Mike Sterling Canadian Community News
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Past, Present and Future?
A number of my friends are interested in time. Iíve had my spells of thinking about it too, the more so as I get older. Tempest Fugit.
Sometimes, but not as often as I would like, equations tell us more than I expect.
When I was a teen I ran across the Lorentz Transformation. It made a deep impression upon me. It opened up an entire world. I read as much as I could about it. Einstein used it in his special relativity mind experiments. I use it sometimes to humble myself relative to the wonderful universe we are part of for the Ďmomentí. Why muse about time? Well, it seems to help me see my place a little better.
I can get some feel for the Big Bang before which time seems to have no meaning. I can get a feel for the ever expanding universe and the how time again ceases to exist as the last tickle of a vibration sluggishly ceases. In this way time seems to run down like a wound clock. What happens beyond those bookends of the universe is a mystery to me.
Normally the Lorentz is taught in high school physics by using 2 dimensional curves. These curves represent mass, directional scaling and what happens to time relative to an object moving from rest and approaching the speed of light. The equations are related to the Pythagorean Theorem as described by Nobel Laurette Richard Feynman in one of his great little books on elementary physics.
Dr. Eli Maor wrote a wonderful book on the Pythagorean Theorem and many of its manifestations.
If you see a square or square root in an equation and no higher terms, you can bet that somewhere deep inside the equation lurks the Theorem of Pythagoras.
Fooling around some years ago, I moved the Lorentz Transformation to the complex number system to examine it. We wonít bore you with what that is. Since I had some tools that were easy to use, it did not take too much Ďtimeí to allow me to pull Lorentz out of 2D and place it in a higher dimension. Curves then became surfaces.
What I found is shown in the embedded picture above. Iíve carved out two views showing both sides of the same shape that resulted. You can see the singularities easily. Thatís where the denominator of fractions goes to zero. The cone-like shapes reach to infinity when that happens.
Thinking about time, we find that we live in and around the saddle as shown in the upper shape. So, if you, the reader, start moving away and proceed up the rightmost cone at increasing speed, your mass will increase and your relationship to time will be different than mine as I remain static at the saddle. All this can of course be deduced from the Lorentz Transformation in 2D.
This nasty fact of time dilation influences GPS and satellites. Corrections are made for static humans on earth and a whirling space object. Without this correction, we might drive our cars into a river directed by a faulty GPS.
If I remain at the saddle and you journey toward the right peak, any distance, your biological and even any old mechanical clocks you might have on hand will slow down relative to mine. You wonít notice it nor will I, but the clocks will. When you return, you have not aged as I have. Our clocks are not the same. If wrinkles are a marker of timeís passing, you will not have as many as I have, if per chance we started as equals. I have a lot of wrinkles caused by time and the earth whirling around the sun as it creates radiation by fusion.
But, there are two other things of interest in the picture. First you see that we have a dual world that Iíll call the pseudo-past, present and future. I donít know what else to call them. These worlds show up as an artifact of the Lorentz in the complex number system as we see that square roots of negative numbers work just fine. So instead of having just one peak, we have two due to our ability to handle roots of negative numbers.
One of the reasons for the great outburst of activity and interest in the complex number system in the 1800ís was due to the beauty, but more importantly, it allowed solutions to problems that were untenable and unreachable before due to the ever occurring nasty square roots of negative numbers issue.
So the topology of future and past are mirror images of each other in the picture. Future and past terminology is stretching things a bit. But, speed, mass and time are closely tied. Very neat, but somehow expected.
The second thing to note is that there is a lot of surface not at or near the saddle point or not on a path up the conical like shapes in a shortest distance manner. What does this mean? What goes on there? Can we wander at will on that gently curved area near, but not on the saddle?
I have no idea? What does it mean to meander around the right conical shape at a modest speed not approaching the speed of light? If you wander ever to the right, but avoid the cone, you are still living in present and are on a path to the future, but you donít seem to have the restrictions of the cone to worry about. It looks like journeys can be taken at modest speed, not going into the past and not restricted by an ever increasing mass as you struggle to go faster. This is most probably an artifact of the method used to show the Lorentz and not reality.
If, however, I stay put at the saddle and you dash off on letís say a 45 degree angle with ever increasing velocity, your path will produce a singularity that is not shown.
So you are trapped if you try to reach the speed of light. But, it would be nice to tip toe out there.
What are those regions? As you can see, they are gently curved and they go on forever. It does not look like they can be reached because we will run into the singularities again in every direction.
Ah, well, I think I will have another cup of coffee.
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Monday, December 28, 2015