingly, it takes almost 25 minutes before a word is spoken in 2001.

In the late 1920s Dirac unified special relativity and plank’s quantum effects, unravel the vacuum and explained what is really taking place in empty space. This is regarded by many, as one of the greatest achievements in mathematics and physics in the 20^th century leading to a new picture of [nothing].

2001’s appeal for Dirac helps give us an insight how he managed this achievement. As Kubrick himself said 2001 is a demonstration that a really good film script can be made without many words but with the power of visual imagery. Dirac had a very strong sense of what his equations were telling him visually.

Making Meaning of Time

Jack Dikian

August 2001

Introduction

When thinking about a world where the linearity of time is no longer maintained – that is, a world where the 24 hour measure of time is no longer divided in equal intervals; the time that elapses between 3pm and 3.15pm is different from the time that elapses between say, 4pm and 4.15pm.

It become very clear that the word “time” is used as a means of measuring the passing of time. So a clock measures intervals of time, but not of time itself. The immediate question, therefore, is what is it that we are measuring? We can use all sorts of instruments and fall back on common descriptors such as;

“the years are going by so quickly”,

“Time is like a river”,

“Time flies by when we are having fun”

These don’t, however, tell us what it is that we are measuring. We are still left without an understanding of what exactly is an interval of time.

The Psychology

Despite the preciseness of our instruments we are still measuring physical time in contrast to a more subjective personal time. Psychologically, time seems to be over before we know it. The past only exists as memory and we can’t remember the future. So, therefore we are always in the moment. Interestingly enough, unlike other spatial dimensions of perception, time is not directly available to any sensory system but can only be sensed through the unfolding of events in the world we live in.

Before examining the mathematics and/or the physics of what we call time it may be useful to briefly look at the psychological construction of time that we use to make meaning of our world.

The orderly nature of our world is born in part from the fact that events’ spatial characteristics are structured in and over time. Speech, body movements and walking gaits in which the sequence of words or actions unfold with a characteristic rhythm and tempo over a given time span. This spatio and temporal structure influences how an event is perceived and remembered, as well as providing some guidance over the accuracy with which the event’s velocity and total duration are subsequently judged. We routinely decide (or to be more précis estimate) whether there is sufficient time to cross a street ahead of an approaching car.

The activities we undertake in everyday life also vary in cyclic patterns over the course of a day or a course of a seasion and provides a scheme that serves to coordinate those individuals in any particular society. Different cultures have different conceptualizations and attitudes toward time which can be reflected in the overall pace of life. In any individual, the temporal perspective and their relative orientation toward the past, present, and future is central to one’s mental well-being and the degree of ego strength displayed in coping with life’s difficulties. As well, in the clinical analysis of timing disorders mechanisms mediating circadian-driven behaviors are quite distinguished from those governing the internal clock and sensitivity to an event’s duration and overall velocity.

The Physics

Firstly, we all tend to perceive time as a smooth flowing and continuous motion. We in the usual course of life never experience time in chunks, slices, or something characterized by broken gaps. This is somewhat consistent with our perception of other natural phenomena such as energy, and the space around us. However, at the quantum level of matter energy is not released continuously - there is a limit to how small a change in energy an atom can experience - it is released in discrete quanta by the emission of photons. This lends itself to the question;- are individual frames of time so small that it only gives the appearance of being smooth and continuous.

In experiments using instruments that can 'slice' moments in time to a small enough granularity in order to capture a chemical reaction and dealing with time at the Femtosecond level, that is one thousandth of one trillionth of a second, (1/1,000,000,000,000,000th) there seems to be no indication of time passing in discreet steps. Time still appears to flow smoothly. It is off course entirely possible that if it does move in discrete steps that we have not yet reached a level small enough to observe it. The smallest measurement of time that can have any meaning in contemporary physics is a 'Planck Time', and is equal to 10-43 seconds.

That is a decimal point followed by 43 zeros and then 1 (0.00000000000000000000000000000000000000000001)

Interestingly, there is nothing in the laws of physics to suggest that time actually flows from the past through the present and into the future. At the subatomic level there is no distinction between the past and the future. By studying these particles it would be impossible to determine the order in which the events took place. At this level there is no way to distinguish the past from the future by simply looking at each particle pairs.

However, we are clearly able to discern the arrow of time or the direction of time at the macrosporic level. We would never see for example, a bottle hitting the floor and breaking before it actually falls off the table. So it seems at the macro level (the level we experience reality) there is a natural inbuilt arrow of time pointing from the past to the future. The distinction between the past and future is expressed mathematically by thermodynamics.

The second law of thermodynamics states that the entropy of a closed system always increases, entropy being the measure of disorder. In other words disorder will always increase and if left to its own devices, a system will run to disorder, and not order, giving us an arrow of time.

One of the most peculiar qualities of time is the fact that it is measured by motion and may possibly therefore emerge through motion. Our perception of “time” may therefore be mostly an illusion. Our memory creates the illusion of the past. Our conscious perception of events gives us the feeling of the present. The future is a mental construct patterned on the memory of experience. The concept of time emerges as our mind tries to make sense of the world we are in.

In general relativity matter produces curvature in space-time, and this concept can be extended to moving objects showing that moving objects can also curve space-time. In an expanding universe, time is related to the expansion of space. Slower time is associated with slower expansion. This extra dimension is not a time dimension in which we can travel. Time is just thought of as the presence of motion and forces and is caused by the expansion of space. Expansion of space, here, is the prime mover behind all the action, it imparts time as the presence of motion and forces to matter in an area of space.

The relationship of the clock and speed of light is fixed at a basic level in a given region of space. Clocks may measure a different speed of light if located in a region of space which is different from where light is being observed. For example a clock on earth measuring speed of light in a region next to a distant neutron star where time is slower will measure a slower speed. While a clock located on the surface of neutron star will measure a faster speed of light in distant space.

Moving objects produce moving wave like curvature in space. Once within the vicinity of the moving object mass-less particles like photons are picked up and carried forward by the time differential which is present in these waves. Also the velocity of light is decreased by the slower time within the moving wave. Within the moving object like earth light is pulled along in the direction of motion and it also slows down to conform to the slower time around the earth.

The faster we travel through the dimensions of space, the slower we travel through the dimension of time, and vice versa. Thus an astronaut zooming along at light speed has used up all their speed 'allocation' in the space dimensions, and as a consequence does not travel through time. This would seem to suggest that the speed of light really is the limiting speed within the universe, and if we had no motion at all through space then we would be travelling at light speed through time. The more movement we make across space the less we make in the direction of time.